# Modern Control Systems

131

Skills Check

graph. Thus, in Chapter 2, we have obtained a useful mathematical model for feedback control systems by developing the concept of a transfer function of a linear system and the relationship among system variables using block diagram and signal-flow graph models. We considered the utility of the computer simulation of linear and nonlinear systems to determine the response of a system for several conditions of the system parameters and the environment. Finally, we continued the development of the Disk Drive Read System by obtaining a model in transfer function form of the motor and arm. CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 2.79 as specified in the various problem statements.

w Controller R{s)

o) w *

Gc(s)

+ • Y(s)

V*

N{s)

FIGURE 2.79 Block diagram for the Skills Check. In the following TVue or False and Multiple Choice problems, circle the correct answer. 1. Very few physical systems are linear within some range of the variables. 2. The s-plane plot of the poles and zeros graphically portrays the character of the natural response of a system. 3. The roots of the characteristic equation are the zeros of the closed-loop system. 4. A linear system satisfies the properties of superposition and homogeneity. 5. The transfer function is the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all initial conditions equal to zero. 6. Consider the system in Figure 2.79 where

True or False True or False True or False True or False

True or False

s + ** s* + 60.y + 500 If the input R(s) is a unit step input, Td(s) = 0, and N(s) = 0, the final value of the output Y(s) is: a. yss = limy(t) = 100

Gc(*) = 10, H(s) = l,

f-»CO

b. yss = lira y(t) = 1 t—*oo

c. yss = lim y(t) = 50 f~»0O

d. None of the above

and

G(s) =

132

Chapter 2

Mathematical Models of Systems

7. Consider the system in Figure 2.79 with s +4 52 - 125 - 65 When all initial conditions are zero, the input R(s) is an impulse, the disturbance Td(s) ~ 0, and the noise N(s) = 0, the output y(t) is a. y(t) = 10e~5' + 10e~3' b. y(t) = e'* + 10e~' c. y(t) = 10e~3' - 10e_5r d. y(t) = 20e -8 ' + 5e~15' 8. Consider a system represented by the block diagram in Figure 2.80. Gc(s) = 20, H{s) = 1, and G(s) =

R(s)

FIGURE 2.80 Block diagram with an internal loop. The closed-loop transfer function T(s) = Y(s)/R(s)

is

s2 + 55s + 50 b. T(s) =

10 s2 + 555 + 10

c. T(s) =

,

v

2

10

5 + 505 + 55

d. None of the above Consider the block diagram in Figure 2.79 for Problems 9 through 11 where Gc(s) = 4, H(s) = 1, and G{s) = 9. The closed-loop transfer function T(s) = Y(s)/R(s) TV ^

a. T(s) =

5 0

s2 + 5s + 50

20 s2 + 105 + 25

b. T(s) =

50 s2 + 55 + 56

c. T(s) =

20 s2 + 105 - 15

d. T(s) =

is:

5 s2 + 10s + 5'

Skills Check 10. The closed-loop unit step response is: 20 20 _,5' _ t2e-5t a. y{t) = i r + 25 25 b. y{t) = 1 + 20re~5' ,s 20 20 5t °~ - 4te~5t - 3 M = 25-25e d. y(t) = 1 - 2e~5' - Ate'5' 11. The final value of y(t) is: a. y„ = lim y(t) = 0.8 r-*oo

b. y w = lim y(f) = 1.0 f-»oo

c. y„ = Kmy(t) = 2.0 /->oo

d. yss = lim y(t) = 1.25 /—»00

12. Consider the differential equation y + 2y + y = u where y(0) = y(0) = 0 and u(t) is a unit step. The poles of this system are: a. s-i = - 1 , ¾ = - 1 b. 5! = 1/, 52 = - 1 ; C

S-i = - 1 , 5 2 = —2

d. None of the above 13. A cart of mass m = 1000 kg is attached to a truck using a spring of stiffness k = 20,000 N/m and a damper of constant 6 = 200 Ns/m, as shown in Figure 2.81. The truck moves at a constant acceleration of a = 0.7 m/s2.

FIGURE 2.81 Truck pulling a cart of mass m. The transfer function between the speed of the truck and the speed of the cart is: 50 552 + s + 100 20 + s b. T(s) ^ + 105 + 25 100 + 5 c. T(s) = 2 5s + s + 100 d. None of the above

a. T(s) =

133

134

Chapter 2

Mathematical Models of Systems

14. Consider the closed-loop system in Figure 2.79 with Gc(s) = 15, H(s) = 1, and

G(s) =

1000 s3 + 50s2 + 45005 + 1000'

Compute the closed-loop transfer function and the closed-loop zeros and poles. 15000 a. T(s) = - 3 , 5 ! = -3.70,5 2 3 = -23.15 ± 61.59/ 5 + 5052 + 45005 + 16000 ' ' . ^, N 15000 „„„ , n nn b. 7/(5) = — , Si = -3.70,5, = -86.29 5052 + 45005 + 16000 1 c. T(5) = -r3 5 ,5 X = -3.70,5 2 , = -23.2 ± 63.2/ ' 5 + 5052 + 45005 + 16000 d. 7/(5) = 1_292 s = _ 3 > 7 0 s = -23.2, s3 = -63.2 53 + 5052 + 45005 + 16000 15. Consider the feedback system in Figure 2.79 with K(s + 0.3) 1 Gc(s) = — -, H{s) = 2s, and G(s) = (5 - 2)(52 + 10s + 45)' Assuming R(s) = 0 and N(s) = 0, the closed-loop transfer function from the disturbance 7/rf(5) to the output Y(s) is: Y(s) a. T (s) d

1 53 + 8^ + ( 2 ^ + 25)5 + (0.6K - 90)

Y(s) Td(s)

100 53 + 852 + (2K + 25)5 + (0.6K - 90)

Y(s)

1 852 + (2K + 25)5 + (0.6K - 90)

b.

Td(s) d.

Y(s)

K(s + 0.3)

Td(s)

s4 + 853 + {2K + 25)52 + {0.6K - 90)5

In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Actuator b. Block diagrams c Characteristic equation d. Critical damping

e. Damped oscillation f. Damping ratio g. DC motor

An oscillation in which the amplitude decreases with time. A system that satisfies the properties of superposition and homogeneity. The case where damping is on the boundary between underdamped and overdamped. A transformation of a function f(t) from the time domain into the complex frequency domain yielding F(s). The device that provides the motive power to the process. A measure of damping. A dimensionless number for the second-order characteristic equation. The relation formed by equating to zero the denominator of a transfer function.

135

Exercises h. Laplace transform i. Linear approximation j . Linear system k. Mason loop rule 1. Mathematical models m. Signal-flow graph n. Simulation o. Transfer function

Unidirectional, operational blocks that represent the transfer functions of the elements of the system. A rule that enables the user to obtain a transfer function by tracing paths and loops within a system. An electric actuator that uses an input voltage as a control variable. The ratio of the Laplace transform of the output variable to the Laplace transform of the input variable. Descriptions of the behavior of a system using mathematics. A model of a system that is used to investigate the behavior of a system by utilizing actual input signals. A diagram that consists of nodes connected by several directed branches and that is a graphical representation of a set of linear relations. An approximate model that results in a linear relationship between the output and the input of the device.

EXERCISES Exercises are straightforward applications of the concepts of the chapter. E2.1 A unity, negative feedback system has a nonlinear function y = /(e) = e2, as shown in Figure E2.1. For an input r in the range of 0 to 4, calculate and plot the openloop and closed-loop output versus input and show that the feedback system results in a more linear relationship.

Spring breaks I fc Displacement "™' (em)

H—I—h

Springf compresses

FIGURE E2.3

Close switch for closed loop FIGURE E2.1 Open and closed loop.

Spring behavior.

E2.4 A laser printer uses a laser beam to print copy rapidly for a computer. The laser is positioned by a control input r(t), so that we have 4(s + 50)

E2.2 A thermistor has a response to temperature represented by R

= /^-01-^

where R0 = 10,000 ft, R = resistance, and T = temperature in degrees Celsius. Find the linear model for the thermistor operating at T = 20°C and for a small range of variation of temperature. Answer: AR = -135AF E2.3 The force versus displacement for a spring is shown in Figure E2.3 for the spring-mass-damper system of Figure 2.1. Graphically find the spring constant for the equilibrium point of y = 0.5 cm and a range of operation of ±1.5 cm.

7(5)-

s2 + 30s + 200

R(s).

The input r(t) represents the desired position of the laser beam. (a) If r(t) is a unit step input, find the output y(t). (b) What is the final value of y{t)l Answer: (a) y{t) = 1 + 0.6
136

Chapter 2

Mathematical Models of Systems

«2

+

^Wv-

R(s)

r~\ _ .

Gito

—•

G2W

Hs)

i

+ 0-

WW

TT FIGURE E2.5 E2.6

(a)

A noninverting amplifier using an op-amp.

Filter

A nonlinear device is represented by the function y = /{X)

A

= e*,

where the operating point for the input x is x0 = 1. Determine a linear approximation valid near the operating point.

E2.7

E2.8

A lamp's intensity stays constant when monitored by an optotransistor-controlled feedback loop. When the voltage drops, the lamp's output also drops, and optotransistor Q\ draws less current. As a result, a power transistor conducts more heavily and charges a capacitor more rapidly [24]. The capacitor voltage controls the lamp voltage directly. A block diagram of the system is shown in Figure E2.7. Find the closed-loop transfer function, I(s)!R(s) where I{s) is the lamp intensity, and R(s) is the command or desired level of light.

\

Iris

Opaque tube

fb) FIGURE E2.7

N

Lamp controller.

R(S) KG,(*)G 2 (s)/j

1 + G1(s)H?,(s) +
A control engineer, N. Minorsky, designed an innovative ship steering system in the 1930s for the U.S. Navy. The system is represented by the block diagram shown in Figure E2.8, where Y(s) is the ship's course, /?(.?) is the desired course, and A(s) is the rudder angle [16]. Find the transfer function Y(s)IR(s).

+ //*(*)] +

A four-wheel antilock automobile braking system uses electronic feedback to control automatically the brake force on each wheel [15]. A block diagram model of a brake control system is shown in Figure E2.9, where iy(s) and FR(s) are the braking force of the front and rear wheels, respectively, and R{s) is the desired automobile response on an icy road. Find Ff(s)/R(s).

H2{s)

R(s)•

ky4

G?iW

ffjW

//,(•')

FIGURE E2.8

Ship steering system.

KCh(s)GAs)/s

A

«-

G2(s)

-*•

I s

Y(s)

137

Exercises H2(s)

R(s\ •

G2(.t)

*• FAx)

G3(s)

*• Fsis)

Plunger

G,(s)

H2(s) FIGURE E2.9

Brake control system.

Damping orifice E2.10 One of the most potentially beneficial applications of an automotive control system is the active control of the suspension system. One feedback control system uses a shock absorber consisting of a cylinder filled with a compressible fluid that provides both spring and damping forces [17].The cylinder has a plunger activated by a gear motor, a displacement-measuring sensor, and a piston. Spring force is generated by piston displacement, which compresses the fluid. During piston displacement, the pressure unbalance across the piston is used to control damping. The plunger varies the internal volume of the cylinder. This feedback system is shown in Figure E2.10. Develop a linear model for this device using a block diagram model.

Piston travel

Piston rod FIGURE E2.10

E2.ll A spring exhibits a force-versus-displacement characteristic as shown in Figure E2.ll. For small deviations from the operating point x0, find the spring constant when x0 is (a) -1.4; (b) 0; (c) 3.5.

Shock absorber.

FIGURE E2.11

E2.12 Off-road vehicles experience many disturbance inputs as they traverse over rough roads. An active suspension system can be controlled by a sensor that looks "ahead" at the road conditions. An example of a simple suspension system that can accommodate the bumps is shown in Figure E2.12. Find the appropriate

Spring characteristic.

gain Kx so that the vehicle does not bounce when the desired deflection is R{s) = 0 and the disturbance is Us). Answer: K^K^ = 1

Bump disturbance Preview of disturbance 1

'j( r;

<• *i

s

Desired «»> J deflection + S

J

._ FIGURE E2.12

dynamics

K2

+ ..

T<~>

Active suspension system.

G(.v)

m

Bounce of auto or

horizontal

138

Chapter 2

Mathematical Models of Systems

E2.13 Consider the feedback system in Figure E2.13. Compute the transfer functions Y(s)/Td(s) and Y(s)/N(s). E2.14 Find the transfer function

-

V2 =

Yj(s)

r(t) = desired platform position

for the multivariate system in Figure E2.14.

p{t) = actual platform position

E2.15 Obtain the differential equations for the circuit in Figure E2.15 in terms of ^ and i2.

V\{l) = amplifier input voltage v2(t) = amplifier output voltage

E2.16 The position control system for a spacecraft platform is governed by the following equations: dp

dt

dt

lV\.

The variables involved are as follows:

R2(s)

d2p

= 06*

0(/) = motor shaft position Sketch a signal-flow diagram or a block diagram of the system, identifying the component parts and determine the system transfer function P(s)/R(s).

—f + 2 - p + 4p = B 2 Vi=r-

p

W

Ms)

FIGURE E2.13 Feedback system with measurement noise, A/(s), and plant disturbances, Td(s).

Hl(5)

«.(*)

+

Gfr)

G7(s)

r~\

+v

G2(s)

G%(s)

G9{s)

+x~ +

1

Gjlis)

"•

Gb{s)

W

k

i k

R2(s)

G4{s)

-iF—

G5{s)

k

H2(s)

FIGURE E2.14 Multivariate system.

Ki(.V)

139

Exercises

E2.21 A high-precision positioning slide is shown in Figure E2.21. Determine the transfer function Xp(s)/Xm(s) when the drive shaft friction is bd = 0.7, the drive shaft spring constant is kd = 2, mc = 1, and the sliding friction is bs = 0.8.

FIGURE E2.15

Electric circuit.

Sliding friction, b.

E2.17 A spring develops a force /represented by the relation / = kx2, where x is the displacement of the spring. Determine a linear model for the spring when x0 = j -

FIGURE E2.21

Precision slide.

E2.18 The output y and input x of a device are related by y = x + 1.4x3. (a) Find the values of the output for steady-state operation at the two operating points x0 = 1 and x0 = 2. (b) Obtain a linearized model for both operating points and compare them.

E2.22 The rotational velocity &> of the satellite shown in Figure E2.22 is adjusted by changing the length of the beam L. The transfer function between
E2.19 The transfer function of a system is Y(s) _

AZ-(.v)

2{s + 4) (s + 5)(s + 1)2

15(.f + 1) The beam length change is AL(i) = 1/s. Determine the response of the rotation co(t).

R(s) ~ s2 + 9s + 14' Determine y{t) when r(t) is a unit step input. Answer: y(t) = 1.07 + l i e - * - 2.57e-7', t s 0

«(r) = 1.6 + 0.025e~5' - 1.625«-' - 1.5te-'

E2.20 Determine the transfer function VQ(s)/V{s) of the operational amplifier circuit shown in Figure E2.20. Assume an ideal operational amplifier. Determine the transfer function when /?, = R2 = 100 kfl, Cx = 10 jttF, and C2 = 5 fiF. C,

-1(i^l' Rotation

•t FIGURE E2.20

*

o+ FIGURE E2.22

-o —

Op-amp circuit.

E2.23 Determine the closed-loop transfer function T(s) = Y(s)/R(s) for the system of Figure E2.23.

140

Chapter 2

Mathematical Models of Systems E2.26 Determine the transfer function X2(s)/F(s) for the system shown in Figure E2.26. Both masses slide on a frictionless surface, and k = 1 N/m. X2(s) 1 Answer: F(s) s2(s2 + 2)

/?(.v) O — • •

FIGURE E2.23 Control system with three feedback loops.

MA/W-

/•*(/) E2.24 The block diagram of a system is shown in Figure E2.24. Determine the transfer function T(s) = Y(s)/R(s).

FIGURE E2.26 Two connected masses on a frictionless surface. R(s)

10 s+ 1

• • Y(s)

E2.27 Find the transfer function Y(s)/Td(s) for the system shown in Figure E2.27. Answer:

Y(s) Td(s)

G^s) 1 + G,(s)G2(s)H(s)

•O

TAs)

FIGURE E2.24 Multiloop feedback system.

> * E2.25 An amplifier may have a region of deadband as shown in Figure E2.25. Use an approximation that uses a cubic equation y = ax3 in the approximately linear region. Select a and determine a linear approximation for the amplifier when the operating point is JC = 0.6.

FIGURE E2.25 An amplifier with a deadband region.

C,(s)

- & *

G2{s)

H(s)

FIGURE E2.27 System with disturbance.

Yis)

141

Problems E2.28 Determine the transfer function \&(s)/V(s) for the op-amp circuit shown in Figure E2.28 [1]. Let /?j = 167 kfl, R2 = 240 kH, R3 = 1 kH, RA = 100 kH, and C = 1 /iF. Assume an ideal op-amp. E2.29 A system is shown in Fig. E2.29(a). (a) Determine G(s) and H(s) of the block diagram shown in Figure E2.29(b) that are equivalent to those of the block diagram of Figure E2.29(a).

(b) Determine Y(s)/R(s) for Figure E2.29(b). E2.30 A system is shown in Figure E2.30. (a) Find the closed-loop transfer function Y(s)/R(s) 10 when G(s) = - : . s2 + 2s + 10 (b) Determine Y(s) when the input R(s) is a unit step. (c) Compute y(t).

3r o—*—WV—"• + R,

-o +

±

_L

FIGURE E2.28 Op-amp circuit.

1

-k>

• ns)

/?(*)

s+ 10 FIGURE E2.30 Unity feedback control system.

(a)

R(s)

_n

• n.v)

E2.31 Determine the partial fraction expansion for V(s) and compute the inverse Laplace transform. The transfer function V(s) is given by: V(s)

400 s2 + Ss + 400

(b) FIGURE E2.29 Block diagram equivalence.

PROBLEMS Problems require an extension of the concepts of the chapter to new situations. P2.1 An electric circuit is shown in Figure P2.1. Obtain a set of simultaneous integrodifferential equations representing the network. P2.2 A dynamic vibration absorber is shown in Figure P2.2. This system is representative of many situations involving the vibration of machines containing unbalanced components. The parameters M2 and kl2 may be chosen so that the main mass Mi does not vibrate in the steady state when F(t) = a sin(a>0f)- Obtain the differential equations describing the system.

FIGURE P2.1 Electric circuit.

142

Chapter 2

Mathematical Models of Systems for the fluid-flow equation, (b) What happens to the approximation obtained in part (a) if the operating point is Pi - P2 = 0?

.J

Force

m

L

<;*!

••

J

>

"

AT,

| v .<')

P2.7 Obtain the transfer function of the differentiating circuit shown in Figure P2.7.

T>'2<'>

ftf2

P2.6 Using the Laplace transformation, obtain the current I2(s) of Problem P2.1. Assume that all the initial currents are zero, the initial voltage across capacitor C\ is zero, v{t) is zero, and the initial voltage across C2 is 10 volts.

1

FIGURE P2.2 Vibration absorber. P2.3 A coupled spring-mass system is shown in Figure P2.3. The masses and springs are assumed to be equal. Obtain the differential equations describing the system.

+ V2[s)

VAs)

-»• I ' I I O

Force ru)

v,(r) M

k

-*• A',(0

FIGURE P2.7 A differentiating circuit.

I

WWWH u k

Hi

nzn

FIGURE P2.3 Two-mass system. P2.4 A nonlinear amplifier can be described by the following characteristic: "o(') = J 4

I 4

P2.8 A bridged-T network is often used in AC control systems as a filter network [8]. The circuit of one bridged-T network is shown in Figure P2.8. Show that the transfer function of the network is V&) Kn(i')

1 + IR^Cs +

RiRjpV

1 + (2«, + R2)Cs + i ? , i ? 2 c V

Sketch the pole-zero diagram when Rx = 0.5,¾ = 1, and C = 0.5. «m < 0'

Tlie amplifier will be operated over a range of ±0.5 volts around the operating point for vin. Describe the amplifier by a linear approximation (a) when the operating point is sjj,, = 0 and (b) when the operating point is win = 1 volt. Obtain a sketch of the nonlinear function and the approximation for each case.

wv

P2.5 Fluid flowing through an orifice can be represented by the nonlinear equation Q = K(P, - A)" 2 , where the variables are shown in Figure P2.5 and K is a constant [2]. (a) Determine a linear approximation

FIGURE P2.8 Bridged-T network. P2.9 Determine the transfer function Xi(s)/F(s) for the coupled spring-mass system of Problem P2.3. Sketch the s-plane pole-zero diagram for low damping when M = l,b/k = l,and 4

FIGURE P2.5 Flow through an orifice.

2-

0.1.

P2.10 Determine the transfer function Yi{s)jF(s) for the vibration absorber system of Problem P2.2. Determine

143

Problems

the necessary parameters M2 and &12 so that the mass Ml does not vibrate in the steady state when F(t) — a sin(&)o t). P2.ll For electromechanical systems that require large power amplification, rotary amplifiers are often used

[8,19]. An amplidyne is a power amplifying rotary amplifier. An amplidyne and a servomotor are shown in Figure P2.ll. Obtain the transfer function 9(s)/Vc(s), and draw the block diagram of the system. Assume vd = k2iq and vq = k{ic.

Control field

it = Constant

lr/ 4

\ 2 1

Am plidyne

FIGURE P2.11

P

\

Ms)

FIGURE P2.12

**

Amplidyne and armature-controlled motor.

Process

A: K

V

f-

P2.12 For the open-loop control system described by the block diagram shown in Figure P2.12, determine the value of K such that y(t) - * 1 as t —» oo when r(r) is a unit step input. Assume zero initial conditions. Controller

( Motor 1

1 -—s+20

• YU)

Open-loop control system.

P2.13 An electromechanical open-loop control system is shown in Figure P2.13. The generator, driven at a constant speed, provides the field voltage for the motor. The motor has an inertia Jm and bearing friction />„,. Obtain

+ o-vV\A-|

the transfer function BL{s)fVf{s) and draw a block diagram of the system. The generator voltage »„ can be assumed to be proportional to the field current if. P2.14 A rotating load is connected to a field-controlled DC electric motor through a gear system. The motor is assumed to be linear. A test results in the output load reaching a speed of 1 rad/s within 0.5 s when a constant 80 V is applied to the motor terminals. The output steady-state speed is 2.4 rad/s. Determine the transfer function 0{s)/Vf(s) of the motor, in rad/V. The inductance of the field may be assumed to be negligible (see Figure 2.18). Also, note that the application of 80 V to the motor terminals is a step input of 80 V in magnitude. P2.15 Consider the spring-mass system depicted in Figure P2.15. Determine a differential equation to describe the motion of the mass m. Obtain the system response x(t) with the initial conditions A(0) = Xg and i(0) = 0.

Motor

01

N

i

Gear ratio n = —— No

Generator FIGURE P2.13

Motor and generator.

144

Chapter 2

Mathematical Models of Systems Obtain the relationship 7[3(s) between X^(s) and X3(s) by using Mason's signal-flow gain formula. Compare the work necessary to obtain 7^0) by matrix methods to that using Mason's signal-flow gain formula. P2.18 An LC ladder network is shown in Figure P2.18. One may write the equations describing the network as follows:

k, spring constant

FIGURE P2.15

Suspended spring-mass system.

P2.16 Obtain a signal-flow graph to represent the following set of algebraic equations where x\ and x2 are to be considered the dependent variables and 6 and 11 are the inputs: Xi + 1.5¾

=

6,

2JC,

h = (Vi ~ Vu)Yh

Va = (J, - /0)Z2,

I, = K - v2)y3,

v2 = / a z 4 .

Construct a flow graph from the equations and determine the transfer function K(s)/Vi(.r). h

L JTYYV Y,1

VAs)

V

L

0

Z2

11.

4A-,

Determine the value of each dependent variable by using the gain formula. After solving for JCJ by Mason's signal-flow gain formula, verify the solution by using Cramers rule. P2.17 A mechanical system is shown in Figure P2.17, which is subjected to a known displacement x$(t) with respect to the reference, (a) Determine the two independent equations of motion, (b) Obtain the equations of motion in terms of the Laplace transform, assuming that the initial conditions are zero, (c) Sketch a signalflow graph representing the system of equations, (d) FIGURE P2.18 LC ladder network. P2.19 A voltage follower (buffer amplifier) is shown in Figure P2.19. Show that T = vQ/vin = 1. Assume an ideal op-amp. + 0- Friction ,r~ A/, f. ,r" i M3 i *4 i < r *i FIGURE P2.19 6, J ZJ 1 r FIGURE P2.17 Mechanical system. A buffer amplifier. P2.20 The source follower amplifier provides lower output impedance and essentially unity gain. The circuit diagram is shown in Figure P2.20(a), and the small-signal model is shown in Figure P2.20(b).This circuit uses an FET and provides a gain of approximately unity. Assume that R2 » R] for biasing purposes and that Rg » R2. (a) Solve for the amplifier gain, (b) Solve for the gain when gm = 2000 (t£l and Rs = 10 kil where Rs = Ry + R2. (c) Sketch a block diagram that represents the circuit equations. P2.21 A hydraulic servomechanism with mechanical feedback is shown in Figure P2.21 [18]. The power piston has an area equal to A. When the valve is moved a small amount Az, the oil will flow through to the cylinder at a rate p • Az, where p is the port coefficient. The 145 Problems input oil pressure is assumed to be constant. From the (a) 'in -> G gs • •« >*a "it (P*"."* v geometry, we find that Az = &—-—(x - y) - —y. h h (a) Determine the closed-loop signal-flow graph or block diagram for this mechanical system, (b) Obtain the closed-loop transfer function Y{s)/X(s). P2.22 Figure P2.22 shows two pendulums suspended from frictionless pivots and connected at their midpoints by a spring [1]. Assume that each pendulum can be represented by a mass M a t the end of a massless bar of length L. Also assume that the displacement is small and linear approximations can be used for sin 8 and cos 8. The spring located in the middle of the bars is unstretched when fy = 82. The input force is represented by /(r), which influences the left-hand bar only, (a) Obtain the equations of motion, and sketch a block diagram for them, (b) Determine the transfer function T(s) = 8i(s)/F(s). (c) Sketch the location of the poles and zeros of T(s) on the s-plane. >xi Q 02 n/WWW (b) FIGURE P2.20 The source follower or common drain amplifier using an FET. Power cylinder FIGURE P2.22 The bars are each of length L and the spring is located at L/2. P2.23 The small-signal circuit equivalent to a commonemitter transistor amplifier is shown in Figure P2.23. The transistor amplifier includes a feedback resistor Rf. Determine the input-output ratio vcJv-m. Input pressure ->WSr rWV-o-*-^vVv—1 'c -4 6- *(* - v) *.© v bf FIGURE P2.23 ,,«,,© ®A CE amplifier. I Output, v FIGURE P2.21 Hydraulic servomechanism. P2.24 A two-transistor series voltage feedback amplifier is shown in Figure P2.24(a). This AC equivalent circuit 146 Chapter 2 Mathematical Models of Systems neglects the bias resistors and the shunt capacitors. A block diagram representing the circuit is shown in Figure P2.24(b).This block diagram neglects the effect of hn., which is usually an accurate approximation, and assumes that R2 + RL » R\- (a) Determine the voltage gain vjvin. (b) Determine the current gain ia/lbi(c) Determine the input impedance V\Jib\. P2.25 H. S. Black is noted for developing a negative feedback amplifier in 1927. Often overlooked is the fact that three years earlier he had invented a circuit de- 'ft i R R, + M sign technique known as feedforward correction [19], Recent experiments have shown that this technique offers the potential for yielding excellent amplifier stabilization. Black's amplifier is shown in Figure P2.25(a) in the form recorded in 1924. The block diagram is shown in Figure P2.25(b). Determine the transfer function between the output Y(s) and the input R(s) and between the output and the disturbance Td(s). G(s) is used to denote the amplifier represented by fi in Figure P2.25(a). "ir (bl (a; FIGURE P2.24 Feedback amplifier. ns) A'(v) • 1 J G(s) H^\PFIGURE P2.25 Black's amplifier. HS J V . (h) (a) P2.26 A robot includes significant flexibility in the arm members with a heavy load in the gripper [6, 20]. A two-mass model of the robot is shown in Figure. P2.26. Find the transfer function Y(s)IF(s). P2.27 Magnetic levitation trains provide a high-speed, very low friction alternative to steel wheels on steel rails. The train floats on an air gap as shown in Figure P2.27 [25]. The levitation force FL is controlled by the coil current i in the levitation coils and may be approximated by G(s) Pit)' VWWVWA k FIGURE P2.26 robot arm. The spring-mass-damper model of a where z is the air gap. This force is opposed by the downward force F = mg. Determine the linearized 147 Problems relationship between the air gap z and the controlling current near the equilibrium condition. input current i controls the torque with negligible friction. Assume the beam may be balanced near the horizontal ( = 0); therefore, we have a small deviation of . Find the transfer function X(s)/I(s). and draw a block diagram illustrating the transfer function showing ¢(5), X(s), and T(s). P2.30 The measurement or sensor element in a feedback system is important to the accuracy of the system [6]. The dynamic response of the sensor is important. Most sensor elements possess a transfer function T.S + 1 FIGURE P2.27 Cutaway view of train. P2.28 A multiple-loop model of an urban ecological system might include the following variables: number of people in the city (P), modernization (M), migration into the city (C), sanitation facilities (S), number of diseases (D), bacteria/area (B), and amount of garbage/area (G), where the symbol for the variable is given in parentheses. The following causal loops are hypothesized: 1. 2. 3. 4. Suppose that a position-sensing photo detector has T = 4,us and 0.999 < k < 1.001. Obtain the step response of the system, and find the k resulting in the fastest response—that is, the fastest time to reach 98% of the final value. P2.31 An interacting control system with two inputs and two outputs is shown in Figure P2.31. Solve for Yt(s)/Ri(s) and Y2(s)/R1(s) when R2 = 0. KM Yds) P^G^B^D-^P P-*M^C-^P P-*-M—S-*D-*P P^>M-*S^B-*D^>P Sketch a signal-flow graph for these causal relationships, using appropriate gain symbols. Indicate whether you believe each gain transmission is positive or negative. For example, the causal link S to B is negative because improved sanitation facilities lead to reduced bacteria/area. Which of the four loops are positive feedback loops and which are negative feedback loops? P2.29 We desire to balance a rolling ball on a tilting beam as shown in Figure P2.29. We will assume the motor Torque motor FIGURE P2.29 Tilting beam and ball. ffji*) . v i—: FIGURE P2.31 i i :—i r,(.v) Interacting System. P2.32 A system consists of two electric motors that are coupled by a continuous flexible belt. The belt also passes over a swinging arm that is instrumented to allow measurement of the belt speed and tension. The basic control problem is to regulate the belt speed and tension by varying the motor torques. An example of a practical system similar to that shown occurs in textile fiber manufacturing processes when yarn is wound from one spool to another at high speed. Between the two spools, the yarn is processed in a way that may require the yarn speed and tension to be controlled within defined limits. A model of the system is shown in Figure P2.32. Find J5(s)/i?j{5), Determine a relationship for the system that will make K independent of jRj. 148 Chapter 2 Mathematical Models of Systems -H2(s) FIGURE P2.32 A model of the coupled motor drives. Speed control input Speed R2(S) O r2[s) Tension Tension control input FIGURE P2.33 -HM Idle speed control system. P233 Find the transfer function for Y(s)/R(s) for the idlespeed control system for a fuel-injected engine as shown in Figure P2.33. P2.34 The suspension system for one wheel of an oldfashioned pickup truck is illustrated in Figure P2.34. The mass of the vehicle is m% and the mass of the wheel is m2-The suspension spring has a spring constant k^ and the tire has a spring constant k2. The damping constant of the shock absorber is b. Obtain the transfer function Y$$s)j'X(s), which represents the vehicle response to bumps in the road. P2.35 A feedback control system has the structure shown in Figure P2.35. Determine the closed-loop transfer function Y(s)/R(s) (a) by block diagram manipulation and (b) by using a signal-flow graph and Mason's signal-flow gain formula, (c) Select the gains /C, and K2 FIGURE P2.34 Pickup truck suspension. so that the closed-loop response to a step input is critically damped with two equal roots at s = -10. (d) Plot the critically damped response for a unit step 149 Problems tf(.v) ' 1 s s +1 YU) K <• ±l 1 ' K2 *•« X FIGURE P2.35 Multiloop feedback system. input. What is the time required for the step response to reach 90% of its final value? P2.36 A system is represented by Figure P2.36. (a) Determine the partial fraction expansion and y{t) for a ramp input, /(f) = t, t > 0. (b) Obtain a plot of y(t) for part (a), and find y(t) for l = 1.0 s. (c) Determine the impulse response of the system v(/) for ( 2 0. (d) Obtain a plot of y(t) for part (c) and find y(i) for ( = 1.0 s. FIGURE P2.37 Two-mass system. -0.5 m24 s* + 9s2 + 26s + 24 RU) • FIGURE P2.36 • * n.v) 4 Q- A third-order system. FIGURE P2.38 P237 A two-mass system is shown in Figure P2.37 with an input force u(t). When m | = m2 = l a n d ^ i = K2 = 1, find the set of differential equations describing the system. P238 A winding oscillator consists of two steel spheres on each end of a long slender rod, as shown in Figure P2.38. The rod is hung on a thin wire that can be twisted many revolutions without breaking. The device will be wound up 4000 degrees. How long will it take until the motion decays to a swing of only 10 degrees? Assume that the thin wire has a rotational spring constant of 2 X 10~ 4 Nm/rad and that the Winding oscillator. viscous friction coefficient for the sphere in air is 2 X 10~4 N m s/rad. The sphere has a mass of 1 kg. P2.39 For the circuit of Figure P2.39, determine the transform of the output voltage V0(s). Assume that the circuit is in steady state when t < 0. Assume that the switch moves instantaneously from contact 1 to contact 2 at t = 0. P2.40 A damping device is used to reduce the undesired vibrations of machines. A viscous fluid, such as a heavy oil, is placed between the wheels, as shown in 1 P 1 2H /YYY\ -If ~^^77Q »«?2fi 6V© FIGURE P2.39 Model of an electronic circuit. < £ > 0.5¾ ©lOe-aV 4ft 150 Chapter 2 Mathematical Models of Systems . Outer wheel Shaft . Inner wheel ./,,0, / % shown in Figure P2.42. As the mirror rotates, a friction force is developed that is proportional to its angular speed. The friction constant is equal to 0.06 N s/rad, and the moment of inertia is equal to 0.1 kg m2. The output variable is the velocity cu(r). (a) Obtain the differential equation for the motor, (b) Find the response of the system when the input motor torque is a unit step and the initial velocity at J = 0 is equal to 0.7. Fluid, b Mirror FIGURE P2.40 Bar code Cutaway view of damping device. Figure P2.40. When vibration becomes excessive, the relative motion of the two wheels creates damping. When the device is rotating without vibration, there is no relative motion and no damping occurs. Find B^s) and 02(s). Assume that the shaft has a spring constant K and that b is the damping constant of the fluid. The load torque is T. Reflected light P2.41 The lateral control of a rocket with a gimbaled enMicrocomputer gine is shown in Figure P2.41. The lateral deviation from the desired trajectory is h and the forward rocket speed is V. The control torque of the engine is £. FIGURE P2.42 Optical scanner. and the disturbance torque is Ttf. Derive the describing equations of a linear model of the system, and P2.43 An ideal set of gears is shown in Table 2.5, item 10. draw the block diagram with the appropriate transfer Neglect the inertia and friction of the gears and asfunctions. sume that the work done by one gear is equal to that of the other. Derive the relationships given in item 10 of Table 2.5. Also, determine the relationship between Aclua] Desired the torques Tm and TL. trajectory trajectory P2.44 An ideal set of gears is connected to a solid cylinder load as shown in Figure P2.44. The inertia of the motor shaft and gear G2 is Jm. Determine (a) the inertia of the load JL and (b) the torque T at the motor shaft. Assume the friction at the load is bL and the friction at the motor shaft is bm. Also assume the density of the load disk is p and the gear ratio is n. Hint: The torque at the motorshaft is given by T = T\ + Tm. Engine FIGURE P2.41 Rocket with gimbaled engine. P2.42 In many applications, such as reading product codes in supermarkets and in printing and manufacturing, an optical scanner is utilized to read codes, as FIGURE P2.44 Motor, gears, and load. P2.45 To exploit the strength advantage of robot manipulators and the intellectual advantage of humans, a class of manipulators called extenders has been examined Problems 151 [22]. The extender is defined as an active manipulator worn by a human to augment the human's strength. The human provides an input U(s), as shown in Figure P2.45. The endpoint of the extender is P(s). Determine the output P(s) for both U(s) and F(s) in the form P(s) = T^Uis) + T2(s)F(s). r '• B . G(s) H(s) , i Pis) 1 £(*) J I Performance ,, filter f | B(s) ,+ >r r K. -V GiW Load J h(t) w •v. Human P2.47 The water level h{t) in a tank is controlled by an open-loop system, as shown in Figure P2.47. A DC motor controlled by an armature current ;'„ turns a shaft, opening a valve. The inductance of the DC motor is negligible, that is, La = 0. Also, the rotational friction of the motor shaft and valve is negligible, that is, b = 0. The height of the water in the tank is Gc{s) j *— K(s) L *-n stability t controller V+ i i J [1.60(f) ~ h(t)]dt. the motor constant is K,„ = 10, and the inertia of the motor shaft and valve is J - 6 X KT3 kgm 2 . Determine (a) the differential equation for h(t) and v(t) and (b) the transfer function H(s)IV(s). P2.48 The circuit shown in Figure P2.48 is called a leadlag filter. (a) Find the transfer function V2(s)/\{(s). Assume an ideal op-amp. (b) Determine V2(s)/V^s) when l?j = 100Hl, R2 = 200 k£l, Q = 1 /JLF, and C2 = 0.1 fiF. (c) Determine the partial fraction expansion for P2.49 A closed-loop control system is shown in Figure P2.49. (a) Determine the transfer function FIGURE P2.45 Model of extender. P2.46 A load added to a truck results in a force F on the support spring, and the tire flexes as shown in Figure P2.46(a).The model for the tire movement is shown in Figure P2.46(b). Determine the transfer function X,(s)/F(s). T{s) = Y(s)/R(s). (b) Determine the poles and zeros of T(s). (c) Use a unit step input, .SKY) = 1/s, and obtain the partial fraction expansion for Y(s) and the value of the residues. Force of material placed in truck bed Truck vehicle mass *i r Shock absorber FIGURE P2.46 Truck support model. la) (h) 152 Chapter 2 Mathematical Models of Systems Amplifier Valve FIGURE P2.47 Open-loop control system for the water level of a tank. (e) Predict the final value of y(t) for the unit step input. )l—i r^Mh -AAAr R(s) ViM tf(.v) ' FIGURE P2.49 •*• i'(-v) V2(s) FIGURE P2.50 FIGURE P2.48 14,000 ,! 3 + 4 5 ^ + 31005 + 500) Third-order feedback system. Lead-lag filter. 6205 s(s-+ 13*+ 1281) •*• Y(s) P2.51 Consider the two-mass system in Figure P2.51. Find the set of differential equations describing the system. Unity feedback control system. (d) Plot y(t) and discuss the effect of the real and complex poles of T(s). Do the complex poles or the real poles dominate the response? P2.50 A closed-loop control system is shown in Figure P2.50. (a) Determine the transfer function T(s) = Y(s)/R(s). (b) Determine the poles and zeros of T(s). (c) Use a unit step input, R(s) = l/s, and obtain the partial fraction expansion for Y(s) and the value of the residues. (d) Plot y(() and discuss the effect of the real and complex poles of T(s). Do the complex poles or the real poles dominate the response? FIGURE P2.51 one damper. Two-mass system with two springs and 153 Advanced Problems ADVANCED PROBLEMS AP2.1 An armature-controlled DC motor is driving a load. The input voltage is 5 V. The speed at ( = 2 seconds is 30 rad/s, and the steady speed is 70 rad/s when t—*oo. Determine the transfer function I GdU) AP2.2 A system has a block diagram as shown in Figure AP2.2. Determine the transfer function T(s) = g(f) L Gc(s) A'(.v) • It is desired to decouple Y(s) from R\(s) by obtaining T(s) = 0. Select C 5 ( J ) in terms of the other Gj(s) to achieve decoupling. -»o- G(0 *- n-o H(s) FIGURE AP2.3 input. Feedback system with a disturbance Hi(s) R,(s) K_}~*" G'W G,W • /,(.0 C5(.v) JV ^o GeC) .,+ «-.(.0 • GiW • • K,(.0 sy*) 4 FIGURE AP2.2 Interacting control system. AP2.3 Consider the feedback control system in Figure AP2.3. Define the tracking error as E(t) = R(s) - Y(s). heat flow of the heating element. The system parameters are C„ £?, S, and Rr The thermal heating system is illustrated in Table 2.5. (a) Determine the response of the system to a unit step q(s) = 1/s. (b) As t—*oo. what value does the step response determined in part (a) approach? This is known as the steady-state response, (c) Describe how you would select the system parameters C„ Q, 5, and R, to increase the speed of response of the system to a step input. AP2.5 For the three-cart system illustrated in Figure AP2.5, obtain the equations of motion.The system has three inputs «j, 1¾. and u3 and three outputs JC-,, JC2and v3. Obtain three second-order ordinary differential equations with constant coefficients. If possible, write the equations of motion in matrix form. : l ". . VvWh >-.v. *i • M, ()() VvVA h .V, ' - * • M2 OO *3 V\AAA h M3 ()() (a) Determine a suitable H(s) such that the tracking error is zero for any input R(s) in the absence of a FIGURE AP2.5 Three-cart system with three inputs and disturbance input (that is, when Tlt(s) = 0). (b) Using three outputs. H{s) determined in part (a), determine the response Y(s) for a disturbance T,j(s) when the input R(s) = 0. AP2.6 Consider the hanging crane structure in Figure (c) Is it possible to obtain Y(s) = 0 for an arbitrary AP2.6. Write the equations of motion describing the disturbance T^(s) when G,i(s) ¥> 0? Explain your motion of the cart and the payload. The mass of the answer. cart is M, the mass of the payload is m, the massless rigid connector has length L, and the friction is modAP2.4 Consider a thermal heating system given by eled as Ft, = —b'x where x is the distance traveled by g(') _ 1 the cart. q(s) C,s + (QS + l/R.Y AP2.7 Consider the unity feedback system described in the block diagram in Figure AP2.7. Compute analytically where the output 3"(.?) is the temperature difference the response of the system to an impulse disturbance. due to the thermal process, the input q(s) is the rate of 154 Chapter 2 Mathematical Models of Systems FIGURE AP2.6 (a) Hanging crane supporting the Space Shuttle Atlantis (Image Credit: NASA/Jack Pfaller) and (b) schematic representation of the hanging crane structure. (a) (b) Td(s) Controller , EM FIGURE AP2.7 Unity feedback control system with controller Gc(s) - K. o K Determine a relationship between the gain K and the minimum time it takes the impulse disturbance response of the system to reach y(r) < 0.1. Assume that K > 0. For what value of K does the disturbance response first reach at y{t) = 0.1 at r = 0.05? AP2.8 Consider the cable reel control system given in Figure AP2.8. Find the value of A and K such that the percent overshoot is P.O. £ 10% and a desired velocity of 50 m/s in the steady state is achieved. Compute the closed-loop response v(f) analytically and confirm that the steady-state response and P. O. meet the specifications. , /?(,)= - ^ FIGURE AP2.8 Cable reel control system. Amplifier Desired velocity + K • > J » * Plant K Measured velocity + 1 +( s + 20 •+• Y(s) AP2.9 Consider the inverting operational amplifier in Figure AP2.9. Find the transfer function VJ,s){Vls), Show that the transfer function can be expressed as G(s) V,(s) = K, + — + K&, where the gains KP, Kh and KD are functions of Cj, C2, JRI, and R2, This circuit is a proportional-integral-derivative (PID) controller (more on PID controllers in Chapter 7). Reel dynamics Motor 200 s+ 1 Tachometer 1 0.25i + 1 Torque 1 j+8 Actual cable velocity • V{s) Design Problems R2 155 c-, Hf l-WV v,w< -° K,M FIGURE AP2.9 An inverting operational amplifier circuit representing a PID controller. DESIGN PROBLEMS CDP2.1 We want to accurately position a table for a machine as shown in Figure CDP2.1. A traction-drive motor with a capstan roller possesses several desirable characteristics compared to the more popular ball screw. The traction drive exhibits low friction and no backlash. However, it is susceptible to disturbances. Develop a model of the traction drive shown in Figure CDP2.1(a) for the parameters given in Table CDP2.1. The drive uses a DC armature-controlled motor with a capstan roller attached to the shaft.The drive bar moves the linear slide-table. The slide uses an air bearing, so its friction is negligible. We are considering the open-loop model, Figure CDP2.1(b), and its transfer function in this problem. Feedback will be introduced later. Table CDP2.1 Typical Parameters for the Armature-Controlled DC Motor and the Capstan and Slide Ms M,, •'m r bm Kn K„ Traction drive motor and capstan roller Linear slide (a) V„(-v) • G(s) X(s) Rm L'm Mass of slide Mass of drive bar Inertia of roller, shaft, motor and tachometer Roller radius Motor damping Torque constant Back emf constant Motor resistance Motor inductance 5.693 kg 6.96 kg 10.91 • lfT3 kg m2 31.75-10 - ½ 0.268 N ms/rad 0.8379 N m/amp 0.838 Vs/rad 1.36 Q, 3.6 mH the closed-loop transfer function Y(s)!R() is exactly equal to 1. DP2.2 The television beam circuit of a television is represented by the model in Figure DP2.2. Select the unknown conductance G so that the voltage v is 24 V. Each conductance is given in Siemens (S). DP2.3 An input r(t) = t, t a 0, is applied to a black box with a transfer function G(s). The resulting output response, when the initial conditions are zero, is y(0 = e - ' - ^ - 2 ' - ^ + | / , / ^ 0 . (b) FIGURE CDP2.1 (a) Traction drive, capstan roller, and linear slide, (b) The block diagram model. DP2.1 A control system is shown in Figure DP2.1. The transfer functions G2(s) and H2(s) are fixed. Determine the transfer functions G{(s) and //](.?) so that Determine G(s) for this system. DP2.4 An operational amplifier circuit that can serve as a filter circuit is shown in Figure DP2.4. Determine the transfer function of the circuit, assuming an ideal op-amp. Find vt)(t) when the input is Uj(f) = At, t >0. 156 Chapter 2 Mathematical Models of Systems W.v) -*• Y(s) FIGURE DP2.1 Selection of transfer functions. ff. Reference JT M. A/W R2 b v> C -o + "< Television beam circuit. DP2.5 Consider the clock shown in Figure DP2.5. The pendulum rod of length L supports a pendulum disk. Assume that the pendulum rod is a massless rigid thin rod and the pendulum disc has mass m. Design the length of the pendulum, L, so that the period of motion is 2 seconds. Note that with a period of 2 seconds each "tick" and each "tock" of the clock represents 1 second, as desired. Assume small angles, Pendulum rod Pendulum disk ^. FIGURE DP2.5 (a) Typical clock (photo courtesy of SuperStock) and (b) schematic representation of the pendulum. a ©20A f: "1 FIGURE DP2.2 «1 FIGURE DP2.4 Operational amplifier circuit. analysis so that sin Computer Problems 157 JM COMPUTER PROBLEMS CP2.1 Consider the two polynomials /?(X = s.21 + 7s + 10 and Forcing function q(s) = s + 2. Compute the following (a) p(s)q(s) (b) poles and zeros of G(s) = (C) /7(-1) in Pis) Plant s +2 A'f v i • FIGURE CP2.2 ,v+ 1 .I' Mass CP2.2 Consider the feedback system depicted in Figure CP2.2. (a) Compute the closed-loop transfer function using the series and feedback functions. (b) Obtain the closed-loop system unit step response with the step function, and verify that final value of the output is 2/5. Controller Spring < * constant j + 3 m A negative feedback control system. CP2.3 Consider the differential equation y + 4y + 3y = u, where y(0) = y(0) = 0 and u(t) is a unit step. Determine the solution y(t) analytically and verify by coplotting the analytic solution and the step response obtained with the step function. CP2.4 Consider the mechanical system depicted in Figure CP2.4.The input is given by/(i). and the output is y(t). Determine the transfer function from f(t) to y(t) and, using an m-file, plot the system response to a Controller 0dU) Desired • attitude FIGURE CP2.5 "> , k(s + a) s+b Friction constant b J, Mass displacement y(?) FIGURE CP2.4 A mechanical spring-mass-damper system. unit step input. Let in = 10, k = 1, and b = 0.5. Show that the peak amplitude of the output is about 1.8. CP2.5 A satellite single-axis attitude control system can be represented by the block diagram in Figure CP2.5. The variables k, a, and b are controller parameters, and J is the spacecraft moment of inertia. Suppose the nominal moment of inertia is J = 10.8E8 (slug ft2), and the controller parameters are k = 10.8E8, a = 1, and 6 = 8. (a) Develop an m-file script to compute the closedloop transfer function T(s) = 0(s)/0,i(s). (b) Compute and plot the step response to a 10° step input. (c) The exact moment of inertia is generally unknown and may change slowly with time. Compare the step response performance of the spacecraft when / i s reduced by 20% and 50%. Use the controller parameters k = 10.8E8, a = 1, and b = 8 and a 10° step input. Discuss your results. CP2.6 Consider the block diagram in Figure CP2.6. (a) Use an m-file to reduce the block diagram in Figure CP2.6, and compute the closed-loop transfer function. Spacecraft 1 J*2 0(1) altitude A spacecraft single-axis attitude control block diagram. 158 Chapter 2 Mathematical Models of Systems + r~\ ~i\. Ris) • 1 s+ 1 s s2 + 2 \r s2 + • 4s+ 2 *• ¥(s) 50 s2 + 2s + 1 s2 + 2 ,93 + 14 FIGURE CP2.6 A multiple-loop feedback control system block diagram. (b) Generate a pole-zero map of the closed-loop transfer function in graphical form using the pzmap function. (c) Determine explicitly the poles and zeros of the closed-loop transfer function using the pole and zero functions and correlate the results with the pole-zero map in part (b). CP2.7 For the simple pendulum shown in Figure CP2.7, the nonlinear equation of motion is given by 0(0 + ™ sin 6 ^ ^ ^ . ^ Plot the response of the system when R(s) is a unit step for the parameter z = 5,10, and 15. CP2.9 Consider the feedback control system in Figure CP2.9, where (a) (b) (c) 0. Create an m-file to plot both the nonlinear and the linear response of the simple pendulum when the initial angle of the pendulum is 0(0) = 30° and explain any differences. ^ X(s) (20/z)(s + z) B{s) ~ s2 + 3s + 20" 0, where L = 0.5 m, m = 1 kg, and g = 9.8 m/s~. When the nonlinear equation is linearized about the equilibrium point 6 = 0, we obtain the linear time-invariant model, 0 + j6 CP2.8 A system has a transfer function ^ ^ (d) G(s) = ^ ^ and H(s) s +2 s+ 1 Using an m-file, determine the closed-loop transfer function. Obtain the pole-zero map using the pzmap function. Where are the closed-loop system poles and zeros? Are there any pole-zero cancellations? If so, use the minreal function to cancel common poles and zeros in the closed-loop transfer function. Why is it important to cancel common poles and zeros in the transfer function? lite) • G(s) •*• Y(s) H(s) 4 FIGURE CP2.9 Control system with nonunity feedback. FIGURE CP2.7 Simple pendulum. CP2.10 Consider the block diagram in Figure CP2.10. Create an m-file to complete the following tasks: (a) Compute the step response of the closed-loop system (that is, R(s) = Vs and 7",,(i) = 0) and plot the steady-state value of the output Y(s) as a function of the controller gain 0 < K s 10. (b) Compute the disturbance step response of the closed-loop system (that is, R(s) = 0 and 159 Terms and Concepts Td(s) — lis) and co-plot the steady-state value of the output Y (s) as a function of the controller gain 0 < K < 10 on the same plot as in (a) above. (c) Determine the value of K such that the steadystate value of the output is equal for both the input response and the disturbance response. ''» FIGURE CP2.10 Block diagram of a unity feedback system with a reference input R[s) and a disturbance input Td(s). m *> Y(s) R(s) ANSWERS TO SKILLS CHECK True or False: (1) False; (2) True; (3) False; (4) True; (5) True Multiple Choice: (6) b; (7) a; (8) b; (9) b; (10) c; (11) a; (12) a; (13) c; (14) a; (15) a Word Match (in order, top to bottom): e, j , d, h, a, f, c, b, k, g, o, 1, n, m, i TERMS AND CONCEPTS Across-Variable A variable determined by measuring the difference of the values at the two ends of an element. closed or otherwise accounted for. Generally obtained by block diagram or signal-flow graph reduction. Actuator The device that causes the process to provide the output. The device that provides the motive power to the process. Coulomb damper A type of mechanical damper where the model of the friction force is a nonlinear function of the mass velocity and possesses a discontinuity around zero velocity. Also know as dry friction. Analogous variables Variables associated with electrical, mechanical, thermal, and fluid systems possessing similar solutions providing the analyst with the ability to extend the solution of one system to all analogous systems with the same describing differential equations. Assumptions Statements that reflect situations and conditions that are taken for granted and without proof. In control systems, assumptions are often employed to simplify the physical dynamical models of systems under consideration to make the control design problem more tractable. Block diagrams Unidirectional, operational blocks that represent the transfer functions of the elements of the system. Branch A unidirectional path segment in a signal-flow graph that relates the dependency of an input and an output variable. Characteristic equation The relation formed by equating to zero the denominator of a transfer function. Closed-loop transfer function A ratio of the output signal to the input signal for an interconnection of systems when all the feedback or feedfoward loops have been Critical damping The case where damping is on the boundary between underdamped and overdamped. Damped oscillation An oscillation in which the amplitude decreases with time. Damping ratio A measure of damping. A dimensionless number for the second-order characteristic equation. DC motor An electric actuator that uses an input voltage as a control variable. Differential equation An equation including differentials of a function. Error signal The difference between the desired output R{s) and the actual output Y(s); therefore E{s) = R(s) - Y(s). Final value The value that the output achieves after all the transient constituents of the response have faded. Also referred to as the steady-state value. Final value theorem The theorem that states that lim y(t) = lim .sY(.y), where Y(s) is the Laplace transform of y(t). Chapter 3 State Variable Models CHECK In this section, we provide three sets of problems to test your knowledge:True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. In the following True or False and Multiple Choice problems, circle the correct answer, 1. The state variables of a system comprise a set of variables that describe the future response of the system, when given the present state, all future excitation inputs, and the mathematical model describing the dynamics. True or False 2. The matrix exponential function describes the unforced response of the system and is called the state transition matrix. True or False 3. The outputs of a linear system can be related to the state variables and the input signals by the state differential equation. True or False 4. A time-invariant control system is a system for which one or more of the parameters of the system may vary as a function of time. True or False 5. A state variable representation of a system can always be written in diagonal form. True or False 6. Consider a system with the mathematical model given by the differential equation: 2 d3y .Jy 5-4+ 10-d—r1y + 5— + 2v = u 3 dt dt dt A state variable representation of the system is: "l " - 2 - 1 -0.4" a. x = 1 0 0 X+ 0 u 0 1 0 _0_ 0 0 0.2]x y = " - 5 - 1 -0.7~ "-1~ b. x = 1 0 0 X+ 0 u 0 1 0 0_ y = [0 0 0.2]x c. x = "-2 -f 1 0_ y = [i x+ "l u 0 0]x ~r " - 2 - 1 -0.4" 1 0 0 X+ 0 u 0 1 0 _0_ y = [1 0 0.2]x d. x = For Problems 7 and 8, consider the system represented by X = A3i + B u, where A = and B = 7. The associated state-transition matrix is: a. (f,0) = [5f] "1 5r" b. ¢(¢,0) = 10 1 215 Skills Check "1 5/' .1 1 . " l 5/ d. #(/,0) = 0 1 0 0 c ¢(/,0) = /2~ t 1_ 8. For the initial conditions x t (0) = .*2(0) = 1, the response x(t) for the zero-input response is: a. x{(t) = (1 + t),x2(t) = l f o r / > 0 b. xx(t) = (5 + t),x2(t) = /for/ > 0 c. jci(f) = (5/ + 1),JC2(/) = 1 for/ > 0 d. xx(t) = x2(t) = l f o r / > 0 9. A single-input, single-output system has the state variable representation 0 1 x + _ -5 -10 y = [0 I0]x The transfer function of the system T(s) = Y(s)/U(s) is "50 TV ^ + 5s2 + 50s -50 a. T{s) = s2 + 10^ + 5 b. T(s) = -5 s + 5 -50 c T(s) = d. T(s) = 2 5 3 s + 55 + 5 10. The differential equation model for two first-order systems in series is x(t) + 4x{t) + 3x(t) = u(t), where u(t) is the input of the first system and x(t) is the output of the second system. The response x(t) of the system to a unit impulse «(/) is: a. x(t) = e~' - 2e~2' b. x(t) = -e~ , X l -t l ,-3/ c. x(t) = — 2 e ' - —e 2 d. x(t) = e~' - e~3' 11. A first-order dynamic system is represented by the differential equation Sx(t) + x(t) = u(t). The corresponding transfer function and state-space representation are x = -0.2* + 0.5« rfc^ — and G{S) 1 + 55 y = OAx b. G W 10 = 1 + 5s CI c\ C{S) "'5 + 5 d. None of the above and and x = -0.2JC + u y =x x — — 5x + u y = x 216 Chapter 3 State Variable Models Consider the block diagram in Figure 3.43 for Problems 12 through 14: Controller +^ RU) Process o Ea{s) •O FIGURE 3.43 10 s+ 10 +>Y{s) Block diagram for the Skills Check. 12. The effect of the input R(s) and the disturbance Td(s) on the output Y(s) can be considered independently of each other because: a. This is a linear system, therefore we can apply the principle of superposition. b. The input R(s) does not influence the disturbance Td(s). c. The disturbance Td(s) occurs at high frequency, while the input R(s) occurs at low frequency. d. The system is causal. 13. The state-space representation of the closed-loop system from R(s) to Y(s) is: x = - 1 0 * + lOKr a. y= x x = -(10 + lOK)x + r b. y = 10* . 1 - = - ( 1 0 + IOJQJC + lOKr c v= x d. None of the above 14. The steady-state error E{s) = Y{s) - R(s) due to a unit step disturbance T(t(s) = l/.v is: a. ess = lim e(t) = oo f->0O b. ess = Iime(/) = 1 1 c. e^ = lime(r) = /-»00 K + 1 d. ess = lim e(t) = K + I /—»oo 15. A system is represented by the transfer function 5(5 + 10) = T(s) = 3 R{s) 5 + 10s2 + 205 + 50' A state variable representation is: 50" -10 -20 0 x+ i 1 0 _0_ 0 1 0 _ y = [0 5 50]x ~r "-10 1 0 y = [l 0 b. x = -20 0 1 50]x 50" "l" 0 x+ 0 0 _ _0_ Exercises 217 •10 c x = y = [0 5 d. x y -10 0 [0 5]x -20 0 1 50]x -20" 1 x -50 0 x + 0 + In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. State vector b. State of a system c. Time-varying system d. Transition matrix e. State variables f. State differential equation g. Time domain The differential equation for the state vector x = Ax + Bw. The matrix exponential function that describes the unforced response of the system. The mathematical domain that incorporates the time response and the description of a system in terms of time, t. Vector containing alln state variables, xx, x2,---, xn. A set of numbers such that the knowledge of these numbers and the input function will, with the equations describing the dynamics, provide the future state of the system. A system for which one or more parameters may vary with time. The set of variables that describe the system. EXERCISES E3.1 For the circuit shown in Figure E3.1 identify a set of state variables. A= ° .-1 ' -2_ Find the characteristic roots of the system. Answer: - 1 , - 1 E3.4 Obtain a state variable matrix for a system with a differential equation d*y FIGURE E3.1 dt*j RLC circuit. E3.2 A robot-arm drive system for one joint can be represented by the differential equation [8] dv(t) dt -kxv(t) + 4 d2y ^dt2 + 6i; dy + 8 y = 20u W- E3.5 A system is represented by a block diagram as shown in Figure E3.5. Write the state equations in the form of Equations (3.16) and (3.17). - k2y(t) + k3i(t), where v(t) = velocity, y(t) = position, and i(t) is the control-motor current. Put the equations in state variable form and set up the matrix form for kx = k2 = 1. E33 A system can be represented by the state vector differential equation of Equation (3.16), where U(s) FIGURE E3.5 Y(s) Block diagram. 218 Chapter 3 State Variable Models E3.6 A system is represented by Equation (3.16), where "0 1 0 0 A= 0 1 A= 6 -5 (a) Find the matrix ¢(0- (b) For the initial conditions (a) Find the roots of the characteristic equation. (b) Find the state transition matrix ¢(0- JC,(0) = -t2(0) = l,findx(0. Answer: (a) .s = - 3 , —2 Answer: (b) .Vj = 1 + t, x2 = 1, t > 0 E3.7 Consider the spring and mass shown in Figure 3.3 where M = 1 kg, k = 100 N/m, and b = 20 Ns/m. (a) Find the state vector differential equation, (b) Find the roots of the characteristic equation for this system. 1 0 Answer: (a) x -20 J 100 Y + 3e - 2 ' - 2e~ e"3' - e~2 (b) ¢ ( 0 - 6 e - 3 ' + 6 E3.ll Determine a state variable representation for the system described by the transfer function "o" _lj 7\s) = R(s) 4(5 + 3) (s + 2)(5 + 6)' (b) s = -10, -10 E3.8 The manual, low-altitude hovering task above a E3.12 Use a state variable model to describe the circuit moving landing deck of a small ship is very demandof Figure E3.12. Obtain the response to an input unit step when the initial current is zero and the initial ing, particularly in adverse weather and sea condicapacitor voltage is zero. tions. The hovering condition is represented by the matrix 0 0 0 1 0 -6 0 1 -3 •,;e Find the roots of the characteristic equation. E3.9 A multi-loop block diagram is shown in Figure E3.9.The state variables are denoted by .Vj and .v2. (a) Determine a state variable representation of the closed-loop system where the output is denoted by y(t) and the input is /-(0- (b) Determine the characteristic equation. FIGURE E3.12 C = 800/XF: RLC series circuit. E3.13 A system is described by the two differential equations cfy dt *l 1 s 1 ' + Y{s) i — 1 2 FIGURE E3.9 — dt J '+ + y — 2u + a%v = 0, and 4— -1^ Ws) * J0 L = 0.2 H •- _ T T Y Y \ • A M x2 1 s 4 Multi-loop feedback control system. E3.10 A hovering vehicle control system is represented by two state variables, and [13] by + Au = 0, where w and y are functions of time, and u is an input u(t). (a) Select a set of state variables, (b) Write the matrix differential equation and specify the elements of the matrices, (c) Find the characteristic roots of the system in terms of the parameters a and b. Answer: (c) s = - 1 / 2 ± V l - Aab/2 E3.14 Develop the state-space representation of a radioactive material of mass M to which additional radioactive material is added at the rate ; ( 0 = Ku(t), where K is a constant. Identify the state variables. 219 Exercises E3.15 Consider the case of the two masses connected as shown in Figure E3.15. The sliding friction of each mass has the constant b. Determine a state variable matrix differential equation. where R, Lu L2 and C are given constants, and -¾ and vh are inputs. Let the state variables be defined as *! = I'I, x2 = l2, and x3 = v. Obtain a state variable representation of the system where the output is x3. E3.19 A single-input, single-output system has the matrix equations and y = [10 0]x. Determine the transfer function G(s) = Y(s)/U(s). FIGURE E3.15 Two-mass system. E3.16 Two carts with negligible rolling friction are connected as shown in Figure E3.16. An input force is u(t). The output is the position of cart 2. that is, y{t) = q(t). Determine a state space representation of the system. H(0 Inpul ' force FIGURE E3.16 Hi m2 Hi Answer: G(s) = -z 1° s + 4s + 3 E3.20 For the simple pendulum shown in Figure E3.20, the nonlinear equations of motion are given by e + f- sin e + —e = o, L m where g is gravity, L is the length of the pendulum, m is the mass attached at the end of the pendulum (we assume the rod is massless), and k is the coefficient of friction at the pivot point. (a) Linearize the equations of motion about the equilibrium condition 6 = 0", (b) Obtain a state variable representation of the system. The system output is the angle 6. Two carts with negligible rolling friction. E3.17 Determine a state variable differential matrix equation for the circuit shown in Figure E3.17: ZZ&ZZZX&& Pivot point Massless rod FIGURE E3.17 in, mass RC circuit. E3.18 Consider a system represented by the following differential equations: rf/, R>i + Lt — + v = v„ L di2 ^ + V = V " dv ix + i2 = C ~dt FIGURE E3.20 Simple pendulum. E3.21 A single-input, single-output system is described by x(r) = 0 I -1 v(0 = [o i]x« X(f) + 0 M(0 220 Chapter 3 State Variable Models Obtain the transfer function G(s) = Y(s)/U(s) and determine the response of the system to a unit step input. E3.22 Consider the system in state variable form x = Ax + Bit y = Cx + Du x = ax + bu y — cx + du where a, b, c, and d are scalars such that the transfer function is the same as obtained in (a). E3.23 Consider a system modeled via the third-order differential equation with A = 'x\t) + 3x(0 + 3x(0 + x{t) "3 2' ,B = b 4J "1' L-iJ , C = [1 = u(t) + 2ii(t) + 4ii(t) + (t). 0], and D = [0]. Develop a state variable representation and obtain a block diagram of the system assuming the output is x(t) and the input is u(t). (a) Compute the transfer function G(s) = Y(s)/U(s). (b) Determine the poles and zeros of the system, (c) If possible, represent the system as a first-order system PROBLEMS P3.1 An RLC circuit is shown in Figure P3.1. (a) Identify a suitable set of state variables, (b) Obtain the set of first-order differential equations in terms of the state variables, (c) Write the state differential equation. -A/Wv(t) + Voltage ^ y source P33 An RLC network is shown in Figure P3.3. Define the state variables as x^ = iL and x2 = vc. Obtain the state differential equation. Partial answer: A = L R 0 •1/C ~j[ L FIGURE P3.1 \/L -l/(RC) RLC circuit. © P3.2 A balanced bridge network is shown in Figure P3.2. (a) Show that the A and B matrices for this circuit are - 2 / ( ( ^ + R2)C) 0 0 -2RlR2/((Rl + R2)L)_ 1/C B = 1/(/^ + R2) IR2/L FIGURE P3.3 P3.4 The transfer function of a system is 1/C -R2JL] (b) Sketch the block diagram. The state variables are (xh x2) = (vc, iL). RLC circuit. T(s) Y(s) s2 + 2s + 10 R(s) ~ s* + 4s2 + 6s + 10' Sketch the block diagram and obtain a state variable model. P3.5 A closed-loop control system is shown in Figure P3.5. (a) Determine the closed-loop transfer function T(s) = Y(s)IR(s). (b) Sketch a block diagram model for the system and determine a state variable model. P3.6 Determine the state variable matrix equation for the circuit shown in Figure P3.6. Let Xj = V\, x2 = V2, and JC3 = FIGURE P3.2 Balanced bridge network. i. P3.7 An automatic depth-control system for a robot submarine is shown in Figure P3.7.The depth is measured 221 Problems Controller -N 3 RU) — K I FIGURE P3.5 Closed-loop system. > Voltage Velocity s+ l s +6 1 .9-2 R[s) Desired depth 1 Hs) Position Actuator + /~N S K G(s) - m Depth 1 s Pressure measurement ii^, FIGURE P3.6 RLC circuit. FIGURE P3.7 by a pressure transducer. The gain of the stern plane actuator is K = 1 when the vertical velocity is 25 m/s. The submarine has the transfer function r•<^ {s Cis) = + Submarine depth control. Module 1 ) 2 7TT and the feedback transducer is H(s) = 2s + 1. Determine a state variable representation for the system. P3.8 The soft landing of a lunar module descending on the moon can be modeled as shown in Figure P3.8. Define the state variables as Xi = y, JC2 = dyldt, x3 = m and the control as u = dmldt. Assume that g is the gravity constant on the moon. Find a state variable model for this system. Is this a linear model? P3.9 A speed control system using fluid flow components is to be designed. The system is a pure fluid control system because it does not have any moving mechanical parts. The fluid may be a gas or a liquid. A system is desired that maintains the speed within 0.5% of the desired speed by using a tuning fork reference and a valve actuator. Fluid control systems are insensitive and reliable over a wide range of temperature, electromagnetic and nuclear radiation, Lunar surface FIGURE P3.8 Lunar module landing control. acceleration, and vibration. The amplification within the system is achieved by using a fluid jet deflection amplifier. The system can be designed for a 500-kW steam turbine with a speed of 12,000 rpm. The block diagram of the system is shown in Figure P3.9. In dimensionless units, we have ¢ = 0 . 1 , / = 1, and 7-,,(.0 Ris) Speed • reference FIGURE P3.9 Steam turbine control. "> fc Filter Valve actuator 10 5+10 1 s Tuning fork and error detector - A.] •« Disturbance 4- Turbine 1 Js + b ^ (0(X) Speed 222 Chapter 3 State Variable Models Ky = 0.5. (a) Determine the closed-loop transfer function (o(s) Tis) =m (b) Determine a state variable representation, (c) Determine the characteristic equation obtained from the A matrix. P3.10 Many control systems must operate in two dimensions, for example, the x- and the y-axes. A two-axis control system is shown in Figure P3.10, where a set of state variables is identified.The gain of each axis is Ki and K2, respectively, (a) Obtain the state differential equation, (b) Find the characteristic equation from the A matrix, (c) Determine the state transition matrix for Ki = 1 and K2 = 2. P3.ll A system is described by x = Ax + Bu where A = 1 L 2 -2~ ,B = -3 J V LuJ and X](0) = x2(0) = 10. Determine x{(t) and x2(t). _ T(s) = R(s) = T(s) = 8(5 + 5) = = ^ R.O—+ +—Or, K2O "*—02 y,(s) 1—*- >Uv) (b) U(s + 4) s 3 + io.v2 + 31s + 16' P3.16 The dynamics of a controlled submarine are significantly different from those of an aircraft, missile, or surface ship. This difference results primarily from the moment in the vertical plane due to the buoyancy effect. Therefore, it is interesting to consider the control (a) FIGURE P3.10 Two-axis system. (a) Signal-flow graph, (b) Block diagram model. s + 50 sA + 12s3 + IO52 + 34s + 50' P3.15 Obtain a block diagram and a state variable representation of this system. y(s) R(s) P3.12 A system is described by its transfer function Y(s) R~(s) (a) Determine a state variable model. (b) Determine ( 0 , the state transition matrix. P3.13 Consider again the RLC circuit of Problem P3.1 when R = 2.5, L = 1/4. and C = 1/6. (a) Determine whether the system is stable by finding the characteristic equation with the aid of the A matrix. (b) Determine the transition matrix of the network, (c) When the initial inductor current is 0.1 amp, vc(0) = 0, and v(t) = 0, determine the response of the system, (d) Repeat part (c) when the initial conditions are zero and v(t) = E, for t > 0, where E is a constant. P3.14 Determine a state variable representation for a system with the transfer function 223 Problems of the depth of a submarine. The equations describing the dynamics of a submarine can be obtained by using Newton's laws and the angles defined in Figure P3.16. To simplify the equations, we will assume that 8 is a small angle and the velocity v is constant and equal to 25 ft/s.The state variables of the submarine, considering only vertical control, are X\ = 6, x2 = dOldt, and x = a, where a is the angle of attack. Thus the state vector differential equation for this system, when the submarine has an Albacore type hull, is Elbow ^i Kr) Current Motor FIGURE P3.18 Corporation.) Wrist 4= k VvWA k,b h An industrial robot. (Courtesy of GCA P3.19 Consider the system described by 0 0.0071 0 x = 1 -0.111 0.07 0 0.12 x + -0.3 0 -0.095 u(t\ +0.072 where u{t) = 8s(t), the deflection of the stem plane, (a) Determine whether the system is stable, (b) Determine the response of the system to a stern plane step command of 0.285° with the initial conditions equal to zero. 0 „ -\^J\ \. FIGURE P3.16 Velocity v Submarine depth control. P3.17 A system is described by the state variable equations 1 3 1 y = [l -1 0 0 x + 0 10_ _4 0 0]x. Determine G(s) = Y(s)/U(s). P3.18 Consider the control of the robot shown in Figure P3.18.The motor turning at the elbow moves the wrist through the forearm, which has some flexibility as shown [16]. The spring has a spring constant k and friction-damping constant b. Let the state variables be *i = 1 0 -2 x(0« where x(t) = [*a(/) x2(t)]T. (a) Compute the state transition matrix (f, 0). (b) Using the state transition matrix from (a) and for the initial conditions JC^O) = 1 and x2(0) = - 1 , find the solution x(r) for t > 0. P3.20 A nuclear reactor that has been operating in equilibrium at a high thermal-neutron flux level is suddenly shut down. At shutdown, the density X of xenon 135 and the density I of iodine 135 are 7 X 1016 and 3 X 1015 atoms per unit volume, respectively. The halflives of I135 and Xej35 nucleides are 6.7 and 9.2 hours, respectively. The decay equations are [15,19] I = Control surface 1 4 -2 i(0 = 0.693 1, ' 6.7 X = 0.693 X - I. " 9.2 Determine the concentrations of I135 and Xei 35 as functions of time following shutdown by determining (a) the transition matrix and the system response. (b) Verify that the response of the system is that shown in Figure P3.20. P3.21 Consider the block diagram in Figure P3.21. (a) Verify that the transfer function is G(s) = Y(s) hxs + h0 + axhx U(s) sz + ais + a0 (b) Show that a state variable model is given by 0 1 y = [l x+ u, 0]x. 4>\ ~ 2 and x2 = (O^COQ, where a>l = Write the state variable equation in matrix form when x3 = a)2Ia>0. P3.22 Determine a state variable model for the circuit shown in Figure P3.22. The state variables are JCJ = I, x2 = V\, and x3 = 1¾. The output variable is P3.23 The two-tank system shown in Figure P3.23(a) is controlled by a motor adjusting the input valve and 224 Chapter 3 State Variable Models X, I 7 1 \ I 1 ?; I § A I 2 v 1 X = Xenon 135 I = Iodine 135 X \io16 \ 1 1015 \ 1 FIGURE P3.20 Nuclear reactor response. 10 15 20 25 Time (t) in hours 30 35 40 -o + K V(s) • fc V J 4 -K % s Y(s) ;Cj Oulput ^o voltage \ «1 FIGURE P3.22 «0 FIGURE P3.21 Model of second-order system. ultimately varying the output flow rate. The system has the transfer function = G(.v) /(.0 Input signal Motor and valve RLC circuit. 1 Valve | 1 ^3¾ I&M S3 + 1 0 J 2 + 31s + 30 (it = 3.V) + Hi + U% <20(.v) Output flow for the block diagram shown in Figure P3.23(b). Obtain a block diagram model and a state variable model. P3.24 It is desirable to use well-designed controllers to maintain building temperature with solar collector space-heating systems. One solar heating system can be described by [10] Q,(s) (a) /CO i n p Ut Input signal Q„U) 1 s+5 am i s+2 Qii-i) 1 s+3 Output flow lb) and dx2 dt 2*2 + «2 + d, FIGURE P3.23 A two-tank system with the motor current controlling the output flow rate, (a) Physical diagram. (b) Block diagram. 225 Problems where x-, = temperature deviation from desired equilibrium, and x2 = temperature of the storage material (such as a water tank). Also, u\ and u2 are the respective flow rates of conventional and solar heat, where the transport medium is forced air. A solar disturbance on the storage temperature (such as overcast skies) is represented by d. Write the matrix equations and solve for the system response when «i = 0, H2 = l,andrf = 1, with zero initial conditions. P3.25 A system has the following differential equation: x + r{t). Determine 4>(r) and its transform ¢(^) for the system. P3.26 A system has a block diagram as shown in Figure P3.26. Determine a state variable model and the state transition matrix ¢(5). 1 s+ 3 25 R(s) 1 s Y(s) 3 25 FIGURE P3.26 Feedback system. P3.27 A gyroscope with a single degree of freedom is shown in Figure P3.27. Gyroscopes sense the angular motion of a system and are used in automatic flight control systems. The gimbal moves about the output axis OB. The input is measured around the input axis OA. The equation of motion about the output axis is obtained by equating the rate of change of angular momentum to the sum of torques. Obtain a statespace representation of the gyro system. vwvw- 1{ '»1 Force O k O 0 0 Rolling friction constant = b FIGURE 3 . 2 8 Two-mass system. P3.29 There has been considerable engineering effort directed at finding ways to perform manipulative operations in space—for example, assembling a space station and acquiring target satellites. To perform such tasks, space shuttles carry a remote manipulator system (RMS) in the cargo bay [4,12,21]. The RMS has proven its effectiveness on recent shuttle missions, but now a new design approach can be considered—a manipulator with inflatable arm segments. Such a design might reduce manipulator weight by a factor of four while producing a manipulator that, prior to inflation, occupies only one-eighth as much space in the cargo bay as the present RMS. The use of an RMS for constructing a space structure in the shuttle bay is shown in Figure P3.29(a), and a model of the flexible RMS arm is shown in Figure P3.29(b), where J is the inertia of the drive motor and L is the distance to the center of gravity of the load component. Derive the state equations for this system. Space structure Spinning wheel « 7 Manipulator (a) Gimbal Load mass I, M B Output axis FIGURE P3.27 Gyroscope. P3.28 A two-mass system is shown in Figure P3.28. The rolling friction constant is /;. Determine a state variable representation when the output variable is yi(t). FIGURE P3.29 Remote manipulator system. P3.30 Obtain the state equations for the two-input and one-output circuit shown in Figure P3.30, where the output is i%. 226 Chapter 3 State Variable Models AW—r R, "'© ©«* FIGURE P3.30 Two-input RLC circuit. P3.31 Extenders are robot manipulators that extend (that is, increase) the strength of the human arm in load-maneuvering tasks (Figure P3.31) [19, 22], The system is represented by the transfer function G(s) 30 s* + 4? + 3 where U(s) is the force of the human hand applied to the robot manipulator, and Y{s) is the force of the robot manipulator applied to the load. Determine a state variable model and the state transition matrix for the system. Extender of the mass of the drug in the gastrointestinal tract is equal to the rate at which the drug is ingested minus the rate at which the drug enters the bloodstream, a rate that is taken to be proportional to the mass present. The rate of change of the mass in the bloodstream is proportional to the amount coming from the gastrointestinal tract minus the rate at which mass is lost by metabolism, which is proportional to the mass present in the blood. Develop a state space representation of this system. For the special case where the coefficients of A are equal to 1 (with the appropriate sign), determine the response when mi(0) = 1 and m2(0) = 0. Plot the state variables versus time and on the xt — x2 state plane. P3.33 The attitude dynamics of a rocket are represented by Y(s) = G(s) U(s) and state variable feedback is used where x% = y(r), x i ~ y(f)> a n d " = ~x2 ~ 0.5;^. Determine the roots of the characteristic equation of this system and the response of the system when the initial conditions are .^(0) = 0 and x2(0) = 1. The input U{s) is the applied torques, and Y(s) is the rocket attitude. P3.34 A system has the transfer function Y(s) R(s) G ripper FIGURE P3.31 Extender for increasing the strength of the human arm in load maneuvering tasks. P3.32 A drug taken orally is ingested at a rate /-.The mass of the drug in the gastrointestinal tract is denoted by /Hi and in the bloodstream by m2. The rate of change 1 T(s) = s-1 + 6s2 + lis + 6 (a) Construct a state variable representation of the system. (b) Determine the element <£n(/) of the state transition matrix for this system. P3.35 Determine a state-space representation for the system shown in Figure P3.35. The motor inductance is negligible, the motor constant is K„, = 10, the back electromagnetic force constant is Kh = 0.0706, the Reservoir FIGURE P3.35 One-tank system. *• ¢0 227 Advanced Problems motor friction is negligible. The motor and valve inertia is J = 0.006, and the area of the tank is 50 m2. Note that the motor is controlled by the armature current i„. Let X\ = h, x2 = 6, and x 3 = dd/dt. Assume that qx = 800, where 6 is the shaft angle. The output flow is qo - 50h(t). P3.36 Consider the two-mass system in Figure P3.36. Find a state variable representation of the system. Assume the output is x. L : _/-,,'. .^;';..i.i.jy * T P3.37 Consider the block diagram in Figure P3.37. Using the block diagram as a guide, obtain the state variable model of the system in the form Using the state variable model as a guide, obtain a third-order differential equation model for the system. i i k2 4T 1^ M2 x = Ax + Bu y = Cx + DH 1 T nit) FIGURE P3.36 Two-mass system with two springs and one damper. U(s) FIGURE P3.37 A block diagram model of a third-order system. ADVANCED PROBLEMS AP3.1 Consider the electromagnetic suspension system shown in Figure AP3.1. An electromagnet is located at the upper part of the experimental system. Using the electromagnetic force /, we want to suspend the iron ball. Note that this simple electromagnetic suspension system is essentially unworkable. Hence feedback control is indispensable. As a gap sensor, a standard induction probe of the type of eddy current is placed below the ball [20]. Assume that the state variables are JCJ = x, x2 = dxldt, and .r3 = i. The electromagnet has an inductance L = 0.508 H and a resistance R = 23.2 il. Use a Taylor series approximation for the electromagnetic force. The current is i± = / 0 + i, where / 0 = 1.06 A is the operating point and i is the variable. The mass m is equal to 1.75 kg. The gap is xg = XQ + x, where XQ = 4.36 mm is the operating point and x is the variable. The electromagnetic force 228 Chapter 3 State Variable Models AP3.4 Front suspensions have become standard equipment on mountain bikes. Replacing the rigid fork that attaches the bicycle's front tire to its frame, such suspensions absorb bump impact energy, shielding both frame and rider from jolts. Commonly used forks, however, use only one spring constant and treat bump impacts at high and low speeds—impacts that vary greatly in severity—essentially the same. A suspension system with multiple settings that are adjustable while the bike is in motion would be attractive. One air and coil spring with an oil damper is available that permits an adjustment of the damping constant to the terrain as well as to the rider's weight [17]. The suspension system model is shown in Figure AP3.4, where b is adjustable. Select the appropriate value for b so that the bike accommodates (a) a large bump at high speeds and (b) a small bump at low speeds. Assume that k%= \ and ki = 2. Gap sensor FIGURE AP3.1 Electromagnetic suspension system. is / = k(itlxg)2, where k = 2.9 x ItT4 N nr/A 2 . Determine the matrix differential equation and the equivalent transfer function X(s)IV(s). AP3.2 Consider the mass m mounted on a massless cart, as shown in Figure AP3.2. Determine the transfer function Y(s)/U(s), and use the transfer function to obtain a state-space representation of the system. Mass u FIGURE AP3.2 IT f Mass m Iv *2 FIGURE AP3.4 JJ > £ Shock absorber. AP3.5 Figure AP3.5 shows a mass A/ suspended from another mass m by means of a light rod of length L. Obtain a state variable model using a linear model assuming a small angle for 9. Assume the output is the angle, 6. u Mass on cart. AP3.3 The control of an autonomous vehicle motion from one point to another point depends on accurate control of the position of the vehicle [16]. The control of the autonomous vehicle position Y(s) is obtained by the system shown hi Figure AP3.3. Obtain a state variable representation of the system. K(.v) Input Controller Vehicle dynamics 2i-2 + 6.v + 5 s •1 1 (s+ l)(.s+ 2) FIGURE AP3.3 Position control. Position FIGURE AP3.5 Mass suspended from cart. AP3.6 Consider a crane moving in the x direction while the mass m moves in the z direction, as shown in 229 Advanced Problems >• x FIGURE AP3.6 A crane moving in the x-direction while the mass moves in the z-direction. Figure AP3.6. The trolley motor and the hoist motor are very powerful with respect to the mass of the trolley, the hoist wire, and the load m. Consider the input control variables as the distances D and R. Also assume that 8 < 50°. Determine a linear model, and describe the state variable differential equation. AP3.7 Consider the single-input, single-output system described by (a) Obtain a state variable model of the closed-loop system with input r(t) and output y(t). (b) Determine the characteristic roots of the system and compute K such that the characteristic values are all co-located at Si = - 2 , ¾ = - 2 , and S3 = - 2 . (c) Determine analytically the unit step-response of the closed-loop system. x(r) = Ax(0 + B«(r) Top View y(t) = Cx(r) where ,B = C=[2 1]. Assume that the input is a linear combination of the states, that is, u{t) = -Kx(0 + r(t), where r(t) is the reference input. The matrix K = [AT] K2] is known as the gain matrix. Substituting u(t) into the state variable equation gives the closed-loop system Platen x(r) = [A - BK]x(0 + B/-(r) Linear motor Side View Motor with lead screw y(t) = cx(?) (a) The design process involves finding K so that the eigenvalues of A-BK are at desired locations in the left-half plane. Compute the characteristic polynomial associated with the closed-loop system and determine values of K so that the closed-loop eigenvalues are in the left-half plane. AP3.8 A system for dispensing radioactive fluid into capsules is shown in Figure AP3.8(a).The horkontal axis moving the tray of capsules is actuated by a linear motor. The .v-axis control is shown in Figure AP3.8(b). R(s) ~\ » f K 1 —• s(s2 + 6s+ 12) (b) FIGURE AP3.8 Automatic fluid dispenser. Y(s) .v-position Chapter 3 230 State Variable Models DESIGN PROBLEMS CDP3.1 The traction drive uses the capstan drive system f - r \ shown in Figure CDP2.1. Neglect the effect of the / 1 1 ¾ motor inductance and determine a state variable model for the system. The parameters are given in Table CDP2.1. The friction of the slide is negligible. DP3.1 A spring-mass-damper system, as shown in Figure 3.3, is used as a shock absorber for a large high-performance motorcycle. The original parameters selected are m — 1 kg, b = 9 N s/m. and k = 20 N/m. (a) Determine the system matrix, the characteristic roots, and the transition matrix (?)• The harsh initial conditions are assumed to be v(0) = 1 and dy!dt\,=Q — 2. (b) Plot the response oiy(t) and dyldt for the first two seconds. (c) Redesign the shock absorber by changing the spring constant and the damping constant in order to reduce the effect of a high rate of acceleration force d2yldr on the rider. The mass must remain constant at'l kg. DP3.2 A system has the state variable matrix equation in phase variable form 0 -a 1 x + -b_ v = [1 «(') 0]x. It is desired that the canonical diagonal form of the differential equation be -5 0 K, v = [-2 2]z. Determine the parameters a, b, and d to yield the required diagonal matrix differential equation. DP3.3 An aircraft arresting gear is used on an aircraft carrier as shown in Figure DP3.3. The linear model of each energy absorber has a drag force f0 = KQXT>. It is desired to halt the airplane within 30 m after engaging the arresting cable [13]. The speed of the aircraft on landing is 60 m/s. Select the required constant KD, and plot the response of the state variables. DP3.4 The Mile-High Bungi Jumping Company wants you to design a bungi jumping system (i.e., a cord) so that the jumper cannot hit the ground when his or her mass is less than 100 kg, but greater than 50 kg. Also, the company wants a hang time (the time a jumper is moving up and down) greater than 25 seconds, but less than 40 seconds. Determine the characteristics of the cord. The jumper stands on a platform 90 m above the ground, and the cord will be attached to a fixed beam secured 10 m above the platform. Assume that the jumper is 2 m tall and the cord is attached at the waist (1 m high). A',(0) = .r2(0) = .v, (0) = 0 n V\AAAA Energy absorber piston mass = m, = 5 r L__j J: Kat-WvWr •Et; Moving carriage mass = m-, = 10 *i Aircraft carrier runway Cable 2 spring constant * 2 = 1000 FIGURE DP3.3 Aircraft arresting gear. «l3 -vvwvj— Cable 1 spring constant h = 500 S^ AW dx\/dt = 60 m/s at x = 0, t = 0 h = 30 m 231 Computer Problems DP3.5 Consider the single-input, single-output system described by x(0 = Ax(0 + Bu(t) y(t) = Cx(0 u(t) = -Kx(r) + r(/), where r(t) is the reference input. Determine K = [K\ K2\ so that the closed-loop system x(/) = [A - BK]x(/) + Br(t) where r "o , C = [1 3 ,B = J [ij ~0 A = |_-2 y(t) = Cx(0 0]. Assume that the input is a linear combination of the states, that is, possesses closed-loop eigenvalues at /j and r2. Note that if /*! = cr 4- jw is a complex number, then r2 = a — jw is its complex conjugate. COMPUTER PROBLEMS CP3.1 Determine a state variable representation for the following transfer functions (without feedback) using the SS function: (a) G{s) = 1 s + 10 (b) G(s) = s2 + 5s + 3 s2 + 8s + 5 (c) G(s) V()(.v) 5 + 1 s7, + 3s2 + 3s + 1 CP3.2 Determine a transfer function representation for the following state variable models using the tf function: (a) A = (b) A = (c) A = r ,B = V L J [lj 0 2 8 1 2 5 0 1 0 4 C = [l o" 4 ,B = -7_ 0] "-1" 0 , C = [0 1 0] 1_ r ,B = "oi ,C = [-2 [-1 - 2 J LJ 1]. CP33 Consider the circuit shown in Figure CP3.3. Determine the transfer function Vo(s)/Vm(s). Assume an ideal op-amp. (a) Determine the state variable representation when 7?i = 1 kXl, R2 = 10 kfl, C, = 0.5 mF, and C2 = 0.1 mF. (b) Using the state variable representation from part (a), plot the unit step response with the step function. FIGURE CP3.3 An op-amp circuit. CP3.4 Consider the system 0 1 o" "o" 1 x + 0 H, 0 0 5_ _1_ -3 -2 y = [1 0 0]x. (a) Using the tf function, determine the transfer function Y(s)/U(s). (b) Plot the response of the system to the initial condition x(0) = [0 - 1 i f for 0 ^ t < 10. (c) Compute the state transition matrix using the expm function,and determine x(r) at/ = 10 for the initial condition given in part (b). Compare the result with the system response obtained in part (b). CP3.5 Consider the two systems Xl 0 0 4 y = [l 1 0 -5 0 o" ~o" 1 xt + 0 -8_ _4_ 0] Xl 0) 232 Chapter 3 State Variable Models and CP3.7 Consider the following system: 0.5000 0.5000 6.3640 0.5000 -0.5000 -0.7071 = [0.7071 0.7071 0.7071 x2 + - 8.000 _ -0.7071 0 0 0 4 0]x2. y = [l (a) Using the tf function, determine the transfer function Y(s)/U(s) for system (1). (b) Repeat part (a) for system (2). (c) Compare the results in parts (a) and (b) and comment. CP3.6 Consider the closed-loop control system in Figure CP3.6. x(0) Using the Isim function obtain and plot the system response (for xx(t) and JC2(/)) when u(t) = 0. CP3.8 Consider the state variable model with parameter K given by (a) Determine a state variable representation of the controller. (b) Repeat part (a) for the process. (c) With the controller and process in state variable form, use the series and feedback functions to compute a closed-loop system representation in state variable form and plot the closed-loop system impulse response. 3 ~ o l o1 0 0 1 x+ . - 2 -K - 2 J i o o]x. To" 0 M, L1. Plot the characteristic values of the system as a function of K in the range 0 ^ K < 100. Determine that range of K for which all the characteristic values he in the left half-plane. Process Controller 5+3 0]x with ((2) i. 1 -3 Jx + L-2 — • I s2 + 2s + 5 ** Y(s) FIGURE CP3.6 A closed-loop feedback control system. m ANSWERS TO SKILLS CHECK True or False: (1) True; (2) True; (3) False; (4) False; (5) False Multiple Choice: (6) a; (7) b; (8) c; (9) b; (10) c; (11) a; (12) a; (13) c; (14) c; (15) c Word Match (in order, top to bottom): f, d, g, a, b, c, e TERMS AND CONCEPTS Canonical form A fundamental or basic form of the state variable model representation, including phase variable canonical form, input feedforward canonical form, diagonal canonical form, and Jordan canonical form. Diagonal canonical form A decoupled canonical form displaying the n distinct system poles on the diagonal of the state variable representation A matrix. Fundamental matrix See Transition matrix. Input feedforward canonical form A canonical form described by n feedback loops involving the an coefficients of the nth order denominator polynomial of the transfer function and feedforward loops obtained by feeding forward the input signal. 233 Terms and Concepts Jordan canonical form A block diagonal canonical form for systems that do not possess distinct system poles. Matrix exponential function An important matrix function, defined as eAf = I + At + (A/)2/2! + • • • + (A()k/kl + • • •, that plays a role in the solution of linear constant coefficient differential equations. Output equation The algebraic equation that relates the state vector x and the inputs u to the outputs y through the relationship y = Cx + Du. Phase variable canonical form A canonical form described by n feedback loops involving the an coefficients of the nth order denominator polynomial of the transfer function and m feedforward loops involving the b,„ coefficients of the /nth order numerator polynomial of the transfer function. Phase variables The state variables associated with the phase variable canonical form. Physical variables The state variables representing the physical variables of the system. State differential equation The differential equation for the state vector: x = Ax + Bu. State of a system A set of numbers such that the knowledge of these numbers and the input function will, with the equations describing the dynamics, provide the future state of the system. State-space representation A time-domain model comprising the state differential equation x = Ax + Bu and the output equation, y = Cx + Du. State variables The set of variables that describe the system. State vector The vector containing all n state variables, Xi, X2, . . . , Xn. Time domain The mathematical domain that incorporates the time response and the description of a system in terms of time t. Time-varying system A system for which one or more parameters may vary with time. Transition matrix ¢ ( / ) The matrix exponential function that describes the unforced response of the system. Skills Check Table 4.5 279 System Response of the System Shown in Figure 4.38(a) Open Loop* Rise time (s) (10% to 90% of final value) Percent overshoot (%) Final value of y(t) due to a disturbance, T(i(s) = l/s Percent steady-state error for unit step input Percent change in steady-state error due to 10% decrease in K Closed Loop K= 1 K= 1 K=8 K= 10 3.35 0 1.0 0 10% 1.52 4.31 0.50 50% 5.3% 0.45 33 0.11 11% 1.2% 0.38 40 0.09 9% 0.9% *Response only when K = 1 exactly. the gain is increased. Also, the feedback system demonstrates excellent reduction of the steady-state error as the gain is increased. Finally, Figure 4.38(b) shows the response for a unit step disturbance (when R(s) = 0) and shows how a larger gain will reduce the effect of the disturbance. Feedback control systems possess many beneficial characteristics. Thus, it is not surprising that there is a multitude of feedback control systems in industry, government, and nature. m SKILLS CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 4.39 as specified in the various problem statements. W Controller . Ea(s) R(s) FIGURE 4.39 K>) * Process + Gc(s) ^ o G(.v) + Y(s) Block diagram for the Skills Check. In the following True or False and Multiple Choice problems, circle the correct answer. 1. One of the most important characteristics of control systems is their transient response. 2. The system sensitivity is the ratio of the change in the system transfer function to the change of a process transfer function for a small incremental change. 3. A primary advantage of an open-loop control system is the ability to reduce the system's sensitivity. 4. A disturbance is a desired input signal that affects the system output signal. True or False True or False True or False True or False 280 Chapter 4 Feedback Control System Characteristics 5. An advantage of using feedback is a decreased sensitivity of the system to variations in the parameters of the process. 6. The loop transfer function of the system in Figure 4.39 is True or False Gc(s)G{s) = 5°, . W v ' TS + 10 The sensitivity of the closed-loop system to small changes in T is: TS a. ST7{s) = TS + 60 b. STT(s) = c. ST/„\ T(s) T d. S T(s) = T TS + 10 TS + 60 - TS + 10 7. Consider the two systems in Figure 4.40. *(•*> — * 0 K, + Y(s) 0.0099 (i) + /f(.v) /~\ 1 K,l * Y(s) A 0.09 4—1 (H) FIGURE 4.40 Two feedback systems with gains /C, and K2. These systems have the same transfer function when K\ = K2 = 100. Which system is most sensitive to variations in the parameter K{> Compute the sensitivity using the nominal values Kx = K2 = 100. a. System (i) is more sensitive and £, = 0.01 b. System (ii) is more sensitive and Sjc, = 0.1 c System (ii) is more sensitive and S£ 1 = 0.01 d. Both systems are equally sensitive to changes in Kx. 8. Consider the closed-loop transfer function A, + kA2 T(s) = A 3 + *A 4 ' where A l5 A2, A3, and A4 are constants. Compute the sensitivity of the system to variations in the parameter k. k{A2A3 - AXAA) 3 Sk * (A 3 + kAt){Ax + kA2) b * Sk k(A2A3 + A,A4) ' (A 3 + kA4)(Ai + kA2) 281 Skills Check = ** * k(Aj + kA2) (A3 + kA4) k^JcAd_ Consider the block diagram in Figure 4.39 for Problems 9-12 where Gc(s) = Kx and K W, - s + KXK2 9., The closed-loop transfer function is: KK\ a. 7 » = s + KX(K + K2) KKX b. T(s) = s + KX{K + K2) KKX c. T(s) = s - KX{K + K2) KKX J TY«.\ — 2 s + KxKs + KXK2 10. The sensitivity Sk, of the closed-loop system to variations in #1 is: a. Sl(s) b - S = (s + Kt(K + K2)f 2s " ^ = s + KX(K + K2) c Sl(s) = T _ = d. SUs) 5 + KX{K + K2) *,(* + / ¾ ) {s + K,(K + K2)Y 11. The sensitivity STK of the closed-loop system to variations in K is: s + KxK2 a. STK{s) = 5 + Kx(K + K2) Ks b. £(*) (5 + ^ ( K + K 2 )) 2 c S'K(s) T d. S K(s) = * + #1*2 K,(s + A ^ ) (5 + * , ( * + K2))2 12. The steady-state tracking error to a unit step input R(s) = \/s with Td{s) = 0 is: K a. €cX K + K 2 b. e, d * €ss K2 K + K2 K2 K}(K + K2) K + K2 282 Chapter 4 Feedback Control System Characteristics Consider the block diagram in Figure 4.39 for Problems 13-14 with Gc(s) = K and W 5 + 1 13. The sensitivity S[ is: 1 s + Kb + 1 s +1 b. Si = s + Kb + 1 s +1 c Sl = s + Kb + 2 s d. Sl = s + Kb+ 2 14. Compute the minimal value of K so that the steady-state error due to a unit step disturbance is less than 10%. a. K = 1 - 7 b b. K = b c. £ = 10 - | o d. The steady-state error is oo for any K 15. A process is designed to follow a desired path described by r(t) = (5 - t + 0.5t2)u{t) where r(t) is the desired response and u{t) is a unit step function. Consider the unity feedback system in Rgure 4.39. Compute the steady-state error (E(s) = R(s) — Y(s) with Td(s) = 0) when the loop transfer function is 10(5 + 1) 5^(5 + 5) a. e„ = lim e(f) - * oo r—»oo b. es, = lime(0 = 1 J-.00 c. eiS - lim e(r) = 0.5 t—tac d. evs = lime(f) = 0 r-»oo In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Instability b. Steady-state error c. System sensitivity d. Components An unwanted input signal that affects the system output signal. The difference between the desired output, R(s), and the actual output, Y(s). A system without feedback that directly generates the output in response to an input signal. The error when the time period is large and the transient response has decayed leaving the continuous response. 283 Exercises e. Disturbance signal f. Transient response g. Complexity h. Error signal i. Closed-loop system j . Loss of gain k. Open-loop system The ratio of the change in the system transfer function to the change of a process transfer function (or parameter) for a small incremental change. The response of a system as a function of time. A system with a measurement of the output signal and a comparison with the desired output to generate an error signal that is applied to the actuator. A measure of the structure, intricateness, or behavior of a system that characterizes the relationships and interactions between various components. The parts, subsystems, or subassemblies that comprise a total system. An attribute of a system that describes a tendency of the system to depart from the equilibrium condition when initially displaced. A reduction in the amplitude of the ratio of the output signal to the input signal through a system, usually measured in decibels. EXERCISES E4.1 A closed-loop system is used to track the sun to obtain maximum power from a photovoltaic array. The tracking system may be represented by Figure 4.3 with H(s) = 1 and G(s) 100 TS + r where r = 3 seconds nominally, (a) Calculate the sensitivity of this system for a small change in r. (b) Calculate the time constant of the closed-loop system response. Answers: S = -3s/(3s + 101); r,. = 3/101 seconds E4.2 A digital audio system is designed to minimize the effect of disturbances as shown in Figure E4.2. As an approximation, we may represent G(s) = K2(a) Calculate the sensitivity of the system due to K2(b) Calculate the effect of the disturbance noise Td{s) on V0. (c) What value would you select for K\ to minimize the effect of the disturbance? E4.3 A robotic arm and camera could be used to pick fruit, as shown in Figure E4.3(a). The camera is used to close the feedback loop to a microcomputer, FIGURE E4.2 Digital audio system. which controls the arm [8,9]. The transfer function for the process is G(s) = K (s + 5)2 (a) Calculate the expected steady-state error of the gripper for a step command A as a function of K. (b) Name a possible disturbance signal for this system. A Answers: (a) e,v = 1 + K/25 E4.4 A magnetic disk drive requires a motor to position a read/write head over tracks of data on a spinning disk, as shown in Figure E4.4. The motor and head may be represented by the transfer function where r = 0.001 second. The controller takes the difference of the actual and desired positions and generates an error. This error is multiplied by an amplifier K. (a) What is the steady-state position error for a 284 Chapter 4 Feedback Control System Characteristics .Gripper u> Compute the steady-state error to a unit step input as a function of the parameter p. E4.6 A unity feedback system has the loop transfer function 1 OK L(s) = Gc(s)G(s) = ^ - T 7 . s(s + b) Determine the relationship between the steady-state error to a ramp input and the gain K and system parameter b. For what values of K and b can we guarantee that the magnitude of the steady-state error to a ramp input is less than 0.1? E4.7 Most people have experienced an out-of-focus slide projector. A projector with an automatic focus adjusts for variations in slide position and temperature disturbances [11]. Draw the block diagram of an autofocus system. and describe how the system works. An unfocused slide projection is a visual example of steady-state error. (b) E4.8 Four-wheel drive automobiles are popular in regions where winter road conditions are often slippery due to snow and ice. A four-wheel drive vehicle with antilock brakes uses a sensor to keep each wheel rotating to maintain traction. One system is shown in Figure E4.8. Find the closed-loop response of this system as it attempts to maintain a constant speed of the wheel. Determine the response when R{s) = A/s. Camera • Y(s) Gripper position Desired gripper positior FIGURE E4.3 Robot fruit picker. step change in the desired input? (b) Calculate the required K in order to yield a steady-state error of 0.1 mm for a ramp input of 10 cm/s. Answers: esi = 0; K = 100 Desired position Motor FIGURE E4.4 Disk drive control. E4.5 A feedback system has the closed-loop transfer function given by s1 +1 ps + 10 7(.y) = - ;3 ir . S + 2ps- + 4.v + (3-p) Compute the sensitivity of the closed-loop transfer function to changes in the parameter p, where p > 0, 5{s + 3) s(s + 15) Wheel speed FIGURE E4.8 Four-wheel drive auto. E4.9 Submersibles with clear plastic hulls have the potential to revolutionize underwater leisure. One small submersible vehicle has a depth-control system as illustrated in Figure E4.9. (a) Determine the closed-loop transfer function 7(s) = Y{s)/R(s). (b) Determine the sensitivity Sjjj and S'K. (c) Determine the steady-state error due to a disturbance T,i(s) - 1/s. (d) Calculate the response y(t) for a step input R(s) = 1/s when/C = K2 = landl < Jfj < 10. Select Kt for the fastest response. E4.10 Consider the feedback control system shown in Figure E4.10. (a) Determine the steady-state error for a step input in terms of the gain, K. (b) Determine the overshoot for the step response for 40 =£ K < 400. (c) Plot the overshoot and the steady-state error versus K. E4.ll Consider the closed-loop system in Figure E4.ll, where K 14 G(s) = s + 10 and H(s) = + 5s + 6 285 Exercises Disturbance T/s) R(s) Desired depth +^ E(s) + ^ - . t Yls) • Actual depth K, Sensor FIGURE E4.9 Depth control system. R(s) 9" Controller Process K{s + 50) s + 200 46.24 s2 + 16.7s + 72.9 - • Y(s) Sensor FIGURE E4.10 Feedback control system. 425 s + 425 + Y(s) R(s) r—*>Y(s) N(s) FIGURE E4.11 feedback. FIGURE E4.12 Closed-loop system with nonunity feedback and measurement noise. Closed-loop system with nonunity (a) Compute the transfer function T(s) = Y(s)/R(s). (b) Define the tracking error to be E(s) = R(s) - Y(s). Compute E(s) and determine the steady-state tracking error due to a unit step input, that is, let R(s) = l/s. (c) Compute the transfer function Y(s)/Td(s) and determine the steady-state error of the output due to a unit step disturbance input, that is, let Td(s) = l/s. (d) Compute the sensitivity S'K. E4.12 In Figure E4.12, consider the closed-loop system with measurement noise N(s), where G(s) = 100 Gc(s) = Ki. .v + 100' and H(s) Ki 5 + 5' In the following analysis, the tracking error is defined to be £(.v) = R(s) - Y(s): (a) Compute the transfer function T(s) = Y(s)/R(.s) and determine the steady-state tracking error due to a unit step response, that is, let R(s) = l/s and assume that A/ (s) = 0. (b) Compute the transfer function Y(s)/N(s) and determine the steady-state tracking error due to a unit step disturbance response, that is, let N(s) = l/s and assume that R(s) = 0. Remember, in this case, the desired output is zero. (c) If the goal is to track the input while rejecting the measurement noise (in other words, while minimizing the effect of N(s) on the output), how would you select the parameters K\ and /C2? E4.13 A closed-loop system is used in a high-speed steel rolling mill to control the accuracy of the steel strip thickness. The transfer function for the process shown in Figure E4.13 can be represented as 1 s{s + 20) Calculate the sensitivity of the closed-loop transfer function to changes in the controller gain K. C(s) = 286 Chapter 4 Feedback Control System Characteristics R(s) O Y(s) Desired lliickness Actual thickness (a) W Controller sJ ,' Process + ir FIGURE E4.13 Control system for a steel rolling mill. (a) Signal flow graph, (b) Block diagram. R(s) Desired thickness ^ , K - < i n.v) Actual thickness G(5) (b) E4.15 Reconsider the unity feedback system discussed in E4.14. This time select K = 120 and Kx = 10. The closed-loop system is depicted in Figure E4.15. (a) Calculate the steady-state error of the closedloop system due to a unit step input, R(s) = l/s, with Td(s) = 0. Recall that the tracking error is defined as E(s) = R(s) - Y(s). (b) Calculate the steady-state response, yss = limy(t), f when Ta(s) = l/s and R(s) = 0. ^°° E4.14 Consider the unity feedback system shown in Figure E4.14. The system has two parameters, the controller gain K and the constant X, in the process. (a) Calculate the sensitivity of the closed-loop transfer function to changes in Kx. (b) How would you select a value for K to minimize the effects of external disturbances, T(i{s)7 7 » Controller FIGURE E4.14 Closed-loop feedback system with two parameters, K and Kv + /?(.V) ' K *> Y(s) ^ 7-(/(.v) FIGURE E4.15 Closed-loop feedback system withK = 120 and /C, = 10. 287 Problems PROBLEMS P4.1 The open-loop transfer function of a fluid-flow system can be written as AQ 2 (*) AQi(s) G(s) = P4.2 1 = TS + V where T = RC, Risa constant equivalent to the resistance offered by the orifice so that 1/R = '/2&.tfo 1/2 ' and C = the cross-sectional area of the tank. Since A / / = R AQ2, we have the following for the transfer function relating the head to the input change: Gi(s) = M(s) 7 R RCs + 1' A £>,(*) G(s) For a closed-loop feedback system, a float-level sensor and valve may be used as shown in Figure P4.1. Assuming the float is a negligible mass, the valve is controlled so that a reduction in the flow rate, A(2i, is proportional to an increase in head, AH, or A(?i = -KAH. Draw a closed-loop flow graph or block diagram. Determine and compare the openloop and closed-loop systems for (a) sensitivity to changes in the equivalent coefficient R and the feedback coefficient K, (b) the ability to reduce the effects of a disturbance in the level A H ( s ) , and (c) the steady-state error of the level (head) for a step change of the input A<2,(.s). + ^ AG, P4.3 6 H Q2 + A 0 , • FIGURE P4.1 S2 + 2((1),,5 One of the most important variables that must be controlled in industrial and chemical systems is temperature. A simple representation of a thermal control system is shown in Figure P4.3 [14].The temperature 2T of the process is controlled by the heater with a resistance R. An approximate representation of the dynamic linearly relates the heat loss from the process to the temperature difference 9i - STe. This relation holds if the temperature difference is relatively small and the energy storage of the heater and the vessel walls is negligible. Also, it is assumed that the voltage e/( applied to the heater is proportional to e^ired o r eh = kEb = kaEbe(t), where ka is the constant of the Tank level control. Wave effect + Ea(s) ffjU) — K 3 FIGURE P4.2 Ship stabilization system. The effect of the waves is a torque Td(s) on the ship. (a) + (it),, where &>„ = 3 rad/s and £ = 0.20. With this low damping factor £, the oscillations continue for several cycles, and the rolling amplitude can reach 18° for the expected amplitude of waves in a normal sea. Determine and compare the open-loop and closedloop system for (a) sensitivity to changes in the actuator constant K„ and the roll sensor K\, and (b) the ability to reduce the effects of step disturbances of the waves. Note that the desired roll d^s) is zero degrees. d§b 0i It is important to ensure passenger comfort on ships by stabilizing the ship's oscillations due to waves [13]. Most ship stabilization systems use fins or hydrofoils projecting into the water to generate a stabilization torque on the ship. A simple diagram of a ship stabilization system is shown in Figure P4.2. The rolling motion of a ship can be regarded as an oscillating pendulum with a deviation from the vertical of degrees and a typical period of 3 seconds. The transfer function of a typical ship is (b) 288 Chapter 4 Feedback Control System Characteristics Environment FIGURE P4.3 Temperature control system. "desired actuator. Then the linearized open-loop response of the system is ST(s) = TS , E(s) + + 1 TS + 1' where T = MC/(PA), M = mass in tank, A = surface area of tank, p = heat transfer constant, C = specific heat constant, k\ = a dimensionality constant, and etf, = output voltage of thermocouple. Determine and compare the open-loop and closedloop systems for (a) sensitivity to changes in the constant K = k\kaEb\ (b) the ability to reduce the effects of a step disturbance in the environmental temperature A2Tt.(i'); and (c) the steady-state error of the temperature controller for a step change in the input, e desired . P4.4 A control system has two forward paths, as shown in Figure P4.4. (a) Determine the overall transfer function 7(5) = Y(s)/R(s). (b) Calculate the sensitivity, S£, using Equation (4.16). (c) Does the sensitivity depend on U(s) or M(*)? P4.5 Large microwave antennas have become increasingly important for radio astronomy and satellite tracking. A large antenna with a diameter of 60 ft, for example, is subject to large wind gust torques. A proposed antenna is required to have an error of less than 0.10° in a 35 mph wind. Experiments show that this wind force exerts a maximum disturbance at the antenna of 200,000 ft lb at 35 mph, or the equivalent to 10 volts at the input T(i(s) to the amplidyne. One problem of driving large antennas is the form of the system transfer function that possesses a structural resonance. The antenna servosystem is shown in Figure P4.5. The transfer function of the antenna, drive motor, and amplidyne is approximated by ,.? G(s) = s(.r + 2£w„s + (o2„y M(s) R{s) Inpul + U(s) f~\ Q(s) n.v) + ^~ Output FIGURE P4.4 Two-path system. TM) /?(.v) FIGURE P4.5 Antenna control system. Antenna, drive motor, and amplidyne G(s) 0(s) - • Position (radians) 289 Problems where £ = 0.707 and (o„ = 15. The transfer function of the power amplifier is approximately k Gi(-v) TS + r where T = 0.15 second, (a) Determine the sensitivity of the system to a change of the parameter ka. (b) The system is subjected to a disturbance Td(s) = 10/s. Determine the required magnitude of ku in order to maintain the steady-state error of the system less than 0.10° when the input R(s) is zero, (c) Determine the error of the system when subjected to a disturbance T,j(s) = 10/s when it is operating as an open-loop system (ks = 0) with R(s) = 0. P4.6 An automatic speed control system will be necessary for passenger cars traveling on the automatic highways of the future. A model of a feedback speed control system for a standard vehicle is shown in Figure P4.6. The load disturbance due to a percent grade A.T,j(s) is also shown. The engine gain Kt. varies within the range of 10 to 1000 for various models of automobiles. The engine time constant re is 20 seconds, (a) Determine the sensitivity of the system to changes in the engine gain Kc. (b) Determine the effect of the load torque on the speed, (c) Determine the constant percent grade &Td(s) = Ad/s for which the vehicle stalls (velocity V(s) = 0) in terms of the gain factors. Note that since the grade is constant, the steady-state solution is sufficient. Assume that R(s) = 30/skm/hr and that KeKx » 1. When Kg/Ki = 2, what percent grade Ad would cause the automobile to stall? P4.7 A robot uses feedback to control the orientation of each joint axis. The load effect varies due to varying load objects and the extended position of the arm.The system will be deflected by the load carried in the gripper. Thus, the system may be represented by Figure P4.7, where the load torque is Tj(s) = D/s. Assume R(s) = 0 at the index position. (a) What is the effect of 7 ^ ) on K(s)? (b) Determine the sensitivity of the closed loop to k2- (c) What is the steady-state error when R(s) = l/s and Td(s) = 0? P4.8 Extreme temperature changes result in many failures of electronic circuits [1J. Temperature control feedback systems reduce the change of temperature by using a heater to overcome outdoor low temperatures. A block diagram of one system is shown in Figure P4.8. The effect of a drop in environmental temperature is a step decrease in Td(s). The actual temperature of the electronic circuit is Y(s). The dynamics of the electronic circuit temperature change are represented by the transfer function. G(S) = -i: . s2 + 20s + 180 (a) Determine the sensitivity of the system to K. (b) Obtain the effect of the disturbance T,i(s) on the output Y(s). Load torque A M R(s) Speed setting FIGURE P4.6 Automobile speed control. Throttle controller O G,(.v) = Engine and vehicle K. G(s) = T„S + 1 ATi T).V + 1 Tachometer K,= \ Load disturbance angle FIGURE P4.7 Robot control system. V(s) Speed 290 Chapter 4 Feedback Control System Characteristics W Heater control K 0.U+ 1 /?(*•) Electronic circuit +i G(s) 1 • >'(-v) + FIGURE P4.8 Temperature control system. P4.9 A useful unidirectional sensing device is the photoemitter sensor [15]. A light source is sensitive to the emitter current flowing and alters the resistance of the photosensor. Both the light source and the photoconductor are packaged in a single four-terminal device. This device provides a large gain and total isolation. A feedback circuit utilizing this device is shown in Figure P4.9(a), and the nonlinear resistance-current characteristic is shown in Figure P4.9(b) for the Raytheon CK1116. The resistance curve can be represented by the equation 0.175 log i o * = (/ - 0.005) l/2 ' where i is the lamp current. The normal operating point is obtained when v0 = 35 V, and vin = 2.0 V. /-IN Constant current ^- •/ source = / (a) Determine the closed-loop transfer function of the system, (b) Determine the sensitivity of the system to changes in the gain, K. P4.10 For a paper processing plant, it is important to maintain a constant tension on the continuous sheet of paper between the wind-off and wind-up rolls. The tension varies as the widths of the rolls change, and an adjustment in the take-up motor speed is necessary, as shown in Figure P4.10. If the wind-up motor speed is uncontrolled, as the paper transfers from the wind-off roll to the wind-up roll, the velocity u 0 decreases and the tension of the paper drops [10, 14]. The threeroller and spring combination provides a measure of the tension of the paper. The spring force is equal to kty, and the linear differential transformer, rectifier, and amplifier may be represented by e() = —k2y. Dd 0 FIGURE P4.9 Photosensor system. 1 2 3 4 5 6 7 8 9 10 Lamp current (mA) (b) (a) Wind-off Wind-up roll t'o(') (o0U) Rectifier FIGURE P4.10 Paper tension control. Linear differential transformer Amplifier J Motor 291 Problems Water Desired consistency = /?(.?) Consistency measurement To paper making (a) Ris) >0 G(.v) • • Y(s) H{s) (b) Therefore, the measure of the tension is described by the relation 2T(s) = kty, where y is the deviation from the equilibrium condition, and T(s) is the vertical component of the deviation in tension from the equilibrium condition. The time constant of the motor is T = La/Rir and the linear velocity of the wind-up roll is twice the angular velocity of the motor, that is, Vo(t) = 2w()(/). The equation of the motor is then [r.vw0(.y) + WQ(S)] + k3&T(s), where AT = a tension disturbance, (a) Draw the closed-loop block diagram for the system, including the disturbance AT(.v). (b) Add the effect of a disturbance in the wind-off roll velocity AVy(s) to the block diagram, (c) Determine the sensitivity of the system to the motor constant Km. (d) Determine the steadystate error in the tension when a step disturbance in the input velocity, A Vx(s) = A/s, occurs. P4.ll One important objective of the paper-making process is to maintain uniform consistency of the stock output as it progresses to drying and rolling. A diagram of the thick stock consistency dilution control system is shown in Figure P4.11(a). The amount of water added determines the consistency. The block diagram of the system is shown in Figure P4.11(b). Let H(s) = 1 and G,(.v) = U(s) M(s) FIGURE P4.11 Paper-making control. Eo(s) Gt(s) K 8.v + r G(s) = 1 3.v + 1' Determine (a) the closed-loop transfer function T(s) = Y(s)/R(s), (b) the sensitivity Si, and (c) the steady-state error for a step change in the desired consistency R(s) = A/s. (d) Calculate the value of K required for an allowable steady-state error of 2%. P4.12 Two feedback systems are shown in Figures P4.12(a) and (b). (a) Evaluate the closed-loop transfer functions Tj and T2 for each system, (b) Compare the sensitivities of the two systems with respect to the parameter K\ for the nominal values of K\ = K2 = 1. "> K-, s- 1 , J * 5+4 •*n.v) 6 (a) R(s) — K 3 — * • +4 A> s- -2 (b) FIGURE P4.12 Two feedback systems. 1 •+Y{s) 292 Chapter 4 Feedback Control System Characteristics • Y(s) «(.?) FIGURE P4.13 Closed-loop system. Y(s) - • Flighi speed 10(.y + 4) s(s + n)(s+ 1) R(s) FIGURE P4.14 Hypersonic airplane speed control. P4.13 One form of a closed-loop transfer function is = (S) P4.15 The steering control of a modern ship may be represented by the system shown in Figure P4.15 [16,20]. (a) Find the steady-state effect of a constant wind force represented by T,i(s) = \/s for K = 10 and K = 25. Assume that the rudder input R(s) is zero, without any disturbance, and has not been adjusted, (b) Show that the rudder can then be used to bring the ship deviation back to zero. P4.16 Figure P4.16 shows the model of a two-tank system containing a heated liquid, where T() is the temperature of the fluid flowing into the first tank and T2 is the temperature of the liquid flowing out of the second tank. The system of two tanks has a heater in the first tank with a controllable heat input Q. The time constants are r^ = 10 s and r2 = 50 s. (a) Determine T2(s) in terms of T0(s) and T2ll(s). (b) If T2d(s), the desired output temperature, is changed instantaneously from T2fi(s) = Ajs to T2(l(s) = 2A/s, where G.Gv) + kG2(s) G3(.v) + kG4(sY (a) Use Equation (4.16) to show that [1] T k(G2G3 - G,G4) (G 3 + AG4)(G, + AG2) (b) Determine the sensitivity of the system shown in Figure P4.13, using the equation verified in part (a). P4.14 A proposed hypersonic plane would climb to 1()0,0()0 feet, fly 3800 miles per hour, and cross the Pacific in 2 hours. Control of the aircraft speed could be represented by the model in Figure P4.14. Find the sensitivity of the closed-loop transfer function T(s) to a small change in the parameter a. Wind disturbance W Rudder input _^J _ 4 i K 1 A k + ^ t Y(s) Ship 75 .r + 10.v + 75 from prescribed course FIGURE P 4 . 1 5 Ship steering w I (7,5+ 1)(7,.9+ 1) Q(s) C,.(.v) FIGURE P 4 . 1 6 Two-tank temperature control. 1/100 (7,.9+ 1)(7,5+ 1) r ,k> T (s •* 1^ E(s) ' JVJ ) ^ — 7"2«/(.v) Advanced Problems 293 Tt)(s) — A/s, determine the transient response of T2(t) when Cc(s) = K = 500. (c) Find the steadystate error e„ for the svstem of part (b). where B(s) = TM(s) - 7\(.v). The model of the control system is shown in part (c), where K, 1 0,, Kf = K, = l,J = 0.1 30. Rf and /) = 1. (a) Determine the response 9{t) of the system to a step change in 0d(l) when K = 20. (b) Assuming 0,/(() = 0. find the effect of a load disturbance Tti() = A/s. (c) Determine the steady-state error esi when the input is r(t) = t,t > 0. (Assume that P4.I7 A robot gripper. shown in part (a) of Figure P4.17, is to be controlled so that it closes to an angle t) byusing a DC motor control system, as shown in part (b). W = 0.) Potentiometer Difference amplifier (b) (a) '/;,<*) Power amplifier W + K o *\ I \(Js + />) -*- m (0 FIGURE P4.17 Robot gripper control. ADVANCED PROBLEMS AP4.1 A tank level regulator control is shown in Figure AP4.1(a). It is desired to regulate the level h in response to a disturbance change q. The block diagram shows small variable changes about the equilibrium conditions so that the desired /7,,(/) = 0. Determine the equation for the error E{s), and determine the steady-state error for a unit step disturbance when (a) G(.v) = K and (b) G(s) = K/s. AP4.2 The shoulder joint of a robotic arm uses a DC motor with armature control and a set of gears on the output shaft. The model of the system is shown in Figure AP4.2 with a disturbance torque Tj(s) which represents the effect of the load. Determine the steady-slate error when the desired angle input is a step so that 6,i{s) = A/s, Gc(s) = K, and the disturbance input is zero. When flrf(.v) = 0 and the load 294 Chapter 4 Feedback Control System Characteristics /i(/) Controller 0= Orifice Capacitance C «o Constant = R (a) :i Controller Hd{s) = 0 Desired height variation D V 1 FIGURE AP4.1 A tank level regulator. Error £ w > J? G(.v) H(s) Height variation +U /?Cv + 1 (b) Load disturbance Controller W FIGURE A P 4 . 2 Robot joint control. Desired angle of rotation ; n _ jJ . w G,(.v) "lO - W Las + /?„ A-. + l rt 5(7.9 + />) ** effect is TrI() = M/s, determine the steady-state error when (a) Ge(s) = K and (b) Gc(s) = K/S. AP4.3 A machine tool is designed to follow a desired path so that r(t) = (1 - /)«(/), where u(t) is the unit step function. The machine tool control system is shown in Figure AP4.3. (a) Determine the steady-state error when r(t) is the desired path as given and Td(s) = 0. (b) Plot the error e{t) for the desired path for part (a) forO < t =s 10 seconds. (c) If the desired input is r(t) = 0, find the steadystate error when Td(s) = l/s. (d) Plot the error e(t) for part (c) for 0 < r < 10 seconds. Load effect Motor and tool Controller R(s) Tool command FIGURE AP4.3 Machine tool feedback. 0(s) Actual angle 4 + 2.v 7 + ob/ C 10 s(s + 5) -• Y(s) Tool position Advanced Problems 295 Power Amplifier Integrator Error Control — • ( > ) voltage * I s — • FIGURE AP4.4 DC motor with feedback. TJs) K Tachometer /:,= 1 Surgical disturbance 7"(/(.v) Patient 1 (s + 2)2 FIGURE AP4.5 Blood pressure control. AP4.4 ter Km (a) An armature-controlled DC motor with tachomefeedback is .shown in Figure AP4.4. Assume that = 1 0 . / = l.andtf = 1. Determine the required gain, K, to restrict the steady-state error to a ramp input (v(t) = t for t > 0) to 0.1 (assume that Td(s) = 0). (b) For the gain selected in part (a), determine and plot the error, e(t), due to a ramp disturbance for 0 < t < 5 seconds. AP4.5 A system that controls the mean arterial pressure during anesthesia has been designed and tested [12]. The level of arterial pressure is postulated to be a proxy for depth of anesthesia during surgery. A block diagram of the system is shown in Figure AP4.5, where the impact of surgery is represented by the disturbance Td(s). (a) Determine the steady-state error due to a disturbance Ttl(s) = \/s (let R(s) = 0). (b) Determine the steady-state error for a ramp input /-(0 = t, t > 0 (let Td(s) = 0). (c) Select a suitable value of K less than or equal to 10, and plot the response y(t) for a unit step disturbance input (assume r(t) = 0). AP4.6 A useful circuit, called a lead network, which we discuss in Chapter 10, is shown in Figure AP4.6. (a) Determine the transfer function G(s) =V{)(s)/ V(s). (b) Determine the sensitivity of G(s) with respect to the capacitance C. Y(s) w Actual blood pressure K R FIGURE AP4.6 A lead network. (c) Determine and plot the transient response u 0 (0 for a step input V(s) = 1/s. AP4.7 A feedback control system with sensor noise and a disturbance input is shown in Figure AP4.7.The goal is to reduce the effects of the noise and the disturbance. Let R(s) = 0. (a) Determine the effect of the disturbance on Y(s). (b) Determine the effect of the noise on Y(s). (c) Select the best value for K when 1 < K s= 100 so that the effect of steady-state error due to the disturbance and the noise is minimized. Assume 7*rf(s) = A/s, and N(s) = B/s. AP4.8 The block diagram of a machine-tool control system is shown in Figure AP4.8. (a) Determine the transfer function T(s) =Y(s)/R(s). (b) Determine the sensitivity Sj. (c) Select K when 1 < K < 50 so that the effects of the disturbance and s£ are minimized 296 Chapter 4 Feedback Control System Characteristics Disturbance T/s) Controller Dynamics K K(s) " • Y(s) .y+ 1 Sensor K, = i FIGURE AP4.7 Feedback system with noise. N{5) Sensor noise W Controller Machine —i t K b .v + 2 1^ l FIGURE AP4.8 Machine-tool control. K ft l • Y(s) —- ' Laser sensor DESIGN PROBLEMS CDP4.1 A capstan drive for a table slide is described in f £> CDP2.1. The position of the slide x is measured with a yH|1^ capacitance gauge, as shown in Figure CDP4.1, which is very linear and accurate. Sketch the model of the feedback system and determine the response of the system when the controller is an amplifier and H(s) = 1. Determine the step response for several selected values of the amplifier gain Gc(s) = Ka. DP4.1 A closed-loop speed control system is subjected to a disturbance due to a load, as shown in Figure DP4.1. The desired speed is (t) for the step disturbance for selected values of gain so W R(s) FIGURE CDP4.1 The model of the feedback system with a capacitance measurement sensor. The tachometer may be mounted on the motor (optional), and the switch will normally be open. Controller Motor and slide G{{s) GAs) Tachometer Switch normally open Capacitance sensor His) = l t—*- 297 Design Problems Load disturbance Controller LOlt(s) FIGURE DP4.1 Speed control system. Desired speed irv-* X I K G(s) —rA-* that 10 < /C < 25. Determine a suitable value for the gain K. I s +4 io(s) Actual speed [17]. The laser allows the ophthalmologist to apply heat to a location in the eye in a controlled manner. Many procedures use the retina as a laser target. The retina is the thin sensory tissue that rests on the inner surface of the back of the eye and is the actual transducer of the eye, converting light energy into electrical pulses. On occasion, this layer will detach from the wall, resulting in death of the detached area from lack of blood and leading to partial or total blindness in that eye. A laser can be used to "weld" the retina into its proper place on the inner wall. Automated control of position enables the ophthalmologist to indicate to the controller where lesions should be inserted. The controller then monitors the retina and controls the laser's position so that each lesion is placed at the proper location. A wide-angle video-camera system is required to monitor the movement of the retina, as shown in Figure DP4.4(a). If the eye moves during the irradiation, the laser must be either redirected or turned off. The positioncontrol system is shown in Figure DP4.4(b). Select an appropriate gain for the controller so that the transient response to a step change in r{t) is satisfactory and the effect of the disturbance due to noise in the system is minimized. Also, ensure that the steady-state error for a step input command is zero. To ensure acceptable transient response, require that K < 10. DP4.2 The control of the roll angle of an airplane is achieved by using the torque developed by the ailerons. A linear model of the roll control system for a small experimental aircraft is shown in Figure DP4.2, where G(.v) = . .v2 + 45 + 9 The goal is to maintain a small roll angle 6 due to disturbances. Select an appropriate gain KKX that will reduce the effect of the disturbance while attaining a desirable transient response to a step disturbance, with 0f/(/) = 0. To obtain a desirable transient response, let KK: < 35. DP4.3 The speed control system of Figure DP4.1 is altered so that G(s) = l/(.v + 5) and the feedback is Ku as shown in Figure DP4.3. (a) Determine the range of / ( | allowable so that the steady state is ess ^ 1 %. (b) Determine a suitable value for K^ and K so that the magnitude of the steady-state error to a wind disturbance Td{i) - 2t mrad/s, 0 < t < 5 s, is less than 0.1 mrad. DP4.4 Lasers have been used in eye surgery for more than 25 years. They can cut tissue or aid in coagulation TAs) 0(s) Roll angle FIGURE DP4.2 Control of the roll angle of an airplane. TAs) 1 K IO.I(S) k FIGURE DP4.3 Speed control system. -ti*i G(s) —• 1 s+5 Tachome ter Speed 298 Chapter 4 Feedback Control System Characteristics Controller Camera Laser Argon laser Ophthalmologist t! Fiber optics - • Y(s) position FIGURE DP4.4 Laser eye surgery system. DP4.5 An op-amp circuit can be used to generate a short pulse. The circuit shown in Figure DP4.5 can generate the pulse vn(t) = 5e~10'", r > 0, when the input v{i) is a unit step [6]. Select appropriate values for the resistors and capacitors. Assume an ideal op-amp. K FIGURE DP4.5 Op-amp circuit. DP4.6 A hydrobol is under consideration for remote exploration under the ice of Europa. a moon of the giant planet Jupiter. Figure DP4.6(a) shows one artistic version of the mission. The hydrobot is a self-propelled underwater vehicle that would analyze the chemical composition of the water in a search for signs of life. An important aspect of the vehicle is a controlled vertical descent lo depth in the presence of underwater currents. A simplified control feedback (h) system is shown in Figure DP4.6(b). The parameter ./ > 0 is the pitching moment of inertia, (a) Suppose that GJs) = K, For what range of if is the system stable? (b) What is the steady-state error to a unit step disturbance when G(.(s) = Kl (c) Suppose that Gt(s) = Kp + Kj>s. For what range of Kp and Ko is the system stable? (d) What is the steady-slate error to a unit step disturbance when Gc(s) = Kr + K0s'l DP4.7. Interest in unmanned underwater vehicles (UUVs) has been increasing recently, with a large number of possible applications being considered. These include intelligence-gathering, mine detection, and surveillance applications. Regardless of the intended mission, a strong need exists for reliable and robust control of the vehicle. The proposed vehicle is shown in Figure DP4.7 (a) [28]. We want to control the vehicle through a range of operating conditions. The vehicle is 30 feet long with a vertical sail near the front.The control inputs are stern plane, rudder, and shaft speed commands. In this case, we wish to control the vehicle roll by using the stern planes. The control system is shown in Figure DP4.7(b), where R(x) = 0, the desired roll angle, and T^s) = l/.v. Suppose that the controller is G,.(s) = K(s + 2). 299 Design Problems U) TM FIGURE DP4.6 (a) Europa exploration under the ice. (Used with permission. Credit: NASA.) (b) Feedback system. —; BA.s) t > Bis) (b) (a) TM) R(x) = 0 FIGURE DP4.7 Control of an underwater vehicle. (a) Design the controller gain K such that the maximum roll angle error due the unit step disturbance input is less than 0.05. (b) Compute the steady-state roll angle error to the disturbance input and explain the result. ^ >'(.v) -*• Roll (h) DP4.8. A new suspended, mobile, remote-controlled video camera system to bring three-dimensional mobility to professional football is shown in Figure DP4.8(a) [29]. The camera can be moved over the field, as well as up and down. The motor control on each pulley is 300 Chapter 4 Feedback Control System Characteristics Pulley (a) TAs) A'(.v) *• « 5 ) FIGURE DP4.8 Remote-controlled TV camera. (b) represented by the system in Figure DP4.8(b). where the nominal values are T-^ = 20 ms and r2 - 2 ms. (a) Compute the sensitivity SZ and the sensitivity S„. (b) Design the controller gain K such that the steadystate tracking error to a unit step disturbance is less than 0.05, COMPUTER PROBLEMS CP4.1 Consider a unity feedback system with — . s2 + Zs + 10 Obtain the step response and determine the percent overshoot. What is the steady-state error? CP4.2 Consider the transfer function (without feedback) 4 1 s + 2s + 20 When the input is a unit step, the desired steady-state value of the output is one. Using the step function, show that the steady-state error to a unit step input is 0.8. CP4.3 Consider the closed-loop transfer function G(s) = Til) SK s2 + 15s + K Obtain the family of step responses for it = 10. 200, and 500. Co-plot the responses and develop a table of results that includes the percent overshoot, settling time, and steady-state error. CP4.4. Consider the feedback system in Figure CP4.4. Suppose that the controller is G,.(.?) = K = 10. w Controller Plant + ., Rtt) FIGURE CP4.4 Unity feedback system with controller gain K. *0—-* -fO 1 .1(.1+ 1.91) -*• K(.v) 301 Computer Problems (a) Develop an m-file to compute the closed-loop transfer function T(s) — Y(s)/R(s) and plot the unit step response, (b) In the same m-file, compute the transfer function from the disturbance T,i(s) to the output Y(s) and plot the unit step disturbance response. (c) From the plots in (a) and (b) above, estimate the steady-state tracking error to the unit step input and the steady-state tracking error to the unit step disturbance input, (d) From the plots in (a) and (b) above, estimate the maximum tracking error to the unit step input and the maximum tracking error to the unit step disturbance input. At approximately what times do the maximum errors occur? CP4.5 Consider the closed-loop control system shown in Figure CP4.5. Develop an m-file script to assist in the search for a value of k so that the percent overshoot to a unit step input is greater than 1%, but less than 10%. The script should compute the closed-loop transfer function T{s) = Y(s)/R(s) and generate the step response. Verify graphically that the steady-state error to a unit step input is zero. value is used for design purposes only, since in reality the value is not precisely known. The objective of our analysis is to investigate the sensitivity of the closedloop system to the parameter a. (a) When a = 1, show analytically that the steadystate value of y{t) is equal to 2 when r{t) is a unit step. Verify that the unit step response is within 2 % of the final value after 4 seconds. (b) The sensitivity of the system to changes in the parameter a can be investigated by studying the effects of parameter changes on the transient response. Plot the unit step response for a = 0.5,2, and 5. Discuss the results. CP4.7 Consider the torsional mechanical system in Figure CP4.7(a). The torque due to the twisting of the shaft is -k6; the damping torque due to the braking device is -bd; the disturbance torque is rrf(f); the input torque is r(t); and the moment of inertia of the mechanical system is / . T h e transfer function of the torsional mechanical system is CP4.6 Consider the closed-loop control system shown in Figure CP4.6. The controller gain is K =2. The nominal value of the plant parameter is a = 1. The nominal Controller FIGURE CP4.5 A closed-loop negative feedback control system. FIGURE CP4.6 A closed-loop control system with uncertain parameter a. K(.v) • Q— «.v) • Controller Process K 1 s—a r(i). Input torque (a) *• >'(.?) ',!IS) Controller >> Braking device s + (b/J)s + k/J -*• Y(s) s+k tjit). Disturbance torque FIGURE CP4.7 (a) A torsional mechanical system. (b) The torsional mechanical system feedback control system. 2 Process 10 . 1/7 GO) *o + J• + * > (b) Mechanical system , 1/./ b k 9{s) 302 Chapter 4 Feedback Control System Characteristics A closed-loop control system for the system is shown in Figure CP4.7(b). Suppose the desired angle 6(t = 0°, k = 5,b = 0.9, a n d / = 1. (a) Determine the open-loop response 6(t) of the system for a unit step disturbance (set r(t) = 0). (b) With the controller gain KQ = 50, determine the closed-loop response, 6(t) to a unit step disturbance. (c) Plot the open-loop versus the closed-loop response to the disturbance input. Discuss your results and make an argument for using closed-loop feedback control to improve the disturbance rejection properties of the system. CP4.8 A negative feedback control system is depicted in Figure CP4.8. Suppose that our design objective is to find a controller Gc(s) of minimal complexity such that our closed-loop system can track a unit step input with a steady-state error of zero. (a) As a first try, consider a simple proportional controller Ge(s) = K, where K is a fixed gain. Let K = 2. Plot the unit step response and determine the steady-state error from the plot. (b) Now consider a more complex controller Gc(s) = K{) + -±, where KQ = 2 and K^ - 20. This controller is known as a proportional, integral (PI) controller. Plot the unit step response, and determine the steady-state error from the plot. (c) Compare the results from parts (a) and (b). and discuss the trade-off between controller complexity and steady-state tracking error performance. CP4.9 Consider the closed-loop system in Figure CP4.9, whose transfer function is 10* 5 and H(s) .v + 50' s + 100 (a) Obtain the closed-loop transfer function T(s) = Y(s)/R(s) and the unit step response; that is, let R(s) = l/.v and assume that N(s) = 0. G(.v) FIGURE CP4.8 A simple singleloop feedback control system. A'(.v) FIGURE CP4.9 Closed-loop system with nonunity feedback and measurement noise. (b) Obtain the disturbance response when N(s) 100 s2 + 100 is a sinusoidal input of frequency a> = 10 rad/s. Assume that R(s) = 0. (c) In the steady-state, what is the frequency and peak magnitude of the disturbance response from part(b)? CP4.10 Consider the closed-loop system is depicted in Figure CP4.1().The controller gain K can be modified to meet the design specifications. (a) Determine the closed-loop transfer function T(s) = Y(s)/R(s). (b) Plot the response of the closed-loop system for K = 5,10, and 50. (c) When the controller gain is K = 10, determine the steady-state value of y{t) when the disturbance is a unit step, that is, when Tti(s) = I/s and R(s) = 0. CP4.11 Consider the non-unity feedback system is depicted in Figure CP4.11. (a) Determine the closed-loop transfer function r(.v) = Y(s)/R(s). (b) For AT = 10,12, and 15, plot the unit step responses. Determine the steady-state errors and the settling times from the plots. For parts (a) and (b), develop an m-file that computes the closed-loop transfer function and generates the plots for varying K. Controller Process Gc(s) 10 s+ 10 • Y(s) 303 Terms and Concepts W Controller FIGURE CP4.10 Closed-loop feedback system with external disturbances. R(s) • Q ^ fc J * s+l s+\5 tO 1 .s2+.v+6.5 Controller Process K 20 s2+4.5s+64 i FIGURE CP4.11 Closed-loop system with a sensor in the feedback loop. m Process -+Y(s) • YU) Sensor 1 .v+l ANSWERS TO SKILLS CHECK True or False: (1) True; (2) True; (3) False; (4) False; (5) True Multiple Choice: (6) a; (7) b; (8) a; (9) b; (10) c; (ll)a;(12)b;(13)b;(14)c;(15)c Word Match (in order, top to bottom): e, h, k, b, c, f, i,g,d,a,j TERMS AND CONCEPTS Closed-loop system A system with a measurement of the output signal and a comparison with the desired output to generate an error signal that is applied to the actuator. Complexity A measure of the structure, intricateness, or behavior of a system that characterizes the relationships and interactions between various components. Components The parts, subsystems, or subassemblies that comprise a total system. Disturbance signal An unwanted input signal that affects the system's output signal. Error signal The difference between the desired output R(s) and the actual output Y(a). Therefore, E(s) = R(s) - Y(s). Instability An attribute of a system that describes a tendency of the system to depart from the equilibrium condition when initially displaced. Loop gain The ratio of the feedback signal to the controller actuating signal. For a unity feedback system we have L(x) = Gc(s)G(s). Loss of gain A reduction in the amplitude of the ratio of the output signal to the input signal through a system, usually measured in decibels. Open-loop system A system without feedback that directly generates the output in response to an input signal. Steady-state error The error when the time period is large and the transient response has decayed, leaving the continuous response. System sensitivity The ratio of the change in the system transfer function to the change of a process transfer function (or parameter) for a small incremental change. Tracking error See error signal. Transient response tion of time. The response of a system as a func- 365 Skills Check In the following Thie or False and Multiple Choice problems, circle the correct answer. 1. In general, a third-order system can be approximated by a second-order system's dominant roots if the real part of the dominant roots is less than 1/10 of the real part of the third root. 2. The number of zeros of the forward path transfer function at the origin is called the type number. 3. The rise time is defined as the time required for the system to settle within a certain percentage of the input amplitude. 4. For a second-order system with no zeros, the percent overshoot to a unit step is a function of the damping ratio only. True or False 5. A type-1 system has a zero steady-state tracking error to a ramp input. True or False True or False True or False True or False Consider the closed-loop control system in Figure 5.50 for Problems 6 and 7 with Us) = G.MGM = ^ . 6. The steady-state error to a unit step input R(s) = \/s is: a. ess = lime(0 = 1 b. ess = \ime(t) = 1/2 c e^ = hme(f) = 1/6 d. ess = lime(f) = oo 7. The percent overshoot of the output to a unit step input is: a. P.O. = 9% b. P.O. = 1% c. P.O. = 20% d. No overshoot Consider the block diagram of the control system shown in Figure 5.50 in Problems 8 and 9 with the loop transfer function L(s) = Gc(s)G(s) = K s(s + 10)' 8. Find the value of K so that the system provides an optimum ITAE response. a. K = 1.10 b. K = 12.56 c K = 51.02 d. K = 104.7 9. Compute the expected percent overshoot to a unit step input. a. P.O. = 1.4% b. P.O. = 4.6% c. P.O. = 10.8% d. No overshoot expected 10. A system has the closed-loop transfer function T(s) given by T(s) = Y(s) R(s) 2500 (s + 20)(.92 + 105 + 125)' 366 Chapter 5 The Performance of Feedback Control Systems Using the notion of dominant poles, estimate the expected percent overshoot. a. P.O. » 5% b. P.O. « 20% c. P.O. » 50% d. No overshoot expected 11. Consider the unity feedback control system in Figure 5.50 where L{s) = Gc{s)G{s) = K s(s + 5)' The design specifications are: i. Peak time Tn < 1.0 ii. Percent overshoot P.O. < 10%. With K as the design parameter, it follows that a. Both specifications can be satisfied. b. Only the first specification Tp < 1.0 can be satisfied. c. Only the second specification P.O. ^ 10% can be satisfied. d. Neither specification can be satisfied. 12. Consider the feedback control system in Figure 5.51 where G(s) = K s + 10 W R(s) O )£"(5>. 1 * Controller Process 1 s G(s) • • Y(s) Measurement FIGURE 5.51 Feedback system with integral controller and derivative measurement. The nominal value of K — 10. Using a 2% criterion, compute the settling time, Ts for a unit step disturbance, TX-s) = l/s. a. Ts = 0.02 s b. Ts = 0.19 s c. Ts = 1.03 s d. Ts = 4.83 s 13. A plant has the transfer function given by G(5) = (1 + 5)(1 + 0.55) and is controlled by a proportional controller Gc(s) - K, as shown in the block diagram in Figure 5.50. The value of K that yields a steady-state error E(s) = Y(s) - R(s) with a magnitude equal to 0.01 for a unit step input is: a. K = 49 b. K = 99 367 Skills Check c. K = 169 d. None of the above In Problems 14 and 15, consider the control system in Figure 5.50, where G{s) = -£ — and Gc(s) = — ^ r r . c w (s + 5)(s + 2) s + 50 14. A second-order approximate model of the loop transfer function is: (3/25) A: a. Gc(s)G(s) = 2 s + Is + 10 (1/25)# b. Gc(s)G(s) = 2 s + Is + 10 (3/25)X c. Gc(s)G(s) = 2 5 + 7^ + 500 6K d. Gc(s)G(s) = 2 5 + Is + 10 15. Using the second-order system approximation (see Problem 14), estimate the gain K so that the percent overshoot is approximately P.O. « 15%. a. K = 10 b. A = 300 c. A = 1000 d. None of the above w In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Unit impulse b. Rise time c. Settling time d. Type number e. Percent overshoot f. Position error constant, Kp g. Velocity error constant, Kv h. Steady-state response i. Peak time j . Dominant roots The time for a system to respond to a step input and rise to a peak response. The roots of the characteristic equation that cause the dominant transient response of the system. The number N of poles of the transfer function, G(s), at the origin. The constant evaluated as lim sG(s). An input signal used as a standard test of a system's ability to respond adequately. The time required for the system output to settle within a certain percentage of the input amplitude. A set of prescribed performance criteria. A system whose parameters are adjusted so that the performance index reaches an extremum value. A quantitative measure of the performance of a system. The time for a system to respond to a step input and attain a response equal to a percentage of the magnitude of the input. k. Test input signal The amount by which the system output response proceeds beyond the desired response. 1. Acceleration error The constant evaluated as lim s2G(s). constant, Ka m. Transient response The constant evaluated as lim G(s). A—»0 368 Chapter 5 The Performance of Feedback Control Systems n. Design The constituent of the system response that exists a long specifications time following any signal initiation. o. Performance index The constituent of the system response that disappears with time. p. Optimum control A test input consisting of an impulse of infinite amphtude system and zero width, and having an area of unity. EXERCISES E5.1 A motor control system for a computer disk drive must reduce the effect of disturbances and parameter variations, as well as reduce the steady-state error. We want to have no steady-state error for the head-positioning control system, which is of the form shown in Figure 5.18. (a) What type number is required? (How many integrations?) (b) If the input is a ramp signal, and we want to achieve a zero steady-stale error, what type number is required? that the system provides an optimum ITAE response. (b) Using Figure 5.8, determine the expected overshoot to a step input of I(s). Answer: K = 100; 4.6% E5.2 The engine, body, and tires of a racing vehicle affect the acceleration and speed attainable [9]. The speed control of the car is represented by the model shown in Figure E5.2. (a) Calculate the steady-state error of the car to a step command in speed, (b) Calculate overshoot of the speed to a step command. Answer: (a) e5S = /1/11: (b) P.O. = 36% Engine and tires R(s) Speed command + •T j _ <. Plfil I P C P«i O 240 • (s + 4)0 + 6) (a) ,. n*i Speed U E5.3 New passenger rail systems that could profitably compete with air travel are under development. Two of these systems, the French TGV and the Japanese Shinkansen, reach speeds of 160 mph [17]. The Transrapid, a magnetic levitation train, is shown in Figure E5.3(a). The use of magnetic levitation and electromagnetic propulsion to provide contactless vehicle movement makes the Transrapid technology radically different. The underside of the carriage (where the wheel trucks would be on a conventional car) wraps around a guideway. Magnets on the bottom of the guideway attract electromagnets on the "wraparound," pulling it up toward the guideway. This suspends the vehicles about one centimeter above the guideway. The levitation control is represented by Figure E5.3(b). (a) Using Table 5.6 for a step input, select K so Gap dynamics Coil current k{J) -i K s(s + 14) > film spacta (b) FIGURE E5.3 Levitated train control. E5.4 A feedback system with negative unity feedback has a loop transfer function L(s) = Gc(s)G(s) 2(5 + 8) s(s + 4)' (a) Determine the closed-loop transfer function T(s) = Y(s)/R(s). (b) Find the time response, y(t), for a step input r(t) = A for t > 0. (c) Using Figure 5.13(a), determine the overshoot of the response. (d) Using the final-value theorem, determine the steady-state value of y(t). Answer: (b) y(r) = I - 1.07c-3' sin(V7< + 1.2) 369 Exercises FIGURE E5.5 Feedback system with proportional controller Gc(s) = K. + Y(s) R(s) E5.5 Consider the feedback system in Figure E5.5. Find K such that the closed-loop system minimizes the ITAE performance criterion for a step input. E5.6 Consider the block diagram shown in Figure E5.6 [16]. (a) Calculate the steady-state error for a ramp input. (b) Select a value of K that will result in zero overshoot to a step input. Provide the most rapid response that is attainable. Plot the poles and zeros of this system and discuss the dominance of the complex poles. What overshoot for a step input do you expect? E5.8 A control system for positioning the head of a floppy disk drive has the closed-loop transfer function T(s) = 11.1(5 + 18) (s + 20)(52 + 4s + 10) Plot the poles and zeros of this system and discuss the dominance of the complex poles. What overshoot for a step input do you expect? E5.9 A unity negative feedback control system has the loop transfer function L(s) = Gc(s)G(s) = R(s) Position feedback FIGURE E5.6 feedback. Block diagram with position and velocity E5.7 Effective control of insulin injections can result in better lives for diabetic persons. Automatically controlled insulin injection by means of a pump and a sensor that measures blood sugar can be very effective. A pump and injection system has a feedback control as shown in Figure E5.7. Calculate the suitable gain K so that the overshoot of the step response due to the drug injection is approximately 7%. R(s) is the desired blood-sugar level and Y(s) is the actual bloodsugar level. (Hint: Use Figure 5.13a.) Answer: K = 1.67 Pump K s(s + V2K)' (a) Determine the percent overshoot and settling time (using a 2% settling criterion) due to a unit step input. (b) For what range of K is the settling time less than 1 second? E5.10 A second-order control system has the closed-loop transfer function T(s) = Y(s)/R(s). The system specifications for a step input follow: (1) Percent overshoot P.O. < 5%. (2) Settling time Ts < 4s. (3) Peak time Tp < Is. Show the permissible area for the poles of T(s) in order to achieve the desired response. Use a 2% settling criterion to determine settling time. E5.ll A system with unity feedback is shown in Figure E5.ll. Determine the steady-state error for a step and a ramp input when G(s) = 5(s + 8) s(s + 1)(5 + 4)(5 + 10)' Human body Insulin k Sensor FIGURE E5.7 Blood-sugar level control. K 1 S+2 s(s+ 1) n.v) Blood-sugar level 370 Chapter 5 The Performance of Feedback Control Systems Ris) *• tt.v) FIGURE E5.11 Unity feedback system. Disturbance 7(/(.v) Controller FIGURE E5.12 Speed control of a Ferris wheel. R(s) Desired speed *+ 9 E5.12 We are all familiar with the Ferris wheel featured at state fairs and carnivals. George Ferris was born in Galesburg, Illinois, in 1859; he later moved to Nevada and then graduated from Rensselaer Polytechnic Institute in 1881. By 1891, Ferris had considerable experience with iron, steel, and bridge construction. H e conceived and constructed his famous wheel for the 1893 Columbian Exposition in Chicago [8]. To avoid upsetting passengers, set a requirement that the steady-state speed must be controlled to within 5 % of the desired speed for the system shown in Figure E5.12. (a) Determine the required gain K to achieve the steady-state requirement. (b) For the gain of part (a), determine and plot the error e(f) for a disturbance 1}(s) = \/s. Does the speed change more than 5%? (Set R(s) = 0 and recall that E(s) = R(s) - T(s).) E5.13 For the system with unity feedback shown in Figure E 5 . l l , determine the steady-state error for a step and a ramp input when G(s) 20 s2 + 14^ + 50* Answer: ess = 0.71 for a step and e ss = oo for a ramp. E5.14 A feedback system is shown in Figure E5.14. (a) Determine the steady-state error for a unit step when K = 0.4 and Gp(s) = 1. /?(.v) • Y{s) V(s) Speed of rotation 4- + 6 (s + 2)(5 + 4) A: i Wheel and motor dynamics (b) Select an appropriate value for Gp(s) so that the steady-state error is equal to zero for the unit step input. E5.15 A closed-loop control system has a transfer function T(s) as follows: Y(s) R(s) T(s) = 2500 (s + 50)(^ 2 + 10s + 50) Ploty(*) for a step input R(s) when (a) the actual T(s) is used, and (b) using the relatively dominant complex poles. Compare the results. E5.16 A second-order system is R(s) (10/Z)(5 + Z) T(S) (5 + 1)(5 + 8) Consider the case where 1 < z < 8. Obtain the partial fraction expansion, and plot y(t) for a step input r(t) for z = 2 , 4 , and 6. E5.17 A closed-loop control system transfer function T(s) has two dominant complex conjugate poles. Sketch the region in the left-hand 5-plane where the complex poles should be located to meet the given specifications. (a) (b) (c) (d) (e) 0.6 < £ < 0.8, o)lt < 10 0.5 < £ ^ 0.707, o>„ > 10 C ^ 0.5, 5 < (on < 10 £ < 0.707, 5 < (o„ < 10 £ > 0.6, o>„ < 6 E5.18 A system is shown in Figure E5.18(a). The response to a unit step, when K = 1, is shown in Figure E5.18(b). Determine the value of K so that the steadystate error is equal to zero. Answer: K = 1.25. E5.19 A second-order system has the closed-loop transfer function i \& i Y(s) FIGURE E5.14 Feedback system. 7(5) 7 + 2£a>„5 + ft>2, 5 2 + 3.1755 + l' 371 Problems RU) • G(s) -CVt!) ES.20 Consider the closed-loop system in Figure E5.19, where Gc(s)G(s) = ^2 - ^ - a n d H(s) = Ka. (a) s + 03s (a) Determine the closed-loop transfer function T(s) = Y(s)/R(s). (b) Determine the steady-state error of the closed-loop system response to a unit ramp input, R() = 1/j2. (c) Select a value for Ka so that the steady-state error of the system response to a unit step input, R(s) = L/i, is zero. (b) FIGURE E5.18 Feedback system with prefilter. RU) o s+ 1 s2 + 3s -*> n.v> (a) Determine the percent overshoot P.O., the time to peak £, and the settling time T. of the unit step response, R(s) = 1/s. To compute the settling time, use a 2% criterion. (b) Obtain the system response to a unit step and ver- FIGURE E5.20 Nonunity closed-loop feedback control ify the results in part (a). system with parameter Ka. PROBLEMS P5.1 An important problem for television systems is the jumping or wobbling of the picture due to the movement of the camera. This effect occurs when the camera is mounted in a moving truck or airplane. The Dynalens system has been designed to reduce the effect of rapid scanning motion; see Figure P5.1. A maximum scanning motion of 25°/s is expected. Let Kg ~ K, = I and assume that Tg is negligible, (a) Determine the error of the system E(s). (b) Determine the necessary loop gain KaKnK, when a l°/s steady-state error is allowable, (c) The motor time constant is 0.40 s. Determine the necessary loop gain so that the settling time (to within 2% of the final value of v!:) is less than or equal to 0.03 s. P5.2 A specific closed-loop control system is to be designed for an underdamped response to a step input.The specifications for the system are as follows: 10% < percent overshoot < 20%, Settling time < 0.6 s. Torque motor Camera (a) Camera Rate gyro speed K. Amplifier K -\ » K* "i + I (a) Identify the desired area for the dominant roots of the system, (b) Determine the smallest value of a third root /•; if the complex conjugate roots are to represent the dominant response, (c) The closed-loop system transfer function T(s) is third-order, and the feedback has a unity gain. Determine the forward transfer function G(s) = Y(s)/E(s) when the settling time to within 2% of thefinalvalue is 0.6 s and the percent overshoot is 20%. FIGURE P5.1 Motor m - ST„, + I Tachometer ,i < (b) Camera wobble control. Bellows speed 372 Chapter 5 The Performance of Feedback Control Systems P53 A laser beam can be used to weld, drill, etch, cut. and mark metals, as shown In Figure P5.3(a) [14]. Assume we have a work requirement for an accurate laser to mark a parabolic path with a closed-loop control system, as shown in Figure P5.3(b). Calculate the necessary gain to result in a steady-state error of 5 mm for r(() = t2 cm. P5.4 The loop transfer function of a unity negative feedback system (see Figure E5.ll) is Us) = G&ycm = . Mirror Laser cavity Focusing lens ^YY Nozzle assembly I A system response to a step input is specified as follows: peak time Tp = 1.1 s, percent overshoot P.O. = 5%. (a) Determine whether both specifications can be met simultaneously, (b) If the specifications cannot be met simultaneously, determine a compromise value for K so that the peak tune and percent overshoot specifications are relaxed by the same percentage. P5.5 A space telescope is to be launched to carry out astronomical experiments [8]. The pointing control system is desired to achieve 0.01 minute of arc and track solar objects with apparent motion up to 0.21 arc minute per second. The system is illustrated in Figure P5.5(a). The control system is shown in Beam Workpiece (a) *• ru His) lb) FIGURE P5.3 Laser beam control. Sim -lighl ~L. fc ^^^3 ^ Space Slllll lb X - B e mF W ^ 4 w .. Tracking and data relay satellite system - & * 0 ,--^ / Ground, station ^ - - > H J i (a) Process Controller FIGURE P5.5 (a) The space telescope, (b) The space telescope pointing control system. «(v) Input + >n - «*) Ar2(r,.? + 1) 2 r 2 i- + 1 s (b angle : 373 Problems Figure P5.5(b). Assume that r, = 1 second and T 2 = 0 (an approximation), (a) Determine the gain K = K]K2 required so that the response to a step command is as rapid as reasonable with an overshoot of less than 5%. (b) Determine the steady-state error of the system for a step and a ramp input, (c) Determine the value of K\K2 for an ITAE optimal system for (1) a step input and (2) a ramp input. P5.6 A robot is programmed to have a tool or welding torch follow a prescribed path [7, 11]. Consider a robot tool that is to follow a sawtooth path, as shown in Figure P5.6(a). The transfer function of the plant is G(s) 20 30 Time (s) (a) A'(.v) 75(5 + 1) Xr -> mi G(s) V trajectory s(s + 5)(s + 25) (b) for the closed-loop system shown in Figure 5.6(b). Calculate the steady-state error. P5.7 Astronaut Bruce McCandless II took the first untethered walk in space on February 7,1984, using the gas-jet propulsion device illustrated in Figure P5.7(a). FIGURE P5.7 (a) Astronaut Bruce McCandless II is shown a few meters away from the earth-orbiting space shuttle. He used a nitrogenpropelled handcontrolled device called the manned maneuvering unit. (Courtesy of National Aeronautics and Space Administration.) (b) Block diagram of controlle-. FIGURE P5.6 Robot path control. The controller can be represented by a gain K2, as shown in Figure P5.7(b). The moment of inertia of the equipment and man is 25 kg m~. (a) Determine the (a) Gas jet controller Astronaut Desired position Force jRT «1 K2 K, (b) i Is Velocity 1 31 Position (meters) 374 Chapter 5 The Performance of Feedback Control Systems necessary gain /C3 to maintain a steady-state error equal to 1 cm when the input is a ramp r(i) = l (meters). (b) With this gain Kr„ determine the necessary gain KXK.2 in order to restrict the percent overshoot to 10%. (c) Determine analytically the gain K^K2 in order to minimize the ISE performance index for a step input. P5.8 Photovoltaic arrays (solar cells) generate a DC voltage that can be used to drive DC motors or that can be converted to AC power and added to the distribution network. It is desirable to maintain the power out of the array at its maximum available as the solar incidence changes during the day. One such closed-loop system is shown in Figure P5.8. The transfer function for the process is where K - 20. Find (a) the time constant of the closed-loop system and (b) the settling time to within 2% of the final value of the system to a unit step disturbance. Disturbance 7',,(.v) Slope of power curve at maximum power "t/**N v' FIGURE P5.8 Solar cell control. P5.9 The antenna that receives and transmits signals to the Telstar communication satellite is the largest horn antenna ever built. The microwave antenna is 177 ft long, weighs 340 tons, and rolls on a circular track. A photo of the antenna is shown in Figure P5.9. The Telstar satellite is 34 inches in diameter and moves about 16,000 mph at an altitude of 2500 miles. The antenna must be positioned accurately to 1/10 of a degree, because the microwave beam is 0.2° wide and highly attenuated by the large distance. If the antenna is following the moving satellite, determine the K„ necessary for the system. P5.10 A speed control system of an armature-controlled DC motor uses the back emf voltage of the motor as a feedback signal, (a) Draw the block diagram of this system (see Equation (2.69)). (b) Calculate the steadystate error of this system to a step input command setting the speed to a new level. Assume that Ra= La = J = b = \, the motor constant is Km = 1, and Kh = 1. (c) Select a feedback gain for the back emf signal to yield a step response with an overshoot of 15%. P5.ll A simple unity feedback control system has a process transfer function —— = 6(5 = —. E(s) s The system input is a step function with an amplitude A. The initial condition of the system at time f0 is y(t0) = Q, where y(t) is the output of the system. The performance index is defined as FIGURE 5.9 A model of the antenna for the Telstar System at Andover, Maine. (Photo courtesy of Bell Telephone Laboratories, inc.) e\t) dt. 375 Problems (a) Show that / = ( / 1 - Q)2/{2K). (b) Determine the gain K that will minimize the performance index /. Is this gain a practical value? (c) Select a practical value of gain and determine the resulting value of the performance index. P5.12 Train travel between cities will increase as trains are developed that travel at high speeds, making the travel time from city center to city center equivalent to airline travel time. The Japanese National Railway has a train called the Bullet Express that travels between Tokyo and Osaka on theTokaido line. This train travels the 320 miles in 3 hours and 10 minutes, an average speed of 101 mph [17]. This speed will be increased as new systems are used, such as magnetically levitated systems to float vehicles above an aluminum guideway. To maintain a desired speed, a speed control system is proposed that yields a zero steady-state error to a ramp input. A third-order system is sufficient. Determine the optimum system transfer function T(s) for an ITAE performance criterion. Estimate the settling time (with a 2% criterion) and overshoot for a step input when &>„ = 10. P5.13 We want to approximate a fourth-order system by a lower-order model. The transfer function of the original system is GH(s) = s 3 + Is2 + 24s + 24 s + 10.53 + 35s2 + 50s + 24 s3 + 7s2 + 24s + 24 (s + l)(s + 2)(s + 3)(s + 4)' 4 Show that if we obtain a second-order model by the method of Section 5.8, and we do not specify the poles and the zero of GL(s), we have 0.2917s + 1 0.399s2 + 1.375s + 1 _ 0.731 (s + 3.428) ~ (s + 1.043)(s + 2.4)* Gds) = P5.14 For the original system of Problem P5.13, we want to find the lower-order model when the poles of the second-order model are specified as - 1 and - 2 and the model has one unspecified zero. Show that this low-order model is = dS) 0.986s + 2 s 2 + 3s + 2 = 0.986(s + 2.028) (s+ 1)(5 + 2) • P5.15. Consider a unity feedback system with loop transfer function L(s) = Gc(s)G(s) = K(s + 1) (s + 4)(s2 + s + 10)* Determine the value of the gain K such that the percent overshoot to a unit step is minimized. P5.16 A magnetic amplifier with a low-output impedance is shown in Figure P5.16 in cascade with a low-pass filter and a preamplifier. The amplifier has a high-input impedance and a gain of 1 and is used for adding the signals as shown. Select a value for the capacitance C so that the transfer function V0(s)/Vin(s) has a damping ratio of 1/ v 2 . The time constant of the magnetic amplifier is equal to 1 second, and the gain is K = 10. Calculate the settling time (with a 2% criterion) of the resulting system. « V„{s) V-mis) * Amplifier FIGURE P5.16 Feedback amplifier. P5.17 Electronic pacemakers for human hearts regulate the speed of the heart pump. A proposed closed-loop system that includes a pacemaker and the measurement of the heart rate is shown in Figure P5.17 [2,3]. The transfer function of the heart pump and the pacemaker is found to be G(s) = K s(s/12 + 1) Design the amplifier gain to yield a system with a settling time to a step disturbance of less than 1 second. The overshoot to a step in desired heart rate should be less than 10%. (a) Find a suitable range of K. (b) If the nominal value of K is K = 10, find the sensitivity of the system to small changes in K. (c) Evaluate the sensitivity of part (b) at DC (set s = 0). (d) Evaluate the magnitude of the sensitivity at the normal heart rate of 60 beats/minute. 376 Chapter 5 The Performance of Feedback Control Systems '/>•) Desired heart rate + Pacemaker K + i., • , hf +^ Heart o Actual heart rate \_ s Rate measurement sensor FIGURE P5.17 Heart pacemaker. Km= I P5.18 Consider the original third-order system given in Example 5.9. Determine a first-order model with one pole unspecified and no zeros that will represent the third-order system. (b) The closed-loop system has a percent overshoot of less than 5%. + P5.19 A closed-loop control system with negative unity feedback has a loop transfer function L{s) = Gc(s)G(s) = ms) • r si P5.20 A system is shown in Figure P5.20. (a) Determine the steady-state error for a unit step input in terms of K and 2^, where E(s) = R(s) - Y(s). (b) Select Ki so that the steady-state error is zero. R(s) FIGURE P5.20 System with pregain, Kv P5.21 Consider the closed-loop system in Figure P5.21. Determine values of the parameters k and a so that the following specifications are satisfied: (a) The steady-state error to a unit step input is zero, •• Y{s) 1 s+a s(s2 + 6s + 12) (a) Determine the closed-loop transfer function T(s). (b) Determine a second-order approximation for T(s). (c) Plot the response of T(s) and the secondorder approximation to a unit step input and compare the results. l s + 2k FIGURE P5.21 Closed-loop system with parameters k and a. P5.22 Consider the closed-loop system in Figure P5.22, where Ge(s)G(s) = s + 0.2K and H(s) = 2s + T (a) If r = 2.43, determine the value of K such that the steady-state error of the closed-loop system response to a unit step input, R(s) = l/s, is zero. (b) Determine the percent overshoot P.O. and the time to peak Tp of the unit step response when K is as in part (a). tf(.v)- O s + 0.2K •** Y(s) 2s + T FIGURE P5.22 Nonunity closed-loop feedback control system. 377 Advanced Problems ADVANCED PROBLEMS AP5.1 A closed-loop transfer function is T(s) = Y(s) 108(* + 3) = R(s) (s + 9)(52 + 8.v + 36)' (a) Determine the steady-state error for a unit step input R(s) = \/s. (b) Assume that the complex poles dominate, and determine the overshoot and settling time to within 2% of the final value. (c) Plot the actual system response, and compare it with the estimates of part (b). AP5.2 A closed-loop system is shown in Figure AP5.2. Plot the response to a unit step input for the system for T, = 0,0.05,0.1, and 0.5. Record the percent overshoot, rise time, and settling time (with a 2% criterion) as TZ varies. Describe the effect of varying TC. Compare the location of the zero - 1 / r , with the location of the closed-loop poles. AP5.3 A closed-loop system is shown in Figure AP5.3. Plot the response to a unit step input for the system with Tp = 0, 0.5, 2, and 5. Record the percent overshoot, rise time, and settling time (with a 2% criterion) as Tp varies. Describe the effect of varying rp. Compare the location of the open-loop pole -\/rp with the location of the closed-loop poles. s(s + 2){rps+ 1) • • Y(s) FIGURE AP5.3 System with a variable pole in the process. AP5.4 The speed control of a high-speed train is represented by the system shown in Figure AP5.4 [17]. Determine the equation for steady-state error for K for a unit step input r(t). Consider the three values for K equal to 1,10, and 100. (a) Determine the steady-state error. (b) Determine and plot the response y{t) for (i) a unit step input R(s) - \(s and (ii) a unit step disturbance input Tlt(s) = 1/s. (c) Create a table showing overshoot, settling time (with a 2% criterion), ess for r(t), and ly/tjmax for the three values of K. Select the best compromise value. *- K(.v) FIGURE AP5.2 System with a variable zero. Disturbance Y(s) Speed R(s) FIGURE AP5.4 Speed control. AP5.5 A system with a controller is shown in Figure AP5.5. The zero of the controller may be varied. Let a = 0, 10,100. (a) Determine the steady-state error for a step input r(t) for a = 0 and a ^ 0. (b) Plot the response of the system to a step input disturbance for the three values of a. Compare the results and select the best value of the three values of a. Disturbance tf(.v) FIGURE AP5.5 System with control parameter a. i{~) Controller Plant s+a 50(.9 + 2) O (s + 3)(s + 4) •*• Yis) 378 Chapter 5 The Performance of Feedback Control Systems AP5.6 The block diagram model of an armature-currentcontrolled DC motor is shown in Figure AP5.6. (a) Determine the steady-state tracking error to a ramp input r(/) = t,t > 0, in terms of K, Kh, and (b) Let Km = 10 and /¾ = 0.05, and select K so that steady-state tracking error is equal to 1. (c) Plot the response to a unit step input and a unit ramp input for 20 seconds. Are the responses acceptable? DC motor ^ O— Rls K m + Yis) s + 0.01 Kb FIGURE AP5.6 DC motor control. AP5.7 Consider the closed-loop system in Figure AP5.7 with transfer functions ^ / ^ G s 1 0 0 c( ) = s + 100 77^ a , "d ^/ x G s K ( ) = s(s + 50)' (b) Determine the actual settling time and percent overshoot to a unit step for the values of K in part (a). (c) Co-plot the results of (a) and (b) and comment. AP5.8 A unity negative feedback system (as shown in Figure E5.ll) has the loop transfer function where 1000 < K ^ 5000. (a) Assume that the complex poles dominate and estimate the settling time and percent overshoot to a unit step input for K = 1000, 2000, 3000, 4000, and 5000. Controller Gc(s) «(.v) — • G(s) = G{s) FIGURE AP5.7 Closed-loop system with unity feedback. Controller Ms) KP+ K(s + 2) i2 + h + \ Determine the gain K that minimizes the damping ratio £ of the closed-loop system poles. What is the minimum damping ratio? AP5.9. The unity negative feedback system in Figure AP5.9 has the process given by Process L FIGURE AP5.9 Feedback control system with a proportional plus integral controller. L{s) = Gc(s)G(s) = 1 s(s + 15)(.y + 25) The controller is a proportional plus integral controller with gains Kp and /C/.The objective is to design the controller gains such that the dominant roots have a damping ratio £ equal to 0.707. Determine the resulting peak time and settling time (with a 2% criterion) of the system to a unit step input. Plant s{s + 15)(5 + 25) - • Yis) 379 Design Problems DESIGN PROBLEMS CDP5.1 The capstan drive system of the previous problems r C> (see CDP1.1-CDP4.1) has a disturbance due to changes \^vj in the part that is being machined as material is removed. The controller is an amplifier Gc(s) — K„. Evaluate the effect of a unit step disturbance, and determine the best value of the amplifier gain so that the overshoot to a step command r(t) = A,t > 0 is less than 5%, while reducing the effect of the disturbance as much as possible. DP5.1 The roll control autopilot of a jet fighter is shown in Figure DPS. 1. The goal is to select a suitable K so that the response to a unit step command 4>d{t) = A, t 2: 0, will provide a response dj>{t) that is a fast response and has an overshoot of less than 20%. (a) Determine the closed-loop transfer function <£(s)/<£rf(s)- (b) Determine the roots of the characteristic equation for K = 0.7, 3, and 6. (c) Using the concept of dominant roots, find the expected overshoot and peak time for the approximate second-order system, (d) Plot the actual response and compare with the approximate results of part (c). (e) Select the gain K so that the percentage overshoot is equal to 16%. What is the resulting peak time? Aileron actuator Aircraft dynamics K s+ 7 «I> (/ (.v) • l — • 12.2 s (s + 2.2) (.V) Roll angle Gyro FIGURE DP5.1 Roll angle control. K A — I . DP5.2 The design of the control for a welding arm with a long reach requires the careful selection of the parameters [13]. The system is shown in Figure DP5.2, where £ = 0.6, and the gain K and the natural frequency (on can be selected, (a) Determine K and co„ so that the response to a unit step input achieves a peak time for the first overshoot (above the desired level of I) that is less than or equal to 1 second and the overshoot is less than 5%. (Hint: Try 0.2 < K/a>„ < 0.4.) (b) Plot the response of the system designed in part (a) to a step input. Actuator and amplifier co}, K R(s) x + 2£ DP5.3 Active suspension systems for modern automobiles provide a comfortable firm ride. The design of an active suspension system adjusts the valves of the shock absorber so that the ride fits the conditions. A small electric motor, as shown in Figure DP5.3, changes the valve settings [13]. Select a design value Amplifier FIGURE DP5.3 Active suspension system. /?(v) Command \r _ K J Y[s) 2 s L FIGURE DP5.2 Welding tip position control. Arm dynamics position for K and the parameter q in order to satisfy the ITAE performance for a step command R(s) and a settling time (with a 2% criterion) for the step response of less than or equal to 0.5 second. Upon completion of your design, predict the resulting overshoot for a step input. Electric motor 1 sis + g) position 380 Chapter 5 The Performance of Feedback Control Systems DP5.4 The space satellite shown in Figure DP5.4(a) uses a control system to readjust its orientation, as shown in Figure DP5.4(b). (a) Determine a second-order model for the closedloop system. (b) Using the second-order model, select a gain K so that the percent overshoot is less than 15% and the steady-state error to a step is less than 12%. ( c ) Verify your design by determining the actual performance of the third-order system. (a) tf(.v) - FIGURE DP5.4 Control of a space satellite. -r GC(S) GOO K s + 70 10 Cs + 3%s + 7) Y(s) orientation (b ) DPS.5 A deburring robot can be used to smooth off machined parts by following a preplanned path (input command signal). In practice, errors occur due to robot inaccuracy, machining errors, large tolerances, and tool wear. These errors can be eliminated using force feedback to modify the path online [8,11]. While force control has been able to address the problem of accuracy, it has been more difficult to solve the contact stability problem. In fact, by closing the force loop and introducing a compliant wrist force sensor (the most common type of force control), one can add to the stability problem. A model of a robot deburring system is shown in Figure DP5.5. Determine the region of stability for the system for K\ and K%, Assume both adjustable gains are greater than zero. DP5.6 The model for a position control system using a DC motor is shown in Figure DPS .6. The goal is to select X| and K2 so that the peak time is T„ s 0.5 X4{s) Position input FM Desired force FIGURE DP5.5 Deburring robot. ^ O • Actual force 381 Design Problems tt Rii) • s(s+ 1) L [31].The control of x may be achieved with a D C motor and position feedback of the form shown in Figure DP5.7(b), with the D C motor and load represented by + Y(s) G(s) 1 + K2s FIGURE DP5.6 where K = 2 and p = 2. Design a proportional plus derivative controller Position control robot. GC(S) = Kp + KDs second and the overshoot P.O. for a step input is P.O. < 2 % . DPS.7 A three-dimensional cam for generating a function of two variables is shown in Figure DP5.7(a). Both x and y may be controlled using a position control system FIGURE DP5.7 (a) Threedimensional cam and (b) x-axis control system. K s(s + p)(s + 4 ) ' to achieve a percent overshoot P.O. £ 5 % to a unit step input and a settling time 7^ :£ 2 seconds. DP5.8. Computer control of a robot to spray-paint an automobile is accomplished by the system shown in R{s) • H »n ^ r 1 ~.' S&ot • GM G{s) Cb) (a) Line conveyor —Q—1.¾¾¾¾¾¾^ Table encoder Computer Input • (a) T/s) «(.v) *• m FIGURE DP5.8 Spray-paint robot. (h) 382 Chapter 5 The Performance of Feedback Control Systems Figure DP5.8(a) [7]. We wish to investigate the system when K = 1,10, and 20. The feedback control block diagram is shown in Figure DP5.8(b). (a) For the three values of K, determine the percent overshoot, the settling time (with a 2% criterion), and the steady-state error for a unit step input. Record your results in a table, (b) Choose one of the three values of K that provides acceptable performance, (c) For the value selected in part (b), determine y(/) for a disturbance Td(s) = \fs when R{s) = 0. COMPUTER PROBLEMS CP5.1 Consider the closed-loop transfer function 15 s + 8s + 15 Obtain the impulse response analytically and compare the result to one obtained using the impulse function. T(s) = CP5.2 A unity negative feedback system has the loop transfer function A- + 10 s2(s + 15)' L(s) = Ge{s)G(s) = Using Isim, obtain the response of the closed-loop system to a unit ramp input, R(s) = 1/52. Consider the time interval 0 s t < 50. What is the steady-state error? CP5.3 A working knowledge of the relationship between the pole locations of the second-order system shown in Figure CP5.3 and the transient response is important in control design. With that in mind, consider the following four cases: 1. (on = 2, £ = 0, 2. a,, = 2, I = 0.1, 3. Ris)' \r _ i FIGURE CP5.4 A negative feedback control system. > - * I K Ris) P S + T Using the impulse and subplot functions, create a plot containing four subplots, with each subplot depicting the impulse response of one of the four cases listed. Compare the plot with Figure 5.17 in Section 5.5, and discuss the results. CP5.4 Consider the control system shown in Figure CP5.4. (a) Show analytically that the expected percent overshoot of the closed-loop system response to a unit step input is about 50%. (b) Develop an m-file to plot the unit step response of the closed-loop system and estimate the percent overshoot from the plot. Compare the result with part (a). CP5.S Consider the feedback system in Figure CP5.5. Develop an m-file to design a controller and prefilter Gc(s) = K 21 1 s+ 2 Controller t( "\ Ea(s) s+z s + p and Gp(s) = Kr S + T such that the ITAE performance criterion is minimized. For (on = 0.45 and £ = 0.59, plot the unit step response and determine the percent overshoot and settling time. Process 5 • • Y(s) s- + 2(a>ns + a>;, FIGURE CP5.3 A simple second-order system. Controller Prefilter FIGURE CP5.5 Feedback control system with controller and prefilter. Ris) 2 s+z s+p Process s2 + 2£ •+ Yis) 383 Computer Problems CP5.6 The loop transfer function of a unity negative feedback system is 25 L(s) = Gc(s)G(s) = s(s + 5)' (b) If we increase the complexity of the controller, we can reduce the steady-state tracking error. With this objective in mind, suppose we replace the constant gain controller with the more sophisticated controller Develop an m-file to plot the unit step response and determine the values of peak overshoot Mp, time to peak Tp, and settling time Ts (with a 2% criterion). K CP5.7 An autopilot designed to hold an aircraft in straight and level flight is shown in Figure CP5.7. (a) Suppose the controller is a constant gain controller given by Gc(s) = 2. Using the Isim function, compute and plot the ramp response for 8(f(t) = at, where a = 0.5°/s. Determine the attitude error after 10 seconds. FIGURE CP5.7 An aircraft autopilot block diagram. Desired attitude This type of controller is known as a proportional plus integral (PI) controller. Repeat the simulation of part (a) with the PI controller, and compare the steadystate tracking errors of the constant gain controller versus the PI controller. Controller Elevator servo G,(.v) -10 s+ 10 Controller R(s) O 10 5+10 - • n.v) 0.5 105 + 0.5 FIGURE CP5.9 Nonunity feedback system. CP5.10 Develop an m-file to simulate the response of the system in Figure CP5.10 to a ramp input R(s) - l/s2. What is the steady-state error? Display the output on an x-y graph. s(s + 3.5s + 6) (Kt) •*• Actual attitude 0(t) 100(5 + 1) *• Actual rate 2 (s + 2s + 100) 5 CP5.9 Develop an m-file that can be used to analyze the closed-loop system in Figure CP5.9. Drive the system with a step input and display the output on a graph. What is the settling time and the percent overshoot? -(-v + 5) 2 Missile dynamics 5 0.1 + 1 Aircraft model Compare the predicted results with the actual unit step response obtained with the step function. Explain any differences. CP5.8 The block diagram of a rate loop for a missile autopilot is shown in Figure CP5.8. Using the analytic formulas for second-order systems, predict Mp„ Tp, and Ts for the closed-loop system due to a unit step input. FIGURE CP5.8 A missile rate loop autopilot. 1 Gc(s) = Ky + — = 2 + - . »—in">1 /?( i , 10 jr(.v+ 15)(5 + 5) tt.v) FIGURE CP5.10 Closed-loop system for m-file. CP5.11 Consider the closed-loop system in Figure CP5.11. Develop an m-file to accomplish the following tasks: (a) Determine the closed-loop transfer function T(s) = Y(s)/R(s), (b) Plot the closed-loop system response to an impulse input R(s) = 1, a unit step input R(s) = l/s, and a unit ramp input R(s) = l/s2. Use the subplot function to display the three system responses. 384 Chapter 5 The Performance of Feedback Control Systems /?(.v) O — * - Y(s) (a) "> R(s) FIGURE CP5.11 h Controller Process o, + i s(s + 2) A single loop unity feedback system. (a) Signal flow graph, (b) Block diagram. I W.V) (b) CP5.12 A closed-loop transfer function is given by T(S) = Y(s) ll{s + 2) R(s) (s + 7)(.v2 + 4s + 22)' (a) Obtain the response of the closed-loop transfer function T(s) = Y(s)/R(s) to a unit step input. El What is the settling time Ts {use a 2% criterion) and percent overshoot P.O.? (b) Neglecting the real pole at s = - 7 , determine the settling time Ts and percent overshoot P.O.. Compare the results with the actual system response in part (a). What conclusions can be made regarding neglecting the pole? ANSWERS TO SKILLS CHECK True or False: (1) True; (2) False; (3) False; (4) True; (5) False Multiple Choice: (6) a; (7) a; (8) c; (9) b; (10) b; (11) a; (12) b; (13) b; (14) a; (15) b Word Match (in order, top to bottom): i, j , d, g, k, c, n, p, o, b, e, 1, f, h, m, a TERMS AND CONCEPTS Peak time The time for a system to respond to a step input and rise to a peak response. as ]\S})[ Gc( )G( )]. The steady-state error for a paraPercent overshoot The amount by which the system outbolic input, /*(/) = At2/2, is equal to A/K„. put response proceeds beyond the desired response. Design specifications A set of prescribed performance Performance index A quantitative measure of the perforcriteria. mance of a system. Dominant roots The roots of the characteristic equation Position error constant, Kp The constant evaluated as that cause the dominant transient response of the jlSJGUsJG^.The steady-state error for a step input system. (of magnitude ,4) is equal to A/{\ + Kp). Optimum control system A system whose parameters are adjusted so that the performance index reaches Rise time The time for a system to respond to a step input and attain a response equal to a percentage of the an extremum value. Acceleration error constant, Ka S s s The constant evaluated 425 Skills Check CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 6.31 as specified in the various problem statements. Controller Process Gc(s) G(s) • Y(S) L FIGURE 6.31 Block diagram for the Skills Check. In the following True or False and Multiple Choice problems, circle the correct answer. 1. A stable system is a dynamic system with a bounded output response for any input. 2. A marginally stable system has poles on the y'co-axis. 3. A system is stable if all poles lie in the right half-plane. 4. The Routh-Hurwitz criterion is a necessary and sufficient criterion for determining the stability of linear systems. 5. Relative stability characterizes the degree of stability. 6. A system has the characteristic equation q(s) = s3 + 4Ks2 + (5 + K)s + 10 = 0. True or False True or False True or False True or False True or False The range of K for a stable system is: a. K > 0.46 b. K < 0.46 c 0 < K < 0.46 d. Unstable for all K 7. Utilizing the Routh-Hurwitz criterion, determine whether the following polynomials are stable or unstable: px(s) = s2 + 10s + 5 = 0, p2(s) = s4 + 53 + 5.s2 + 20.S + 10 0. a. pi(s) is stable, p2(s) is stable b. pi(s) is unstable, p2(s) is stable c pi(^) is stable, p2(s) is unstable d. pi(s) is unstable, ^2(^) is unstable 8. Consider the feedback control system block diagram in Figure 6.31. Investigate closedloop stability for Gc(s) = K(s + I) and G(s) = K = 1 and K = 3. a. Unstable for K = 1 and stable for K = 3 (s + 2)(s - 1) , for the two cases where 426 Chapter 6 The Stability of Linear Feedback Systems b. Unstable for K = 1 and unstable for K = 3 c. Stable for K = 1 and unstable for K = 3 d. Stable for K = 1 and stable for K = 3 9. Consider a unity negative feedback system in Figure 6.31 with loop transfer function where L(s) = Gc(s)G(s) = K (1 + 0.5^)(1 + 0.5* + 0.25.V2)' Determine the value of K for which the closed-loop system is marginally stable. a. K = 10 b. K = 3 c. The system is unstable for all K d. The system is stable for all K 10. A system is represented by x = Ax, where A = 0 0 -5 1 0 -K 0 1 10 The values of K for a stable system are a. K < 1/2 b. K > 1/2 c. K = 111 d. The system is stable for all /C 11. Use the Routh array to assist in computing the roots of the polynomial q{s) = 2s3 + 2s2 + s + 1=0. V2. a. *! = - 1; s2(3 = ± — ; V2. b. st = 1; 52,3 = ± —j c. sx = - 1; 52,3 = 1 ± — / d. sj = - 1; 2,3 = 1 12. Consider the following unity feedback control system in Figure 6.31 where G(s) K(s + 0.3) 1 andGc(.s') = {s - 2)(s2 + 10s + 45) The range of K for stability is a. K < 260.68 b. 50.06 < K < 123.98 c 100.12 < K < 260.68 d. The system is unstable for all K > 0 427 Skills Check In Problems 13 and 14, consider the system represented in a state-space form x= 0 0 -5 1 0 -10 o" ~ 0" 1 x + 0 _20_ 5_ y = [1 0 l]x. 13. The characteristic equation is: a. q(s) = s3 + 5s2 - 10s - 6 b. q(s) = s3 + 5s2 + 10s + 5 c. q(s) = s3 - 5s2 + 10s - 5 d. q(s) = s2 - 5s + 10 14. Using the Routh-Hurwitz criterion, determine whether the system is stable, unstable, or marginally stable. a. Stable b. Unstable c. Marginally stable d. None of the above 15. A system has the block diagram representation as shown in Figure 6.31, where 10 K G(s) = ~— . 2 an( * Gc(s) = _s ^+Qn80, where K is always positive. The limiting gain (s + HrlSy for a stable system is: a. 0 < K < 28875 b. 0 < K < 27075 c 0 < K < 25050 d. Stable for all K > 0 In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. A performance measure of a system. a. Routh-Hurwitz criterion b. Auxiliary polynomial A dynamic system with a bounded system response to a bounded input. The property that is measured by the relative c. Marginally stable real part of each root or pair of roots of the characteristic equation. d. Stable system A criterion for determining the stability of a system by examining the characteristic equation of the transfer function. e. Stability The equation that immediately precedes the zero entry in the Routh array. f. Relative stability g. Absolute stability A system description that reveals whether a system is stable or not stable without consideration of other system attributes such as degree of stability. A system possesses this type of stability if the zero input response remains bounded as t -* oo. 428 Chapter 6 The Stability of Linear Feedback Systems EXERCISES E6.1 A system has a characteristic equation s 3 + Ks2 + (1 + K)s + 6 = 0. Determine the range of K for a stable system. Answer: K > 2 E6.2 A system has a characteristic equation 53 + IQs2 + 2s + 30 = 0. Using the Routh-Hurwitz criterion, show that the system is unstable. E6.3 A system has the characteristic equation s4 + IO53 + 32s2 + 37s + 20 = 0. Using the RouthHurwitz criterion, determine if the system is stable. E6.4 A control system has the structure shown in Figure E6.4. Determine the gain at which the system will become unstable. Answer: K = 20/7 E6.5 A unity feedback system has a loop transfer function L(S) = (s+ l)(s + 3)(s + 6)' where K = 20. Find the roots of the closed-loop system's characteristic equation. E6.6 For the feedback system of Exercise E6.5, find the value of K when two roots lie on the imaginary axis. Determine the value of the three roots. Answer: s = - 1 0 , ±/5.2 K } E6.7 A negative feedback system has a loop transfer function L(s) = E6.9 A system has a characteristic equation + 252 + (K + 1)5 + 8 = 0. Find the range of K for a stable system. A-3 Answer: K > 3 E6.10 We all use our eyes and ears to achieve balance. Our orientation system allows us to sit or stand in a desired position even while in motion. This orientation system is primarily run by the information received in the inner ear, where the semicircular canals sense angular acceleration and the otoliths measure linear acceleration. But these acceleration measurements need to be supplemented by visual signals. Try the following experiment: (a) Stand with one foot in front of another, with your hands resting on your hips and your elbows bowed outward, (b) Close your eyes. Did you experience a low-frequency oscillation that grew until you lost balance? Is this orientation position stable with and without the use of your eyes? E6.ll A system with a transfer function Y(s)/JR(5) is Y(s) 24(5 + 1) A K(s + 2) 5(5-1)- (a) Find the value of the gain when the £ of the closedloop roots is equal to 0.707. (b) Find the value of the gain when the closed-loop system has two roots on the imaginary axis. «(.v) E6.8 Designers have developed small, fast, vertical-takeoff fighter aircraft that are invisible to radar (stealth aircraft). This aircraft concept uses quickly turning jet nozzles to steer the airplane [16]. The control system for the heading or direction control is shown in Figure E6.8. Determine the maximum gain of the system for stable operation. O 3 s+ 1 ~R(s)~ s + 653 + 252 + 5 + 3' Determine the steady-state error to a unit step input. Is the system stable? E6.12. A system has the second-order characteristic equation 52 + as + b = 0, \( *\ K s(s + 4) •+m FIGURE E6.4 Feedforward system. R(s) FIGURE E6.8 Aircraft heading control. •^ *Q J i fc * Controller Aircraft dynamics K {s + 20) s(s + 10)2 Y(s) Heading Exercises FIGURE E6.13 Closed-loop system with a proportional plus derivative controller Gc(s) = KP + Kos. 429 Controller Process Kp + KpS 4 s{s + 2) k where a and b are constant parameters. Determine the necessary and sufficient conditions for the system to be stable. Is it possible to determine stability of a second-order system just by inspecting the coefficients of the characteristic equation? E6.13. Consider the feedback system in Figure E6.13. Determine the range of Kp and KD for stability of the closed-loop system. E6.14 By using magnetic bearings, a rotor is supported contactless. The technique of contactless support for rotors becomes more important in light and heavy industrial applications [14]. The matrix differential equation for a magnetic bearing system is 0 -3 -2 1 -1 -1 0' 0 -2. where x r = [y, dy/dt, i], y = bearing gap, and i is the electromagnetic current. Determine whether the system is stable. Answer: The system is stable. E6.15 A system has a characteristic equation q(s) = s6 + 9s5 + 31.25s4 + 61.25s3 + 67.75s2 + 14.75s + 15 = 0. (a) Determine whether the system is stable, using the Routh-Hurwitz criterion, (b) Determine the roots of the characteristic equation. Answer: (a) The system is marginally stable. (b)s = - 3 , - 4 , - 1 ± 2/, ±0.5; E6.16 A system has a characteristic equation q(s) = s 4 + 9s3 + 45s2 + 87s + 50 = 0. (a) Determine whether the system is stable, using the Routh-Hurwitz criterion, (b) Determine the roots of the characteristic equation. E6.17 The matrix differential equation of a state variable model of a system has A 0 -8 -8 1 -12 -12 •1 • n.v) (a) Determine the characteristic equation, (b) Determine whether the system is stable, (c) Determine the roots of the characteristic equation. Answer: (a) q(s) = s 3 + 7s2 + 36s + 24 = 0 E6.18 A system has a characteristic equation q(s) = s 3 + 20s2 + 5s + 100 = 0. (a) Determine whether the system is stable, using the Routh-Hurwitz criterion, (b) Determine the roots of the characteristic equation. E6.19 Determine whether the systems with the following characteristic equations are stable or unstable: (a) s3 + 4s2 + 6s + 100 = 0, (b) s4 + 6s3 + 10s2 + 17s + 6 = 0, and (c) s2 + 6s + 3 = 0. E6.20 Find the roots of the following polynomials: (a) s 3 + 5s2 + 8s + 4 = 0 and (b) s 3 + 9s2 + 27s + 27 = 0. E6.21 A system has the characteristic equation q(s) = s 3 + 10s2 + 29s + K = 0. Shift the vertical axis to the right by 2 by using s - s„ - 2, and determine the value of gain K so that the complex roots are s = - 2 ± j . E6.22 A system has a transfer function Y(s)IR{s) = T(s) = 1/s. (a) Is this system stable? (b) If r{f) is a unit step input, determine the response y{t). E6.23 A system is represented by Equation (6.22) where r e 1 0 A = 0 1 -k -4 Find the range of k where the system is stable. E6.24 Consider the system represented in state variable form x = Ax -I- Bu y = Cx + DM, where A = 0 0 -k 1 0 -* C = [1 0 0 1 ,B = -k 0], D = [0]. 0 0 1 430 Chapter 6 The Stability of Linear Feedback Systems (a) What is the system transfer function? (b) For what values of k is the system stable? E6.26 Consider the closed-loop system in Figure E6.26, where E6.25 A closed-loop feedback system is shown in Figure E6.25. For what range of values of the parameters K and p is the system stable? R(s) O- Ks+ 1 - • Yis) s\s + p) G(s) = 10 10 and Gc(s) = 1 2s + K' (a) Determine the characteristic equation associated with the closed-loop system. (b) Determine the values of K for which the closedloop system is stable. FIGURE E6.25 Closed-loop system with parameters K and p. R(s)Q (a) W Controller R{s) O E \ «W ) I 2s + K FIGURE E6.26 Closed-loop feedback control system with parameter K. + Y{s) N(s) (b) PROBLEMS P6.1 Utilizing the Routh-Hurwitz criterion, determine the stability of the following polynomials: (a) s2 + 5s + 2 (b) s 3 + 4^2 + &r + 4 (c) 5 3 + 2s2 - 6s + 20 + 53 + 2s2 + 12s + 10 (d) (e) 54 + s* + 3s2 + 25 + X 5 4 3 (t) s + s + 25 + s + 6 5 4 3 2 (g) 5 + .y + 2^ + s + s + K Determine the number of roots, if any, in the righthand plane. If it is adjustable, determine the range of K that results in a stable system. 431 Problems P6.2 P6.3 An antenna control system was analyzed in Problem P4.5, and it was determined that, to reduce the effect of wind disturbances, the gain of the magnetic amplifier, ka, should be as large as possible, (a) Determine the limiting value of gain for maintaining a stable system. (b) We want to have a system settling time equal to 1.5 seconds. Using a shifted axis and the Routh-Hurwitz criterion, determine the value of the gain that satisfies this requirement. Assume that the complex roots of the closed-loop system dominate the transient response. (Is this a valid approximation in this case?) A r c welding is one of the most important areas of application for industrial robots [11]. In most manufacturing welding situations, uncertainties in dimensions of the part, geometry of the joint, and the welding process itself require the use of sensors for maintaining weld quality. Several systems use a vision system to measure the geometry of the puddle of melted metal, as shown in Figure P6.3. This system uses a constant rate of feeding the wire to be melted. (a) Calculate the maximum value for K for the system that will result in a stable system, (b) For half of the maximum value of K found in part (a), determine the roots of the characteristic equation, (c) Estimate the overshoot of the system of part (b) when it is subjected to a step input. Controller Desired diameter + ,-->> Error ^r^ K s +2 Measured diameter FIGURE P6.3 Welder control. P6.4 A feedback control sys tern is shown in Rgure P6.4. The controller and process transfer functions are given by Gc(s) = K and G(s) = and the feedback transfer function is H(s) = l/(s + 20). (a) Determine the limiting value of gain K for a stable system, (b) For the gain that results in marginal stability, determine the magnitude of the imaginary roots, (c) Reduce the gain to half the magnitude of the marginal value and determine the relative stability of the system (1) by shifting the axis and using the Routh-Hurwitz criterion and (2) by determining the root locations. Show the roots are between —1 and —2. P6.5 Determine the relative stability of the systems with the following characteristic equations (1) by shifting the axis in the s-plane and using the Routh-Hurwitz criterion, and (2) by determining the location of the complex roots in the s-plane: (a) s3 + 3s2 + 4* + 2 = 0. (b) s4 + 9s 3 + 305 2 + 42s + 20 = 0. (c) J 3 + 19s 2 + 110* + 200 = 0. P6.6 A unity-feedback control system is shown in Figure P6.6. Determine the relative stability of the Arc current Wire-melting process Vision system 1 Controller Process G(.(s) G(s) • • Y(s) Sensor FIGURE P6.4 Nonunity feedback system. H(s) R(s) FIGURE P6.6 Unity feedback system. Puddle diameter (0.55+ 1)(5+1) 0.005s + 1 R(s) s + 40 s(s + 10) • Y(s) 432 Chapter 6 The Stability of Linear Feedback Systems system with the following transfer functions by locating the complex roots in the s-plane: (a) Gc(s)G(s) = IPs + 2 sz(s + 1) 24 sis3 + 10s2 + 35s + 50) (s + 2){s + 3) (c) Gc(s)G(s) = s(s + 4)(s + 6) (b) Gc(s)G(s) = P6.7 The linear model of a phase detector (phase-lock loop) can be represented by Figure P6.7 [9].The phaselock systems are designed to maintain zero difference in phase between the input carrier signal and a local voltage-controlled oscillator. Phase-lock loops find application in color television, missile tracking, and space telemetry. The filter for a particular application is chosen as F(s) = 10(5 + 10) (s + 1)(5 + 100)' We want to minimize the steady-state error of the system for a ramp change in the phase information signal, (a) Determine the limiting value of the gain KaK = Kv in order to maintain a stable system, (b) A steady-state error equal to 1° is acceptable for a Amplifier Ka F(s) K s Desired velocity • Velocity Power amplifier R(s) FIGURE P6.9 Tape drive control. Voltage-controlled oscillator Filter k FIGURE P6.7 Phase-lock loop system. FIGURE P6.8 Wheelchair control system. ramp signal of 100 rad/s. For that value of gain Kv, determine the location of the roots of the system. P6.8 A very interesting and useful velocity control system has been designed for a wheelchair control system. We want to enable people paralyzed from the neck down to drive themselves in motorized wheelchairs. A proposed system utilizing velocity sensors mounted in a headgear is shown in Figure P6.8.The headgear sensor provides an output proportional to the magnitude of the head movement. There is a sensor mounted at 90° intervals so that forward, left, right, or reverse can be commanded. Typical values for the time constants are TX = 0.5 s, T 3 = 1 s, and T4 = \ s. (a) Determine the limiting gain K = K\K2K^ for a stable system. (b) When the gain K is set equal to one-third of the limiting value, determine whether the settling time (to within 2% of the final value of the system) is less than 4 s. (c) Determine the value of gain that results in a system with a settling time of 4 s. Also, obtain the value of the roots of the characteristic equation when the settling time is equal to 4 s. P6.9 A cassette tape storage device has been designed for mass-storage [1]. It is necessary to control the velocity of the tape accurately. The speed control of the tape drive is represented by the system shown in Figure P6.9. "> fc J * Motor and drive mechanism K 10 s + 100 (s + 20)2 Y(s) Speed 433 Problems (a) Determine the limiting gain for a stable system. (b) Determine a suitable gain so that the overshoot to a step command is approximately 5%. P6.10 Robots can be used in manufacturing and assembly operations that require accurate, fast, and versatile manipulation [10,11]. The open-loop transfer function of a direct-drive arm may be approximated by K(s + 10) G(s)H(s) K(s2 + 305 + 1125) s(s + 3)(s2 + 4s + 8)" 2 (a) Determine the value of gain K when the system oscillates, (b) Calculate the roots of the closed-loop system for the K determined in part (a). P6.ll A feedback control system has a characteristic equation 53 + (1 + K)s2 + 105 + ( 5 + 15K) = 0. The parameter K must be positive. What is the maximum value K can assume before the system becomes unstable? When K is equal to the maximum value, the system oscillates. Determine the frequency of oscillation. P6.12. A system has the third-order characteristic equation 53 + as2 + bs + c = 0, where a, b, and c are constant parameters. Determine the necessary and sufficient conditions for the system to be stable. Is it possible to determine stability of the system by just inspecting the coefficients of the characteristic equation? P6.13. Consider the system in Figure P6.13. Determine the conditions on K,p, and z that must be satisfied for closed-loop stability. Assume that K > 0, £ > 0, and (on> 0. P6.14 A feedback control system has a characteristic equation 56 + 2 / + 1254 + As2 + 2152 + 25 + 10 = 0. Determine whether the system is stable, and determine the values of the roots. Controller FIGURE P6.13 Control system with controller with three parameters K,p, and z. FIGURE P6.17 Elevator control system. R(s) R(s) Desired vertical position ±Q-^ ~ 5(5 + 20)(5 + 105 + 125)(52 + 6O5 + 3400)' (a) As an approximation, calculate the acceptable range of K for a stable system when the numerator polynomial (zeros) and the denominator polynomial (52 + 6O5 + 3400) are neglected, (b) Calculate the actual range of acceptable K, account for all zeros and poles. P6.16 A system has a closed-loop transfer function 1 s3 + 552 + 205 + 6' (a) Determine whether the system is stable, (b) Determine the roots of the characteristic equation, (c) Plot the response of the system to a unit step input. T(s) = P6.17 The elevator in Yokohama's 70-story Landmark Tower operates at a peak speed of 45 km/hr. To reach such a speed without inducing discomfort in passengers, the elevator accelerates for longer periods, rather than more precipitously. Going up, it reaches full speed only at the 27th floor; it begins decelerating 15 floors later. The result is a peak acceleration similar to that of other skyscraper elevators—a bit less than a tenth of the force of gravity. Admirable ingenuity has gone into making this safe and comfortable. Special ceramic brakes had to be developed; iron ones would melt. Computer-controlled systems damp out vibrations. The lift has been streamlined to reduce the wind noise as it speeds up and down [19]. One proposed control system for the elevator's vertical position is shown in Figure P6.17. Determine the range of K for a stable system. Process 5 +Z s+p Controller -^ P6.15 The stability of a motorcycle and rider is an important area for study because many motorcycle designs result in vehicles that are difficult to control [12,13]. The handling characteristics of a motorcycle must include a model of the rider as well as one of the vehicle. The dynamics of one motorcycle and rider can be represented by a loop transfer function (Figure P6.4) * K+ l s(s + 2£a>„) - • n.v) Elevator dynamics 1 2 s{s + 3s + 3) Y(s) Vertical position 434 Chapter 6 The Stability of Linear Feedback Systems P6.18 Consider the case of rabbits and foxes in Australia. The number of rabbits is xt and. if left alone, it would grow indefinitely (until the food supply was exhausted) so that flight [16]. An aircraft taking off in a form similar to a missile (on end) is inherently unstable (see Example 3.4 for a discussion of the inverted pendulum). A control system using adjustable jets can control the vehicle, as shown in Figure P6.19. (a) Determine the range of gain for which the system is stable, (b) Determine the gain K for which the system is marginally stable and the roots of the characteristic equation for this value of K. X\ — KX\. However, with foxes present on the continent, we have .t] = kx-i — ax2, where .¾ is the number of foxes. Now, if the foxes must have rabbits to exist, we have x2 = -hx2 + bxiDetermine whether this system is stable and thus decays to the condition X\{t) = x2(t) = 0 at r = oo. What are the requirements on a, b. h, and k for a stable system? What is the result when k is greater than hi P6.20 A personal vertical take-off and landing (VTOL) aircraft is shown in Figure P6.20(a). A possible control system for aircraft altitude is shown in Figure P6.20(b). (a) For K - 6, determine whether the system is stable, (b) Determine a range of stability, if any, for K > 0. P6.21 Consider the system described in state variable form by P6.19 The goal of vertical takeoff and landing (VTOL) aircraft is to achieve operation from relatively small airports and yet operate as a normal aircraft in level FIGURE P6.19 R(s) Desired vertical path . x(0 = Ax(r) + B«(f) y(t) = Cx(f) Controller Aircraft dynamics K{s + 2) I s{s - 1) 5+10 Actual vertical path Control of a jumpjet aircraft. FIGURE P6.20 (a) Personal VTOL aircraft. (Courtesy of Mirror Image Aerospace at www.skywalkervtol.com) (b) Control system. Rti) K(s2 + 2s+ 1) I s2(s2 + s + 9) (IV) -• m 435 Advanced Problems where 0 , a n d C = [l B -1], -*1 and where kx 5* k2 and both k\ and k2 are real numbers. (a) Compute the state transition matrix ¢ ( / , 0). (b) Compute the eigenvalues of the system matrix A. (c) Compute the roots of the characteristic polynomial. (d) Discuss the results of parts (a)-(c) in terms of stability of the system. ADVANCED PROBLEMS AP6.1 A teleoperated control system incorporates both a person (operator) and a remote machine. The normal teleoperation system is based on a one-way link to the machine and limited feedback to the operator. However. two-way coupling using bilateral information exchange enables better operation [18]. In the case of remote control of a robot, force feedback plus position feedback is useful. The characteristic equation for a teleoperated system, as shown in Figure AP6.1, is sA + 20s 3 + K,s2 + 4s + K2 Operator commands Remote machine Human operator Feedback FIGURE AP6.1 0, where /C, and K2 are feedback gain factors. Determine and plot the region of stability for this system for Kt and K2. AP6.2 Consider the case of a navy pilot landing an aircraft on an aircraft carrier. The pilot has three basic tasks. The first task is guiding the aircraft's approach to the ship along the extended centerline of the runway. The second task is maintaining the aircraft on the correct glidesiope. The third task is maintaining the correct speed. A model of a lateral position control system is shown in Figure AP6.2. Determine the range of stability for K a 0. Model of a teleoperated machine. AP6.3 A control system is shown in Figure AP6.3. We want the system to be stable and the steady-state error for a unit step input to be less than or equal to 0.05 (5%). (a) Determine the range of a that satisfies the error requirement, (b) Determine the range of a that satisfies the stability requirement, (c) Select an a that meets both requirements. AP6.4 A bottle-filling line uses a feeder screw mechanism, as shown in Figure AP6.4. The tachometer feedback is used to maintain accurate speed control. Determine and plot the range of K and /; that permits stable operation. Aircraft Pilot Ailerons and aircraft Controller FIGURE AP6.2 Lateral position control for landing on an aircraft carrier. K •>l J~ Center line I (s- 1 ) ( / + lO.c + 40) position d R(s) FIGURE AP6.3 Third-order unity feedback system. Vis) 1 K(s + 1) s , Y(s) s + a i 2 J 3 + (1 + a)s + (a- 1).5 + (1 -a) 436 Chapter 6 The Stability of Linear Feedback Systems Controller Tachometer feedback (a) Controller R(s)FIGURE AP6.4 Speed control of a bottle-filling line. Motor and Screw 1 (s+$$(s+p) K_ s m Speed (a) System layout. (b) Block diagram. (b) AP6.5 Consider the closed-loop system in Figure AP6.5. A proportional plus derivative controller is used in a system as shown in Figure AP6.6(b), where Suppose that all gains are positive, that is, K j > 0, K2 > 0, K3 > 0, Ki > 0, and K 5 > 0. (a) Determine the closed-loop transfer function and where Kp > 0 and KD > 0. Obtain and plot the T(s) = Y(s)IR(s). relationship between Kp and KD that results in a sta(b) Obtain the conditions on selecting the gains ble closed-loop system. K1% K2, K-j, X 4 ,and K5, so that the closed-loop system is guaranteed to be stable. AP6.7. A human's ability to perform physical tasks is limit(c) Using the results of part (b), select values of the ed not by intellect but by physical strength. If, in an apfive gains so that the closed-loop system is stable, propriate environment, a machine's mechanical power and plot the unit step response. is closely integrated with a human arm's mechanical strength under the control of the human intellect, the reAP6.6. A spacecraft with a camera is shown in Figure sulting system will be superior to a loosely integrated AP6.6(a).The camera slews about 16° in a canted plane combination of a human and a fully automated robot. relative to the base. Reaction jets stabilize the base Extenders are defined as a class of robot manipulaagainst the reaction torques from the slewing motors. tors that extend the strength of the human arm while Suppose that the rotational speed control for the cammaintaining human control of the task [23].The defining era slewing has a plant transfer function characteristic of an extender is the transmission of both I power and information signals. The extender is worn by G(s) = the human; the physical contact between the extender (s + 1)(6- + 2)(j + 4)" Advanced Problems RU) O (a) R(s) • ~\ K{ A \ - 1 * ,+ 1 v "-3 3 2s-4 .1 10 s - 10 *s KA (b) FIGURE AP6.5 Multiloop feedback control system, (a) Signal flow graph, (b) Block diagram. Camera Solar panel Boom (a) V*J Controller Plant + Kp + fCtf FIGURE AP6.6 (a) Spacecraft with a camera. (b) Feedback control system. 6 (s+ 1)(,+ 2)(, + 4) + (b) •+• i t s ) 438 Chapter 6 The Stability of Linear Feedback Systems and the human allows the direct transfer of mechanical power and information signals. Because of this unique interface, control of the extender trajectory can be accomplished without any type of joystick, keyboard, or master-slave system.The human provides a control system for the extender, while the extender actuators provide most of the strength necessary for the task. The human becomes a part of the extender and "feels" a scaled-down version of the load that the extender is carrying. The extender is distinguished from a conventional master-slave system; in that type of system, the human operator is either at a remote location or close to the slave manipulator, but is not in direct physical contact with the slave in the sense of transfer of power. An extender is shown in Figure AP6.7(a) [23]. The block diagram of the system is shown in Figure AP6.7(b). Consider the proportional plus integral controller s Determine the range of values of the controller gains KP and K, such that the closed-loop system is stable. (a) Actuator Rfs) Human — input i -N J * 8 s(2s+ l)(0.05.s + 1) G,(i) FIGURE AP6.7 Extender robot control. Y(s) Output (1-) DESIGN PROBLEMS CDP6.1 The capstan drive system of problem CDP5.1 f C\ uses the amplifier as the controller. Determine the /"TJK maximum value of the gaiu K„ before the system becomes unstable. DP6.1 The control of the spark ignition of an automotive engine requires constant performance over a wide range of parameters [15]. The control system is shown in Figure DP6.1, with a controller gain K to be selected. l 5 K kK FIGURE DP6.1 Automobile engine control. 1 s+5 1 5 1 s+p 1 s >• Y(s) 439 Design Problems The parameter p is equal to 2 for many autos but can equal zero for those with high performance. Select a gain K that will result in a stable system for both values of p. DP6.2 An automatically guided vehicle on Mars is represented by the system in Figure DP6.2. The system has a steerable wheel in both the front and back of the vehicle, and the design requires that H(s) = Ks + 1. Determine (a) the value of K required for stability, (b) the value of K when one root of the characteristic equation is equal to s = - 5 , and (c) the value of the two remaining roots for the gain selected in part (b). (d) Find the response of the system to a step command for the gain selected in part (b). DP6.3 A unity negative feedback system with K(s + 2) G C < « < * > = ,(1 + „ ) ( ! + 2,) has two parameters to be selected, (a) Determine and plot the regions of stability for this system, (b) Select T and K so that the steady-state error to a ramp input is less than or equal to 25% of the input magnitude. (c) Determine the percent overshoot for a step input for the design selected in part (b). DP6.4 The attitude control system of a space shuttle rocket is shown in Figure DP6.4 [17]. (a) Determine Ris) Steering command 9 the range of gain K and parameter m so that the system is stable, and plot the region of stability, (b) Select the gain and parameter values so that the steady-state error to a ramp input is less than or equal to 10% of the input magnitude, (c) Determine the percent overshoot for a step input for the design selected in part (b). DP6.5 A traffic control system is designed to control the distance between vehicles, as shown in Figure DP6.5 [15]. (a) Determine the range of gain K for which the system is stable, (b) If Km is the maximum value of K so that the characteristic roots are on the yw-axis, then let K = K„,/N,where 6 < N < 7. We want the peak time to be less than 2 seconds and the percent overshoot to be less than 18%. Determine an appropriate value for N. DP6.6 Consider the single-input, single-output system as described by yit) = Cx(/) where A = "o 2 L i_ ,B = -2J V ,C [_ij Yls) • • Direction of travel 20 5 + 20 FIGURE DP6.2 Mars guided vehicle control. H(s) Space shuttle rocket Controller /?(.v) +0 "•> t 1 FIGURE DP6.4 Shuttle attitude control. K (s + m)(s + 2) s s2-] Desired distance k Vis) Attitude Throttle, engine, and automobile Controller FIGURE DP6.5 Traffic distance control. x(r) = Ax(t) + Eu(t) 1 K s 5 2 + 105 + 20 Sense r Y(s) Actual distance = [1 0]. 440 Chapter 6 The Stability of Linear Feedback Systems Assume that the input is a linear combination of the states, that is, u(t) = -Kx(t) + /•(/), where /(f) is the reference input. The matrix K = [Kx K2] is known as the gain matrix. If you substitute u(t) into the state variable equation you will obtain the closed-loop system x(0 = [A - BK]x(r) + Br(t) y(t) = Cx(/) For what values of K is the closed-loop system stable? Determine the region of the left half-plane where the desired closed-loop eigenvalues should be placed so that the percent overshoot to a unit step input, R(s) = Vs, is less than P.O. < 5% and the settling time is less than Ts < 4s. Select a gain matrix, K, so that the system step response meets the specifications P.O. < 5%and7*, < 4s. DP6.7 Consider the feedback control system in Figure DP6.7.The system has an inner loop and an outer loop. The inner loop must be stable and have a quick speed of response, (a) Consider the inner loop first. Determine the range of Kx resulting in a stable inner loop. That is, the transfer function Y(s)/U(s) must be stable. (b) Select the value of K\ in the stable range leading to the fastest step response, (c) For the value of Kx selected in (b), determine the range of K2 such that the closed-loop system T(s) = Y(s)/R(s) is stable. DP6.8 Consider the feedback system shown in Figure DP6.8.The process transfer function is marginally stable. The controller is the proportional-derivative (PD) controller Gc(s) = KP + KDs. Determine if it is possible to find values of Kp and KD such that the closed-loop system is stable. If so, obtain values of the controller parameters such that the steady-state tracking error E(s) = R(s) - Y(s) to a unit step input R(s) = 1/s is ess = lim e{t) ^ 0.1 and the damping of the closedloop system is £ = V2/2. FIGURE DP6.7 Feedback system with inner and outer loop. FIGURE DP6.8 A marginally stable plant with a PD controller in the loop. /?(.v) ^¾.) *Q J * Controller Process KP+K0s 4 s2 + 4 + Y(s) i COMPUTER PROBLEMS CP6.1 Determine the roots of the following characteristic equations: (a) q(s) = s3 + 3.v2 + 105 + 14 = 0. (b) q(s) = s4 + 8 ? + 24s2 + 325 + 16 = 0. (c) q(s) = 54 + 2s2 + 1 = 0. CP6.2 Consider a unity negative feedback system with -2 - s + 2 G((s) = K and G(s) = + 2.v + 1 Develop an m-file to compute the roots of the closedloop transfer function characteristic polynomial for K = 1,2, and 5. For which values of K is the closedloop system stable? CP6.3 A unity negative feedback system has the loop transfer function Gc(s)G(s) = s + 1 + 4.v2 + 65 + 10 Computer Problems 441 Develop an m-file to determine the closed-loop transfer function and show that the roots of the characteristic equation are s^ = -2.89 and ^2,3 = -0.55 ± /1.87. to compute the closed-loop transfer function poles for 0 :£ K S 5 and plot the results denoting the poles with the "X" symbol. Determine the maximum range of K for stability with the Routh-Hurwitz method. Compute the roots of the characteristic equation when K is the minimum value allowed for stability. CP6.4 Consider the closed-loop transfer function s* + 2s* + 2s3 + As2 + s + 2 CP6.7 Consider a system in state variable form: (a) Using the Routh-Hurwitz method, determine whether the system is stable. If it is not stable, how many poles are in the right half-plane? (b) Compute the poles of T(s) and verify the result in part (a). (c) Plot the unit step response, and discuss the results. CP6.5 A "paper-pilot" model is sometimes utilized in aircraft control design and analysis to represent the pilot in the loop. A block diagram of an aircraft with a pilot "in the loop" is shown in Figure CP6.5. The variable r represents the pilot's time delay. We can represent a slower pilot with T = 0.6 and a faster pilot with T = 0.1. The remaining variables in the pilot model are assumed to be K = 1, rx - 2, and T2 = 0.5. Develop an m-file to compute the closed-loop system poles for the fast and slow pilots. Comment on the results. What is the maximum pilot time delay allowable for stability? CP6.6 Consider the feedback control system in Figure CP6.6. Using the for function, develop an m-file script k>- FIGURE CP6.5 An aircraft with a pilot in the loop. FIGURE CP6.6 A single-loop feedback control system with parameter K. R(s) X+ (a) Compute the characteristic equation using the poly function, (b) Compute the roots of the characteristic equation, and determine whether the system is stable. (c) Obtain the response plot of y(t) when «(f) is a unit step and when the system has zero initial conditions. CP6.8 Consider the feedback control system in Figure CP6.8. (a) Using the Routh-Hurwitz method, determine the range of K] resulting in closed-loop stability. (b) Develop an m-file to plot the pole locations as a function of 0 < Kx < 30 and comment on the results. CP6.9 Consider a system represented in state variable form x = Ax + Bu y = Cx + DH, Riot model Elevator servo Aircraft model -K(T]S+ 1 ) ( « - 2) (r2i- + 1)( 7* + 2) -10 5+ 10 .v(.v2 + 3s + 6) s* + 5s2 + (K- 3).v + K - • Y{s) Process RU) FIGURE CP6.8 Nonunity feedback system with parameter K\. s(s + 10) Controller 2+ ~ 0~ 0 _12_ • * Y(s) - ( 5 + 6) 1 » 442 Chapter 6 The Stability of Linear Feedback Systems (a) For what values of k is the system stable? (b) Develop an m-file to plot the pole locations as a function of 0 < k < 10 and comment on the results. where " 0 2 A = _-* 1 0 -3 C = [1 2 m 0~ ~-l~ 1 ,B = 0 _1 -2_ 0], D = [0] ANSWERS TO SKILLS CHECK True or False: (1) False; (2) True; (3) False; (4) True; (5) True Multiple Choice: (6) a; (7) c; (8) a; (9) b; (10) b; (11) a; (12) a; (13) b; (14) a; (15) b Word Match (in order, top to bottom): e, d, f, a, b, TERMS AND CONCEPTS Absolute stability A system description that reveals whether a system is stable or not stable without consideration of other system attributes such as degree of stability. Auxiliary polynomial The equation that immediately precedes the zero entry in the Routh array. Marginally stable A system is marginally stable if and only if the zero input response remains bounded as t —* oo. Relative stability The property that is measured by the relative real part of each root or pair of roots of the characteristic equation. Routh-Hurwitz criterion A cri terion for determining the stability of a system by examining the characteristic equation of the transfer function. The criterion states that the number of roots of the characteristic equation with positive real parts is equal to the number of changes of sign of the coefficients in the first column of the Routh array. Stability A performance measure of a system. A system is stable if all the poles of the transfer function have negative real parts. Stable system A dynamic system with a bounded system response to a bounded input. Chapter 7 The Root Locus Method Furthermore, we extended the root locus method for the design of several parameters for a closed-loop control system. Then the sensitivity of the characteristic roots was investigated for undesired parameter variations by defining a root sensitivity measure. It is clear that the root locus method is a powerful and useful approach for the analysis and design of modern control systems and will continue to be one of the most important procedures of control engineering. CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 7.74 as specified in the various problem statements. R(s) tnJ _ Controller Process Gc{s) G(s) * Y(s) i FIGURE 7.74 Block diagram for the Skills Check. In the following True or False and Multiple Choice problems, circle the correct answer. 1. The root locus is the path the roots of the characteristic equation (given by 1 + KG(s) = 0) trace out on the s-plane as the system parameter 0 < K < oo varies. True or False 2. On the root locus plot, the number of separate loci is equal to the number of poles of G(s). True or False 3. The root locus always starts at the zeros and ends at the poles of G(s). True or False 4. The root locus provides the control system designer with a measure of the sensitivity of the poles of the system to variations of a parameter of interest. True or False 5. The root locus provides valuable insight into the response of a system to various test inputs. True or False 6. Consider the control system in Figure 7.74, where the loop transfer function is K(s2 + 5s + 9) L(s) = Gc(s)G(s) = s2{s + 3) Using the root locus method, determine the value of K such that the dominant roots have a damping ratio £ = 0.5. a. K = 1.2 b. K = 4.5 c K = 9.7 d. K = 37.4 523 Skills Check In Problems 7 and 8, consider the unity feedback system in Figure 7.74 with L(s) = Gc(s)G(S) = K(s + 1) \ / s* + 5s + 17.33 7. The approximate angles of departure of the root locus from the complex poles are a. 4>d = ±180° b. 4>d = ±115° c (f>d = ±205° d. None of the above 8. The root locus of this system is given by which of the following 9. A unity feedback system has the closed-loop transfer function given by T(s) = K (s + 45)2 + K Using the root locus method, determine the value of the gain K so that the closed-loop system has a damping ratio £ = V7/2. a. K = 25 b. K = 1250 c K = 2025 d. K = 10500 524 Chapter 7 The Root Locus Method 10. Consider the unity feedback control system in Figure 7.74 where L(s) = Gc(s)G(s) = 10(s + z) \ ' s(s* + 4s + 8) Using the root locus method, determine that maximum value of z for closed-loop stability. a. z = 7.2 b. z = 12.8 c Unstable for all z > 0 d. Stable for all z > 0 In Problems 11 and 12, consider the control system in Figure 7.74 where the model of the process is r({S)) 750 _ ° (s + l)(s + 10)(s + 50)" 11. Suppose that the controller is Gc(s) = K(l + 0.2s) 1 + 0.025* Using the root locus method, determine the maximum value of the gain K for closed-loop stability. a. K = 2.13 b, K = 3.88 c, K = 14.49 d. Stable for all £ > 0 12. Suppose that a simple proportional controller is utilized, that is, Gc(s) = K. Using the root locus method, determine the maximum controller gain K for closed-loop stability. a. K = 0.50 b. K = 1.49 c K = 4.49 d. Unstable for K > 0 13. Consider the unity feedback system in Figure 7.74 where L(s) = Gc(s)G(s) = , + ^ , l 2 ^ .un^s(s + 5)(s + 6s + 17.76) Determine the breakaway point on the real axis and the respective gain, K. a. s = -1.8, K = 58.75 b. s = -2.5, K = 4.59 c. s = 1.4, iC = 58.75 d. None of the above In Problems 14 and 15, consider the feedback system in Figure 7.74, where L(s) = Gc(s)G(s) = K(s + 1 + j)(s + l - y ) s{s + 2j)(s - 2/) 525 Skills Check 14. Which of the following is the associated root locus? 4, . . , , 2 1 < c <0 -X E -1 2 • i-i -20 —2 -4 -4 -3 -2 -1 Real Axis (c) -40 -25 -20 -15 -10 Real Axis (d) -5 15. The departure angles from the complex poles and the arrival angles at the complex zeros are: a. D = ±180°, 4>A = 0° b . D = ±116.6°, 4 u = ±198.4° c. 4>D= ±45.8°, A = ±116.6° d. None of the above In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Parameter design b. Root sensitivity c. Root locus d. Root locus segments on the real axis e. Root locus method The amplitude of the closed-loop response is reduced approximately to one-fourth of the maximum value in one oscillatory period. The path the root locus follows as the parameter becomes very large and approaches co. The center of the linear asymptotes, 526 Chapter 7 The Root Locus Method f. Asymptote centroid g. Breakaway point h. Locus i. Angle of departure j . Number of separate loci k. Asymptote I. Negative gain root locus m. PID tuning n. Quarter amplitude decay o. Ziegler-Nichols PID tuning method The root locus lying in a section of the real axis to the left of an odd number of poles and zeros. The root locus for negative values of the parameter of interest where - c o < K ^ 0. The angle at which a locus leaves a complex pole in the s-plane. A path or trajectory that is traced out as a parameter is changed. The locus or path of the roots traced out on the .s-plane as a parameter is changed. The sensitivity of the roots as a parameter changes from its normal value. The method for determining the locus of roots of the characteristic equation 1 + KG(s) = 0 as 0 < K < co. The process of determining the PID controller gains. The point on the real axis where the locus departs from the real axis of the s-plane. Equal to the number of poles of the transfer function, assuming that the number of poles is greater than or equal to the number of zeros of the transfer function. EXERCISES E7.1 Let us consider a device that consists of a ball rolling on the inside rim of a hoop [11]. This model is similar to the problem of liquid fuel sloshing in a rocket. The hoop is free to rotate about its horizontal principal axis as shown in Figure E7.1. The angular position of the hoop may be controlled via the torque T applied to the hoop from a torque motor attached to the hoop drive shaft. If negative feedback is used, the system characteristic equation is Ks(s + 4) = 0. s- + 2s + 2 (a) Sketch the root locus, (b) Find the gain when the roots are both equal, (c) Find these two equal roots. 1 + Torque Hoop (d) Find the settling time of the system when the roots are equal. E7.2 A tape recorder has a speed control system so that H(s) = 1 with negative feedback and L(s) = Gc(s)G(s) = K s(s + 2)(52 + 4.v + 5) (a) Sketch a root locus for K, and show that the dominant roots are s = -0.35 ± /0.80 when K = 6.5. (b) For the dominant roots of part (a), calculate the settling time and overshoot for a step input. E7.3 A control system for an automobile suspension tester has negative unity feedback and a process [12] L{s) = Gc(s)G(s) = K(s2 + 4s + 8) s2(s + 4) We desire the dominant roots to have a £ equal to 0.5. Using the root locus, show that K = 7.35 is required and the dominant roots are s = —1.3 ± /2.2. E7.4 Consider a unity feedback system with FIGURE E7.1 Hoop rotated by motor. L(s) = Ge(s)G(s) = K(s + 1) 2 s + 4s + 5 527 Exercises (a) Find the angle of departure of the root locus from the complex poles, (b) Find the entry point for the root locus as it enters the real axis. E7.8 Sketch the root locus for a unity feedback system with L{s) = Gc(s)G(s) = Answers: ±225°; -2.4 E7.5 Consider a unity feedback system with a loop transfer function s2 + 2s + 10 Gc(s)G(s) = s4 + 38r1 + 515*' + 2950* + 6000 (a) Find the breakaway points on the real axis, (b) Find the asymptote centroid. (c) Find the values of AT at the breakaway points. E7.6 One version of a space sta tion is shown in Figure E7.6 [28]. It is critical to keep this station in the proper orientation toward the Sim and the Earth for generating power and communications. The orientation controller may be represented by a unity feedback system with an actuator and controller, such as Gc{s)G(s) 15 A s^s2 + 15J + 75)' Sketch the root locus of the system as K increases. Find the value of K that results in an unstable system. Answers: K = 75 Solar power panels Rockets s\s + 9)' (a) Find the gain when all three roots are real and equal, (b) Find the roots when all the roots are equal as in part (a). Answers: K = 27; s = —3 E7.9 The world's largest telescope is located in Hawaii. The primary mirror has a diameter of 10 m and consists of a mosaic of 36 hexagonal segments with the orientation of each segment actively controlled. This unity feedback system for the mirror segments has the loop transfer function L(s) = Gc(s)G(s) K s(s2 + 2s + 5) (a) Find the asymptotes and draw them in the s-plane, (b) Find the angle of departure from the complex poles. (c) Determine the gain when two roots lie on the imaginary axis. (d) Sketch the root locus. E7.10 A unity feedback system has the loop transfer function L(x) = KG(s) = Radar antenna K{s + 1) K(s + 2) ~s(s + 1)' (a) Find the breakaway and entry points on the real axis. (b) Find the gain and the roots when the real part of the complex roots is located at - 2 . (c) Sketch the locus. Answers: (a) -0.59, -3.41; (b) K = 3,.y = - 2 ± / V 5 E7.ll A robot force control system with unity feedback has a loop transfer function [6] Adjuster rockets Us) = KG(s) Space shuttle E7.7 The elevator in a modern office building travels at a top speed of 25 feet per second and is still able to stop within one-eighth of an inch of the floor outside. The loop transfer function of the unity feedback elevator position control is K(s + 8) = C « W ° « = ,(, + 4)(, + 6)(, (s + 2s + 2){s2 + 4s + 5) (a) Find the gain A' that results in dominant roots with a damping ratio of 0.707. Sketch the root locus. (b) Find the actual percent overshoot and peak time for the gain K of part (a). FIGURE E7.6 Space station. L W K(s + 2.5) 2 + 9)- Determine the gain K when the complex roots have a t equal to 0.8. E7.12 A unity feedback system has a loop transfer function K(s + 1) L(s) = KG(s) = --—2 . sis + 6s + 18) (a) Sketch the root locus for K > 0. (b) Find the roots when K - 10 and 20. (c) Compute the rise time, percent overshoot, and settling time (with a 2% criterion) of the system for a unit step input when K = 10 and 20. 528 Chapter 7 The Root Locus Method E7.13 A unity feedback system has a loop transfer function 4{s + z) s(s + l)(.y + 3)' L(s) = Gc(s)G(s) (a) Draw the root locus as z varies from 0 to 100. (b) Using the root locus, estimate the percent overshoot and settling time (with a 2% criterion) of the system at z = 0.6, 2, and 4 for a step input, (c) Determine the actual overshoot and settling time at z = 0.6, 2, and 4. E7.14 A unity feedback system has the loop transfer function L(s) = Gc(s)G(s) = Kjs + 10) s(s + 5) ' (a) Determine the breakaway and entry points of the root locus and sketch the root locus for K > 0. (b) Determine the gain K when the two characteristic roots have a £ of 1/v2. (c) Calculate the roots. E7.15 (a) Plot the root locus for a unity feedback system with loop transfer function L(s) = Gc(s)G(s) Gc(s) Answers: (a) K > 1.67; (b) ess = 0 E7.16 A negative unity feedback system has a loop transfer function Ke' L(s) = Gc(s)G(s) = s+ V where T = 0.1 s. Show that an approximation for the time delay is 2 T Using 20 - s 20 + s' G(s) • Y(s) FIGURE E7.17 Feedback system. (a) When Gc(s) = K, show that the system is always unstable by sketching the root locus, (b) When Gc(s) K(s + 2) s + 20 ' sketch the root locus and determine the range of K for which the system is stable. Determine the value of K and the complex roots when two roots lie on the /w-axis. E7.18 A closed-loop negative unity feedback system is used to control the yaw of the A-6 Intruder attack jet. When the loop transfer function is L(s) = Gc(s)G{s) = K(s + 10)0? + 2) (b) Calculate the range of K for which the system is stable, (c) Predict the steady-state error of the system for a ramp input. ,-0.1 j- R{s) K s(s + 3)(s2 + 2sr + 2) determine (a) the root locus breakaway point and (b) the value of the roots on the /w-axis and the gain required for those roots. Sketch the root locus. Answers: (a) Breakaway: .v = —2.29 (b) jco-axis: s = ±jl.Q9tK = 8 E7.19 A unity feedback system has a loop transfer function L(.v) = Gc(s)G(s) = £- . W s(s + 3)(s2 + 6i- + 64) (a) Determine the angle of departure of the root locus at the complex poles, (b) Sketch the root locus. (c) Determine the gain K when the roots are on the /tt-axis and determine the location of these roots. E7.20 A unity feedback system has a loop transfer function *tv + 1) L(s) = G(.is)Gis) = sis - 2)is + 6) (a) Determine the range of K for stability, (b) Sketch the root locus, (c) Determine the maximum £ of the stable complex roots. Answers: (a) K > 16; (b) t = 0.25 obtain the root locus for the system for K > 0. Determine the range of K for which the system is stable. E7.21 A unity feedback system has a loop transfer function E7.17 A control system, as shown in Figure E7.17, has a process Lis) = Gcis)Gis) - -. ^ . w 53 + 5s2 + 10 Sketch the root locus. Determine the gain K when the complex roots of the characteristic equation have a C approximately equal to 0.66. G(s) 1 s(s - I) - 529 Exercises E7.22 A high-performance missile for launching a satellite has a unity feedback system with a loop transfer function Gc(s)G(s) = K(s2 + I8)(s + 2) (s2 - 2)(5 + 12) ' Sketch the root locus as K varies from 0 < K < oo. E7.23 A unity feedback system has a loop transfer function L(s) = Gc(s)G(s) = Sketch the root locus for 0 Determine the characteristic equation and then sketch the root locus as 0 < k < oo. E7.25 A closed-loop feedback system is shown in Figure E7.25. For what range of values of the parameters K is the system stable? Sketch the root locus as 0 < K < oo. E7.26 Consider the signle-input, single-output system is described by x(/) = Ax(/) + Bu(t) y{t) = Cx(0 4(s? + 1) where s(s + a) a < oo. E7.24 Consider the system represented in state variable form DM, -4 c = [1 /?(,) 1 r°i n Controller Process K 10 .y + 25 1 1 s -.w FIGURE E7.27 Unity feedback system with parameter p. Controller Process 5+10 s 4 s+p Controller FIGURE E7.28 Feedback system for negative gain root locus. *• Sensor FIGURE E7.25 Nonunity feedback system with parameter K. tf(.Y) A'(.v) O -1]. E7.28. Consider the feedback system in Figure E7.28. Obtain the negative gain root locus as - o o < K ^ 0. For what values of K is the system stable? ,B = -k i 0], an d D =[( • -Q^ ) 0 , C = [1 1 E7.27 Consider the unity feedback system in Figure E7.27. Sketch the root locus as 0 < p < oo. where A = 1 ,B = -2 - K Compute the characteristic polynomial and plot the root locus as 0 ^ K < oo. For what values of K is the system stable? x = Ax + Bw y = Cx + 0 3 - K A = *> n.v) Process 5- 1 s(s2 + 2A- + 2) - • >'(.v) Y(s) 530 Chapter 7 The Root Locus Method PROBLEMS P7.1 Sketch the root locus for the following loop transfer functions of the system shown in Figure P7.1 when 0 < K < oo: K s(s + 10)(5 + 8) K (b) Gc(s)G(s) = 2 (s + 2s + 2)(5 + 2) (a) Gc(s)G{s) = (c) Gc(s)G(s) = K(s + 5) s(s + 1 ) ( 5 + 10) 0 < ka < oo. Determine the maximum allowable gain of the amplifier for a stable system. P7.5 Automatic control of helicopters is necessary because, unlike fixed-wing aircraft which possess a fair degree of inherent stability, the helicopter is quite unstable. A helicopter control system that utilizes an automatic control loop plus a pilot stick control is shown in Figure P7.5. When the pilot is not using the control stick, the switch may be considered to be open. The dynamics of the helicopter are represented by the transfer function K(s2 + 45 + 8) (d) Gc(s)G(s) = sz(s + 1) P7.3 A unity feedback system has the loop transfer function K s(s + 2)(5 + 5)' Find (a) the breakaway point on the real axis and the gain K for this point, (b) the gain and the roots when two roots lie on the imaginary axis, and (c) the roots when K = 6. (d) Sketch the root locus. P7.4 The analysis of a large antenna was presented in Problem P4.5. Sketch the root locus of the system as (s + 0.4)(52 - 0.365 + 0.16) (a) With the pilot control loop open (hands-off control), sketch the root locus for the automatic stabilization loop. Determine the gain K2 that results in a damping for the complex roots equal to t, = 0.707. (b) For the gain K2 obtained in part (a), determine the steady-state error due to a wind gust Td(s) = 1/5. (c) With the pilot loop added, draw the root locus as K] varies from zero to 00 when K2 is set at the value calculated in part (a), (d) Recalculate the steady-state error of part (b) when K\ is equal to a suitable value based on the root locus. P7.2 The linear model of a phase detector was presented in Problem P6.7. Sketch the root locus as a function of the gain Kv = KaK. Determine the value of Kv attained if the complex roots have a damping ratio equal to 0.60 [13]. Gc(s)G(s) = 25(5 + 0.03) G(s) = P7.6 An attitude control system for a satellite vehicle within the earth's atmosphere is shown in Figure P7.6. The transfer functions of the system are G(5) = K(s + 0.20) (5 + 0.90)(5 - 0.60)(5 - 0.10) /?(.v) • Yis) FIGURE P7.1 Pilot His) -JK>-+ Control stick Ki Helicopter dynamics G(s) 2 s + 12s + 1 Disturbance Switch Automatic stabilization FIGURE P7.5 Helicopter control. 5+9 t l Y(s) > Pitch attitude 531 Problems FIGURE P7.6 Satellite attitude control. 4j(s) Desired attitude Controller Satellite dynamics Gc(s) G(s) fas) Attitude l P7.8 Consider again the power control system of Problem P7.7 when the steam turbine is replaced by a hydroturbine. For hydroturbines, the large inertia of the water used as a source of energy causes a considerably larger time constant. The transfer function of a hydroturbine may be approximated by and (.v + 2 + ;1.5)(.v + 2 - /1.5) GAs) s + 4.0 (a) Draw the root locus of the system as K varies from 0 to oo. (b) Determine the gain K that results in a system with a settling time (with a 2% criterion) less than -TS + 1 12 seconds and a damping ratio for the complex roots Gr(s) = (T/2)S + r greater than 0.50. P7.7 The speed control system for an isolated power system where T = 1 second. With the rest of the system is shown in Figure P7.7. The valve controls the steam remaining as given in Problem P7.7, repeat parts (a) flow input to the turbine in order to account for load and (b) of Problem P7.7. changes AL(s) within the power distribution network. The equilibrium speed desired results in a generator fre- P7.9 The achievement of safe, efficient control of the spacing of automatically controlled guided vehicles is quency equal to 60 cps. The effective rotary inertia J is an important part of the future use of the vehicles in a equal to 4000 and the friction constant b is equal to 0.75. manufacturing plant [14, 15]. It is important that the The steady-state speed regulation factor R is represystem eliminate the effects of disturbances (such as sented by the equation R ~ (Q equals the speed between vehicles on a guideway. The system can be at no load. We want to obtain a very small R, usually less represented by the block diagram of Figure P7.9. The than 0.10. (a) Using root locus techniques, determine the vehicle dynamics can be represented by regulation R attainable when the damping ratio of the roots of the system must be greater than 0.60. (b) Verify (s + 0.1)(.v2 + 2s + 289) that the steady-state speed deviation for a load torque G(s) = s(s - 0.4)(.9 + 0.8)(4-2 + 1.45J + 361)' change AL(s) = AL/s is, in fact, approximately equal to RMwhen R < 0.1. Load torque Speed governor Reference speed L 1 0.25.v + 1 Steam turbine Valve FIGURE P7.7 G,(s) -- 1 0.25s + 1 1 R Power system control. «(,•) spacing , —' <. Power system -fO— 1 Js + b Controller Engine throttle Vehicle K$$s + 0.5) K, s + 30 G(s) (s + 30) 1 Ato(s) Speed deviation R = re gulation factor Sensor FIGURE P7.9 Guided vehicle control. AZ.(.v) n.v) • Spacing between vehicles 532 Chapter 7 The Root Locus Method (a) Sketch the root locus of the system, (b) Determine all the roots when the loop gain K = K \K^ is equal to 41)00. P7.10 New concepts in passenger airliner design will have the range to cross the Pacific in a single flight and the efficiency to make it economical [16. 29]. These new designs will require the use of temperature-resistant. lightweight materials and advanced control systems. Noise control is an important issue in modern aircraft designs since most airports have strict noise level requirements. One interesting concept is the Boeing Sonic Cruiser depicted in Figure P7.10(a). It would seat 200 to 250 passengers and cruise at just below the speed of sound. The flight control system must provide good handling characteristics and comfortable flying conditions. An automatic control system can be designed for the next generation passenger aircraft. The desired characteristics of the dominant roots of the control system shown in Figure P7.10(b) have a t = 0.707. The characteristics of the aircraft are io„ = 25,1 = 0-30, and T = 0.1. The gain factor Kh however, will vary over the range 0.02 at mediumweight cruise conditions to 0.20 at lightweight descent conditions, (a) Sketch the root locus as a function of the loop gain KiK2. (b) Determine the gain K2 necessary to yield roots with f = 0.707 when the aircraft is in the medium-cruise condition, (c) With the gain K2 as found in part (b), determine the f of the roots when the gain K\ results from the condition of light descent. P7.ll A computer system requires a high-performance magnetic tape transport system [17].The environmental conditions imposed on the system result in a severe test of control engineering design. A direct-drive DC motor system for the magnetic tape reel system is shown in Figure P7.ll, where r equals the reel radius, and J equals the reel and rotor inertia. A complete reversal of the tape reel direction is required in 6 ms, and the tape reel must follow a step command in 3 ms or less. The tape is normally operating at a speed of (a) Controller >o i (s + 2 ) K , 10 (,S + 10)(.5+ 100) 1 + 10 FIGURE P7 10 (a) A passenger jet aircraft of the future. (™ and © Boeing. Used under license.) (b) Control system. Aircraft dynamics Actuator : Rate gyro 1 (b) K , ( T i + 1) • .v2 + 2£u>„s + u>l m rate 533 Problems Tape reels and motors (a) Photocell transducer ff(s) + Desired — • ( " ) " * " position •— 0.5 K, T s + \ - -i/ J^' _; Amplifier Motor Ka r„.s + 1 KTIL s + R/L Xn • _ Tachometer i K, Reel 1 Js r s Tape position Motor back emf K2 FIGURE P7.11 *P (a) Tape control system, (b) Block diagram. (b) 100 in/s. The motor and components selected for this system possess the following characteristics: Kh = 0.40 T] = T„ = 1 IBS KT/(LJ) r = 0.2 A', = 2.0 K2 is adjustable. = 2.0 The inertia of the reel and motor rotor is 2.5 X 10"3 when the reel is empty, and 5.0 x 1(T3 when the reel is full. A series of photocells is used as an errorsensing device. The time constant of the motor is L/R = 0.5 ms. (a) Sketch the root locus for the system when K2 = 10 and J = 5.0 X 10"3, 0 < K„ < oo. (b) Determine the gain K„ thai results in a well-damped system so that the £ of all the roots is greater than or equal to 0.60. (c) With the K„ determined from part (b), sketch a root locus for 0 < K2 < so. P7.12 A precision speed control system (Figure P7.12) is required for a platform used in gyroscope and inertial system testing where a variety of closely controlled speeds is necessary. A direct-drive DC torque motor system was utilized to provide (1) a speed range of 0.017s to 6007s, and (2) 0.1% steady-state error maximum for a step input. The direct-drive DC torque motor avoids the use of a gear train with its attendant backlash and friction. Also, the direct-drive motor has a high-torque capability, high efficiency, and low motor time constants. The motor gain constant is nominally K,„ = 1.8, but is subject to variations up to 50%. The amplifier gain Ka is normally greater than 10 and subject to a variation of 10%. (a) Determine the minimum loop gain necessary lo satisfy the steady-state error requirement, (b) Determine the limiting value of gain for stability, (c) Sketch the rool locus as K„ varies from 0 to co. (d) Determine the roots when Ka = 40, and estimate the response to a step input. P7.13 A unity feedback system has the loop transfer function Us) = GAs)C(s) K s(s + 3)(s + 4.v + 7.84) 2 534 Chapter 7 The Root Locus Method Disturbance Controller Ka(s + 25)(s + 15) Kis) Reference sU + 2) 1 I As ^O 1.6 Tachometer m Speed * FIGURE P7.12 Speed control. (a) Find the breakaway point on the real axis and the gain for this point, (b) Find the gain to provide two complex roots nearest the y&i-axis with a damping ratio of 0.707. (c) A r e the two roots of part (b) dominant? (d) Determine the settling time (with a 2 % criterion) of the system when the gain of part (b) is used. P7.14 The loop transfer function of a single-loop negative feedback system is Us) = G,(s)G{s) = K(s + 2.5)(.5 + 3.2) P7.15 Let us again consider the stability and ride of a rider and high performance motorcycle as outlined in Problem P6.13. The dynamics of the motorcycle and rider can be represented by the loop transfer function G,(s)G(s) = K(s2 + 30s + 625) s(s + 20)(.r + 20s + 200)(.r + 60s + 3400)' Sketch the root locus for the system. Determine the f of the dominant roots when A! = 3 X 10 4 . s*( + !)(.? + 10)(.5+ 30)' This system is called conditionally stable because it is stable only for a range of the gain R such that ky < K < k2. Using the Routh-Hurwitz criteria and the root locus method, determine the range of the gain for which the system is stable. Sketch the root locus fort) < K < oo. P7.16 Control systems for maintaining constant tension on strip steel in a hot strip finishing mill are called "loopers." A typical system is shown in Figure P7.16. The looper is an arm 2 to 3 feet long with a roller on the end; it is raised and pressed against the strip by a motor [18], The typical speed of the strip passing the looper is 2000 ft/min. A voltage proportional to the looper Steel Rolls Rolls Motor (a) SMHO-* -J FIGURE P7.16 Steel mill control system. (b) 535 Problems position is compared with a reference voltage and integrated where it is assumed that a change in looper position is proportional to a change in the steel strip tension. The time constant r of the filter is negligible relative to the other time constants in the system. (a) Sketch the root locus of the control system for 0 < Ka < oo. (b) Determine the gain K„ that results in a system whose roots have a damping ratio of C = 0.707 or greater, (c) Determine the effect of T as 7 increases from a negligible quantity. P7.17 Consider again the vibration absorber discussed in Problems 2.2 and 2.10 as a design problem. Using the root locus method, determine the effect of the parameters M2 and kn- Determine the specific values of the parameters M2 and kn so that the mass Wj does not vibrate when F(t) = a sin(woT). Assume that Mi = l.jfcj = 1. and 6 = 1 . Also assume that kn < 1 and that the term k]2~may be neglected. P7.18 A feedback control system is shown in Figure P7.18. The filter Gc(s) is often called a compensator. and the design problem involves selecting the parameters a and /3. Using the root locus method, determine the effect of varying the parameters. Select a suitable filter so that the time to settle (to within 2% of the final value) is less than 4 seconds and the damping ratio of the dominant roots is greater than 0.60. Filter Process as + 1 4 s(s + 2) /fc + 1 1 FIGURE P7.18 Filter design. P7.19 In recent years, many automatic control systems for guided vehicles in factories have been installed. One system uses a magnetic tape applied to the floor to guide the vehicle along the desired lane [10, 15]. Using transponder tags on the floor, the automatically guided vehicles can be tasked (for example, to speed up or slow down) at key locations. An example of a guided vehicle in a factory is shown in Figure P7.19(a). We have G(s) s2 + As + 100 s(s + 2)(s + 6) and Ka is the amplifier gain. Sketch a root locus and determine a suitable gain K„ so that the damping ratio of the complex roots is 0.707. [a) FIGURE P7.19 (a) An automatically guided vehicle. (Photo courtesy of the Jervis B. Webb Company) (b) Block diagram. R(s) \r *• Y(s) Controller Actuator and vehicle K, Gis) £„(.v) reference m of travel ( b) 536 Chapter 7 The Root Locus Method P7.20 Determine the root sensitivity for the dominant roots of the design for Problem P7.18 for the gain K = 4a//3 and the pole s = -2. P7.21 Determine the root sensitivity of the dominant roots of the power system of Problem P7.7. Evaluate the sensitivity for variations of (a) the poles at s = - 4 , and (b) the feedback gain, 1/7?. P7.22 Determine the root sensitivity of the dominant roots of Problem P7.1(a) when K is set so that the damping ratio of the unperturbed roots is 0.707. Evaluate and compare the sensitivity as a function of the poles and zeros of Gc(s)G(s). P7.23 Repeat Problem P7.22 for the loop transfer function Gc(s)G(s) of Problem P7.1(c). P7.24 For systems of relatively high degree, the form of the root locus can often assume an unexpected pattern. The root loci of four different feedback systems of third order or higher are shown in Figure P7.24. The open-loop poles and zeros of KG(s) are shown, and the form of the root loci as K varies from zero to infinity is presented. Verify the diagrams of Figure P7.24 by constructing the root loci. P7.25 Solid-state integrated electronic circuits are composed of distributed R and C elements. Therefore, feedback electronic circuits in integrated circuit form must be investigated by obtaining the transfer function of the distributed RC networks. It has been shown that the slope of the attenuation curve of a distributed RC network is 10« dB/decade, where n is the order of the RC filter [13]. This attenuation is in contrast with the normal 20n dB/decade for the lumped parameter circuits. (The concept of the slope of an attenuation curve is considered in Chapter 8. If it is unfamiliar, (a) -1.125 K= 1010 +X 5 -15 X * * < )£ -3 K= 1010 (b) FIGURE P7.24 Root loci of four systems. (c) (d) Problems 537 reexamine this problem after studying Chapter 8.) An interesting case arises when the distributed RC network occurs in a series-to-shunt feedback path of a transistor amplifier. Then the loop transfer function may be written as K(s - 1 ) ( 5 + 3) 1/2 L(s) = G (s)G(s) = — Tzrc W cK ' KJ (5 + 1)(5 + 2)1'2 (a) Using the root locus method, determine the locus of roots as K varies from zero to infinity, (b) Calculate the gain at borderline stability and the frequency of oscillation for this gain. P7.26 A single-loop negative feedback system has a loop transfer function L(s) = Gc(s)G(s) = K(s + 2)2 5(.9 + 1 ) ( 5 + 8) P7.27 A unity negative feedback system has a loop transfer function K(s2 + 0.1) sis2 + 2) K(s + /0.3162)(5 /0.3162) 2 s(s + 1) Sketch the root locus as a function of K. Carefully calculate where the segments of the locus enter and leave the real axis. P7.28 To meet current U.S. emissions standards for automobiles, hydrocarbon (HC) and carbon monoxide (CO) emissions are usually controlled by a catalytic converter in the automobile exhaust. Federal standards for nitrogen oxides (NO x ) emissions are met mainly by exhaust-gas recirculation (EGR) techniques. However, as NO x emissions standards were tightened from the R(s) Reference FIGURE P7.28 Auto engine control. L(s) 2 (a) Sketch the root locus for 0 < K < oo to indicate the significant features of the locus, (b) Determine the range of the gain K for which the system is stable. (c) For what value of K in the range K 2: 0 do purely imaginary roots exist? What are the values of these roots? (d) Would the use of the dominant roots approximation for an estimate of settling time be justified in this case for a large magnitude of gain (K > 50)? L(s) = Gc(s)G(s) current limit of 2.0 grams per mile to 1.0 gram per mile, these techniques alone were no longer sufficient. Although many schemes are under investigation for meeting the emissions standards for all three emissions, one of the most promising employs a three-way catalyst—for HC, CO, and NO x emissions—in conjunction with a closed-loop engine-control system. The approach is to use a closed-loop engine control, as shown in Figure P7.28 [19,23]. The exhaust-gas sensor gives an indication of a rich or lean exhaust and compares it to a reference. The difference signal is processed by the controller, and the output of the controller modulates the vacuum level in the carburetor to achieve the best air-fuel ratio for proper operation of the catalytic converter. The loop transfer function is represented by + f\ — i Controller Ks2 + Us + 20 5 3 + 105 2 + 255' Calculate the root locus as a function of K. Carefully calculate where the segments of the locus enter and leave the real axis. Determine the roots when K = 2. Predict the step response of the system when K = 2. P7.29 A unity feedback control system has a transfer function L(s) = Gc(s)G(s) = K(s2 + 105 + 30) s2(s + 10) We desire the dominant roots to have a damping ratio equal to 0.707. Find the gain K when this condition is satisfied. Show that the complex roots are 5 = -3.56 ± y'3.56 at this gain. P7.30 An RLC network is shown in Figure P7.30. The nominal values (normalized) of the network elements are L — C = 1 and R — 2.5. Show that the root sensitivity of the two roots of the input impedance Z{s) to a change in R is different by a factor of 4. + 0- R Z(.v) FIGURE P7.30 RLC network. Carburetor — • Engine i Sensor Uxygen Three-way catalytic converter Exhaust 538 Chapter 7 The Root Locus Method P731 The development of high-speed aircraft and missiles requires information about aerodynamic parameters prevailing at very high speeds. Wind tunnels are used to test these parameters. These wind tunnels are constructed by compressing air to very high pressures and releasing it through a valve to create a wind. Since the air pressure drops as the air escapes, it is necessary to open the valve wider to maintain a constant wind speed. Thus, a control system is needed to adjust the valve to maintain a constant wind speed. The loop transfer function for a unity feedback system is L(S) = GAs)G(s) K(s + 4) s(s + 0.16)(s + p)(s - p)' where p = 7.3 + 9.7831/.Sketch the root locus and show the location of the roots for K = 326 and K = 1350. P7.32 A mobile robot suitable for nighttime guard duty is available. This guard never sleeps and can tirelessly patrol large warehouses and outdoor yards. The steering control system for the mobile robot has a unity feedback with the loop transfer function L(s) = Gc(s)G(s) K(s + 1)(5 + 5) 5(5 + 1.5)(5 + 2)' (a) Find K for all breakaway and entry points on the real axis, (b) Find K when the damping ratio of the complex roots is 0.707. (c) Find the minimum value of the damping ratio for the complex roots and the associated gain K. (d) Find the overshoot and the time to settle (to within 2% of the final value) for a unit step input for the gain, K, determined in parts (b) and (c). P7.33 The Bell-Boeing V-22 Osprey Tiltrotor is both an airplane and a helicopter. Its advantage is the ability to rotate its engines to 90° from a vertical position for takeoffs and landings as shown in Figure P7.33(a), and then to switch the engines to a horizontal position for cruising as an airplane [20].The altitude control system in the helicopter mode is shown in Figure P7.33(b). (a) Determine the root locus as K varies and determine the range of K for a stable system, (b) For K = 280, find the actual y{t) for a unit step input r(i) and the percentage overshoot and settling time (with a 2% criterion), (c) When K = 280 and r(t) = 0, find y(i) for a unit step disturbance, Td(s) = \/s. (d) Add a prefilter between R(s) and the summing node so that GJs) 0.5 ,«2 + 1.55 + 0.5' and repeat pari (b). P7.34 The fuel control for an automobile uses a diesel pump that is subject to parameter variations. A unity negative feedback has a loop transfer function G f (5)G(s) K(s + 2) (s + 1)(5 + 2.5)(s + 4)(5 + 10)' (a) Sketch the root locus as K varies from 0 to 2000. (b) Find the roots for K equal to 400, 500, and 600. (c) Predict how the percent overshoot to a step will vary for the gain K, assuming dominant roots, (d) Find the actual time response for a step input for all three gains and compare the actual overshoot with the predicted overshoot. Y(s) Altitude FIGURE P7.33 (a) Osprey Tiltrotor aircraft, (b) Its control system. (b) 539 Advanced Problems P7.35 A powerful electrohydraulic forklift can be used to lift pallets weighing several tons on top of 35-foot scaffolds at a construction site. The negative unity feedback system has a loop transfer function L(s) = Gc(s)G(s) K (.y + 40) (s + a)(s + b) K(s + 1)2 s(s2 + 1 ) ' FIGURE P7.37 Feedback system. (a) Sketch the root locus for K > 0. (b) Find the gain K when two complex roots have a £ of 0.707, and calculate all three roots, (c) Find the entry point of the root locus at the real axis, (d) Estimate the expected overshoot to a step input, and compare it with the actual overshoot determined from a computer program. P7.38 A unity feedback system has the loop transfer function L{s) = Ge{s)G(s) = ' (a) Sketch the root locus for K > 0. (b) Find the gain and roots when the characteristic equation has two imaginary roots, (c) Determine the characteristic roots when K = 20 and K = 100. (d) For K = 20, estimate the percent overshoot to a step input, and compare the estimate to the actual overshoot determined from a computer program. P7.37 Identify the parameters K, a, and b of the system shown in Figure P7.37. The system is subject to a unit step input, and the output response has an overshoot but ultimately attains the final value of 1. When the closed-loop system is subjected to a ramp input, the output response follows the ramp input with a finite steadystate error. When the gain is doubled to 2K, the output response to an impulse input is a pure sinusoid with a period of 0.314 second. Determine K, a, and b. FIGURE P7.39 Tilt control for a high-speed train. /?(.v) Command tilt >l _ i Controller Dynamics A: 22 ,v + 8s + 22 fc s(s ~ 3) • P7.39 High-speed trains for U.S. railroad tracks must traverse twists and turns. In conventional trains, the axles are fixed in steel frames called trucks.The trucks pivot as the train goes into a curve, but the fixed axles stay parallel to each other, even though the front axle tends to go in a different direction from the rear axle [24]. If the train is going fast, it may jump the tracks. One solution uses axles that pivot independently. To counterbalance the strong centrifugal forces in a curve, the train also has a computerized hydraulic system that tilts each car as it rounds a turn. On-board sensors calculate the train's speed and the sharpness of the curve and feed this information to hydraulic pumps under the floor of each car. The pumps tilt the car up to eight degrees, causing it to lean into the curve like a race car on a banked track. The tilt control system is shown in Figure P7.39. Sketch the root locus, and determine the value of K when the complex roots have maximum damping. Predict the response of this system to a step input R(s). K(s + l)(.v + 2)(s + 3) s\s - 1) K(s + 1) This system is open-loop unstable, (a) Determine the range of K so that the closed-loop system is stable. (b) Sketch the root locus, (c) Determine the roots for K = 10. (d) For K = 10, predict the percent overshoot for a step input using Figure 5.13. (e) Determine the actual overshoot by plotting the response. P7.36 A microrobot with a high-performance manipulator has been designed for testing very small particles, such as simple living cells [6]. The single-loop unity negative feedback system has a loop transfer function L(s) = Gc(s)G(s) = • + • K(.v) S+1 Y(s) 2 lilt ADVANCED PROBLEMS AP7.1 The top view of a high-performance jet aircraft is shown in Figure AP7.1(a) [20]. Sketch the root locus and determine the gain K so that the £ of the complex poles near the y'w-axis is the maximum achievable. Evaluate the roots at this K and predict the response to a step input. Determine the actual response and compare it to the predicted re.sponse. 540 Chapter 7 The Root Locus Method Aileron Elevator Rudder (a) R{s) • „ ns) K(s + 6) s{ + 4){s2 + 4s + 8) . Pitch lb) FIGURE AP7.1 (a) High-performance aircraft, (b) Pitch control system. AP7.2 A magnetically levitated high-speed train "flies" on an air gap above its rail system, as shown in Figure AP7.2(a) [24], The air gap control system has a unity feedback system with a loop transfer function Gc(s)G(s) K(s + l)(s + 3) s(s - i)(s + 4)(.? + sy The feedback control system is illustrated in Figure AP7.2(b). The goal is to select K so that the response for a unit step input is reasonably damped and the settling time is less than 3 seconds. Sketch the root locus, and select K so that all of the complex roots have a f greater than 0.6. Determine the actual response for the selected K and the percent overshoot. Air gap Area of attraction —- - T-shaped guideway (a) W Controller ~\EaU) L FIGURE AP7.2 (a) Magnetically levitated highspeed train. (b) Feedback control system. K{s+ 1) s J, (b) Plant u- s +4 l)(s + 5)(.?+ 10) Y(s) Air gap 541 Advanced Problems AP7.3 A compact disc player for portable use requires a good rejection of disturbances and an accurate position of the optical reader sensor. The position control system uses unity feedback and a loop transfer function L(s) = Gc(s)G(s) = AP7.6 L(s) = Gc(s)G(s) s(s + 1)0 + Py KG(s) = l/±*360° for k = 0 , 1 , 2 , . . . . (s + a) G(s) .r3 + ( 1 + a)s2 + (a -$$s + 1 - a We want the steady-state position error for a step input to be less than or equal to 10% of the magnitude of the input. Sketch the root locus as a function of the parameter a. Determine the range of a required for the desired steady-state error. Locate the roots for the allowable value of a to achieve the required steady-state error, and estimate the step response of the system. AP7.5 A unity feedback system has a loop transfer function Gc(s)G(s) K 5 3 + 10s 2 + 75 - 18' (a) Sketch the root locus and determine K for a stable system with complex roots with £ equal to l / v 2 . (b) Determine the root sensitivity of the complex roots of part (a). (c) Determine the percent change in K (increase or decrease) so that the roots lie on the /w-axis. R(s) ~} l\ FIGURE AP7.8 A position control system with velocity feedback. R(s) FIGURE AP7.9 A unity feedback control system. s + 2^2 + 3s + 1 Sketch the root locus for 0 < K < oo. + L(s) = K(s2 + 3.? + 6) 3 AP7.7 A feedback system with positive feedback is shown in Figure AP7.7. The root locus for K > 0 must meet the condition AP7.4 A remote manipulator control system has unity feedback and a loop transfer function = = Sketch the root locus for K > 0, and select a value for K that will provide a closed step response with settling time less than 1 second. 10 The parameter p can be chosen by selecting the appropriate D C motor. Sketch the root locus as a function of p . Select/? so that the £ of the complex roots of the characteristic equation is approximately 1/V2. Gc(s)G(s) A unity feedback system has a loop transfer function O R(s) r~\ 1 (5 + 4)(.5 + 8) K +V *> Y{s) FIGURE AP7.7 A closed-loop system with positive feedback. AP7.8 A position control system for a D C motor is shown in Figure AP7.8. Obtain the root locus for the velocity feedback constant K, and select K so that all the roots of the characteristic equation are real (two are equal and real). Estimate the step response of the system for the K selected. Compare the estimate with the actual response. AP7.9 A control system is shown in Figure AP7.9. Sketch the root loci for the following transfer functions Gc(sY (a) Gc(s) = K (b) G(.(.v) = K(s + 3) 120 (5 + 2)(.^+ 17) \ Y(s) Position s K Controller Process G,.(s) 1 s(s + 2)(5 + 5) " • Y(s) 542 Chapter 7 The Root Locus Method (c) Gc(s) = w c W K(s + 1) ' s + 20 K(s + 1)(5 + 4) AP7.10 A feedback system is shown in Figure AP7.10. Sketch the root locus as K varies when K > 0. Determine a value for K that will provide a step response with an overshoot less than 5% and a settling time (with a 2% criterion) less than 2.5 seconds. AP7.12 A control system with PI control is shown in Figure AP7.12. (a) Let K,/Kp = 0.2 and determine KP so that the complex roots have maximum damping ratio, (b) Predict the step response of the system with KP set to the value determined in part (a). AP7.13 The feedback system shown in Figure AP7.13 has two unknown parameters K\ and K2. The process transfer function is unstable. Sketch the root locus for 0 < KUK2 < oo. What is the fastest settling time that you would expect of the closed-loop system in response to a unit step input R(s) = 1/s? Explain. 10 X AP7.11 A control system is shown in Figure AP7.11. Sketch the root locus, and select a gain K so that the step response of the system has an overshoot of less than 10% and the settling time (with a 2% criterion) is less than 4 seconds. (s + 2)(s + 5) K s+ K FIGURE AP7.10 A nonunity feedback control system. Controller Process 1 2 K(s + 2) (s + \0)(s + 20) /?(.v) k _ FIGURE AP7.11 A control system with parameter K. s(s2 + 3s + 3.5) Controller "•> Process 1 fc s(s2 + Is + 10) i FIGURE AP7.12 A control system with a PI controller. Kt «(v) O—*- (x + 5)(* - I) 3 • + — o — * — o — • — o «*) -A:, (a) R(s) *1 1 — • (s + 5)(A- - I) 1 FIGURE AP7.13 An unstable plant with two parameters / K (b) 3 Yis) • Y(s) 543 Design Problems AP7.14 A unity feedback control system shown in Figure AP7.14 has the process C{s) 10 Design a PID controller using Ziegler-Nichols methods. Determine the unit step response and the unit disturbance response. What is the maximum percent overshoot and settling time for the unit step input? s(s + 10)0 + 7.5)' w Controller KP+ -^-+ FIGURE AP7.14 Unity feedback loop with PID controller. So$

1

^L

Process

10 S(S + 10)(.5 + 7.5)

DESIGN PROBLEMS CDP7.1 The drive motor and slide system uses the output of a tachometer mounted on the shaft of the motor as shown in Figure CDP4.1 (switch-closed option). The output voltage of the tachometer is vT = K\B. Use the velocity feedback with the adjustable gain K\, Select the best values for the gain K^ and the amplifier gain K„ so that the transient response to a step input has an overshoot less than 5% and a settling time (to within 2% of the final value) less than 300 ms. DP7.1 A high-performance aircraft, shown in Figure DP7.1(a). uses the ailerons, rudder, and elevator to steer through a three-dimensional flight path [20]. The pitch rate control system for a fighter aircraft at 10.000 m and Mach 0.9 can be represented by the system in Figure DP7.1(b), where C(5) =

(a) Sketch the root locus when the controller is a gain. so that Gc(s) - K, and determine K when I for the roots with co„ > 2 is larger than 0.15 (seek a maximum I). (b) Plot the response q(t) for a step input r(i) with K as in (a), (c) A designer suggests an anticipatory controller with Gc(s) = K, + K2s = K(s + 2). Sketch the root locus for this system as K varies and determine a K so that the f of all the closed-loop roots is >0.8. (d) Plot the response q{t) for a step input r(t) with Kasin (c). DP7.2 A large helicopter uses two tandem rotors rotating in opposite directions, as shown in Figure P7.33(a). The controller adjusts the tilt angle of the main rotor and thus the forward motion as shown in Figure DP7.2.The helicopter dynamics are represented by

- 1 8 Q + 0.015)0 + 0.45)

G(s)

(s2 + 1.2s + 12)(52 + 0.01s + 0.0025)'

Rudder

Ailerons

(a)

Pitch rate command + -.

FIGURE DP7 1 (a) Highperformance aircraft, (b) Pitch rate control system.

< ^J

Controller

>

'

GM

lb)

Aircraft

Q(s) Pitch rale

—•

O(s)

H) i'2 + 4.5s + 9'

544

Chapter 7 The Root Locus Method

Helicopter dynamics

Controller

/?(*)

FIGURE DP7.2 Two-rotor helicopter velocity control.

"^

- 91

,

Tilt angle

Gc(s)

G(s)

be represented by the system shown in Figure DP7.3. (a) Sketch the root locus for K and identify the roots for K = 4.1 and 41. (b) Determine the gain K that results in an overshoot to a step of approximately 1%. (c) Determine the gain that minimizes the settling time (with a 2% criterion) while maintaining an overshoot of less than 1 %. DP7.4 A welding torch is remotely controlled to achieve high accuracy while operating in changing and hazardous environments [21]. A model of the welding arm position control is shown in Figure DP7.4, with the disturbance representing the environmental changes. (a) With Ttl(s) = 0, select K\ and K to provide high-quality performance of the position control system. Select a set of performance criteria, and examine the results of your design, (b) For the system in part (a), let R(s) = 0 and determine the effect of a unit stepr rf (s) - 1 /sby obtaining y{t).

and the controller is selected as Gc(s) = K} +

K2

K(s + 1)

(a) Sketch the root locus of the system and determine K when £ of the complex roots is equal to 0.6. (b) Plot the response of the system to a step input r{t) and find the settling time (with a 2% criterion) and overshoot for the system of part (a). What is the steady-state error for a step input? (c) Repeat parts (a) and (b) when the £ of the complex roots is 0.41. Compare the results with those obtained in parts (a) and (b). DP7.3 The vehicle Rover has been designed for maneuvering at 0.25 mph over Martian terrain. Because Mars is 189 million miles from Earth and it would take up to 40 minutes each way to communicate with Earth [22,27], Rover must act independently and reliably- Resembling a cross between a small flatbed truck and an elevated jeep. Rover is constructed of three articulated sections, each with its own two independent, axle-bearing, one-meter conical wheels. A pair of sampling arms—one for chipping and drilling, the other for manipulating fine objects—extend from its front end like pincers. The control of the arms can

DP7.5 A high-performance jet aircraft with an autopilot control system has a unity feedback and control system, as shown in Figure DP7.5. Sketch the root locus and select a gain K that leads to dominant poles. With this gain K, predict the step response of the system. Determine the actual response of the system, and compare it to the predicted response.

Controller

/?(.v)

+Q •>)

Manipulator U(s)

2

k

K(s + 6.5s + 12) .V

FIGURE DP7.3 Mars vehicle robot control system.

Y(s)

1 (5+ 1)(.9 + 2)

7'(/(.v) Controller

R(s)

FIGURE DP7.4 Remotely controlled welder.

K(\ +0.01.S)

Process

kc

H

10 s2(s + 10)

K,s

- • Y(s)

545

Design Problems

Autopilot

l _

FIGURE DP7.5 High-performance jet aircraft.

K{s + 1) s

RU) Leg

position input

FIGURE DP7.6 Automatic control of walking motion.

<2r

Aircraft dynamics 1 0.1)(.s2 + 10.5 + 41)

(s-

Controller

Dynamics

K(s + 2) (5 + 10)

1 . 5 ( 5 - 1)

DP7.6 A system to aid and control the walk of a partially disabled person could use automatic control of the walking motion [25]. One model of a system that is open-loop unstable is shown in Figure DP7.6. Using the root locus, select K for the maximum achievable £ of the complex roots. Predict the step response of the system, and compare it with the actual step response.

1 (v + 1)((),.% + 1)"

YU) Actual leg position

and Gt.(.f) is selected as a PI controller so that the steady-state error for a step input is equal to zero. We then have Gc(s)

KP +

K,

Design the PI controller so that (a) the percent overshoot for a step input is P.O. s 5 % ; (b) the settling time (with a 2 % criterion) is Ts s 6 seconds; (c) the system velocity error constant Kv > 0.9; and (d) the peak time, Tp. for a step input is minimized.

DP7.7 A mobile robot using a vision system as the measurement device is shown in Figure DP7.7(a) [36].The control system is shown in Figure DP7.7(b) where G(s)

*• n.rt

DP7.8 Most commercial op-amps are designed to be unity-gain stable [26]. That is, they are stable when

Motion subsystem Recognition subsystem

(a)

• Y(a)

FIGURE DP7.7 (a) A robot and vision system. (b) Feedback control system.

ib)

546

Chapter 7 The Root Locus Method

* VJs) VM

FIGURE DP7.8 (a) Op-amp circuit. (b) Control system.

(a)

(b)

used in a unity-gain configuration. To achieve higher bandwidth, some op-amps relax the requirement to be unity-gain stable. One such amplifier has a DC gain of 105 and a bandwidth of 10 kHz. The amplifier, G(.v), is connected in the feedback circuit shown in Figure DP7.8(a).The amplifier is represented by the model shown in Figure DP7.8(b). where K„ = 105. Sketch the root locus of the system for K. Determine the minimum value of the DC gain of the closed-loop amplifier for stability. Select a DC gain and the resistors Rx andR2. DP7.9 A robotic arm actuated at the elbow joint is shown in Figure DP7.9(a), and the control system for the

actuator is shown in Figure DP7.9(b). Plot the root locus for K > 0. Select Gp(s) so that the steady-state error for a step input is equal to zero. Using the Gp(s) selected, plot y(t) for K equal to 1, 1.5, and 2.85. Record the rise time, settling time (with a 2% criterion), and percent overshoot for the three gains. We wish to limit the overshoot to less than 6% while achieving the shortest rise time possible. Select the best system fori < / C s 2.85. DP7.10 The four-wheel-steering automobile has several benefits. The system gives the driver a greater degree of control over the automobile. The driver gets a more forgiving vehicle over a wide variety of conditions.

Wrist

Light weight flexible arm

Elbow

(a)

Ris) Position

FIGURE DP7.9 (a) A robotic arm actuated at the joint elbow, (b) Its control system.

+ >

/^\

Gp(s) _

i

Controller

Elbow actuator

K(s+ 1)

2 s(s + 4)

k

.V -

(b)

3 + 2s +

Yti) position

547

Design Problems

The system enables the driver to make sharp, smooth lane transitions. It also prevents yaw, which is the swaying of the rear end during sudden movements. Furthermore, the four-wheel-steering system gives a car increased maneuverability. This enables the driver to park the car in extremely tight quarters. With additional closed-loop computer operating systems, a car could be prevented from sliding out of control in abnormal icy or wet road conditions. The system works by moving the rear wheels relative to the front-wheel-steering angle. The control system takes information about the front wheels* steering angle and passes it to the actuator in the back. This actuator then moves the rear wheels appropriately. When the rear wheels are given a steering angle relative to the front ones, the vehicle can vary its lateral acceleration response according to the loop transfer function Gc(s)G(s) = K

pilot crane control is shown in Figure DP7.11(b). Design a controller that will achieve control of the desired variables when G c (s) = K. DP7.12 A rover vehicle designed for use on other planets and moons is shown in Figure DP7.12(a) [21]. The block diagram of the steering control is shown in Figure DP7.12(b), where G(s) =

1.5

(s + 1)(.5 + 2)0$+ 4)(.s + 10)' 1 + ( 1 + X)Txs + ( 1 + K)T2s2 s[l + (2£/a>„).9 + (\/o>n2)s2] ' (a) where A = 2(//(1 - q), and q is the ratio of rear wheel angle to front wheel steering angle [14]. We will assume that F t = T2 — ) second and a>„ = 4. Design m a unity feedback system, selecting an appropriate set R(s) G(is) G(s) • Sleeiiiig of parameters (A, K, f) so that the steering control angle response is rapid and yet will yield modest overshoot characteristics. In addition, q must be be tween 0 and 1. DP7.11 A pilot crane control is shown in Figure CM DP7.11(a). The trolley is moved by an input F(i) in order to control x(t) and (t) [13]. The model of the FIGURE DP7.12 (a) Planetary rover vehicle, (b) Steering Xn > V I control system. (a) ¢(5) s2 + 10 Desired trolley position FIGURE DP7.11 (a) Pilot crane control system. (b) Block diagram. C,(s) -rO 10 (b) Speed Trolley position 548 Chapter 7 The Root Locus Method (a) When G c (s) = K, sketch the root locus as K varies from 0 to 1000. Find the roots for K equal to 100, 300, and 600. (b) Predict the overshoot, settling time (with a 2% criterion), and steady-state error for a step input, assuming dominant roots, (c) Determine the actual time response for a step input for the three values of the gain K, and compare the actual results with the predicted results. desired roll angle d and the actual angle will drive the hydraulic actuator, which in turn adjusts the deflection of the aileron surface. A simplified model where the rolling motion can be considered independent of other motions is assumed, and its block diagram is shown in Figure DP7.13(b). Assume that Kx = 1 and that the roll rate 4> is fed back using a rate gyro. The step response desired has an overshoot less than 10% and a settling time (with a 2% criterion) less than 9 seconds. Select the parameters Ka and K2- DP7.13 The automatic control of an airplane is one example that requires multiple-variable feedback methods. In this system, the attitude of an aircraft is controlled by three sets of surfaces: elevators, a rud- DP7.14 Consider the feedback system shown in Figure DP7.14. The process transfer function is marginally der, and ailerons, as shown in Figure DP7.I3(a). By stable. The controller is the proportional-derivative manipulating these surfaces, a pilot can set the aircraft (PD) controller on a desired flight path [20]. An autopilot, which will be considered here, is Gc(s) = Kp + KDs. an automatic control system that controls the roll angle (j> by adjusting aileron surfaces. The deflection (a) Determine the characteristic equation of the of the aileron surfaces by an angle 8 generates a closed-loop system. torque due to air pressure on these surfaces. This (b) Let T = KF/KD.Write the characteristic equation causes a rolling motion of the aircraft. The aileron in the form surfaces are controlled by a hydraulic actuator with a n(s) transfer function lis. Ms) 1 + K, d(sY The actual roll angle is measured and compared with the input. The difference between the Aileron (a) Amplifier K„ Actuator 1 s+ 1 1 .V l \ . Rate gyro K2 FIGURE DP7.13 (a) An airplane with a set of ailerons. (b) The block diagram for controlling the roll rate of the airplane. Attitude gyro "1 (b) 4 1 4> Computer Problems 549 (c) Plot the root locus for 0 =£ KD < oo when T = 6. (d) What is the effect on the root locus when 0 < T < VlO? Controller FIGURE DP7.14 A marginally stable plant with a PD controller in the loop. -. E,M) ...-4ot> /?( Kp •+• K[)& (e) Design the PD controller to meet the following specifications: (i) P.O. < 5% (ii) £ < 1 s Process 10 s2+10 • Yis) COMPUTER PROBLEMS CP7.1 Using the riocus function, obtain the root locus for the following transfer functions of the system shown in Figure CP7.1 when 0 < K < oo: 30 (a) G(.v) = -%3 7^ , s + 1%2 + 43i + 30 (b) G(s) = -z s2 + As + 20 s2 + s + 2 (c) G(.v) = s(s2 + 6s + 10) (d) G(s) = G{s) •v5 + 4 / + 653 + 10s2 -I- 65 + 4 s + 4.v5 + 4s4 + s3 + s2 + 10s + 1 6 R(s) >0 » KG(s) I * Y(s) FIGURE CP7.1 A single-loop feedback system with parameter K. CP7.2 A unity negative feedback system has the loop transfer function KG{s) = K s2 - 2s -f 2 s(s2 + 3s + 2) Develop an m-file to plot the root locus and show with the rlocfind function that the maximum value of K for a stable system is K = 0.79. CP7.3 Compute the partial fraction expansion of rw = , ' + « sis2 + 5s + 4) and verify the result using the residue function. CP7.4 A unity negative feedback system has the loop transfer function Gc{s)G(s) = (1 +p)s2 Develop an m-file to obtain the root locus as p varies; 0 < p < 00. For what values of/; is the closed-loop stable? CP7.5 Consider the feedback system shown in Figure CP7.1, where p s + 4s + 10" s+ 1 For what value of K is I = 0.707 for the dominant closed-loop poles? CP7.6 A large antenna, as shown in Figure CP7.6(a), is used to receive satellite signals and must accurately track the satellite as it moves across the sky. The control system uses an armature-controlled motor and a controller to be selected, as shown in Figure CP7.6(b). The system specifications require a steady-state error for a ramp input r(r) = Bi, less than or equal to 0.012?, where B is a constant. We also seek a percent overshoot to a step input of P.O. < 5% with a settling time (with a 2% criterion) of Ts < 2 seconds, (a) Using root locus methods, create an m-file to assist in designing the controller, (b) Plot the resulting unit step response and compute the percent overshoot and the settling time and label the plot accordingly, (c) Determine the effect of the disturbance T^s) = Q/s (where Q is a constant) on the output Y(s). CP7.7 Consider the feedback control system in Figure CP7.7. We have three potential controllers for our system: 1. Gc(s) = K (proportional controller) 2. Gc(s) = K/s (integral controller) 3. Gc(s) = K(\ + 1/s) (proportional, integral (PI) controller) The design specifications are 7, s 10 seconds and P.O. •& 10% for a unit step input. (a) For the proportional controller, develop an m-file to sketch the root locus for 0 < K < 00, and 550 Chapter 7 The Root Locus Method (a) Motor and antenna Controller Gc(s) R{s) • t 10 *LJ FIGURE CP7.6 Antenna position control. (b) FIGURE CP7.7 A single-loop feedback control system with controller Gc(s). -N Controller Process G,(s) 1 s 2 + 5.5 + 6 t J * »• Yds) i- determine the value of K so that the design specifications are satisfied. (b) Repeat part (a) for the integral controller. (c) Repeat part (a) for the PI controller. (d) Co-plot the unit step responses for the closedloop systems with each controller designed in parts (a)-(c). FIGURE CP7.8 A spacecraft attitude control system with a proportionalderivative controller. Position J ( J + 5 ) ( . S + 10) Desired altitude ">l , (e) Compare and contrast the three controllers obtained in parts (a)-(c), concentrating on the steady-state errors and transient performance. CP7.8 Consider the spacecraft single-axis attitude control system shown in Figure CP7.8. The controller is known as a proportional-derivative (PD) controller. Suppose that we require the ratio of Kp/KD = 5.Then, develop PD controller Spacecraft model Kv + KDs 1 Js2 0 Actual altitude 551 Terms and Concepts an m-filc using root locus methods find the values of KD/J and Kp/J so that the settling time 7^. is less than or equal to 4 seconds, and the peak overshoot P.O. is less than or equal to 10% for a unit step input. Use a 2% criterion in determining the settling time. CP7.9 Consider the feedback control system in Figure CP7.9. Develop an m-file to plot the root locus for 0 < K < DO . Find the value of K resulting in a damping ratio of the closed-loop poles equal to 0.707. CP7.10 Consider the system represented in state variable form where A = 0 0 -1 C = [1 1 0 -5 -9 0 1 ,B = 0 _4_ - 2 - k_ ~r 12], and D = [0]. (a) Determine the characteristic equation, (b) Using the Routh-Hurwitz criterion, determine the values of k for which the system is stable, (c) Develop an m-file to plot the root locus and compare the results to those obtained in (b). x = Ax + Bu y = Cx + DH, FIGURE CP7.9 Unity feedback system with parameter K. m • Y{s) ANSWERS TO SKILLS CHECK True or False: (1) True; (2) True; (3) False; (4) True; (5) True Multiple Choice: (6) b; (7) c; (8) a; (9) c; (10) a; (ll)b;(12)c;(13)a;(14)c;(15)b Word Match (in order, top to bottom): k, f, a, d, i, h, c b, e, g, j TERMS AND CONCEPTS Angle of departure The angle at which a locus leaves a Logarithmic sensitivity A measure of the sensitivity of complex pole in the .v-plane. the system performance to specific parameter changes, aT{s)/T(s) Angle of the asymptotes The angle cf>A that the asympTt ^ where T(s) is the system given by S'K(s) = tote makes with respect to the real axis. 9K/K transfer function and K is the parameter of interest. Asymptote The path the root locus follows as the parameter becomes very large and approaches infinity. The Manual PID tuning methods The process of determining number of asymptotes is equal to the number of poles the PID controller gains by trial-and-error with miniminus the number of zeros. mal analytic analysis. Asymptote centroid The center aA of the linear asymp- Negative gain root locus The root locus for negative totes. values of the parameter of interest, where - o o < K < 0. Breakaway point The point on the real axis where the locus departs from the real axis of the ,s-planc. Number of separate loci Equal to the number of poles of the transfer function, assuming that the number of Dominant roots The roots of the characteristic equation poles is greater than or equal to the number of zeros that represent or dominate the closed-loop transient of the transfer function. response. Parameter design A method of selecting one or two Locus A path or trajectory that is traced out as a paraparameters using the root locus method. meter is changed. 552 PID controller Chapter 7 The Root Locus Method A widely used controller used in industry Kj of the form Gc(s) = Kp +— + KDs, where Kp is the proportional gain, Kt is the integral gain, and KD is the derivative gain. PID tuning The process of determining the PID troller gains. Proportional plus deriviative (PD) controller A term controller of the form Gc(s) = Kp + where Kp is the proportional gain and KD is the vative gain. Proportional plus integral (PI) controller contwoKD$, deri-

A two-term

controller of the form Gc(s) Kp + -, where K.} is the proportional gain and Kj is the integral gain. Quarter amplitude decay The amplitude of the closedloop response is reduced approximately to one-fourth of the maximum value in one oscillatory period. Reaction curve The response obtained by taking the controller off-line and introducing a step input. The underlying process is assumed to be a first-order system with a transport delay. Root contours The family of loci that depict the effect of varying two parameters on the roots of the characteristic equation.

Root locus The locus or path of the roots traced out on the 5-plane as a parameter is changed. Root locus method The method for determining the locus of roots of the characteristic equation 1 + KP{s) = 0 as K varies from 0 to infinity. Root locus segments on the real axis The root locus lying in a section of the real axis to the left of an odd number of poles and zeros. Root sensitivity The sensitivity of the roots as a parameter changes from its normal value. The root sensitivity is given by 5^ =

, the incremental change in the <)K/K root divided by the proportional change of the parameter. Ultimate gain The PD controller proportional gain, Kp, on the border of instability when KD = 0 and K{ = 0. Ultimate period The period of the sustained oscillations when Kp is the ultimate gain and KD = 0 and Kf = 0. Ziegler-Nichols PID tuning method The process of determining the PID controller gains using one of several analytic methods based on open-loop and closed-loop responses to step inputs.

Chapter 8

Frequency Response Methods

CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 8.55 as specified in the various problem statements. Controller

Process

Gc(s)

G(s)

+ Y(s)

L

FIGURE 8.55 Block diagram for the Skills Check. In the following True or False and Multiple Choice problems, circle the correct answer. 1. The frequency response represents the steady-state response of a stable system to a sinusoidal input signal at various frequencies. 2. A plot of the real part of G(ja)) versus the imaginary part of G(ja)) is called a Bode plot. 3. A transfer function is termed minimum phase if all its zeros lie in the right-hand s-plane. 4. The resonant frequency and bandwidth can be related to the speed of the transient response. 5. One advantage of frequency response methods is the ready availability of sinusoidal test signals for various ranges of frequencies and amplitudes. 6. Consider the stable system represented by the differential equation x(t) + 3*(r) = uit), where u(t) - sin 3/. Determine the phase lag for this system.

True or False True or False True or False True or False True or False

a. = 0°

b. tj> = -45° c 0 = -60° d. = -180° In Problems 7 and 8, consider the feedback system in Figure 8.55 with the loop transfer function L{s) = G{s)Gc{s) =

80 + 1) s(2 + 5)(2 + 3s)

7. The Bode diagram of this system corresponds to which plot in Figure 8.56? 8. Determine the frequency at which the gain has unit magnitude and compute the phase angle at that frequency: a. o) = 1 rad/s, = -82° b. to = 1.26 rad/s, = -133° c. «M = 1.26 rad/s, = 133° d.
609

Skills Check 50

CQ

0

_L_LL_

-50

L I,

|

s - 100 -90

J

T

135

180 10

10_l

j

|

ill.

^7-

-50

s -

r

s

135 -

102

0- _ 180 10-2

10

102

103

(b) §

1 h

100

T?]f--4iiLi If 1111'

-90

j

50 0

^

I1

^

,,

100 50 0 -50 - 100

20 0

J

-20 -40

0 -45

-90 10-1

102

-90 io- 2

] Cl-

(c)

FIGURE 8.56

102

Bode plot selections.

In Problems 9 and 10, consider the feedback system in Figure 8.55 with the loop transfer function L(s) = G(s)Gc(s) =

50

^ + 125 + 20'

610

Chapter 8

Frequency Response Methods

11. Consider the Bode plot in Figure 8.57.

100 50

-L

PQ

I a -so -100

-150 -90 -135 "SB « -180 -

« -225

-270 10"

10'

10°

10'

102

103

Frequency (rad/s) FIGURE 8.57 Bode plot for unknown system.

Which loop transfer function L(s) = Gc(s)G(s) corresponds to the Bode plot in Figure 8.57? 100 a. L(s) = Gc(s)G(s) = s(s + 5)(s + 6) 24 b. L(s) = Gc(s)G(s) = s(s + 2)(s + 6) 24 s2(s + 6) 10 d. L(s) = Gc(s)G(s) = 2 .s + 0.55 + 10 c. L(s) = Gc(s)G(s) =

12. Suppose that one design specification for a feedback control system requires that the percent overshoot to a step input be less than 10%.The corresponding specification in the frequency domain is a. Mpw < 0.55 b. Mm ^ 0.59 c. Mpw < 1.05 d. Mpoi < 1.27 13. Consider the feedback control system in Figure 8.55 with Gc(s)G(s) =

100 s(s + 11.8)'

611

Skills Check The resonant frequency, a)n and the bandwidth, coh, are: a. dir — 1.59rad/s, cob = 1.86rad/s b . (or = 3.26 rad/s, o)b = 16.64 rad/s c cor = 12.52 rad/s, a>h = 3.25 rad/s d. cor = 5.49 rad/s,
For Problems 14 and 15, consider the frequency response of a process G(j(o) depicted in Figure 8.58.

i

-45 -90

IV

135 180 , io-2

- — 10' 102 Frequency (rad/s)

10u

10"

FIGURE 8.58

r

103

Bode plot for G(/o>).

14. Determine the system type (that is, the number of integrators, JV): a. N = 0 b. N = 1 c

N = 2

d. N > 2 15. The transfer function corresponding to the Bode plot in Figure 8.58 is: 100(5 + 10)(s + 5000) a. G(s) = s(s + 5)(s + 6) b. G(s) = c. G(s) = d. G{s) =

100 (s + l)(s + 20) 100 (s + 1)(5 + 50)(5 + 200) 100(5 + 20)(5 + 5000) (5 + 1)(5 + 50)(5 + 200)

IO4

105

612

Chapter 8

Frequency Response Methods

In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Laplace transform pair b. Decibel (dB) c Fourier transform

The logarithm of the magnitude of the transfer function and the phase are plotted versus the logarithm of o>, the frequency. The logarithm of the magnitude of the transfer function, 2uloglolG(/co)|. A plot of the real part of G(j(o) versus the imaginary part of G(jct)).

d. Bode plot e. Transfer function in the frequency domain f. Decade g. Dominant roots

h. All-pass network

i. Logarithmic magnitude j . Natural frequency

k, Fourier transform pair 1. Minimum phase m. Bandwidth n. Frequency response o. Resonant frequency p. Break frequency q. Polar plot r. Maximum value of the frequency response s, Nonminimum phase

The steady-state response of a system to a sinusoidal input signal. All the zeros of a transfer function lie in the left-hand side of the s-plane. The frequency at which the frequency response has declined 3 dB from its low frequency value. The frequency at which the maximum value of the frequency response of a complex pair of poles is attained. The frequency of natural oscillation that would occur for two complex poles if the damping were equal to zero. Transfer functions with zeros in the right-hand s-plane. The frequency at which the asymptotic approximation of the frequency response for a pole (or zero) changes slope. The transformation of a function of time into the frequency domain. The ratio of the output to the input signal where the input is a sinusoid. The units of the logarithmic gain. A pair of complex poles will result in a maximum value for the frequency response occurring at the resonant frequency. A nonminimum phase system that passes all frequencies with equal gain. A factor of ten in frequency. The roots of the characteristic equation that represent or dominate the closed-loop transient response. A pair of functions, one in the time domain, and the other in the frequency domain, and both related by the Fourier transform. A pair of functions, one in the time domain, and the other in the frequency domain, and both related by the Laplace transform.

613

Exercises

EXERCISES E8.1 Increased track densities for computer disk drives necessitate careful design of the head positioning control [l].The loop transfer function is L(s) = Gc(s)G(s) =

K (s + If

Plot the frequency response for this system when K = 4. Calculate the phase and magnitude at w = 0.5,1,2, 4, and oo. Answer: |L(/0.5)| = 0.94 and /LQ0.5) = -28.1°. E8.2 A tendon-operated robotic hand can be implemented using a pneumatic actuator [8].The actuator can be represented by G(s) =

5000

G(5) =

E8.3 A robotic arm has a joint-control loop transfer function 300(5 + 100) L(s) = Gr(s)G(s) = ,v(.v + 10)(5 + 40)' Show that the frequency equals 28.3 rad/s when the phase angle of L(jco) is -180°. Find the magnitude of L(jco) at that frequency. Answer: \L(j2$.3)\ = -2.5 dB E8.4 The frequency response for a process of the form is shown in Figure E8.5. Determine K, a, and b from the plot. Answer: K = 8, a = If A, b = 1/24 E8.6 Several studies have proposed an extravehicular robot that could move around in a NASA space station and perform physical tasks at various worksites [9]. The arm is controlled by a unity feedback control with loop transfer function L(s) = Gc(s)G(s) = K 5(5/5 + 1)(5/100 + 1)' Draw the Bode diagram for K - 20, and determine the frequency when 20 log|/„(/fc>)|is 0 dB. E8.7 Consider a system with a closed-loop transfer function T(s) = Y(s) R(s) (5 2 + 5 + 1)(5 2 + 0.45 + 4 ) ' This system will have no steady-state error for a step input, (a) Plot the frequency response, noting the two peaks in the magnitude response, (b) Predict the time response to a step input, noting that the system has four poles and cannot be represented as a dominant second-order system, (c) Plot the step response. E8.8 A feedback system has a loop transfer function Ks (s + a)(s2 + 20s + 100) m K{\ + 0.55)(1 + as) 5(1 + 5/8)(1 + bs)(l + 5/36) (s + 70)(5 + 500)' Plot the frequency response of G(ja>). Show that the magnitude of G{ju>) is -17 dB at co — 10 and -27.1 dB at w = 200. Show also that the phase is -138.7° at a) = 700. G(s) = is shown in Figure E8.4. Determine K and a by examining the frequency response curves. E8.5 The magnitude plot of a transfer function L(s) = Gc(s)G(s) = 100(5 - 1) s2 + 255 + 100' 0 O 3 -10 i -90° -20 t -180° FIGURE E8.4 Bode diagram. a> (rad/s) 614 Chapter 8 Frequency Response Methods ' 0 dB/dec ' +20 dB/dec -20 dB/dec 0 dB/dec ^ FIGURE E8.5 Bode diagram. 40 I 1 5 20 , 1 j I 0 -20 -; CM -40 -3- — FIGURE E8.9 Bode diagram. 0.1 \ 10 co (rad/s) (a) Determine the corner frequencies (break frequencies) for the Bode plot, (b) Determine the slope of the asymptotic plot at very low frequencies and at high frequencies, (c) Sketch the Bode magnitude plot. E8.9 The Bode diagram of a system is shown in Figure E8.9. Determine the transfer function G(s). E8.10 The dynamic analyzer shown in Figure E8.10(a) can be used to display the frequency response of a system. Also shown is the signal analyzer used to measure the mechanical vibration in the cockpit of an automobile. Figure E8.10(b) shows the actual frequency response of a system. Estimate the poles and zeros of the device. Note X = 1.37 kHz at the first cursor, and AX = 1.257 kHz to the second cursor. E8.ll Consider the feedback control system in Figure E8.ll. Sketch the Bode plot of G(s) and determine 100 - -90° 1000 the crossover frequency, that is. the frequency when 201og]0|G(/a>)| = 0 d B . E8.12 Consider the system represented in state variable form 0 -2 y = [1 •3 x + - l ] x + [0]» (a) Determine the transfer function representation of the system, (b) Sketch the Bode plot. E8.13 Determine the bandwidth of the feedback control system in Figure E8.I3. E8.14 Consider the nonunity feedback system in Figure E8.14, where the controller gain is /C = 2. Sketch the Bode plot of the loop transfer function. Determine the 615 Exercises (a) X = 1.37kHz Ya = -4.9411 AYa = 4.076 dB AX = 1.275kHz M: FreqResp 20Avg 0%0vlp Unif 10.0 iA dB ; I I \ i \y i i -30.0 2kHz 4kHz (b) FIGURE E8.10 (a) Photo showing the Signal Analyzer 35670A used to analyze mechanical vibration in the cockpit of an automobile, (b) Frequency response. (Courtesy of the Agilent Technologies Foundation.) Controller FIGURE E8.11 Unity feedback system. i 1000 s+2 Process 1 2 s + 10s + 100 t — • K(.v) 616 Chapter 8 Frequency Response Methods Controller Ms) . FIGURE E8.13 Third-order feedback system. Process 100 1 s+ 1 s2 + 10? + 10 Controller, Gc(s) Ri • Y(s) Process, G(s) 1 <>-iO— • Yis) s 2 + 1.4s + 1 Sensor, H(s) FIGURE E8.14 Nonunity feedback system with controller gain K. 10 s + To phase of the loop transfer function when the magnitude 20 logiL(/w)| = 0 dB.Recall that the loop transfer function is L(s) = Gc(s)G(s)H(s). E8.15 Consider the single-input, single-output system described by x(0 = Ax(0 + B«(f) where 0 -6 - K A = 1 ,B = -1 , C = [5 3]. Compute the bandwidth of the system for K - 1,2, and 10. As K increases, does the bandwidth increase or decrease? y(t) = Cx(0 PROBLEMS P8.1 Sketch the polar plot of the frequency response for the following loop transfer functions: (a) Gc(S)G{s) (b) Gc(s)G(s) = ./"WW L (1 + 0.25.v)(I + 3,v) V-, 5(s2 + 1.45+1) R, (s - 1)2 s -8 s2 + 6s + 8 20(s + 8) (d) Gc(s)G(s) = s(s + 2)(s + 4) (c) Gc(s)G(S) = P8.2 Sketch the Bode diagram representation of the frequency response for the transfer functions given in Problem P8.1. P8.3 A rejection network that can be used instead of the twin-T network of Example 8.4 is the bridged-T network shown in Figure P8.3. The transfer function of this network is FIGURE P8.3 Bridged-T network. G(s) = S* + (On r + 2{o)JQ)s + a),2 (can you show this?), where „L)2/4/?i [3]. (a) Determine the pole-zero pattern and, using the vector approach, evaluate the approximate frequency 617 Problems response, (b) Compare the frequency response of the twin-T and bridged-T networks when Q = 10. Gc(s)G(s) P8.4 A control system for controlling the pressure in a closed chamber is shown in Figure P8.4. The transfer function for the measuring element is H(s) = where K = 10. Sketch the Bode diagram of this system. P8.6 The asymptotic log-magnitude curves for two transfer functions are given in Figure P8.6. Sketch the corresponding asymptotic phase shift curves for each system. Determine the transfer function for each system. Assume that the systems have minimum phase transfer functions. P8.7 Driverless vehicles can be used in warehouses, airports, and many other applications. These vehicles follow a wire embedded in the floor and adjust the steerable front wheels in order to maintain proper direction, as shown in Figure P8.7(a) [10]. The sensing coils, mounted on the front wheel assembly, detect an error in the direction of travel and adjust the steering. The overall control system is shown in Figure P8.7(b). The loop transfer function is 150 s2 + 15s + 150 and the transfer function for the valve is Gi(s) = K (1 + s/4)(l + 5)(1 + 5/20)(1 + 5/80)' 1 (0.1.5 + 1)(5/20 4- 1) The controller transfer function is Gc(s) = 2s + 1. Obtain the frequency response characteristics for the loop transfer function GAs)Gl(s)H(S)'[l/sl L(s) = P8.5 The robot industry in the United States is growing at a rate of 30% a year [8]. A typical industrial robot has degrees of freedom. A unity feedback position control system for a force-sensing joint has a loop transfer function Desired pressure K, K s(s + IT) 2 S{S/TT + We want the bandwidth of the closed-loop system to exceed 2TT rad/s. (a) Set Kv = Itr and sketch the Bode diagram, (b) Using the Bode diagram, obtain the logarithmic-magnitude versus phase angle curve. Controller Valve J Infinite pressure source Pressure 1 chamber f 'o l>kl (a) Pjs) O Controller Valve G,(.v) Gt(s) Measurement FIGURE P8.4 (a) Pressure controller, (b) Block diagram model. if His) (b) • Prfis) 618 Chapter 8 Frequency Response Methods v -20dB/dec £ 12 • log (a) 03 20 + 20dB/dec - > log (o 500' -20dB/dec' FIGURE P8.6 Log-magnitude curves. (b) Steering servo i i l JTT, / C0 Steerable wheels Sensing •*-t 1 1 coils / Energizt wire :d guidepath (a) Reference — "> J fc > Motor Controller Vehicle wheels Direction of travel i _ Sensing coils FIGURE P8.7 Steerable wheel control. P8.8 (b) A feedback control system is shown in Figure P8.8.The specification for the closed-loop system requires that the overshoot to a step input be less than 15%. (a) Determine the corresponding specification Mpo) in the frequency domain for the closed-loop transfer function = Ujco). (b) Determine the resonant frequency P8.9 Sketch the logarithmic-magnitude versus phase angle curves for the transfer functions (a) and (b) of Problem P8.1. P8.10 A linear actuator is used in the system shown in Figure P8.10 to position a mass M.The actual position of the mass is measured by a slide wire resistor, and thus H{s) = 1.0. The amplifier gain is selected so that the steady-state error of the system is less than 1 % of the magnitude of the position reference R(s). The actuator has a field coil with a resistance Rf = 0.1 ft 619 Problems FIGURE P8.8 Second-order unity feedback system. Process R(s) O— s(s + 10) • • Y(s) Amplifier R{s) FIGURE P8.10 Linear actuator control. O Measurement His) and Lf = 0.2 H. The mass of the load is 0.1 kg, and the friction is 0.2 N s/m. The spring constant is equal to 0.4 N/m. (a) Determine the gain K necessary to maintain a steady-state error for a step input less than 1 %.That is, Kp must be greater than 99. (b) Sketch the Bode diagram of the loop transfer function, L(s) = G(s)H(s). (c) Sketch the logarithmic magnitude versus phase angle curve for L(jco). (d) Sketch the Bode diagram for the closed-loop transfer function, Y(jco)/R(ju>). Determine Mpa>, a>r, and the bandwidth. P8.ll Automatic steering of a ship would be a particularly useful application of feedback control theory [20]. In the case of heavily traveled seas, it is important to maintain the motion of the ship along an accurate track. An automatic system would be more likely to maintain a smaller error from the desired heading than a helmsman who recorrects at infrequent intervals. A mathematical model of the steering system has been developed for a ship moving at a constant velocity and for small deviations from the desired track. For a large tanker, the transfer function of the ship is G(s) E(s) _ 0.164(.9 + 0.2)(-s + 0.32) S(s) ~ s2(s + 0.25)(^ - 0.009) ' where E(s) is the Laplace transform of the deviation of the ship from the desired heading and S(s) is the Laplace transform of the angle of deflection of the steering rudder. Verify that the frequency response of the ship, E(jto)/8(jw). is that shown in Figure P8.ll. P8.12 The block diagram of a feedback control system is shown in Figure P8.12(a).The transfer functions of the blocks are represented by the frequency response curves shown in Figure P8.12(b). (a) When G3 is disconnected from the system, determine the damping ratio £ of the system, (b) Connect G3 and determine the damping ratio f. Assume that the systems have minimum phase transfer functions. P8.13 A position control system may be constructed by using an AC motor and AC components, as shown in Figure P8.13.The syncro and control transformer may be considered to be a transformer with a rotating winding. The syncro position detector rotor turns with the load through an angle 80. The syncro motor is energized with an AC reference voltage, for example, 115 volts, 60 Hz. The input signal or command is R(s) = &m(s) ana " is applied by turning the rotor of the control transformer. The AC two-phase motor operates as a result of the amplified error signal. The advantages of an AC control system are (1) freedom from DC drift effects and (2) the simplicity and accuracy of AC components. To measure the open-loop frequency response, we simply disconnect X from Y and X' from Y' and then apply a sinusoidal modulation signal generator to the Y — Y' terminals and measure the response at X - X'. (The error (00 - 0,) will be adjusted to zero before applying the AC generator.) The resulting frequency response of the loop transfer function LQGO) = Gc(j(o)G(jco)H(jo)) is 620 Chapter 8 Frequency Response Methods s! i !i 100 J_^r-* 80 i i —.^^ I pv " • " ; i! 1 40 -280 r •^Iphase O 60 1 1 -320 1 I Amplitude ! iv u 60 -360 ^= r 8 .a i -400 1 20 — FIGURE P8.11 Frequency response of ship control system. - -440 , !! 0.002 0.01 0.4 0.1 &> (rad/s) /?(.¥) — H O G, t I flY) (a) 1m Polar plot G,(» Bode plot G2(ja>) Re 10 dB increasing | 0)= 1 -360° FIGURE P8.12 Feedback system. -270° Logarithmic magnitude vs. phase plot 9.54 C3(» -180° (b) -90° 621 Problems 9 Control winding V, P Reference winding AC two-phase motor Load Rotor Svncro cenerator (a) 40 ^ \ - 2 0 dB/dec 5 \ - 4 0 dB/dec o \ -20 -80dB/de\ -40 FIGURE P8.13 (a) AC motor control. (b) Frequency response. 10 100 to (rad/s) 1000 (b) shown in Figure P8.13(b). Determine the transfer function L(;'w). Assume that the system has a minimum phase transfer function. P8.14 A bandpass amplifier may be represented by the circuit model shown in Figure P8.14 [3J. When R{ = R2 = 1 left, C, - 100 pF, C2 = 1 fiF, and K = 100, show that G(s) 100 1000 Hfs (s + 1000)(.y + 10' (a) Sketch the Bode diagram of G(ja>). (b) Find the midband gain (in dB). (c) Find the high and low frequency - 3 dB points. P8.15 To determine the transfer function of a process G(s), the frequency response may be measured using a sinusoidal input. One system yields the data in the following table: co, rad/s 0.1 1 2 4 5 6.3 8 10 12.5 20 31 \G(JOJ)\ 50 5.02 2.57 1.36 1.17 1.03 0.97 0.97 0.74 0.13 0.026 Phase, degrees -90 -92.4 -96.2 -100 -104 -110 -120 -143 -169 -245 -258 Determine the transfer function G(s). P8.16 The space shuttle has been used to repair satellites and the Hubble telescope. Figure P8.16 illustrates how 622 O + Chapter 8 Frequency Response Methods Maximum skewed wing position V W m KV7 m FIGURE P8.14 Bandpass amplifier. a crew member, with his feet strapped to the platform on the end of the shuttle's robotic arm, used his arms to stop the satellite's spin. The control system of the robotic arm has a closed-loop transfer function no 60.2 s2 + 12.15 + 60.2 R(s) (a) Determine the response y(t) to a unit step input, R(s) = 1/s. (b) Determine the bandwidth of the system. FIGURE P8.16 Satellite repair. P8.17 The experimental Oblique Wing Aircraft (OWA) has a wing that pivots, as shown in Figure P8.17. The wing is in the normal unskewed position for low speeds and can move to a skewed position for improved supersonic flight [11]. The aircraft control system loop transfer function is 4(0.5* + l) Gc(s)G(s) s(2s + 1) - ] 2 + l 20' (a) Sketch the Bode diagram, (b) Find the frequency oil when the magnitude is 0 dB, and find the frequency w2 when the phase is -180°. ii f«d ' I l' FIGURE P8.17 The Oblique Wing Aircraft, top and side views. P8.18 Remote operation plays an important role in hostile environments, such as those in nuclear or hightemperature environments and in deep space. In spite of the efforts of many researchers, a teleoperation system that is comparable to the human's direct operation has not been developed. Research engineers have been trying to improve teleoperations by feeding back rich sensory information acquired by the robot to the operator with a sensation of presence. This concept is called tele-existence or telepresence [9]. The tele-existence master-slave system consists of a master system with a visual and auditory sensation of presence, a computer control system, and an anthropomorphic slave robot mechanism with an arm having seven degrees of freedom and a locomotion mechanism. The operator's head movement, right arm movement, right hand movement, and other auxiliary motion are measured by the master system. A specially designed stereo visual and auditory input system mounted on the neck mechanism of the slave robot gathers visual and auditory information from the remote environment. These pieces of information are sent back to the master system and are applied to the specially designed stereo display system to evoke the sensation of presence of the operator. The locomotion control system has the loop transfer function Gc(s)G(s) \2(s + 0.5) 2 s + Us + 30* Obtain the Bode diagram for Gc(jo))G(Ja>) and determine the frequency when 20 ]og\Gc(jw)G(ja>)\ is very close to 0 dB. P8.19 A DC motor controller used extensively in automobiles is shown in Figure P8.19(a). The measured plot of Q(s)/I(s) is shown in Figure P8.19(b). Determine the transfer function of Q(s)/I(s). 623 Problems i(t) o,i—+Q—• Current Amplifier DC motor + (i Sensor (a) j m ._ 1 i 1 ' 0.1 I Hz J. 10 i j i <-> ^ ^X j FIGURE P8.19 (a) Motor controller. (b) Measured plot. 4t>4THJm|! -180 0.1 I Hz 4r I | " - - < — —H-4. 10 (b) P8.20 For the successful development of space projects, robotics and automation will be a key technology. Autonomous and dexterous space robots can reduce the workload of astronauts and increase operational efficiency in many missions. Figure P8.20 shows a concept called a free-flying robot [9,13]. A major characteristic of space robots, which clearly distinguishes them from robots operated on earth, is the lack of a fixed base. Any motion of the manipulator arm will induce reaction forces and moments in the base, which disturb its position and attitude. FIGURE P8.20 A space robot with three arms, shown capturing a satellite. ~h .-1- -30 4} The control of one of the joints of the robot can be represented by the loop transfer function 823(s + 9.8) L(s) = Gc(s)G(s) = -i— —. w ' s2 + 22s + 471 (a) Sketch the Bode diagram of L(j -200s2 s* + 14s2 + 44s + 40' 624 Chapter 8 Frequency Response Methods Note the negative gain in Gc(s)G(s). This system represents the control system for the climb rate. Sketch the Bode diagram and determine gain (in dB) when the phase is -180°. P8.22 The frequency response of a process G(/'w) is shown in Figure P8.22. Determine G(s). P8.23 The frequency response of a process G(j(o) is shown in Figure P8.23. Deduce the type number (number of integrations) for the system. Determine the transfer function of the system, G(s). Calculate the error to a unit step input. P8.24 The Bode diagram of a closed-loop film transport system is shown in Figure P8.24 [17]. Assume that the system transfer function T(s) has two dominant complex conjugate poles, (a) Determine the best second-order model for the system, (b) Determine the system bandwidth, (c) Predict the percent overshoot and settling time (with a 2% criterion) for a step input. P8.25 A unity feedback closed-loop system has a steadystate error equal to .4/10, where the input is r{t) = At2j2. The Bode plot of the magnitude and phase angle versus co is shown in Figure P8.25 for G(ja>). Determine the transfer function G(s). P8.26 Determine the transfer function of the op-amp circuit shown in Figure P8.26. Assume an ideal op-amp. Plot the frequency response when R = 10/:12, R} = 9 kf>. R2 = 1 kH, and C = 1 /xF. P8.27 A unity feedback system has the loop transfer function Sketch the Bode plot of the loop transfer function and indicate how the magnitude 20 log|L(/w)| plot varies as K varies. Develop a table for K = 0.75,2, and 10, and for each K determine the crossover frequency (o>c. for 201og|L(y'a))| = 0 dB), the magnitude at low frequency (20 log|L(/w)| for o> « 1), and for the closed-loop system determine the bandwidth for each K. I0 - 1 Frequency co (rad/s) FIGURE P8.22 101 102 103 Frequency a) (rad/s) Bode plot of G(s). Frequency to (rad/s) FIGURE P8.23 10° Frequency response of G{j(o). Frequency co (rad/s) 104 103 Problems 625 20 « i ! : 10 —M^\|Sf• \ 0 j J -1. 3 -10 •~> "5o - 2 0 5 -30 1i t -40 i V ! •T ... S|J| 0 1 -50 1 j It)1 10° 10" Frequency &» (rad/s) FIGURE P8.24 Bode plot of a closed-film transport system. 100 50 CO -a ^ 0 O 50 liiii lii Ill H *itfc •I-jtf -150 il: j : 1 ' Jllll •; 1 i J ijS<.ii 1 ji. MMJ4 47|!"- 1fl- i LLilb 1 1 | 111 m I; - { LilW if- :i TUTS jlj ; ti -100 Frequency co (rad/s) [111I. ! L-Jiii ntii HIT !. -200 10" 10° |! 10' 1 ! 10 2 I I Ml! 10 3 104 101 10 5 Bode plot of a unity feedback system. R2 A/W A/W Vf.v) FIGURE P8.26 An op-amp circuit. X 10 3 Frequency co (rad/s) Frequency co (rad/s) FIGURE P8.25 10 2 -o + V()(.v) —o + 104 626 Chapter 8 Frequency Response Methods ADVANCED PROBLEMS AP8.1 A spring-mass-damper system is shown in Figure AP8.1(a).The Bode diagram obtained by experimental means using a sinusoidal forcing function is shown in Figure AP8.1(b). Determine the numerical values of m, b, and k. -10 | -20 1 -.—J - , \\ San -30 Spring, k •\ -1- V -50 Tx Damper, b 0.01 FIGURE AP8.1 A spring-massdamper system. 0.1 l v.10 -90° 100 180° (a (rad/s) M AP8.2 A system is shown in Figure AP8.2. The nominal value of the parameter b is 4.0. Determine the sensi- (b) tivity Si and plot 20 l o g | S j | , t h e Bode magnitude diagram for K = 5. • Yis) R(x) FIGURE AP8.2 System with parameter b. AP8.3 As an automobile moves along the road, the vertical displacements at the tires act as the motion excitation to the automobile suspension system [16]. Figure AP8.3 is a schematic diagram of a simplified automobile suspension system, for which we assume the input is sinusoidal. Determine the transfer function X(s)/R(s), and sketch the Bode diagram when M - 1 kg, b = 4 N s/m, and k = 18 N / m . AP8.4 A helicopter with a load on the end of a cable is shown in Figure AP8.4(a).The position control system is shown in Figure AP8.4(b), where the visual feedback is represented by H{s). Sketch the Bode diagram of the loop transfer function L(jco) = G(j(o)H(j(o). AP8.5 A closed-loop system with unity feedback has a transfer function T(s) FIGURE AP8.3 Auto suspension system model. 10(5 + 1) 2 s + 9s + 10* (a) Determine the loop transfer function Gc(s)G(s). (b) Plot the log-magnitude-phase (similar to Figure 8.27), and identify the frequency points for co equal to 627 Advanced Problems G(s) R(x) Xn _ FIGURE AP8.4 A helicopter feedback control system. A i- + 3.¾ + 15 1 5 +1 (b) (a) AP8.6 Consider the spring-mass system depicted in Figure AP8.6. Develop a transfer function model to describe the motion of the mass M = 2 kg, when the input is u(t) and the output is x(t). Assume that the initial conditions are .v(0) = 0 and i(0) = 0. Determine values of k and b such that the maximum steady-state response of the system to a sinusoidal input u{t) = sin(w/)is less than 1 for all co. For the values position H(s) 1,10,50,110, and 500. (c) Is the open-loop system stable? Is the closed-loop system stable? )'(.v) I 2 you selected for k and b, what is the frequency at which the peak response occurs? AP8.7 An op-amp circuit is shown in Figure AP8.7. The circuit represents a lead compensator discussed in more detail in Chapter 10. (a) Determine the transfer function of this circuit. (b) Sketch the frequency response of the circuit when/?! = 10kfl./? 2 = 1 0 H . C ! = 0.1 ^F,and C2 - 1 mF. R, AA/V o- r VW VJs) V,(s) FIGURE AP8.7 FIGURE AP8.6 Suspended springmass system with parameters k and b. Op-amp lead circuit. 628 Chapter 8 Frequency Response Methods DESIGN PROBLEMS CDP8.1 In this chapter, we wish to use a PD controller such that Gc(s) = K(s + 2). The tachometer is not used (see Figure CDP4.1). Plot the Bode diagram for the system when K = 40. Determine the step response of this system and estimate the overshoot and settling time (with a 2% criterion). DP8.1 Understanding the behavior of a human steering an automobile remains an interesting subject [14,15, 16, 21]. The design and development of systems for four-wheel steering, active suspensions, active, independent braking, and "drive-by-wire" steering provide the engineer with considerably more freedom in altering vehicle-handling qualities than existed in the past. The vehicle and the driver are represented by the model in Figure DP8.1, where the driver develops anticipation of the vehicle deviation from the center line. For K = 1, plot the Bode diagram of (a) the loop transfer function Gc(s)G(s) and (b) the closed-loop transfer function T(s). (c) Repeat parts (a) and (b) when K = 50. (d) A driver can select the gain K. Determine the appropriate gain so that Mpa) ^ 2, and the bandwidth is the maximum attainable for the closed-loop system, (e) Determine the steady-state error of the system for a ramp input r(f) = t. DP8.2 The unmanned exploration of planets such as Mars requires a high level of autonomy because of the communication delays between robots in space and their Earth-based stations. This affects all the components of the system: planning, sensing, and mechanism. In particular, such a level of autonomy can be achieved only if each robot has a perception system that can reliably build and maintain models of the environment. The perception system is a major part of the development of a complete system that includes planning and mechanism design. The target vehicle is the Spider-bot, a four-legged walking robot shown in Figure DP8.2(a), being developed at NASA Jet Propulsion Laboratory [18]. The control system of one leg is shown in Figure DP8.2(b). FIGURE DP8.1 Human steering control system. IHs) Desired distance From center line (a) Sketch the Bode diagram for Gc(s)G(s) when K = 20. Determine (1) the frequency when the phase is -180° and (2) the frequency when 201og|GcG| = OdB. (b) Plot the Bode diagram for the closed-loop transfer function T(s) when K = 20. (c) Determine Mpu), a>r, and coB for the closed-loop system when K = 22 and K = 25. (d) Select the best gain of the two specified in part (c) when it is desired that the overshoot of the system to a step input r(t) be less than 5% and the settling time be as short as possible. DP8.3 A table is used to position vials under a dispenser head, as shown in Figure DP8.3(a). The objective is speed, accuracy, and smooth motion in order to eliminate spilling. The position control system is shown in Figure DP8.3(b). Since we want small overshoot for a step input and yet desire a short settling time, we will limit 20 log Mpui to 3 dB for T (/to). Plot the Bode diagram for a gain K that will result in a stable system. Then adjust K until 20 log Mpo) - 3 dB, and determine the closed-loop system bandwidth. Determine the steady-state error for the system for the gain K selected to meet the requirement for M pm. DP8.4 Anesthesia can be administered automatically by a control system. For certain operations, such as brain and eye surgery, involuntary muscle movements can be disastrous. To ensure adequate operating conditions for the surgeon, muscle relaxant drugs, which block involuntary muscle movements, are administered. A conventional method used by anesthesiologists for muscle relaxant administration is to inject a bolus dose whose size is determined by experience and to inject supplements as required. However, an anesthesiologist may sometimes fail to maintain a steady level of relaxation, resulting in a large drug consumption by the patient. Significant improvements may be achieved by introducing the concept of automatic control, which results in a considerable reduction in the total relaxant drug consumed [19]. A model of the anesthesia process is shown in Figure DP8.4. Select a gain K so that the bandwidth of the closed-loop system is maximized while Mpa) ;£ 1.5. Determine the bandwidth attained for your design. Gc(s) Driver -^ Error ^Q K(s + 2) G(s) Vehicle l s2(s + 12) Yi v\ Distance from center line Design Problems 629 rV v-f; (a) FIGURE DP8.2 (a) The Mars-bound Spider-bot. (Photo courtesy of NASA.) (b) Block diagram of the control system for one leg. • Y(s) /?(.?) Dispenser j-axis motor and sensor (a) R(s) i ummand + , _ J t K s2 + 2.9 + 2 Sensor FIGURE DP8.3 Automatic table and dispenser. 5 (.v + 5) (b) Vis) Position 630 Chapter 8 Frequency Response Methods Ris) Desired Controller K 03s + 1 + FIGURE DP8.4 relaxation Model of an anesthesia control system. level Drug input DP8.5 Consider the control system depicted in Figure DP8.5(a) where the plant is a "black box" for which little is known in the way of mathematical models. The only information available on the plant is the frequency response shown in Figure DP8.5(b). Design a controller Gc(s) to meet the following specifications: (i) The crossover frequency is between 10 rad/s and 50 rad/s; (ii) The magnitude of Gc(s)G(s) is greater than 20 dB for to < 0.1 rad/s. (a) Determine p and K such that the unit step response exhibits a zero steady-state error and the percent overshoot meets the requirement P.O. < 5%. (b) For the values of p and K determined in part (a), determine the system damping ratio £ and the natural frequency #. (d) Using the approximate formula shown in Figure 8.26, compute the bandwidth using £ and (on and compare the value to the actual bandwidth from part (c). DP8.6 A single-input, single-output system is described by 0 x(/) = y(t) = [0 x(/) + u(r) l]x(/) R{s) • Controller Black box C,(.v) G(s) • Y(s) 10° 10' Frequency (rad/s) 102 l (a) f) PQ 20 1 u 1 ~40 S I ~60 -80 0 -45 FIGURE DP8.5 (a) Feedback system with "black box" plant, (b) Frequency response plot of the "black box" represented by G{s). • i \ -90 ' -135 -180 10" (h) 631 Computer Problems DP8.7 Consider the system of Figure DPS.7. Consider the controller to be a proportional plus integral plus derivative (PID) given by Design the PID controller gains to achieve (a) an acceleration constant Ka = 2, (b) a phase margin of P.M. > 45°, and (c) a bandwidth 3.0. Plot the response of the closed-loop system to a unit step input. K, Gc(s) = KP + KDs + — . s FIGURE DP8.7 Closed-loop feedback system. -AM #(.s) ? ' Controller Plant Gc(s) 3 s(s2 + 4s + 5) • n.v) COMPUTER PROBLEMS CP8.1 Consider the closed-loop transfer function T(s) = 25 s + s + 25 2 Develop an m-file to, obtain the Bode plot and verify that the resonant frequency is 5 rad/s and that the peak magnitude Mpw is 14 dB. CP8.2 For the following transfer functions, sketch the Bode plots, then verify with the bode function: , x ^,^ 1000 (a) G(s) = (s + 10)(5 + 100) s + 100 (b) G(s) = (s + 2)(s + 25) 100 (c) G(s) = 2 s + 2s + 50 s - 6 (d) G(s) = (s + 3)(52 + 12s + 50) CP83 For each of the following transfer functions, sketch the Bode plot and determine the crossover frequency (that is, the frequency at which 20 log10|G(/w) | = OdB): / , ,-,/ * (a) G(s) 200° (s + I0)(s + 100) 10() (b) G(s) (s + l)(s2 + 10s + 2) (c) G(s) = (d) G(s) = 50(^ + 100) (s + l)(s + 50) Determine the closed-loop system bandwidth. Using the bode function obtain the Bode plot and label the plot with the bandwidth. CP8.5 A block diagram of a second-order system is shown in Figure CP8.5. (a) Determine the resonant peak Mpw the resonant frequency cor, and the bandwidth o)B, of the system from the closed-loop Bode plot. Generate the Bode plot with an m-file for en = 0.1 tow = 1000 rad/susing the logspace function, (b) Estimate the system damping ratio, £, and natural frequency a)n, using Equations (8.36) and (8.37) in Section 8.2. (c) From the closedloop transfer function, compute the actual £ and con and compare with your results in part (b). Rls) O 100 s(s + 6) • Vis) FIGURE CP8.5 A second-order feedback control system. CP8.6 Consider the feedback system in Figure CP8.6. Obtain the Bode plots of the loop and closed-loop transfer functions using an m-file. 100(52 + 145 + 50) (5 + 1)(5 + 2)(5 + 500) CP8.4 A unity negative feedback system has the loop transfer function Gc(s)G(s) = • Ft s) Rls) 54 5(5 + 6)* FIGURE CP8.6 Closed-loop feedback system. 632 Chapter 8 Frequency Response Methods CP8.7 A unity feedback system has the loop transfer function L(s) = Gc(s)G(s) = 1 s(s + 2p)' Generate a plot of the bandwidth versus the parameter p as 0 < p < 1. CP8.8 Consider the problem of controlling an inverted pendulum on a moving base, as shown in Figure CP8.8(a).The transfer function of the system is G(s) = -l/(MbL) 2 s - (Mb + Ms)g/(MhL) The design objective is to balance the pendulum (i.e., 0(f) « 0) in the presence of disturbance inputs. A block diagram representation of the system is depicted in Figure CP8.8(b). Let Ms = 10 kg, Mb =100 kg, L = 1 m, g = 9.81 m/s 2 , a = 5, and b = 10. The design specifications, based on a unit step disturbance, are as follows: 1. settling time (with a 2% criterion) less than 10 seconds, 2. percent overshoot less than 40%, and 3. steady-state tracking error less than 0.1° in the presence of the disturbance. Develop a set of interactive m-file scripts to aid in the control system design.The first script should accomplish at least the following: 1. Compute the closed-loop transfer function from the disturbance to the output with K as an adjustable parameter. 2. Draw the Bode plot of the closed-loop system. 3. Automatically compute and output Mpc0 and cor. As an intermediate step, use M poj and u>r and Equations (8.36) and (8.37) in Section 8.2 to estimate £ and aj„.The second script should at least estimate the settling time and percent overshoot using £ and ion as input variables. If the performance specifications are not satisfied, change K and iterate on the design using the first two scripts. After completion of the first two steps, the final step is to test the design by simulation. The functions of the third script are as follows: 1. plot the response, 9{t), to a unit step disturbance with K as an adjustable parameter, and 2. label the plot appropriately. Utilizing the interactive scripts, design the controller to meet the specifications using frequency response Bode methods. To start the design process, use analytic methods to compute the minimum value of K to meet the steady-state tracking error specification. Use the minimum K as the first guess in the design iteration. Input UO Td(s) Disturbance Pendulum model Controller ft/(.v) = 0 FIGURE CP8.8 (a) An inverted pendulum on a moving base. (b) A block diagram representation. O^ -K(s + a) s+b t/ *\ (b) MuL (M„ + Ms)g - • t)(s) Terms and Concepts CP8.9 Design a filter, G(s), with the following frequency response: 1. For u) < 1 rad/s, the magnitude 20 log]()|G(/a>)|< OdB 2. For 1 < co < 1000 rad/s, the magnitude 201og10 I GOV) | > OdB m 633 3. For co > 1000 rad/s, the magnitude 20 logI0 I GOV) | < OdB Try to maximize the peak magnitude as close to oo - 40 rad/s as possible. ANSWERS TO SKILLS CHECK True or False: (1) True; (2) False; (3) False; (4) True; (5) True Multiple Choice: (6) a; (7) a; (8) b; (9) b; (10) c; (ll)b;(12)c;(13)d;(14)a;(15)d Word Match (in order, top to bottom): d, i, q, n, 1, m, o, j , s, p, c, e, b, r, h, f, g, k, a TERMS AND CONCEPTS All-pass network A nonminimum phase system that frequency domain, denoted by F(s), related by the passes all frequencies with equal gain. Laplace transform as F(s) = %{f{t)}, where .¾ denotes the Laplace transform. Bandwidth The frequency at which the frequency response has declined 3 dB from its low-frequency Logarithmic magnitude The logarithm of the magnitude value. of the transfer function, usually expressed in units of 20dB,thus201og,„|G|. Bode plot The logarithm of the magnitude of the transfer function is plotted versus the logarithm of co, the Logarithmic plot See Bode plot. frequency. The phase 0 of the transfer function is separately plotted versus the logarithm of the frequency. Maximum value of the frequency response A pair of complex poles will result in a maximum value for the freBreak frequency The frequency at which the asymptotic quency response occurring at the resonant frequency. approximation of the frequency response for a pole Minimum phase transfer function All the zeros of a (or zero) changes slope. transfer function lie in the left-hand side of the sCorner frequency See Break frequency. plane. Decade A factor of 10 in frequency (e.g., the range of freNatural frequency The frequency of natural oscillation quencies from 1 rad/s to 10 rad/s is one decade). that would occur for two complex poles if the dampDecibel (dB) The units of the logarithmic gain. ing were equal to zero. Dominant roots The roots of the characteristic equation Nonminimum phase transfer function Transfer functions that represent or dominate the closed-loop transient with zeros in the right-hand s-plane. response. Octave The frequency interval co2 — 2cox is an octave of Fourier transform The transformation of a function of frequencies (e.g., the range of frequencies from time /(f) into the frequency domain. coi = 100 rad/s to co2 = 200 rad/s is one octave). Fourier transform pair A pair of functions, one in the time domain, denoted by /(/), and the other in the fre- Polar plot A plot of the real part of G(jco) versus the imaginary part of G(jco). quency domain, denoted by F( 711 Skills Check Table 9.6 (continued) Root Locus Nichols Diagram Gain margin Comments x, TjOdE -270° -180' Conditionally stable; becomes unstable if gain is t o o low 90° Phase / margin -270°1 -1801 -90° '5 r '4 3 i T 4 r 3 T b T a T 2 T \ Conditionally stable; stable at low gain, becomes unstable as gain is raised, again becomes stable as gain is further increased, and becomes unstable for very high gains Conditionally stable; becomes unstable at high gain m SKILLS CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 9.70 as specified in the various problem statements. 712 Chapter 9 Stability in the Frequency Domain * ( * ) • o Controller Process Gc(s) G(s) + Y{s) FIGURE 9.70 Block diagram for the Skills Check. In the following True or False and Multiple Choice problems, circle the correct answers. 1. The gain margin of a system is the increase in the system gain when the phase is -180° that will result in a marginally stable system. 2. A conformal mapping is a contour mapping that retains the angles on the 5-plane on the transformed F(s)-plane. 3. The gain and phase margin are readily evaluated on either a Bode plot or a Nyquist plot. 4. A Nichols chart displays curves describing the relationship between the open-loop and closed-loop frequency responses. 5. The phase margin of a second-order system (with no zeros) is a function of both the damping ratio £ and the natural frequency, a>„. 6. Consider the closed-loop system in Figure 9.70 where L{s) = Gc(s)G(s) = True or False True or False True or False True or False True or False 3.25(1 + s/6) s(l + s/3)(l + s/8)' The crossover frequency and the phase margin are: a. ID = 2.0 rad/s, P.M. = 37.2° b. o) = 2.5 rad/s, P.M. = 54.9° c. o) = 5.3 rad/s, P.M. = 68.1° d. w = 10.7 rad/s, P.M. = 47.9° 7. Consider the block diagram in Figure 9.70. The plant transfer function is G(s) = 1 (1 + 0.25.y)(0.5.y + 1)' and the controller is Gc(s) s + 0.2 s +5 " Utilize the Nyquist stability criterion to characterize the stability of the closed-loop system. a. The closed-loop system is stable. b. The closed-loop system is unstable. c The closed-loop system is marginally stable. d. None of the above. For Problems 8 and 9, consider the block diagram in Figure 9.70 where 9 G(s) = (s + l)(s2 + Ss + 9)' 713 Skills Check and the controller is the proportional-plus-derivative (PD) controller Ge(s) = K(\ + Tds). 8. When Td = 0, the PD controller reduces to a proportional controller, Gc(s) = K. In this case, use the Nyquist plot to determine the limiting value of K for closed-loop stability. a. K = 0.5 b. K = 1.6 c. K = 2.4 d. K = 4.3 9. Using the value of K in Problem 8, compute the gain and phase margins when Tj = 0.2. a. G.M. = 14 dB, P.M. = IT b. G.M. = 20 dB, P.M. = 64.9° c. G.M. = oo dB, P.M. = 60° d. Closed-loop system is unstable 10. Determine whether the closed-loop system in Figure 9.70 is stable or not, given the loop transfer function L{s) = Gc{s)G{s) = '** . s {4s + 1) In addition, if the closed-loop system is stable, compute the gain and phase margins. a. Stable, G.M. = 24dB,P.M. = 2.5° b. Stable, G.M. = 3 dB, P.M. = 24° c. Stable, G.M. = oo dB, P.M. = 60° d. Unstable 11. Consider the closed-loop system in Figure 9.70, where the loop transfer function is L(S) = Gc(s)G(s) = K {s + 4) '. s' Determine the value of the gain K such that the phase margin is P.M. = 40°. a. K = 1.64 b. K = 2.15 c. K = 2.63 d. Closed-loop system is unstable for all K > 0 12. Consider the feedback system in Figure 9.70, where -0.2s Gc(s) = K and G(s) = -. Notice that the plant contains a time-delay of T = 0.2 seconds. Determine the gain K such that the phase margin of the system is P.M. = 50°. What is the gain margin for the same gain Kl a. K = 8.35, G.M. = 2.6 dB b. K =2.15, G.M. = 10.7 dB Chapter 9 Stability in the Frequency Domain c. K = 5.22, G.M. = oo dB d. K = 1.22, G.M. = 14.7 dB 13. Consider the control system in Figure 9.70, where the loop transfer function is The value of the resonant peak, Mp^ and the damping factor, £, for the closed-loop system are: a. M„ = 0.37, f = 0.707 b. M P w = 1.15,1: = 0.5 c. MPo = 2.55, £ = 0.5 d. MPui = 0.55, t, = 0.25 14. A feedback model of human reaction time used in analysis of vehicle control can use the block diagram model in Figure 9.70 with < « , ) - > * and G W ^ — i - j y . A typical driver has a reaction time of T — 0.3 seconds. Determine the bandwidth of the closed-loop system. a. (ob = 0.5 rad/s b . o>b = 10.6 rad/s c o)h = 1.97 rad/s d. iab = 200.6 rad/s 15. Consider a control system with unity feedback as in Figure 9.70 with loop transfer function Us) = G e W G W = s{s <{ + *\ - . The gain and phase margin are: a. G.M. = oo d B , P . M . = 58.1° b. G.M. = 20.4 dB, P.M. = 47.3° c. G.M. = 6.6 dB, P . M . = 60.4° d. Closed-loop system is unstable In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Time delay The frequency response of the closed-loop transfer function T(ja>). b. Cauchy's theorem A chart displaying the curves for the relationship between the open-loop and closed-loop frequency response. 715 Exercises c. Bandwidth A contour mapping that retains the angles on the s-plane on the F(s)-plane. d. Contour map If a contour encircles Z zeros and P poles of F(s) traversing clockwise, the corresponding contour in the F(s,)-plane encircles the origin of the F(s)-plane N = Z — P times clockwise. e. Nichols chart The amount of phase shift of Gc(ja))G(j f. Closed-loop frequency response Events occurring at time t at one point in the system occur at another point in the system at a later time,/ + T. g. Logarithmic (decibel) measure A feedback system is stable if and only if the contour in the G(s)-plane does not encircle the (—1,0) point when the number of poles of G(s) in the right-hand 5-plane is zero. If G(s) has P poles in the righthand plane, then the number of counterclockwise encirclements of the (-1,0) point must be equal to P for a stable system. h. Gain margin A contour or trajectory in one plane is mapped into another plane by a relation F(s). i. Nyquist stability criterion The increase in the system gain when phase = -180° that will result in a marginally stable system with intersection of the - 1 + /0 point on the Nyquist diagram. j . Phase margin The frequency at which the frequency response has declined 3 dB from its low-frequency value. k. Conformal mapping A measure of the gain margin. EXERCISES E9.1 A system has the loop transfer function L(s) = Gc(s)G(s) = 2(1 + s/\0) s(l + 5s)(l + s/9 + i-781) Plot the Bode diagram. Show that the phase margin is approximately 17.5° and that the gain margin is approximately 26.2 dB. E9.2 A system has the loop transfer function L(s) = Ge(s)G(s) = K(\ + s/5) s(l + 5/2)(1 + 5/10)' where K = 10.5. Show that the system crossover (OdB) frequency is 5 rad/s and that the phase margin is 40°. E9.3 An integrated circuit is available to serve as a feedback system to regulate the output voltage of a power supply. The Bode diagram of the required loop transfer function Gj(J(o)G(J(a) is shown in Figure E9.3 Estimate the gain and phase margins of the regulator. Answer. G.M. = 25 dB, PM, = 75° E9.4 Consider a system with a loop transfer function Gc(s)G(s) = 100 s(s + 10)' 716 Chapter 9 Stability in the Frequency Domain E9.6 A system has a loop transfer function K(s + 100) L(s) = Gc(s)G(s) s(s + 10)(.9 + 40)' When K = 500, the system is unstable. Show that if we reduce the gain to 50, the resonant peak is 3.5 dB. Find the phase margin of the system with K = 50. E9.7 A unity feedback system has a loop transfer function 180° L{s) = Gc(S)G(s) = 10 100 Ik 10k 100k Frequency (Hz) 1M FIGURE E9.3 Power supply regulator. We wish to obtain a resonant peak Mpa) = 3.0 dB for the closed-loop system. The peak occurs between 6 and 9 rad/s and is only 1.25 dB. Plot the Nichols chart for the range of frequency from 6 to 15 rad/s. Show that the system gain needs to be raised by 4.6 dB to 171. Determine the resonant frequency for the adjusted system. Answer: (or = 11 rad/s E9.5 An integrated CMOS digital circuit can be represented by the Bode diagram shown in Figure E9.5. (a) Find the gain and phase margins of the circuit. (b) Estimate how much we would need to reduce the system gain (dB) to obtain a phase margin of 60°. ~-^. Determine the range of K for which the system is stable using the Nyquist plot. E9.8 Consider a unity feedback system with the loop transfer function Us) = Gc(s)G(s) K s{s + l)(.v + 2)' (a) For K = 4, show that the gain margin is 3.5 dB. (b) If we wish to achieve a gain margin equal to 16 dB, determine the value of the gain K. Answer: (b) K = 0.98 E9.9 For the system of E9.8, find the phase margin of the system for K - 5. E9.10 Consider the wind tunnel control system of Problem P7.31 for K = 326. Obtain the Bode diagram and show that the P.M. = 25° and that the G.M. = 10 dB. Also, show that the bandwidth of the closed-loop system is 6 rad/s. E9.ll Consider a unity feedback system with the loop transfer function 10(1 + 0.45) Gt.(s)G(s) = .y(l + 2.v)(l + 0.245 + 0.04.V2) (a) Plot the Bode diagram, (b) Find the gain margin and the phase margin. E9.12 A unity feedback system with the loop transfer function L(s) = Gc(s)G(s) (a) 1 kHz 10 kHz 100 kHz 1 MHz 10 MHz. Frequency (b) -360 FIGURE E9.5 CMOS circuit. K 5(r,5 + l)(r 2 5 + 1)' where T\ = 0.02 and T2 = 0.2 s. (a) Select a gain K so that the steady-state error for a ramp input is 10% of the magnitude of the ramp function A, where r(l) = At,fs: 0. (b) Plot the Bode plot of Gc(s)G(s), and determine the phase and gain margins. (c) Using the Nichols chart, determine the bandwidth wB, the resonant peak M/)aJ, and the resonant frequency cor of the closed-loop system. Answer: (a) K ~ 10 (b) PM. = 32°, G.M. = 15 dB (c) coB = 10.3, Mp = 1.84, o>, - 6.5 717 Exercises E9.13 A unity feedback system has a loop transfer function 150 L(s) = Gc(s)G(s) = s{s + 5)(a) Find the maximum magnitude of the closed-loop frequency response using the Nichols chart, (b) Find the bandwidth and the resonant frequency of this system. (c) Use these frequency measures to estimate the overshoot of the system to a step response. Answers: (a) 7.5 dB, (b) coB = 19, i w3 rad/s fc>4 10 E9.15 Consider a unity feedback system with the loop transfer function L(s) = Gc(s)G(s) = Find the bandwidth of the closed-loop system. Answers: a>B = 6.4 rad/sec E9.16 The pure time delay e~sT may be approximated by a transfer function as _$T _ 1 - Ts/2 V 1 + Ts/2 for 0 < a) < 2/T. Obtain the Bode diagram for the actual transfer function and the approximation for T = 0.2 for 0 < (o < 10.

30

-O.IOdB

-0.25 dB

-0.5 dB

FIGURE E9.14 Nichols chart for G C (/«)G(M

-210

-180

100 s(s + 20)

-150 -120 -90 Loop phase L GCG (degrees)

-60

718

Chapter 9

Stability in the Frequency Domain

E9.17 A unity feedback system has a loop transfer function L(s) = Gc(s)G(s) =

the steady-state error for a ramp input for the gain of part(b)?'

K{s + 2) s3 + 2 r + 155

(a) Plot the Bode diagram and (b) determine the gain K required to obtain a phase margin of 30°. What is

E9.18 An actuator for a disk drive uses a shock mount to absorb vibrational energy at approximately 60 Hz [14]. The Bode diagram of G(.(s)G(s) of the control system is shown in Figure E9.18. (a) Find the expected percent overshoot for a step input for the closed-loop

X: 486.93 Y: -4.5924m Y: 0.0 Trans 1 R#:3 #A: 100 Expand 40.000

LGMAG DB

-10.000

40.000

486 Hz 600.00 Gain crossover

X: 486.93 Y: 36.215 Y: 0.0 Trans 1 R#:3 #A: 100 Expand 180.00

Phase margin

FIGURE E9.18 Bode diagram of the disk drive, Gc(s)G(s).

-180.00

Phase margin at crossover

40.000

LGHZ (h)

486 Hz

600.00

719

Exercises

R(s)

In

Driver

Automobile

2

34 (s + 2)(.v + 8)

Kf.v) Velocity of Automobile

FIGURE E9.20 Automobile control system. system, (b) estimate the bandwidth of the closed-loop system, and (c) estimate the settling time (with a 2% criterion) of the system.

E9.24 A unity feedback system has a loop transfer function L(s) = Gc(s)G(s)

E9.19 A unity feedback system with Gc(s) = K has „-0.1*

G(s) =

s + 10'

Select a gain K so that the phase margin of the system is 50°. Determine the gain margin for the selected gam,K. E9.20 Consider a simple model of an automobile driver following another car on the highway at high speed. The model shown in Figure E9.20 incorporates the driver's reaction time, T. One driver has T = 1 s, and another has T = 1.5 s. Determine the time response y(t) of the system for both drivers for a step change in the command signal R(s) = —1/s, due to the braking of the lead car. E9.21 A unity feedback control system has a loop transfer function L(s) = Gc(s)G(s) =

K s(s + 2)(.v + 50)'

Determine the phase margin, the crossover frequency, and the gain margin when K - 1300. Answers: PM = 16.6°,
K (s + 1)2

K - 1 + TS

where K = \ and T = 1. The polar plot for Gc(j) is shown in Figure E9.24. Determine whether the system is stable by using the Nyquist criterion.

FIGURE E9.24 Polar plot for Gc(s)G(s) = K/(-1 + TS).

E9.25 A unity feedback system has a loop transfer function L(s) = Gc(s)G(s) =

11.7 s(\ + 0.05^)(1 + 0.1s)'

Determine the phase margin and the crossover frequency. Answers: PM. = 27.7°, a>t. = 8.31 rad/s

(a) Using a Bode diagram for K = 10, determine the system phase margin, (b) Select a gain K so that the phase margin is at least 60°.

E9.26 For the system of E9.25, determine Mpa), con and u)B for the closed-loop frequency response by using the Nichols chart.

E9.23 Consider again the system of E9.21 when K = 438. Determine the closed-loop system bandwidth, resonant frequency, and Mpat using the Nichols chart.

E9.27 A unity feedback system has a loop transfer function

L(s) = Gc(s)G(s) =

K

s(s + 6)2

720

Chapter 9

Stability in the Frequency Domain

Determine the maximum gain K for which the phase margin is at least 40° and the gain margin is at least 6 dB. What are the gain margin and phase margin for this value of /C? E9.28 A unity feedback system has the loop transfer function L(s) = Gc(s)G(s) =

E9.30 Consider the system represented in state variable form x = Ax + Bw v = Cx + Du, where 0 -10

K s(s + 0.2)'

(a) Determine the phase margin of the system when K = 0.16. (b) Use the phase margin to estimate £ and predict the overshoot, (c) Calculate the actual response for this second-order system, and compare the result with the part (b) estimate. E9.29 A loop transfer function is L(s) = Gc(s)G(s) =

1

,B =

100 J

"o

[lj

[1000 0], and D = Sketch the Bode plot. E9.31 A closed-loop feedback system is shown in Figure E9.31. Sketch the Bode plot and determine the phase margin.

1 •

R{s)

i- + 2

Y{s)

Using the contour in the .y-plane shown in Figure E9.29, determine the corresponding contour in the F(s)-plane (B = - 1 + /).

FIGURE E9.31

J Jt)

Nonunity feedback system.

i

E9.32 Consider the system described in state variable form by

-j2

£(/) = Ax(0 + Bu(f) C

B

M - Cx(/)

D

where 0

A = A

E

-1

0

1

H

G

F

FIGURE E9.29

Contour in the s-plane.

-\

k

3.2

, C = [2

0].

Compute the phase margin. E9.33 Consider the system shown in Figure E9.33. Compute the loop transfer function L(s), and sketch the Bode plot. Determine the phase margin and gain margin when the controller gain K - 5.

Process

Controller

FIGURE E9.33 Nonunity feedback system with proportional controller K.

l

4

K

.v2 + 2.83* + 4 Sensor 10 s + 10

Problems

721

PROBLEMS P9.1

For the Nyquist plots of Problem P8.1, use the Nyquist criterion to ascertain the stability of the various systems. In each case, specify the values of N, P, and Z.

P9.2

Sketch the Nyquist plots of the following loop transfer functions L(s) - Gc(s)G(s), and determine whether the system is stable by applying the Nyquist criterion: (a) L(s) = Gc(s)G(s)

(b) L(s) =

Gc(s)G(s)

=

whether the system is stable, and find the number of roots (if any) in the right-hand s-plane. The system has no poles of Gc(s)G(s) in the right half-plane, (b) Determine whether the system is stable if the - 1 point lies at the dot on the axis. P9.5

s(s2 + ^ + 6) K(s + 1) s2(s + 6 ) '

A speed control for a gasoline engine is shown in Figure P9.5. Because of the restriction at the carburetor intake and the capacitance of the reduction manifold, the lag T, occurs and is equal to 1 second. The engine time constant re is equal to J/b = 3 s. The speed measurement time constant is T„, = 0.4 s. (a) Determine the necessary gain K if the steady-state speed error is required to be less than 10% of the speed reference setting. (b) With the gain determined from part (a), apply the Nyquist criterion to investigate the stability of the system, (c) Determine the phase and gain margins of the system.

If the system is stable, find the maximum value for K by determining the point where the Nyquist plot crosses the w-axis. P9.3 (a) Find a suitable contour Ys in the s-plane that can be used to determine whether all roots of the characP9.6 A direct-drive arm is an innovative mechanical arm in teristic equation have damping ratios greater than £j, which no reducers are used between motors and their (b) Find a suitable contour Ts in the .v-plane that can loads. Because the motor rotors are directly coupled to be used to determine whether all the roots of the charthe loads, the drive systems have no backlash, small fricacteristic equation have real parts less than s = —
\

/"—\

l\-L'

x^ ^ v ^ 0

P9.7

J

A vertical takeoff (VTOL) aircraft is an inherently unstable vehicle and requires an automatic stabilization system. An attitude stabilization system for the K-16B U.S. Army V T O L aircraft has been designed and is shown in block diagram form in Figure P9.7 [16]. A t 40 knots, the dynamics of the vehicle are approximately represented by the transfer function G(s) = —. . s2 + 0.36

FIGURE P9.4 Nyquist plot of conditionally stable system.

R(s)

Torque

Throttle T,S

FIGURE P9.5 Engine speed control.

+

T,S

r,„s + 1

+1

V(s) Speed

722

Chapter 9

Stability in the Frequency Domain

4-u

i -40

30

:

20

i

v

....

vi \ )

10 0 in

eu -120

. | 4 Hz

10

20

180

60 100 Hz

4 Hz

10

20

60 100 Hz

(a)

18ck» FIGURE P9.6 The MIT arm: (a) frequency response, and (b) position response.

200 ins

82 ms (b)

7'(/(.v)

R(s) Reference

\r _ i

"\

fc

J

*

Actuator and filter

:A

+ v~/

Vehicle Gis)

0(t) Attitude

i

FIGURE P9 7 VTOL aircraft stabilization system.

Rate gyro H(s) = s

The actuator and filter are represented by the transfer function Gc(s) =

Kj(s + 7) s + 3

(a) Obtain the Bode diagram of the loop transfer function L(s) = Gc(s)Gis)H(s) when the gain is K] = 2. (b) Determine the gain and phase margins of this system, (c) Determine the steady-state error for a wind disturbance of Td(s) = l/s. (d) Determine the maximum amplitude of the resonant peak of the closed-loop frequency response and the frequency of the resonance, (e) Estimate the damping ratio of the system from Mpas and the phase margin. P9.8 Electrohydraulic servoniechanisms are used in control systems requiring a rapid response for a large mass. An electrohydraulic servomechanism can provide an output of 100 kW or greater [17]. A photo of a servovalve and actuator is shown in Figure P9.8(a).

The output sensor yields a measurement of actuator position, which is compared with Vm. The error is amplified and controls the hydraulic valve position, thus controlling the hydraulic fluid flow to the actuator. The block diagram of a closed-loop electrohydraulic servomechanism using pressure feedback to obtain damping is shown in Figure P9.8(b) [17, 18]. Typical values for this system are T = 0.02 s; for the hydraulic system they are w2 = 7(2-n-) and £2 = 0.05. The structural resonance coy is equal to 10(27r), and the damping is Ci = 0.05. The loop gain is KAKXK2 = 1.0. (a) Sketch the Bode diagram and determine the phase margin of the system, (b) The damping of the system can be increased by drilling a small hole in the piston so that £2 = 0.25. Sketch the Bode diagram and determine the phase margin of this system. P9.9 The space shuttle, shown in Figure P9.9(a), carries large payloads into space and returns them to earth for reuse [19]. The shuttle uses elevons at the trailing

723

Problems

* (a)

Servovalve and actuator Amplifier

Gain I(s)

KA

— + —L s + 1 CO\

K\

u)2

i

(b)

2

0.30(5 + 0.05)(5 + 1600) (s2 + 0.05^ + 16)(5 + 70)'

The controller Gc(s) can be a gain or any suitable transfer function, (a) Sketch the Bode diagram of the system when GL(s) - 2 and determine the stability margin, (b) Sketch the Bode diagram of the system when G,(J)

\ a>2

Position

K2

edge of the wing and a brake on the tail to control the flight during entry. The block diagram of a pitch rate control system is shown in Figure P9.9(b). The sensor is represented by a gain. H(s) = 0.5, and the vehicle bv the transfer function G(s)

Y(s) i v

t -i-' \ ( « + 1) — + — .v + 1

L

FIGURE P9.8 (a) A servovalve and actuator (courtesy of Moog, Inc., Industrial Division). (b) Block diagram.

CO]

s 2

= KP + Ki/s

and

K,/KP = 0.5.

The gain Kf> should be selected so that the gain margin is 10 dB. P9.10 Machine tools are often automatically controlled as shown in Figure P9.10. These automatic systems are often called numerical machine controls [9]. On each axis, the desired position of the machine tool is

compared with the actual position and is used to actuate a solenoid coil and the shaft of a hydraulic actuator. The transfer function of the actuator (see Table 2.7) is Ga{s) =

K„ Y(S)

5( V V + 1)"

where K„ = 1 and ra = 0.4 s. The output voltage of the difference amplifier is £,,(5) = Kx{X{s) -

Xd(s)l

where xlt(t) is the desired position input. The force on the shaft is proportional to the current /, so that F = K2i(t). where K2 = 3.0. The spring constant K, is equal to 1.5,7? = 0.1, and L = 0.2. (a) Determine the gain K] that results in a system with a phase margin of 30°. (b) For the gain Kx of part (a), determine M[m, con and the closed-loop system bandwidth, (c) Estimate the percent overshoot of the transient response to a step input X(t(s) = l/.y, and the settling time (to within 2% of the final value).

724

FIGURE P9.9 (a) The Earthorbiting space shuttle against the blackness of space. The remote manipulator robot is shown with the cargo bay doors open in this top view, taken by a satellite, (b) Pitch rate control system. (Courtesy of NASA.)

Chapter 9

Stability in the Frequency Domain

(a)

»—O

R(s

Controller

Vehicle

Gc(s)

G(s)

K(.v) Pilch rate

Sensor H{s)

(b)

WvVvVv^ FIGURE P9.10 Machine tool control.

Spring K

P9.ll A control system for a chemical concentration control system is shown in Figure P9.ll. The system receives a granular feed of varying composition, and we want to maintain a constant composition of the output mixture by adjusting the feed-flow valve. The transfer function of the tank and output valve is

Supply G(s) =

55 + r

and that of the controller is Gc(s) = Kx + — . s

725

Problems

Feed

V^

Desired concentration

Controller Gc(s)

Measurement of concentration

Conveyor *

«=$=» FIGURE P9.11 Chemical concentration control. Ris) • Controller Conveyor Gt.(s) e~'T The transport of the feed along the conveyor requires a transport (or delay) time, T = 1.5 s. (a) Sketch the Bode diagram when Ki = K2 = 1, and investigate the stability of the system, (b) Sketch the Bode diagram when K\ = 0.1 and K2 = 0.04, and investigate the stability of the system, (c) When K} = 0, use the Nyquist criterion to calculate the maximum allowable gain K2 for the system to remain stable. P9.12 A simplified model of the control system for regulating the pupillary aperture in the human eye is shown in Figure P9.12 [20].The gain K represents the pupillary gain, and r is the pupil time constant, which is 0.5 s.The time delay 7" is equal to 0.5 s.The pupillary gain is equal to 2. Tank Gis) — • Composition (a) Assuming the time delay is negligible, sketch the Bode diagram for the system. Determine the phase margin of the system, (b) Include the effect of the time delay by adding the phase shift due to the delay. Determine the phase margin of the system with the time delay included. P9.13 A controller is used to regulate the temperature of a mold for plastic part fabrication, as shown in Figure P9.13.The value of the delay time is estimated as 1.2 s. (a) Using the Nyquist criterion, determine the stability of the system for Ka - K — 1. (b) Determine a suitable value for Kn for a stable system that will yield a phase margin greater than 50° when K — 1. Light intensity lis) Pupil K • > \iefe :rerence FIGURE P9.12 Human pupil aperture control. Pupil area | X w ^s-' (1 + TS)7, e sT Nerve impulses + ^ Optic nerve 1 - • Y(s) 726 Chapter 9 Stability in the Frequency Domain Heating dynamics Controller FIGURE P 9 . 1 3 Temperature controller. K(.s) Desired — temperature + >n > *„(/v + Human reaction time FIGURE P 9 . 1 4 Automobile steering control. Desired direction of travel i/"~Y_ ^- esT (a) Using the Nichols chart, determine the magnitude of the gain K that will result in a system with a peak magnitude of the closed-loop frequency response Mput less than or equal to 2 dB. (b) Estimate the damping ratio of the system based on (1) Mpo) and (2) the phase margin. Compare the results and explain the difference, if any. (c) Determine the closed-loop 3-dB bandwidth of the system. P9.15 Consider the automatic ship-steering system discussed in Problem P8.ll. The frequency response of the open-loop portion of the ship steering control system is shown in Figure P8.ll. The deviation of the tanker from the straight track is measured by radar and is used to generate the error signal, as shown in Figure P9.15. This error signal is used to control the rudder angle 8(s). (a) Is this system stable? Discuss what an unstable ship-steering system indicates in terms of the transient response of the system. Recall that the system under consideration is a ship attempting to follow a straight track. (b) Is it possible to stabilize this system by lowering the gain of the transfer function G(.v)? (c) Is it possible to stabilize this system? Suggest a suitable feedback compensator. Temperature Vehicle and front wheels Control stick K(.v) K 5(0.15- + 1) i P9.14 Electronics and computers arc being used to control automobiles. Figure P9.14 is an example of an automobile control system, the steering control for a research automobile. The control stick is used for steering. A typical driver has a reaction time of T = 0.2 s. Y{s) esT i) of travel (d) Repeat parts (a), (b), and (c) when switch S is closed. Rudder angle 5(,) + A\ Desired constant heading Ship G(s) Heading X. E(s ) iU Derivative feedback Ks Switch S FIGURE P9.15 Automatic ship steering. P9.16 An electric carrier that automatically follows a tape track laid out on a factory floor is shown in Figure P9.16(a) [15]. Closed-loop feedback systems are used to control the guidance and speed of the vehicle. The cart senses the tape path by means of an array of 16 phototransistors. The block diagram of the steering system is shown in Figure P9.16(b). Select a gain K so that the phase margin is approximately 30°. P9.17 The primary objective of many control systems is to maintain the output variable at the desired or reference condition when the system is subjected to a disturbance 727 Problems (a) Phototransistor array FIGURE P9.16 (a) An electric carrier vehicle (photo courtesy of Control Engineering Corporation). (b) Block diagram. R(s) •(") Motor and cart dynamics I (s/10+ l){s2 + s + 2) Yis) Cart headim (b) U(s) R( v.^O + Yis) FIGURE P9.17 Chemical reactor control. [22]. A typical chemical reactor control scheme is shown in Figure P9.17. The disturbance is represented by U(s), and the chemical process by G3 and G4. The controller is represented by G{ and the valve by G2. The feedback sensor is H(s) and will be assumed to be equal to l .We will assume that G2, G3, and GA are all of the form G,(s) = Ki 1 + TiS' where T-, = T4 = 4 s and #3 = K4 = 0.1. The valve constants are K2 = 20 and T2 = 0.5 s. We want to maintain a steady-state error less than 5% of the desired reference position. 728 Chapter 9 Stability in the Frequency Domain P9.19 In the United States, billions of dollars are spent annually for solid waste collection and disposal. One system, which uses a remote control pick-up arm for collecting waste bags, is shown in Figure P9.19. The loop transfer function of the remote pick-up arm is (a) When G$$s) = Kh find the necessary gain to satisfy the error-constant requirement. For this condition, determine the expected overshoot to a step change in the reference signal r(f). (b) If the controller has a proportional term plus an integral term so that Gi(s) - K\(l + l/s), determine a suitable gain to yield a system with an overshoot less than 30%,but greater than 5%. For parts (a) and (b), use the approximation of the damping ratio as a function of phase margin that yields £ = 0.01 <£pm. For these calculations, assume that U(s) = 0. (c) Estimate the settling time (with a 2% criterion) of the step response of the system for the controller of parts (a) and (b). (d) The system is expected to be subjected to a step disturbance U(s) — A/s. For simplicity, assume that the desired reference is r(t) = 0 when the system has settled. Determine the response of the system of part (b) to the disturbance. L(s) = Gc(s)G(s) = (a) Plot the Nichols chart and show that the gain margin is approximately 32 dB. (b) Determine the phase margin and the Mpo> for the closed loop. Also, determine the closed-loop bandwidth. P9.20 The Bell-Boeing V-22 Osprey Tiltrotor is both an airplane and a helicopter. Its advantage is the ability to rotate its engines to a vertical position, as shown in Figure P7.33(a), for takeoffs and landings and then switch the engines to a horizontal position for cruising as an airplane. The altitude control system in the helicopter mode is shown in Figure P9.20. (a) Obtain the frequency response of the system for K = 100. (b) Find the gain margin and the phase margin for this system, (c) Select a suitable gain K so that the phase margin is 40°. (Decrease the gain above K = 100.) (d) Find the response y(t) of the system for the gain selected in part (c). P9.18 A model of an automobile driver attempting to steer a course is shown in Figure P9.18, where K = 5.3. (a) Find the frequency response and the gain and phase margins when the reaction time T is zero, (b) Find the phase margin when the reaction time is 0.1 s. (c) Find the reaction time that will cause the system to be borderline stable (phase margin = 0°). R(s) course FIGURE P 9 . 1 8 Automobile and driver control. FIGURE P9.19 Waste collection system. "toJ *L Predicted course 0.5 s(2s + l)(s + 4)' Steering Auto K i s Predictor ,r + 0.8.9 + 0.32 Driver I Auto Lateral displacement 729 Problems W RW — H Q - Controller Dynamics K(s + 1.5s + 0.5) I (20.?+ 1)(105+ 1)(0.5.5+ I) 2 n.v) Altitude FIGURE P9.20 Tiltrotor aircraft control. P9.21 Consider a unity feedback system with the loop transfer function Gc(s)G(s) K s(s + 1)(5- + 4)' (a) Sketch the Bode diagram for K = 4. Determine (b) the gain margin, (c) the value of K required to provide a gain margin equal to 12 dB, and (d) the value of K to yield a steady-state error of 2 5 % of the magnitude A for the ramp input r{t) - At J > 0. Can this gain be utilized and achieve acceptable performance? P9.22 The Nichols diagram for Gc(jio)G(jcS) of a closedloop system is shown in Figure P9.22.The frequency for each point on the graph is given in the following table: Point 1 to 1 8 2.0 2.6 3.4 -0.10 dB 180 5.2 6.0 7.0 8.0 Determine (a) the resonant frequency, (b) the bandwidth, (c) the phase margin, and (d) the gain margin. (e) Estimate the overshoot and settling time (with a 2 % criterion) of the response to a step input. 30 FIGURE P9.22 Nichols chart. 4.2 -150 -120 Loop phase G, G (degrees) 730 Chapter 9 Stability in the Frequency Domain P9.23 A closed-loop system has a loop transfer function L(s) = Gc(s)G(s) = (a) Determine the gain K so that the phase margin is 60° when T - 0.2. (b) Plot the phase margin versus the time delay T for K as in part (a). K s(s + 8)(.v + 12)' (a) Determine the gain K so that the phase margin is 60°. (b) For the gain K selected in part (a), determine the gain margin of the system. P9.24 A closed-loop system with unity feedback has a loop transfer function K(s + 20) L(s) = Gc(s)G(s) = (a) Determine the gain K so that the phase margin is 45°. (b) For the gain K selected in part (a), determine the gain margin, (c) Predict the bandwidth of the closed-loop system. P9.25 A closed-loop system has the loop transfer function L(s) = Gc(s)G(s) = P9.26 A specialty machine shop is improving the efficiency of its surface-grinding process [21]. The existing machine is mechanically sound, but manually operated. Automating the machine will free the operator for other tasks and thus increase overall throughput of the machine shop. The grinding machine is shown in Figure P9.26(a) with all three axes automated with motors and feedback systems. The control system for the y-axis is shown in Figure P9.26(b). To achieve a low steady-state error to a ramp command, we choose K = 10. Sketch the Bode diagram of the open-loop system and obtain the Nichols chart plot. Determine the gain and phase margin of the system and the bandwidth of the closed-loop system. Estimate the £ of the system and the predicted overshoot and settling time (with a 2% criterion). P9.27 Consider the system shown in Figure P9.27. Determine the maximum value of K = Kma,, for which the — . t v-axis motor (b) (a) FIGURE P9.26 Surface-grinding wheel control system Process Controller 4 K Ris) r + 2s + 4 L FIGURE P9.27 Sensor Nonunity feedback system with proportional controller K. 1 s+ 1 • K(.v) » Y{s) 731 Advanced Problems closed-loop system is stable. Plot the phase margin as a function of the gain 1 =s K < Kmax. Explain what happens to the phase margin as K approaches P9.28 Consider the feedback system shown in Figure P9.28 with the process transfer function given as G(s) = FIGURE P 9 . 2 8 A unity feedback system with a proportional controller in the loop. 1 s(s + 1)' The controller is the proportional controller GM = KP. (a) Determine a value of KP such that the phase margin is approximately P.M. «s 45°. (b) Using the P.M. obtained, predict the percent overshoot of the closed-loop system to a unit step input. (c) Plot the step response and compare the actual percent overshoot with the predicted percent overshoot. Controller R(x) Q Process Ea(s) I s(s+ I) - • Y(s) ADVANCED PROBLEMS AP9.1 Operational spacecraft undergo substantial mass property and configuration changes during their lifetime [25]. For example, the inertias change considerably during operations. Consider the orientation control system shown in Figure AP9.1. (a) Plot the Bode diagram, and determine the gain and phase margins when con2 = 15,267. (b) Repeat part (a) when a),,2 = 9500. Note the effect of changing w„ 2 by 3 8 % . Orientation command pC j • > J fc * AP9.2 Anesthesia is used in surgery to induce unconsciousness. One problem with drug-induced unconsciousness is large differences in patient responsiveness. Furthermore, the patient response changes during an operation. A model of drug-induced anesthesia control is shown in Figure AP9.2. The proxy for unconsciousness is the arterial blood pressure. (a) Plot the Bode diagram and determine the gain margin and the phase margin when T = 0.05 s. (b) Repeat Controller Dynamics 775 (.v + 10) s(s + 2) 305.3 5 " + 105 + Y(s) rientatii angle Sensor FIGURE AP9.1 Spacecraft orientation control. 5+5 s+ 1 Rti) Desired blood pressure FIGURE AP9.2 Control of blood pressure with anesthesia. Controller O Body dynamics •ic- 2(.v + 5) Sensor s +2 Actual blood pressure 732 Chapter 9 R(s) Desired depth Stability in the Frequency Domain O Controller Process 66(1 +0.1.T) 1 + 0,01 s K (1 +0.015)(1 + \.5s) -• Actual head depth Sensor FIGURE AP9.3 Weld bead depth control. FIGURE AP9.4 Paper machine control. 1 + 0.2* Desired paper weight/area 1 Controller Paper machine K(s + 40) s + 15 I s(s + 10) part (a) when T = 0.1 s. Describe the effect of the 100% increase in the time delay T. (c) Using the phase margin, predict the overshoot for a step input for parts (a) and (b). AP9.3 Welding processes have been automated over the past decades. Weld quality features, such as final metallurgy and joint mechanics, typically are not measurable online for control. Therefore, some indirect way of controlling the weld quality is necessary. A comprehensive approach to in-process control of welding includes both geometric features of the bead (such as the cross-sectional features of width, depth, and height) and thermal characteristics (such as the heataffected zone width and cooling rate). The weld bead depth, which is the key geometric attribute of a major class of welds, is very difficult to measure directly, but a method to estimate the depth using temperature measurement has been developed [26]. A model of the weld control system is shown in Figure AP9.3. (a) Determine the phase margin and gain margin for the system when K = 1. (b) Repeat part (a) when K = 1.5. (c) Determine the bandwidth of the system for K = 1 and K = 1.5 by using the Nichols chart. (d) Predict the settling time (with a 2% criterion) of a step response for K = 1 and K = 1.5. AP9.4 The control of a paper-making machine is quite complex [27].The goal is to deposit the proper amount of fiber suspension (pulp) at the right speed and in a uniform way. Dewatering, fiber deposition, rolling, and drying then take place in sequence. Control of the paper weight per unit area is very important. For the control system shown in Figure AP9.4. select K so that the phase margin P.M. > 45° and the gain margin G.M. ^ 10 dB. Plot the step response for the selected Y(s) Actual paper weight/area gain. Determine the bandwidth of the closed-loop system. AP9.5 NASA is planning many Mars missions with rover vehicles. A typical rover is a solar-powered vehicle which will see where it is going with TV cameras and will measure distance to objects with laser range finders. It will be able to climb a 30° slope in dry sand and will carry a spectrometer that can determine the chemical composition of surface rocks. It will be controlled remotely from Earth. For the model of the position control system shown in Figure AP9.5, determine the gain K that maximizes the phase margin. Determine the overshoot for a step input with the selected gain. Controller /?(.*) FIGURE AP9.5 K(s + 0.4) 1- + 3 Plant — • 1 s2(s + 6) >'(.*) Position control system of a Mars rover. AP9.6 The acidity of water draining from a coal mine is often controlled by adding lime to the water. A valve controls the lime addition and a sensor is downstream. For the model of the system shown in Figure AP9.6, determine K and the distance D to maintain stability. We require D > 2 meters in order to allow full mixing before sensing. AP9.7 Building elevators are limited to about 800 meters. Above that height, elevator cables become too thick 733 Advanced Problems and too heavy for practical use. One solution is to eliminate the cable. The key to the cordless elevator is the linear motor technology now being applied to the development of magnetically levitated rail transportation systems. Under consideration is a linear synchronous motor that propels a passenger car along the tracklike guideway running the length of the elevator shaft. The motor works by the interaction of an electromagnetic field from electric coils on the guideway with magnets on the car [28]. If we assume that the motor has negligible friction, the system may be represented by the model shown in Figure AP9.7. Determine K so that the phase margin of the system is 45°. For the gain K selected, determine the system bandwidth. Also calculate the maximum value of the output for a unit step disturbance for the selected gain. AP9.8 A control system is shown in Figure AP9.8. The gain K is greater than 500 and less than 3000. Select a gain that will cause the system step response to have an overshoot of less than 20%. Plot the Nichols diagram, and calculate the phase margin. AP9.9 Consider again the system shown in Figure AP7.12 which uses a PI controller. Let and determine the gain KP that provides the maximum phase margin. AP9.10 A multiloop block diagram is shown in Figure AP9.10. Delay Motor and valve FIGURE AP9.6 Mine water acidity control. o R(s) = 0 Desired ' acidity Disturbance FIGURE AP9.7 -U o K(s + 4) " • Actual acidity Elevator and linear motor Controller K(.v) Desired elevator position ns) ,-sD/A (.s2+ 10i + 100) 6 Y(s) •*• Elevator position Elevator position control. Kjs+ 1)2 (.v+ IOX.v + 25) tf(.v) • • Y(s) s(s2 + 3.2.v + 3.56) FIGURE AP9.8 Gain selection. Ms) FIGURE AP 9.10 Multiloop feedback control system. •(*) W> * K •*- K(.v) 734 Chapter 9 Stability in the Frequency Domain electric power is transmitted inductively across the skin through a transmission system. The batteries and the transmission system limit the electric energy storage and the transmitted peak power. We desire to drive the EVAD in a fashion that minimizes its electric power consumption [33]. The EVAD has a single input, the applied motor voltage, and a single output, the blood flow rate. The control system of the EVAD performs two main tasks: It adjusts the motor voltage to drive the pusher plate through its desired stroke, and it varies the EVAD blood flow to meet the body's cardiac output demand. The blood flow controller adjusts the blood flow rate by varying the EVAD beat rate. A model of the feedback control system is shown in Figure AP9.11(b). The motor, pump, and blood sac can be modeled by a (a) Compute the transfer function T(s) = Y(s)/R(s). (b) Determine K such that the steady-state tracking error to a unit step input R(s) = \js is zero. Plot the unit step response. (c) Using K from part (b). compute the system bandwidth cjb. AP9.11. Patients with a cardiological illness and less than normal heart muscle strength can benefit from an assistance device. An electric ventricular assist device (EVAD) converts electric power into blood flow by moving a pusher plate against a flexible blood sac. The pusher plate reciprocates to eject blood in systole and to allow the sac to fill in diastole. The EVAD will be implanted in tandem or in parallel with the intact natural heart as shown in Figure AP9.11(a). The EVAD is driven by rechargeable batteries, and the Compliance chamber Assist pump Energy transmission system Battery pack Controller (a) FIGURE AP 9.11 (a) An electric ventricular assist device for cardiology patients. (b) Feedback control system. Controller R(s) Desired flow G((.v) V(s) Motor voltage Motor, pump, and blood sac G(s) - e sT Y(s) Blood flow rate L rale (b) 735 Design Problems nominal time delay with T = 1 s. The goal is to achieve a step response with zero steady-state error and percent overshoot P.O. < 10%. Consider the controller GAs) = ^TWy For the nominal time delay of T = 1 s, plot the step response and verify that steady-state tracking error and percent overshoot specifications are satisfied. Determine the maximum time delay, 7", possible with the PID controller that continues to stabilize the closedloop system. Plot the phase margin as a function of time delay up to the maximum allowed for stability. DESIGN PROBLEMS CDP9.1 The system of Figure CDP4.1 uses a controller r^i G,(.y) = K0. Determine the value of Ku so that the \^Vj phase margin is 70°. Plot the response of this system to a step input. DP9.1 A mobile robot for toxic waste cleanup is shown in Figure DP9.1(a) [23]. The closed-loop speed control is represented by Figure 9.1 with H{s) = 1. The Nichols chart in Figure DP9.1(b) shows the plot of Gc{jco) G(jio)/K versus co. The value of the frequency at the points indicated is recorded in the following table: Point 1 2 3 4 5 co 5 10 20 50 2 (a) Determine the gain and phase margins of the closed-loop system when K - \. (b) Determine the resonant peak in dB and the resonant frequency for K =-1, (c) Determine the system bandwidth and estimate the settling time (with a 2% criterion) and percent overshoot of this system for a step input. (d) Determine the appropriate gain K so that the overshoot to a step input is 30%, and estimate the settling time of the system. DP9.2 Flexible-joint robotic arms are constructed of lightweight materials and exhibit lightly damped open-loop dynamics [15]. A feedback control system for a flexible arm is shown in Figure DP9.2. Select K so that the system has maximum phase margin. Predict the overshoot for a step input based on the phase margin attained, and compare it to the actual overshoot for a step input. Determine the bandwidth of the closed-loop system. Predict the settling time (with a 2% criterion) of the system to a step input and compare it to the actual settling time. Discuss the suitability of this control system. DP9.3 An automatic drug delivery system is used in the regulation of critical care patients suffering from cardiac failure [24]. The goal is to maintain stable patient status within narrow bounds. Consider the use of a drug delivery system for the regulation of blood pressure by the infusion of a drug. The feedback control system is shown in Figure DP9.3. Select an appropriate gain K that maintains narrow deviation for blood pressure while achieving a good dynamic response. DP9.4 A robot tennis player is shown in Figure DP9.4(a), and a simplified control system for 62(t) is shown in Figure DP9.4(b).The goal of the control system is to attain the best step response while attaining a high Kv for the system. Select KP\ = 0.4 and Kt,2 = 0.75, and determine the phase margin, gain margin, bandwidth. percent overshoot, and settling time for each case. Obtain the step response for each case and select the best value for K. DP9.5 An electrohydrauiic actuator is used to actuate large loads for a robot manipulator, as shown in Figure DP9.5 [17]. The system is subjected to a step input, and we desire the steady-state error to be minimized. However, we wish to keep the overshoot less than 10%. Let T = 0.8 s. (a) Select the gain K when Gc(s) - K, and determine the resulting overshoot, settling time (with a 2% criterion), and steady-state error, (b) Repeat part (a) when Gc(s) = Ky + K2/s by selecting Kx and K2. Sketch the Nichols chart for the selected gains K\ and K2. DP9.6 The physical representation of a steel strip-rolling mill is a damped-spring system [8]. The output thickness sensor is located a negligible distance from the output of the mill, and the objective is to keep the thickness as close to a reference value as possible. Any change of the input strip thickness is regarded as a disturbance. The system is a nonunity feedback system, as shown in Figure DP9.6. Depending on the maintenance of the mill, the parameter varies as 80 < b < 300. Determine the phase margin and gain margin for the two extreme values of b when the normal value of (a) V, 12 -210 -180 -150 -120 Loop phase LGCG (degrees) (b) -60 Design Problems Controller Q- R(s) FIGURE DP9.2 Control of a flexible robot arm. Flexible arm K(s + 0.6) s(s2 + 9s + 12) Patient's physiological dynamics Kfs) •*• Blood pressure Ke -I0.v 4().v + 1 •0— /?(.¥) FIGURE DP9.3 Automatic drug delivery. (a) FIGURE DP9.4 (a) An articulated two-link tennis player robot. (b) Simplified control system. I>u - >J*+4 * f\ \r *L 1 s(s+ 1) (b) + R(s) FIGURE DP9.5 Electrohydraulic actuator. r~\ __- i Controller Actuator Gv(s) e-*T 105 + 1 • Yis) 738 Chapter 9 Stability in the Frequency Domain Controller Strip mill 333.3 s 2 + bs + 10,000 Ris) I » Y(s) Sensor FIGURE DP9.6 Steel strip-rolling mill. 3 s+ 3 the gain is K - 170. Recommend a reduced value for K so that the phase margin is greater than 40° and the gain margin is greater than 8 dB for the range of b. out of the second tank. The block diagram model is shown in Figure DP9.9(b). The system of the two tanks has a heater in tank 1 with a controllable heat input Q. The time constants are n = 1 0 s and T2 = 50 s. DP9.7 Vehicles for lunar construction and exploration work will face conditions unlike anything found on Earth. Furthermore, they will be controlled via remote control. A block diagram of such a vehicle and the control are shown in Figure DP9.7. Select a suitable gain K when T = 0.5 s. The goal is to achieve a fast step response with an overshoot of less than 20%. (a) Determine T2(s) in terms of TQ(s) and T2li(s). (b) If T2t](s), the desired output temperature, is changed instantaneously from T2ci(s) = A/s to T2(i(s) - 2A/s, determine the transient response of T2(t) when Gc(s) = K = 500. Assume that, prior to the abrupt temperature change, the system is at steady state. (c) Find the steady-state error ess for the system of part (b), where E(s) = T2l,{sjT2(s). (d) Let Gc(s) = K/s and repeat parts (b) and (c). Use a gain K such that the percent overshoot is less than 10%. (e) Design a controller that will result in a system with a settling time (with a 2 % criterion) of Ts < 150 s and a percent overshoot of less than 10%, while maintaining a zero steady-state error when DP9.8 The control of a high-speed steel-rolling mill is a challenging problem. The goal is to keep the strip thickness accurate and readily adjustable. The model of the control system is shown in Figure DP9.8. Design a control system by selecting K so that the step response of the system is as fast as possible with an overshoot less than 0.5% and a settling time (with a 2 % criterion) less than 4 seconds. Use the root locus to select K, and calculate the roots for the selected K. Describe the dominant root(s) of the system. DP9.9 A two-tank system containing a heated liquid has the model shown in Figure DP9.9(a), where r ( ) is the temperature of the fluid flowing into the first tank and T2 is the temperature of the liquid flowing R(s) + "> Q fc Gc(s) = KP + —, Controller Vehicle K I s(s+ I) 2 Signal transmission delay esT FIGURE DP9.7 Lunar vehicle control. Dynamics FIGURE DP9.8 Steel-rolling mill control. R(s) Desired thickness O K x(s + 25)(s + 100s + 2600) 2 Y(s) - • Thickness of strip •• tt.v) 739 Design Problems Liquid in Valve Valve Liquid oul (a) • Ty(s) FIGURE DP9.9 Two-tank temperature control. T2cl(s) (b) (f) Prepare a table comparing the percent overshoot, settling time, and steady-state error for the designs of parts (b) through (e). DP9.10 Consider the system is described in state variable form by DP9.11. The primary control loop of a nuclear power plant includes a time delay due to the need to transport the fluid from the reactor to the measurement point as shown in Figure DP9.11. The transfer function of the controller is x(0 = Ax(/) + B«(f) y(t) = Cx(r) where A - L 0 1 ,B = 2 3 "o" [lj (b) Design the gain matrix K to meet the following specifications: (i) the closed-loop system is stable; (ii) the system bandwidth [1 0]. Assume that the input is a linear combination of the states, that is, u(t) = -Kx(r) + r(f), where r{t) is the reference input and the gain matrix is K = [K\ K2]. Substituting u(t) into the state variable equation yields the closed-loop system x(0 = [A - BK]x(0 + Br(/) y{t) = Cx(/) (a) Obtain the characteristic equation associated withA-BK. Gc(s) ~KP + K, The transfer function of the reactor and time delay is G(s) - TS + r where T = 0.4 s and T = 0.2 s. Using frequency response methods, design the controller so that the overshoot of the system is P.O. ss 10%. With this controller in the loop, estimate the percent overshoot and settling time (with a 2% criterion) to a unit step. Determine the actual overshoot and settling time and compare with the estimated values. * 740 Chapter 9 Stability in the Frequency Domain Temperature measurement FIGURE DP9.10 Nuclear reactor control. + Temperature COMPUTER PROBLEMS CP9.1 Consider a unity negative feedback control system with L(s) = Gc(s)G(S) = - 141 s + 2s 4- 12 Verify that the gain margin is co and that the phase margin is 10°. CP9.2 Using the nyquist function, obtain the polar plot for the following transfer functions: 5 (a) G(s) = S + 5' 50 s2 + 10s + 25 15 (c) G(s) = 3 A' + 3s2 + 3.y + 1 (b) G(s) = CP9.3 Using the nichols function, obtain the Nichols chart with a grid for the following transfer functions: (a) G(s) = (b) G(s) = s + 0.2' 1 s2 + 2s + r 6 (c) G(s) = 3 2 s + 6s + l b + 6* Determine the approximate phase and gain margins from the Nichols charts and label the charts accordingly. CP9.4 A negative feedback control system has the loop transfer function L(s) = Gc(s)G(s) = f~-. (a) When T = 0.2 s, find K such that the phase margin is 40° using the margin function, (b) Obtain a plot of phase margin versus T for K as in part (a), with 0 < T < 0.3 s. CP9.5 Consider the paper machine control in Figure AP9.4. Develop an m-file to plot the bandwidth of the closed-loop system as K varies in the interval 1 < K < 50. CP9.6 A block diagram of the yaw acceleration control system for a bank-to-turn missile is shown in Figure CP9.6. The input is yaw acceleration command (in g's), and the output is missile yaw acceleration (in g's). The controller is specified to be a proportional, integral (PI) controller. The nominal value of b{) is 0.5. (a) Using the margin function, compute the phase margin, gain margin, and system crossover frequency (0 dB), assuming the nominal value of b(). (b) Using the gain margin from part (a), determine the maximum value of 60 for a stable system. Verify your answer with a Routh-Hurwitz analysis of the characteristic equation. CP9.7 An engineering laboratory has presented a plan to operate an Earth-orbiting satellite that is to be controlled from a ground station. A block diagram of the proposed system is shown in Figure CP9.7. It takes T seconds for a signal to reach the spacecraft from the ground station and the identical delay for a return signal. The proposed ground-based controller is a proportional-derivative (PD) controller, where Gr(s) = Kr + Kos. 741 Answers to Skills Check FIGURE CP9.6 A feedback control system for the yaw acceleration control of a bank-to-tum missile. o desired PI controller Bank-to-turn missile 10(5 + 3) -b(){s2 - 2500) - • «, ( 5 - 3)(.52 + 505+ 1000) t—• o FIGURE CP9.7 A block diagram of a ground-controlled satellite. (a) Assume no transmission time delay (i.e., T = 0), and design the controller to the following specifications: (1) percent overshoot less than 20% to a unit step input and (2) time to peak less than 30 seconds. (b) Compute the phase margin with the controller in the loop but assuming a zero transmission time delay. Estimate the amount of allowable time delay for a stable system from the phase margin calculation. (c) Using a second-order Pade approximation to the time delay, determine the maximum allowable delay Tmw for system stability by developing a mfile script that employs the pade function and computes the closed-loop system poles as a function of the time delay T. Compare your answer with the one obtained in part (b). CP9.9 For the system in CP9.8, use the nichols function to obtain the Nichols chart and determine the phase margin and gain margin. CP9.10 A closed-loop feedback system is shown in Figure CP9.10. (a) Obtain the Nyquist plot and determine the phase margin. Assume that the time delay T = 0 s. (b) Compute the phase margin when T = 0.05 s. (c) Determine the minimum time delay that destabilizes the closed-loop system. Time Delay + —. _i CP9.8 Consider the system represented in state variable form 0 -1 y = [8 i -15 J 0]x + [0]« 1 5+ 1 K(.v) 10 s "o ; 1 + _M Using the nyquist function, obtain the polar plot. El e~*T R(s) FIGURE CP9.10 delay. Nonunity feedback system with a time ANSWERS TO SKILLS CHECK True or False: (1) True; (2) True; (3) True; (4) True; (5) False Multiple Choice: (6) b; (7) a; (8) d; (9) a; (10) d; (11) b; (12) a; (13) b; (14) c; (15) a Word Match (in order, top to bottom): f, e, k, b, j , a, i, d, h, c, g 742 Chapter 9 Stability in the Frequency Domain TERMS AND CONCEPTS Bandwidth The frequency at which the frequency response has declined 3 dB from its low-frequency value. Cauchy's theorem If a contour encircles Z zeros and P poles of F(s) traversing clockwise, the corresponding contour in the F(.v)-plane encircles the origin of the F(s)-plane N = Z - P times clockwise. Closed-loop frequency response The frequency response of the closed-loop transfer function T{jui). Conformal mapping A contour mapping that retains the angles on the .v-plane on the -F(s)-plane. Contour map A contour or trajectory in one plane is mapped into another plane by a relation F(s). Gain margin The increase in the system gain when phase = -180° that will result in a marginally stable system with intersection of the - 1 + /0 point on the Nyquist diagram. Logarithmic (decibel) measure A measure of the gain 1 1 margin defined as 20 logio( 1/d), where - = •: r d when the phase shift is -180°. \L(ju))\ Nichols chart A chart displaying the curves for the relationship between the open-loop and closed-loop frequency response. Nyquist stability criterion A feedback system is stable if, and only if, the contour in the L(s)-plane does not encircle the (—1,0) point when the number of poles of L(s) in the right-hand .y-plane is zero. If L(s) has P poles in the right-hand plane, then the number of counterclockwise encirclements of the (-1,0) point must be equal to P for a stable system. Phase margin The amount of phase shift of the L(jto) at unity magnitude that will result in a marginally stable system with intersections of the —1 + /0 point on the Nyquist diagram. Principle of the argument See Cauchy's theorem. Time delay A time delay T, so that events occurring at time t at one point in the system occur at another point in the system at a later time t + T. 806 Chapter 10 The Design of Feedback Control Systems Table 10.8 Type of Controller PD Operational Amplifier Circuits for Compensators Gc(s) = Vo(s) VI(S) Gc R3Rx K (RiCiS + 1) + o- A/W *3 AMr U PI G = R4R2(R2C2s + 1) R,R,(R2C2s) «1 rsHih -t-o- a< *3 4—VW B Lead or lag Gc = R4R2(RyClS + 1) R3Rx(R2C2s + 1) * o+ ie Lead if RXCX > R2C2 Lag if Rxd < R2C2 C, + o- AA/v—• R, «3 -o + £ m SKILLS CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 10.43 as specified in the various problem statements. , + /-\ J FIGURE 10.43 Controller Process Gc(s) G(s) Block diagram for the Skills Check. - • y\s) 807 Skills Check In the following True or False and Multiple Choice problems, circle the correct answer. 1. A cascade compensator network is a compensator network that is placed in parallel with the system process. 2. Generally, a phase-lag network speeds up the transient response. 3. The arrangement of the system and the selection of suitable components and parameters is part of the process of control system design. 4. A deadbeat response of a system is a rapid response with minimal overshoot and zero steady-state error to a step input. 5. A phase-lead network can be used to increase the system bandwidth. 6. Consider the feedback system in Figure 10.43, where True or False True or False True or False True or False True or False W s(s + 400)(5 + 20)' A phase lag compensator is designed for the system to give additional attenuation at higher frequencies. The compensator is 1+0-253 G (s) < ~ TTzP When compared with the uncompensated system (that is, Gc(s) = 1), the compensated system utilizing the lag compensator: a. Increases the phase lag near the cross-over frequency. b. Decreases the phase margin. c Provides additional attenuation at higher frequencies. d. All of the above. 7. A position control system can be analyzed using the feedback system in Figure 10.43, where the process transfer function is G ^ = s(s + 1)(0.45 + 1)' A phase-lag compensator that provides a phase margin of P.M. « 30° is: 1 + s a. Gc(s) = 1 + 1065 1 + 265 b. Gc.(s) = 1 + 1155 1 + 1065 c. Gc(s) = 1 + 1185 d. None of the above 8. Consider a unity feedback system in Figure 10.43, where 1450 G(s) s{s + 3)(5 + 25) A lead compensator is introduced into the feedback loop, where 1 + 0.35 Cc(5) = ITO03? The peak magnitude and the bandwidth of the closed-loop frequency response are: a. MPia = 1.9dB;&),, = 12.1 rad/s b. MPu = 12.8 d B ; ^ = 14.9 rad/s c. MPo> = 5.3 dB; OJ,, = 4.7 rad/s d. MPu = 4.3 dB;wb = 24.2 rad/s 808 Chapter 10 The Design of Feedback Control Systems 9. Consider the feedback system in Figure 10.43, where the plant model is 500 GW = 7(7715) and the controller is a proportional-plus-integral (PI) controller given by Gc{s) = KP + -L. s Selecting Kj — 1, determine a suitable value of KP for a percent overshoot of approximately 20%. a. KP = 0.5 b. KP = 1.5 c. KP = 2.5 d. KP = 5.0 10. Consider the feedback system in Figure 10.43, where 1 G(,S) = s{\ + 5/8)(1 + 5/20)' The design specifications are: K0 a 100, G.M. ^ 10 dB, P.M. > 45°, and the crossover frequency, o)c s 10 rad/s. Which of the following controllers meets these specifications? (1 + 5)(1 + 20s) a. Gc(s) (1 + ^/0.01)(1 + s/50) 100(1 + 5)(1 + 5/5) b. Gc(s) (1 + 5/0.1)(1 + 5/50) 1 + 1005 c Gc{s) 1 + 1205 d. Gc(s) = 100 11. Consider a feedback system in which a phase-lead compensator Gc{s) = TTOMS is placed in series with the plant G(S) = 500 ( 5 + 1)(5 + 5 ) ( 5 + 10)' The feedback system is a negative unity feedback control system shown in Figure 10.43. Compute the gain and phase margin. a. G.M. = oo dB, P.M. = 60° b. G.M. = 20.5 dB,P.M. - 47.8° c G.M. = 8.6 dB, P.M. = 33.6° d. Closed-loop system is unstable. 12. Consider the feedback system in Figure 10.43, where 1 " s{s + 10)(5 + 15)' Which of the following represents a suitable lag compensator that achieves a steady-state error less than 10% for a ramp input and a damping ratio of the closed-loop system dominant roots of £ « 0.707. (S) a * GM = 2850(5 + 1) (105 + 1) 809 Skills Check b ' Gc{s) = c. Gc(s) 1 0 0 ( 5 + 1 ) ( , + 5) (s + 10)(, + 50) 10 , + 1 d. Closed-loop system cannot track a ramp input for any Gc(s). 13. A viable lag-compensation for a unity negative feedback system with plant transfer function { } (s + 8)(5 + 14)(, + 20) that satisfies the design specifications: (i) percent overshoot P.O. £ 5%; (ii) rise time Tr ss 20 seconds, and (hi) position error constant Kp > 6, is which of the following: a. Gc(s) b. Gc(s) c. G c (,) d. Gc(s) 14. Consider s + 0.074 , + 0.074 = s + 1 20, + 1 = 100, + 1 = 20 the feedback system depicted in Figure 10.43, where G(s) = -. V ' s(s + 4) 2 A suitable compensation Gc(s) for this system that satisfies the specifications: (i) P.O. < 20%, and (ii) velocity error constant Kv a 10, is which of the following: a. Gc(s) (5+1) 160(105 + 1) b. Gc(s) = 2005 + 1 24(, + 1) c. Gc(s) = , + 4 d. None of the above 15. Using a Nichols chart, determine the gain and phase margin of the system in Figure 10.43 with loop gain transfer function L{s) = Gc(s)G(s) = a. b. c. d. G.M. G.M. G.M. G.M. = = = 8 + \ • 2 ' s^s* + 25 + 4) 20.4 dB, P.M. = 58.1° oo dB, P.M. = 47° 6 dB, P.M. = 45° codB,P.M. = 23° In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Deadbeat response A system with a rapid response, minimal overshoot, and zero steady-state error for a step input. 810 Chapter 10 The Design of Feedback Control Systems b. Phase lead compensation c. PI controller A network that provides a positive phase angle over the frequency range of interest. d. Lead-lag network A network with the characteristics of both a lead network and a lag network. e. Design of a control system A network that provides a negative phase angle and a significant attenuation over the frequency range of interest. f. Phase lag compensation An additional component or circuit that is inserted into the system to compensate for a performance deficiency. g. Integration network A compensator network placed in cascade or series with the system process. h. Compensator Controller with a proportional term and an integral term. i. Compensation A transfer function, Gp(s), that filters the input signal R(s) prior to calculating the error signal. j . Phase-lag network The arrangement or the plan of the system structure and the selection of suitable components and parameters. k. Cascade compensation network The alteration or adjustment of a control system in order to provide a suitable performance. 1. Phase-lead network m. Prefilter A network that acts, in part, like an integrator. A widely-used compensator that possesses one zero and one pole with the pole closer to the origin of the s-plane. A widely-used compensator that possesses one zero and one pole with the zero closer to the origin of the s-plane. EXERCISES ElO.l A negative feedback control system has a transfer function G(s) = K 5 + 2 We select a compensator Gc(s) + a in order to achieve zero steady-state error for a step input. Select a and K so that the overshoot to a step is approximately 5% and the settling time (with a 2% criterion) is approximately 1 second. Answer: K = 6, a = 5.6 E10.2 A control system with negative unity feedback has a process G(s) 400 s(s + 40)' and we wish to use proportional plus integral compensation, where Gc(s) = KP + —. s Note that the steady-state error of this system for a ramp input is zero, (a) Set K/ = 1 and find a suitable value of KP so the step response will have an overshoot of approximately 20%. (b) What is the expected 811 Exercises settling time (with a 2% criterion) of the compensated system? Answer: KP = 0.5 E10.3 A unity negative feedback control system in a manufacturing system has a process transfer function E10.5 Consider a unity feedback system with the transfer function K s(s + 2)(s + 4) We desire to obtain the dominant roots with co„ = 3 and £ = 0.5. We wish to obtain a Kv = 2.7. Show that we require a compensator G(s) = G(s) = s +V and it is proposed that we use a compensator to achieve a 5% overshoot to a step input.The compensator is [4] Gc(s) m K I 1 + 1 which provides proportional plus integral control. Show that one solution is K = 0.5 and T = 1. E10.4 Consider a unity negative feedback system with K s(s + 5)(s + 10)' G(s) where K is set equal to 100 in order to achieve a specified Kv = 2. We wish to add a lead-lag compensator Gc(s) = G c (5) (s + 0.15)(5 + 0.7) (s + 0.015)(^ + 7)" Show that the gain margin of the compensated system is 28.6 dB and that the phase margin is 75.4°. 7.53(5 + 2.2) (s + 16.4) Determine the value of K that should be selected. Answer: K = 22 E10.6 Consider again the wind tunnel control system of Problem P7.31. When K = 326, find T(s) and estimate the expected overshoot and settling time (with a 2% criterion). Compare your estimates with the actual overshoot of 60% and a settling time of 4 seconds. Explain the discrepancy in your estimates. E10.7 NASA astronauts retrieved a satellite and brought it into the cargo bay of the space shuttle, as shown in Figure E10.7(a). A model of the feedback control system is shown in Figure. E10.7(b). Determine the value of K that will result in a phase margin of 40° when T = 0.6 s. Answer: K = 26.93 (a) Ris)- FIGURE E10.7 Retrieval of a satellite. "> * e~sT s(s + 20) K Visual feedback (b) K(.v) • Robot arm position 812 Chapter 10 The Design of Feedback Control Systems E10.8 A negative unity feedback system has a plant G(s) + 1)' S(TS Gc(s) = KP + Kt/s, so that the dominant roots of the characteristic equation have I equal to 1/V2. Plot y(t) for a step input. E10.9 A control system with a controller is shown in Figure E10.9. Select KP and Kf so that the overshoot to a step input is equal to 5% and the velocity constant Kv is equal to 5. Verify the results of your design. E10.10 A control system with a controller is shown in Figure E10.10. We will select Kf = 2 in order to provide a reasonable steady-state error to a step [8], Find KP to obtain a phase margin of 60°. Find the peak time and percent overshoot of this system. A unity feedback system has G( Answer: Gc(s) = Determine the peak magnitude and the bandwidth of the closed-loop frequency response using (a) the Nichols chart, and (b) a plot of the closed-loop frequency response. 2.3 dB, (oB = 22 Answer: M, E10.12 The control of an automobile ignition system has unity negative feedback and a loop transfer function Ge(s)G(s), where Controller FIGURE E10.9 Design of a controller. i fc KP+- 1 + 7.55 1 + 1105 E10.15 A unity feedback control system has a plant transfer function G(s) = 40 5(5 + 2 ) ' We desire to attain a steady-state error to a ramp r(f) = At of less than 0.05/1 and a phase margin of 30°. We desire to have a crossover frequency coc of 10 rad/s. Process 1 5-+ 1 (5 + l)(s + 2) FIGURE E10.10 Design of a PI controller. 5 5(5 + 1)(0.255 + 1) Determine a compensator lag network Gc(s) that will provide a phase margin of 45°. 1 + 0.255 1 + 0.025s' — i~>o Gc(s) = KP + K,/s. E10.14 A robot will be operated by NASA to build a permanent lunar station. The position control system for the gripper tool is shown in Figure 10.1(a), where //(5) = I, and G(s) A lead network is selected so that K(S) and E10.13 The design of Example 10.3 determined a lead network in order to obtain desirable dominant root locations using a cascade compensator Gc(s) in the system configuration shown in Figure 10.1 (a). The same lead network would be obtained if we used the feedback compensation configuration of Figure 10.1(b). Determine the closed-loop transfer function T(s) - Y(s)/R(s) of both the cascade and feedback configurations, and show how the transfer function of each configuration differs. Explain how the response to a step R(s) will be different for each system. 1350 ^ "s(s + 2)(s + 30)' Gc(s) = K s(s + 5) A designer selects Kf/KP ~ 0.5 and asks you to determine KKP so that the complex roots have a £ of 1/V2. where T = 2.8 ms. Select a compensator E10.ll G(s) 2257 - • n.v) 813 Exercises Use the methods of Section 10.9 to determine whether a lead or a lag compensator is required. E10.19 A unity feedback control system has the plant transfer function E10.16 Consider again the system and specifications of Exercise E10.15 when the required crossover frequency is 2 rad/s. E10.17 Consider again the system of Exercise 10.9. Select KP and Kf so that the step response is deadbeat and the settling time (with a 2% criterion) is less than 2 seconds. E10.18 The nonunity feedback control system shown in Figure E10.18 has the transfer functions G s ( ) =— s —^2v and H W = 10 - Design a compensator Gt.(s) and prefilter Gp(s) so that the closed-loop system is stable and meets the following specifications: (i) a percent overshoot to a unit step input of less than 10%, (ii) a settling time of less than 2 seconds, and (iii) zero steady-state tracking error to a unit step. "\ fc Gp(s) G,(.v) FIGURE E10.18 Nonunity feedback system with a prefilter. 1 s(s - 5)' Design a PID controller of the form Ki GJs) = Kp + KDs + — s so that the closed-loop system has a settling time less than 1 second to a unit step input. E10.20 Consider the system shown in Figure E 10.20. Design the proportional-derivative controller Gc(s) = KP + KDs such that the system has a phase margin of 40° < P.M. =s 60°. E 10.21 Consider the unity feedback system shown in Figure E10.21. Design the controller gain, K, such that the maximum value of the output y(t) in response to a unit step disturbance T(i(s) = l/s is less than 0.1. Controller Prefilter His) G(s) Plant G(s) —• • n.s) H(s) Controller + „ EJLs) Rt \ FIGURE E10.20 Unity feedback system with PD controller. ^O Process 1 Kp + KpS I s(s - 2) * l'(s) T./(.< Ris) FIGURE E10.21 Closed-loop feedback system with a disturbance input. ^) , Controller Process K l s(s + 4.4) J ' • Yl.$$ 814 Chapter 10 The Design of Feedback Control Systems PROBLEMS P10.1 The design of a lunar excursion module (LEM) is an interesting control problem. The attitude control system for the lunar vehicle is shown in Figure PI 0.1. The vehicle damping is negligible, and the attitude is controlled by gas jets. The torque, as a first approximation, will be considered to be proportional to the signal V(s) so that T(s) = K2V(s). The loop gain may be selected by the designer in order to provide a suitable damping. A damping ratio of £ = 0.6 with a settling time (with a 2 % criterion) of less than 2.5 seconds is required. Using a lead network compensation, select the necessary compensator Gc(s) by using (a) frequency response techniques and (b) root locus methods. a suitable performance, a compensator Gc(s) is inserted immediately following the photocell transducer. Select a compensator Gc(s) so that the overshoot of the system for a step input is less than 2 5 % . We will assume thatTi = T„ = 0. P10.3 A simplified version of the attitude rate control for the F-94 or X-15 type aircraft is shown in Figure PI0.3. When the vehicle is flying at four times the speed of sound (Mach 4) at an altitude of 100,000 ft, the parameters are [26] P10.2 A magnetic tape recorder transport for modern computers requires a high-accuracy, rapid-response control system. The requirements for a specific transport are as follows: (1) The tape must stop or start in 10 ms, and (2) it must be possible to read 45,000 characters per second. This system was discussed in Problem P7.ll. We desire to set J = 5 x 10" 3 , and K1 is set on the basis of the maximum error allowable for a velocity input. In this case, we desire to maintain a steady-state speed error of less than 5 % . We will use a tachometer in this case and set Ka = 50,000 and K2 = l . T o provide C - Actuator Compensation V{s) # 1 = 1.0, Vehicle T(s) Gc(s) Reference = 1.0, " • Js2 Allil :: FIGURE P10.1 Attitude control system for a lunar excursion module. Compensation Gc(s) Km Hydraulic actuator Aircraft 1 *W(V+1) —• s ,-/7,- L Rate gyro FIGURE P10.3 Aircraft attitude control. 1 Amplifier !<{•<)• FIGURE P10.4 Nuclear reactor rod control. &a Clutches Compensation —• Gc(s) —• K.j TS+ l —• Gears Load n 1 Js2 Y{s) Rod position 815 Problems The clutch housings are geared through parallel gear trains, and the direction of the servo output is dependent on the clutch that is energized. The time constant of a 200-W clutch is T = 1/40 s. The constants are such that K i-n/J — 1. We want the maximum overshoot for a step input to be in the range of 10% to 20%. Design a compensating network so that the system is adequately stabilized. The settling time (with a 2% criterion) of the system should be less than or equal to 2 seconds. R(s) = A/s. For the system with the compensation added, estimate the settling time of the system. P10.8 A numerical path-controlled machine turret lathe is an interesting problem in attaining sufficient accuracy [2, 23]. A block diagram of a turret lathe control system is shown in Figure PI0.8. The gear ratio is n = 0.1. J - 10 -3 , and b — 10 -2 . It is necessary to attain an accuracy of 5 x 10"4 in., and therefore a steady-state position accuracy of 2.5% is specified for a ramp input. Design a cascade compensator to be inserted before the silicon-controlled rectifiers in order to provide a response to a step command with an overshoot of less than 5%. A suitable damping ratio for this system is 0.7. The gain of the silicon-controlled rectifiers is KR = 5. Design a suitable lag compensator by using the (a) Bode diagram method and (b) s-plane method. P10.5 A stabilized precision rate table uses a precision tachometer and a DC direct-drive torque motor, as shown in Figure P10.5. We want to maintain a high steady-state accuracy for the speed control. To obtain a zero steady-state error for a step command design, select a proportional plus integral compensator. Select the appropriate gain constants so that the system has an overshoot of approximately 10% and a settling time (with a 2% criterion) less than 1.5 seconds. P10.6 Repeat Problem P10.5 by using a lead network compensator, and compare the results. P10.7 A chemical reactor process whose production rate is a function of catalyst addition is shown in block diagram form in Figure P10.7 [10]. The time delay is T = 50 s, and the time constant T is approximately 40 s. The gain of the process is K = 1. Design a compensation by using Bode diagram methods in order to provide a suitable system response. We want to have a steady-state error less than 0.10/1 for a step input "> P10.9 The Avemar ferry, shown in Figure PI0.9(a), is a large 670-ton ferry hydrofoil built for Mediterranean ferry service. It is capable of 45 knots (52 mph) [29]. The boat's appearance, like its performance, derives from the innovative design of the narrow "wavepiercing" hulls which move through the water like racing shells. Between the hulls is a third quasihull which gives additional buoyancy in rough seas. Loaded with 900 passengers and crew, and a mix of cars, buses, and freight cars trucks, one of the boats can carry almost its own weight. The Avemar is capable of operating in Amplifier Motor and load K, 3.75 (A-+ 0.15)(0.15s + 1) fc Gc(s) i FIGURE P10.5 Stabilized rate table. Catalyst input /?(v) Gc{s) G(s) = FIGURE P10.7 Chemical reactor control. Silicon-controlled rectifiers Input FIGURE P10.8 Path-controlled turret lathe. KR 0.1A-+ I K(.v) •*• Production output Kc 2 (TS + 1) Motor l s(Jx + b) )\s) Speed Gears Tool slide n —r—^mm'ii 816 Chapter 10 The Design of Feedback Control Systems (a) TAs) Compensator FIGURE P10.9 (a) The Avemar ferry built for ferry service between Barcelona and the Balearic Islands. (b) A block diagram of the lift control system. Gc(s) Amplifier — • K Actuator Foil and vehicle dynamics s 50 s + 80.? + 2500 —• 2 (Hs) !—• Pilch angle (b) seas with waves up to 8 ft in amplitude at a speed of 40 knots as a result of an automatic stabilization control system. Stabilization is achieved by means of flaps on the main foils and the adjustment of the aft foil. The stabilization control system maintains a level flight through rough seas. Thus, a system that minimizes deviations from a constant lift force or, equivalently, that minimizes the pitch angle 8 has been designed. A block diagram of the lift control system is shown in Figure P10.9(b).The desired response of the system to wave disturbance is a constant-level travel of the craft. Establish a set of reasonable specifications and design a compensator Gc(s) so that the performance of the system is suitable. Assume that the disturbance is due to waves with a frequency o) = 6 rad/s. Select a lead-lag compensator so that the percent overshoot for a step input is less than 5% and the settling time (with a 2% criterion) is less than 1 second. It also is desired that the acceleration constant Ka be greater than 7500 (see Table 5.5). P10.12 A unity feedback system has a plant P10.10 A unity feedback system of the form shown in Figure 10.1(a) has a plant It is required that the error for a ramp input be 0.5% of the magnitude of the ramp input (Kv = 200). P10.13 Materials testing requires the design of control systems that can faithfully reproduce normal specimen operating environments over a range of specimen parameters [23]. From the control system design viewpoint, a materials-testing machine system can be considered a servomechanism in which we want to have the load waveform track the reference signal. The system is shown in Figure P10.13. G(s) s(s2 + 6s + 10) (a) Determine the step response when Gc(s) = l,and calculate the settling time and steady state for a ramp input r(t) = t, t > 0. (b) Design a lag network using the root locus method so that the velocity constant is increased to 10. Determine the settling time (with a 2% criterion) of the compensated system. PlO.ll A unity feedback control system of the form shown in Figure 10.1(a) has a plant 160 G(s) = — . s G{s) = 20 s{\ + 0.15)(1 + 0.05^)' Select a compensator Gc(s) so that the phase margin is at least 75°. Use a two-stage lead compensator Gc(s) = K(\ + s/cox)(i + s/o)?,) (1 + s/(o2)(l + s/a>4) ' (a) Determine the phase margin of the system with Gc(s) = K, choosing K so that a phase margin of 50° is achieved. Determine the system bandwidth for this design. (b) The additional requirement introduced is that the velocity constant Kr be equal to 2.0. Design a lag 817 Problems <-K> •> Referenc FIGURE P10.13 Materials testing machine system. Controller Process Gc(s) 4 s(s + DCs + 2) fc network so that the phase margin is 50° and Kv = 2. P10.14 For the system described in Problem P10.13, the goal is to achieve a phase margin of 50° with the additional requirement that the time to settle (to within 2% of the final value) be less than 4 seconds. Design a lead network to meet the specifications. As before, we require Kv = 2. P10.15 A robot with an extended arm has a heavy load, whose effect is a disturbance, as shown in Figure P10.15 [22]. Let R(s) = 0 and design Gc(s) so that the effect of the disturbance is less than 20% of the openloop system effect. P10.16 A driver and car may be represented by the simplified model shown in Figure P10.16 [17].The goal is to have the speed adjust to a step input with less than 10% overshoot and a settling time (with a 2% criterion) of 1 second. Select a proportional plus integral (PI) controller to yield these specifications. For the selected controller, determine the actual response (a) for Gp(s) = 1 and (b) with a prefilter Gp(s) that removes the zero from the closed-loop transfer function T(s). P10.17 A unity feedback control system for a robot submarine has a plant with a third-order transfer function [20]: K s(s + 10)(5 + 50) G(s) We want the overshoot to be approximately 7.5% for a step input and the settling time (with a 2% criterion) of the system be 400 ms. Find a suitable phase-lead Vis) • Load wave form compensator by using root locus methods. Let the zero of the compensator be located at s = - 1 5 , and determine the compensator pole. Determine the resulting system Kv. P10.18 NASA is developing remote manipulators that can be used to extend the hand and the power of humankind through space by means of radio. A concept of a remote manipulator is shown in Figure P10.18(a) [11, 22]. The closed-loop control is shown schematically in Figure P10.18(b). Assuming an average distance of 238,855 miles from Earth to the Moon, the time delay T in transmission of a communication signal is 1.28 seconds. The operator uses a control stick to control remotely the manipulator placed on the Moon to assist in geological experiments, and the TV display to access the response of the manipulator. The time constant of the manipulator is ^ second. (a) Set the gain Kx so that the system has a phase margin of approximately 30°. Evaluate the percentage steady-state error for this system for a step input. (b) To reduce the steady-state error for a position command input to 5%, add a lag compensation network in cascade with K\. Plot the step response. P10.19 Tliere have been significant developments in the application of robotics technology to nuclear power plant maintenance problems. Thus far, robotics technology in the nuclear industry has been used primarily on spent-fuel reprocessing and waste management. Today, the industry is beginning to apply the technology to such areas as primary containment inspection, reactor maintenance, facility decontamination, ' / > ) = I/.V Disturbance Hit) FIGURE P10.15 Robot control. Hix) FIGURE P10.16 Speed control of an automobile. \c _ i > Gc(s) l Y(x) Speed 818 Chapter 10 The Design of Feedback Control Systems TV display TV camera Control to signaltransmitting antenna Receiving antenna Remote lanipulator Control stick Moon's surface FIGURE P10.18 (a) Conceptual diagram of a remote manipulator on the Moon controlled by a person on the Earth, (b) Feedback diagram of the remote manipulator control system with T = transmission time delay of the video signal. (a) Man's desired action Xn V *i Position of manipulator TS + 1 Video return signal (b) Manipulator/arm Remotely controlled robot for nuclear plants. I e-sT e and accident recovery activities. These developments suggest that the application of remotely operated devices can significantly reduce radiation exposure to personnel and improve maintenanceprogram performance. Currently, an operational robotic system is under development to address particular operational problems within a nuclear power plant. This device, IRIS (Industrial Remote Inspection System), is a generalpurpose surveillance system that conducts particular inspection and handling tasks with the goal of significantly reducing personnel exposure to high radiation FIGURE P10.19 Remote manipulator Transmitted signal fields [12]. The device is shown in Figure P10.19. The open-loop transfer function is Ke~sT GU) = (, + 1)(, + 3)' (a) Determine a suitable gain K for the system when T = 0.5 s, so that the overshoot to a step input is less than 30%. Determine the steady-state error, (b) Design a compensator s +o Surveillance camera Communication 819 Problems to improve the step response for the system in part (a) so that the steady-state error is less than 12%. Assume the closed-loop system of Figure 10.1(a). P10.23 A system of the form of Figure 10.1 (a) with unity feedback has G(s) = P10.20 An uncompensated control system with unity feedback has a plant transfer function We desire the steady-state error to a step input to be approximately 5% and the phase margin of the system to be approximately 45°. Design a lag network to meet these specifications. K s(s/2 + 1)(.9/6 + 1) G(s) We want to have a velocity error constant of Kv - 20. We also want to have a phase margin of approximately 45° and a closed-loop bandwidth greater than a) = 4 rad/s. Use two identical cascaded phase-lead networks to compensate the system. P10.21 For the system of Problem P10.20, design a phaselag network to yield the desired specifications, with the exception that a bandwidth equal to or greater than 2 rad/s will be acceptable. P10.22 For the system of Problem P10.20, we wish to achieve the same phase margin and Kv, but in addition, we wish to limit the bandwidth to less than 10 rad/s but greater than 2 rad/s. Use a lead-lag compensation network to compensate the system. The leadlag network could be of the form Gc(s) = (1 + s/l0a)(\ + s/b) (1 + s/a){\ + s/lQb)' where a is to be selected for the lag portion of the compensator, and b is to be selected for the lead portion of the compensator. The ratio a is chosen to be 10 for both the lead and lag portions. Desired position Rls) Reference P10.24 The stability and performance of the rotation of a robot (similar to waist rotation) presents a challenging control problem. The system requires high gains in order to achieve high resolution; yet a large overshoot of the transient response cannot be tolerated. The block diagram of an electrohydraulic system for rotation control is shown in Figure P10.24 [15]. The armrotating dynamics are represented by G(s) = Controller Arm Gc(s) G(s) _ i [X K(s + 20) s I$

K,

FIGURE P10.25 Airgap control of train.

80 s(s2/4900 + s/70 + 1)

We want to have K„ = 20 for the compensated system. Design a compensator that results in an overshoot to a step input of less than 10%. P10.25 The possibility of overcoming wheel friction, wear, and vibration by contactless suspension for passenger-carrying mass-transit vehicles is being investigated throughout the world. One design uses a magnetic suspension with an attraction force between the vehicle and the guideway with an accurately controlled airgap. A system is shown in Figure P10.25, which incorporates feedback compensation.

L

FIGURE P10.24 Robot position control.

K (s + 5)2

K2(s + b) s + 200

Actual position

l s

Y(s) v AtLual

air gap

820

Chapter 10 The Design of Feedback Control Systems

Using root locus methods, select a suitable value for K\ and h so the system has a damping ratio for the underdamped roots of £ = 0.50. Assume, if appropriate, that the pole of the air gap feedback loop (s = -200) can be neglected.

We desire to achieve a steady-state error for a ramp input of 10% and a damping ratio of the dominant roots of 0.707. Determine a suitable lag compensator, and determine the actual overshoot and the time to settle (to within 2% of the final value).

P10.26 A computer uses a printer as a fast output device. We desire to maintain accurate position control while moving the paper rapidly through the printer. Consider a system with unity feedback and a transfer function for the motor and amplifier of

P10.29 A liquid-level control system (see Figure 9.32) has a loop transfer function

0.15 s(s + l)(5s + 1)"

G(s) =

P10.27 An engineering design team is attempting to control a process shown in Figure PI0.27. The system has a controller Gc(s), but the design team is unable to select Gc(s) appropriately. It is agreed that a system with a phase margin of 50° is acceptable, but Gc(s) is unknown. Determine Gc(s). First, let Gc(s) = K and find (a) a value of K that yields a phase margin of 50° and the system's step response for this value of K. (b) Determine the settling time, percent overshoot, and the peak time, (c) Obtain the system's closed-loop frequency response, and determine Mpa> and the bandwidth. The team has decided to let K(s + 12) (s + 20)

and to repeat parts (a), (b), and (c). Determine the gain K that results in a phase margin of 50° and then proceed to evaluate the time response and the closedloop frequency response. Prepare a table contrasting the results of the two selected controllers for Gc(s) by comparing settling time (with a 2% criterion), percent overshoot, peak time, Mpo), and bandwidth. P10.28 An adaptive suspension vehicle uses a legged locomotion principle. The control of the leg can be represented by a unity feedback system with [12] K s(s + 10)(5 + 14)

G(s)

Controller A'f

FIGURE P10.27 Controller design.

.,—tr>

Gc(s)

Gc(s)G(s)H(s),

where H(s) = 1, Gc(s) is a compensator, and the plant is G(s)

Design a lead network compensator so that the system bandwidth is 0.75 rad/s and the phase margin is 30°. Use a lead network with a = 10.

Ge(s)

L(s) =

10e-*r s (s + 10) 2

where T - 50 ms. Design a compensator so that Mpu does not exceed 3.5 dB and
10 s(s + 10)

••

Yu

821

Problems

R(s)

•O

Deviation from guide path

Vehicle

Controller Gc(s)

s(TtS+

l)(T2S+

- • n.v)

I)

FIGURE P10.30 Steering control for vehicle.

P10.31 For the system of Problem PI0.30, use a phase-lag compensator and attempt to achieve a phase margin of approximately 50°. Determine the actual overshoot and peak time for the compensated system. P10.32 When a motor drives a flexible structure, the structure's natural frequencies, as compared to the bandwidth of the servodrive, determine the contribution of the structural flexibility to the errors of the resulting motion. In current industrial robots, the drives are often relatively slow, and the structures are relatively rigid, so that overshoots and other errors are caused mainly by the servodrive. However, depending on the accuracy required, the structural deflections of the driven members can become significant. Structural flexibility must be considered the major source of motion errors in space structures and manipulators. Because of weight restrictions in space, large arm lengths result in flexible structures. Furthermore, future industrial robots should require lighter and more flexible manipulators.

To investigate the effects of structural flexibility and how different control schemes can reduce unwanted oscillations, an experimental apparatus was constructed consisting of a DC motor driving a slender aluminum beam. The purpose of the experiments was to identify simple and effective control strategies to deal with the motion errors that occur when a servomotor is driving a very flexible structure [13]. The experimental apparatus is shown in Figure PI0.32(a), and the control system is shown in Figure P10.32(b).The goal is that the system will have a Kvof 100. (a) When Gc.(s) = K, determine K and plot the Bode diagram. Find the phase margin and gain margin. (b) Using the Nichols chart, find a)r, Mp„„ and o)B. (c) Select a compensator so that the phase margin is greater than 35° and find and toB for the compensated system. P10.33. Consider the extender robot presented in AP6.7. The block diagram of the system is shown in Figure P10.33 [14]. The goal is that the compensated system

Potentiometer Strain gauge

Motor frame

Accelerometer

Motor

(a)

Rig)

FIGURE P10.32 Flexible arm control.

* Q

Gt.(s)

s + 500 s(s + 0.0325)(52 + 2.57.-,- + 6667)

(b)

1 — • Y(s)

822

Chapter 10 The Design of Feedback Control Systems

FIGURE P10.33

His) Human input

Actuator

o

Gc(s)

8 s{2s+ l)(0.05s+ l)

Y(s) Output

Extender robot control.

will have a velocity constant K „ equal to 80, so that the settling time (with a 2 % criterion) will be 1.6 seconds, and that the overshoot will be 16%, so that the dominant roots have a f of 0.5. Determine a lead-lag compensator using root locus methods. P10.34 A magnetically levitated train is operating in Berlin, Germany. The M-Bahn 1600-m line represents the current state of worldwide systems. Fully automated trains can run at short intervals and operate with excellent energy efficiency. The control system for the levitation of the car is shown in Figure PI 0.34. Select a compensator so that the phase margin of the system is 45° < P.M. :< 55°. Predict the response of the system to a step command, and determine the actual step response for comparison. P10.35 A unity feedback system has the loop transfer function L(s) =

Gc(s)G(s)

Ks + 0.54 s(s + 1.76)

where T is a time delay and K is the controller proportional gain. The block diagram is illustrated in Figure P10.35. The nominal value of K =2. Plot the phase margin of the system for 0 < T s 2 s when K = 2.

Compensator

What happens to the phase margin as the time delay increases? What is the maximum time delay allowed before the system becomes unstable? P10.36 A system's open-loop transfer function is a pure time delay of 0.5 s, so that G(s) = e~s^2. Select a compensator Gc(s) so that the steady-state error for a step input is less than 2 % of the magnitude of the step and the phase margin is greater than 30°. Determine the bandwidth of the compensated system and plot the step response. P10.37 A unity feedback system of the form shown in Figure 10.1(a) has G(s)

1 (s + 2)(s + 8)

Design a compensator Gc(s) so that the overshoot for a step input R(s) is less than 5% and the steady-state error is less than 1 %. Determine the bandwidth of the system. P10.38

A unity feedback system has a plant G(s) =

40 s(s + 2)

Process

mi FIGURE P10.34 Magnetically levitated train control.

Input

*\)~

Gc(s)

command

Controller

FIGURE P10.35 Unity feedback system with a time delay and PI controller.

K+

0.54

x2(.v + 10)

Time Delay

• • Levitation distance

Process 1 s + 1.76

•+YU

823

We desire to have a phase margin of 30° and a relatively large bandwidth. Select the crossover frequency ct>c = 10 rad/s, and design a lead compensator. Verify the results. P10.39 A unity feedback system has a plant G(s)

40 s(s + 2)'

We desire that the phase margin be equal to 30°. For a ramp input r(t) ~ t, we want the steady-state error to be equal to 0.05. Design a lag compensator to satisfy the requirements. Verify the results. P10.40 For the system and requirements of Problem P10.39, determine the required compensator when the steady-state error for the ramp input must be equal to 0.02. P10.41 Repeat Example 10.12 when we want the 100% rise time Tr=l second.

compensator determined in Equation (10.46), select an appropriate prefilter. Compare the response of the system with and without the prefilter. P10.43 Consider the system shown in Figure P10.43 and let R(s) = 0 and T(i(s) = 0. Design the controller Gc(s) = Ksuch that, in the steady-state, the response of the system y(t) is less than - 4 0 dB when the noise N(s) is a sinusoidal input at a frequency of cd > 100 rad/s. P10.44 A unity feedback system has a loop transfer function L(s) = Gc(s)G(s)

K(s2 + 2s + 20) s(s + 2 ) 0 2 + 2s + 1)'

Plot the percent overshoot of the closed-loop system response to a unit step input for K in the range 0 < K < 100. Explain the behavior of the system response for K in the range 0.129 < K s 69.872.

P10.42 Consider again the design for Example 10.4. Using a system as shown in Figure 10.22 and the

7",/(.v)

FIGURE P10.43 Unity feedback system with proportional controller and measurement noise.

• Yts) N(.\)

ADVANCED PROBLEMS APlO.l A three-axis pick-and-place application requires the precise movement of a robotic arm in threedimensional space, as shown in Figure APlO.l for joint 2. The arm has specific linear paths it must follow to avoid other pieces of machinery. The overshoot for a step input should be less than 13%. (a) Let Gc(s) = K, and determine the gain K that satisfies the requirement. Determine the resulting settling time (with a 2% criterion), (b) Use a lead network and reduce the settling time to less than 3 seconds.

AP10.2 The system of Advanced Problem APlO.l is to have a percent overshoot less than 13%. In addition, we desire that the steady-state error for a unit ramp input will be less than 0.125 (Kv = 8) [24]. Design a lag network to meet the specifications. Check the resulting percent overshoot and settling time (with a 2% criterion) for the design. AP10.3 The system of Advanced Problem AP 10.1 is required to have a percent overshoot less than 13% with a steadystate error for a unit ramp input less than 0.125(Kv = 8).

824

Chapter 10 The Design of Feedback Control Systems Motor 2

Motor 1

Motor 4

(a)

_vV

l S(S+ 1)(5 + 4)

Gc(s)

Risi —

- • Vis)

— i i

FIGURE AP10.1 Pick-and-place robot.

(b)

Design a proportional plus integral (PI) controller to meet the specifications. AP10.4 A DC motor control system with unity feedback has the form shown in Figure AP10.4. Select K1 and K2 so that the system response has a settling time (with a 2% criterion) less than 1 second and an overshoot less than 5% for a step input. AP10.5 A unity feedback system is shown in Figure AP10.5. We want the step response of the system to have an overshoot of about 10% and a settling time (with a 2% criterion) of about 4 seconds.

R[s)

\r •v.

k •>

fc

(a) Design a lead compensator Gc(s) to achieve the dominant roots desired, (b) Determine the step response of the system when Gp(s) = 1. (c) Select a prefilter Gp(s), and determine the step response of the system with the prefilter. AP10.6 Consider again Example 10.12 when we wish to minimize the settling time of the system while requiring that K < 52. Determine the appropriate compensator that will minimize the settling time. Plot the system response.

8 s(s + 8)

Ki

\

\

FIGURE AP10.4 Motor control system.

Velocity feedback

K 2s

Y(s) Position

825

GJs)

Rls) FIGURE AP10.5 Unity feedback with a prefilter.

o-+

AP10.7 A system has the form shown in Figure 10.22, with G(5) =

10 s(s+ 1)(5-+ 10)

G,(.v)

of the system is shown in Figure AP10.8(b).The plant dynamics are represented by

I

G(s)

s(s + 2)(s + 8)

A lead compensator is used, with Gc(s) =

K(s + 3) s + 28

Determine K so that the complex roots have £ = 1/V2. The prefilter is GP(s) =

(5)

Arm dynamics Gt.(.v)

G(s)

i

FIGURE AP10.8 (a) Manutec robot. (b) Block diagram.

(b)

100 s(s + 5)(s + 10)'

We desire that the system have a small steady-state error for a ramp input so that Kv = 100. For stability purposes, we desire a gain margin of 10 dB or greater and a phase margin of 40° or greater. Determine a lead-lag compensator that meets these specifications. Assume the system is of the form shown in Figure 10.1(a) with H(s) = 1.

(a)

R(s)

250 s(s + 2)(s + 40)(5 + 45)'

The percentage overshoot for a step input should be less than 20% with a rise time less than \ second and a settling time (with a 2% criterion) less than 1.2 seconds. Also, we desire that for a ramp input K „ ^ 10. Determine a suitable lead compensator. AP10.9 Tire plant dynamics of a chemical process are represented by

s + p

(a) Determine the overshoot and rise time for Gp(s) = 1 and for p = 3. (b) Select an appropriate value for p that will give an overshoot of 1 % and compare the results. AP10.8 The Manutec robot has large inertia and arm length resulting in a challenging control problem, as shown in Figure AP10.8(a). The block diagram model

•*• Y(s)

826

Chapter 10 The Design of Feedback Control Systems

DESIGN PROBLEMS CDP10.1 The capstan-slide system of Figure CDP4.1 f _\> uses a PD controller. Determine the necessary values °f tne g a m constants of the PD controller so that the \S^ deadbeat response is achieved. Also, we want the settling time (with a 2% criterion) to be less than 250 ms. Verify the results. DP10.1 In Figure DP10.1, two robots are shown cooperating with each other to manipulate a long shaft to insert it into the hole in the block resting on the table. Long part insertion is a good example of a task that can benefit from cooperative control. The unity feedback control system of one robot joint has the process transfer function

The specifications require a steady-state error for a unit ramp input of 0.02, and the step response has an overshoot of less than 15% with a settling time (with a 2% criterion) of less than 1 second. Determine a leadlag compensator that will meet the specifications, and plot the compensated responses for the ramp and step inputs.

FIGURE DP10.1

Two robots cooperate to insert a shaft.

DP10.2 The heading control of the traditional bi-wing aircraft, shown in Figure DP10.2(a), is represented by the block diagram of Figure DP 10.2(b). (a) Determine the minimum value of the gain K when Gc{s) = K, so that the steady-state effect of a unit step disturbance Td(s) = 1/sis less than or equal to 5% of the unit step (y(oo) = 0.05).

(b) Determine whether the system using the gain of part (a) is stable. (c) Design a compensator using one stage of lead compensation, so that the phase margin is 30°. (d) Design a two-stage lead compensator so that the phase margin is 55°. (e) Compare the bandwidth of the systems of parts (c) and (d). (f) Plot the step response y(t) for the systems of parts (c) and (d) and compare percent overshoot settling time (with a 2% criterion), and peak time. DP10.3 NASA has identified the need for large deployable space structures, which will be constructed of lightweight materials and will contain large numbers of joints or structural connections. This need is evident for programs such as the space station. These deployable space structures may have precision shape requirements and a need for vibration suppression during in-orbit operations [16]. One such structure is the mast flight system, which is shown in Figure DP10.3(a). The intent of the system is to provide an experimental test bed for controls and dynamics.The basic element in the mast flight system is a 60.7-m-long truss beam structure, which is attached to the shuttle orbiter. Included at the tip of the truss structure are the primary actuators and collocated sensors. A deployment/retraction subsystem, which also secures the stowed beam package during launch and landing, is provided. The system uses a large motor to move the structure and has the block diagram shown in Figure DPI0.3(b).The goal is an overshoot to a step response of less than or equal to 16%; thus, we estimate the system I as 0.5 and the required phase margin as 50°. Design for 0.1 < K < 1 and record overshoot, rise time, and phase margin for selected gains. DP10.4 A high-speed train is under development in Texas [21] with a design based on the French Train a Grande Vitesse (TGV). Train speeds of 186 miles per hour are foreseen. To achieve these speeds on tight curves, the train may use independent axles combined with the ability to tilt the train. Hydraulic cylinders connecting the passenger compartments to their wheeled bogies allow the train to lean into curves like a motorcycle. A pendulum like device on the leading bogie of each car senses when it is entering a curve and feeds this information to the hydraulic system. Tilting does not make the train safer, but it does make passengers more comfortable. Consider the tilt control shown in Figure DP10.4. Design a compensator Gc(s) for a stepinput command so that the overshoot is less than 5% and the settling time (with a 2% criterion) less than

827

Design Problems

(a)

Engine

Controller FIGURE DP10.2 (a) Bi-wing aircraft. (Source: The illustrated London News, October 9, 1920.) (b) Control system.

Gc(s)

K(.v)

— •

Wind disturbance TM)

100 (.v + 10)

Aircraft dynamics 40 s(s + 20)

(b)

Mast

Deployer/retractor

Shuttle

(a)

Ms)

FIGURE DP10.3 Mast flight system.

• Q

s(s+ 1.5)(5 + 3.9)

(b)

- • Yis)

- • K(.v)

828

Chapter 10 The Design of Feedback Control Systems

Desired

FIGURE DP10.4

till

O

12

Gc(s)

}'(.v) Tilt

s(s + 10)(i- + 70)

High-speed train feedback control system.

0.6 second. We also desire that the steady-state error for a velocity (ramp) input be less than 0.15A, where r(t) = At.t > 0. Verify the results for the design. DP10.5 High-performance tape transport systems are designed with a small capstan to pull the tape past the read/write heads and with take-up reels turned by DC motors. The tape is to be controlled at speeds up to 200 inches per second, with start-up as fast as possible, while preventing permanent distortion of the tape. Since we wish to control the speed and the tension of the tape, we will use a DC tachometer for the speed sensor and a potentiometer for the position sensor. We will use a DC motor for the actuator. Then the linear model for the system is a unity feedback system with Y(s) E(s)

= G{s) =

K(s + 4000) s(s + 1000)(5 + 3000)(.v + px)(s + pi) where p , = +2000 + /2000, and Y(s) is position. The specifications for the system are (1) settling time of less than 12 ms, (2) an overshoot to a step position command of less than 10%, and (3) a steadystate velocity error of less than .5%. Determine a compensator scheme to achieve these stringent specifications. DP10.6 The past several years have witnessed a significant engine model-building activity in the automotive industry in a category referred to as "control-oriented" or "control design" models. These models contain representations of the throttle body, engine pumping phenomena, induction process dynamics, fuel system, engine torque generation, and rotating inertia. The control of the fuel-to-air ratio in an automobile carburetor became of prime importance in

the 1980s as automakers worked to reduce exhaustpollution emissions. Thus, auto engine designers turned to the feedback control of the fuel-to-air ratio. Operation of an engine at or near a particular air-to-fuel ratio requires management of both air and fuel flow into the manifold system. The fuel command is considered the input and the engine speed is considered the output [9,10]. The block diagram of the system is shown in Figure DP10.6, where T - 0.066 second. A compensator is required to yield zero steady-state error for a step input and an overshoot of less than 10%. We also desire that the settling time (with a 2% criterion) not exceed 10 seconds. DP10.7 A high-performance jet airplane is shown in Figure DP10.7(a), and the roll-angle control system is shown in Figure DP10.7(b). Design a controller Gc(s) so that the step response is well behaved and the steady-state error is zero. DP10.8 A simple closed-loop control system has been proposed to demonstrate proportional-integral (PI) control of a windmill radiometer [27]. The windmill radiometer is shown in Figure DP10.8(a) and the control system is shown in Figure DPI0.8(b). The variable to be controlled is the angular velocity ta of the windmill radiometer whose vanes turn when exposed to infrared radiation. An experimental setup using a reflexive photoelectric sensor and basic electronic circuitry makes possible the design and implementation of a high performance control system. The transfer function of the light source and radiometer is G(s)

where T = 20 s. Design a PI controller so that the system achieves a deadbeat response with a settling time less than 25 s. TJs)

FIGURE DP10.6 Engine control system.

TS + V

Design Problems

(a)

K(5)

FIGURE DP10.7 Roll-angle control of a jet airplane.

(b)

Gc(s)

G(s)

*- V'(.v)

FIGURE DP10.8 (a) Radiometric windmill, (b) Control

system.

(b)

(a)

DP10.9 The feedback control system shown in Figure DPI 0.9 has the transfer function 60 0(1) = 2 (s + 4s + 6)(s + 10)' Design a PID compensator Gcl(s) and a lead-lag compensator GC2(s) such that, in each case, the closed-loop system is stable in the presence of a time-delay T = 0.1 s. Discuss the capability of each compensator to insure stability in the presence of an increase in the time-delay uncertainty of up to 0.2 second.

+

fils)

r~\

Controller

Plant

Gc(s)

G{s)

- ii Time delay e-*T FIGURE DP10.9 time-delay.

Feedback control system with a

830

Chapter 10 The Design of Feedback Control Systems

DPIO.IO A unity feedback system has the process transfer function G(*>-

s + 1.59 s(s + 3.7)(.v2 + 2.4s + 0.43)

Design the controller Gc(s) such that the Bode magnitude plot of the loop transfer function L(s) = greater than 20 dB for « < 0.01 rad/s Gc(s)G(s)is and less than - 2 0 dB for w < 10 r a d / s . The desired shape of the loop transfer function Bode plot magnitude is illustrated in Figure DPIO.IO. Explain why we would want the gain to be high at low-frequency and the gain to be low at high-frequency.

D P l O . l l Modern microanalytical systems used for polymerase chain reaction (PCR) requires fast, damped tracking response. The control of the temperature of the PCR reactor can be represented as shown in Figure D P l O . l l . The controller is chosen to be PID controller, denoted by Gc(s), with a prefilter, denoted by Gp(s). The transfer function is [30] G(s) =

45 (5 + 2.9)(5 + 0.14)'

It is required that the percent overshoot P.O. < 1 % and the settling time Ts < 3 seconds to a unit step input. Design a controller Gc(s) and prefilter Gp(s) to achieve the control specifications.

High frequency loop gain requirement

FIGURE DP10.10 Bode plot loop shaping

10 _ ' 10° Frequency (rad/s)

requirements.

Controller, GJs)

Prefilter Desired temperature

Gp(s)

Ea(s)

KP + K,js + * i

JftJTl

FIGURE DP10.11 Polymerase chain reaction control system.

Reactor, G(s) 45

(s+ 2.9)(s + 0.14)

Actual - • temperature >'(.v)

831

Computer Problems COMPUTER PROBLEMS CPlO.l Consider the control system in Figure CPlO.l, where

requirement has been satisfied and that the steadystate error is zero. CP10.4 A fighter aircraft has the transfer function 6 _ -I0(s + 1)(5 + 0.01) 8 ~ (s2 + 2s + 2)(5 2 + 0.025 + 0.0101)'

Develop an m-file to show that the phase margin is approximately 50° and that the percent overshoot to a unit step input is 18%.

Gc(s)

fl(.v)

G{s)

* Y(s)

FIGURE CP10.1 A feedback control system with compensation.

CP10.2 A negative feedback control system is shown in Figure CP10.2 Design the proportional controller Gc(s) = K so that the system has a 40° phase margin. Develop an m-file to obtain a Bode plot and verify that the design specification is satisfied. CP10.3 Consider the system in Figure CPlO.l, where G{s) =

where 6 is the pitch rate (rad/s) and 5 is the elevator deflection (rad). The four poles represent the phugoid and short-period modes. The phugoid mode has a natural frequency of 0.1 rad/s, and the short period mode is 1.4 rad/s. The block diagram is shown in Figure CP10.4. (a) Let the lead compensator be G„ = K

where \z\ < \p\- Using Bode plot methods, design the lead compensator to meet the following specifications: (1) settling time (with a 2% criterion) to a unit step less than 2 seconds, and (2) percent overshoot less than 10%. (b) Simulate the closed-loop system with a step input of 10°/second, and show the time history of $. CP10.5 The pitch attitude motion of a rigid spacecraft is described by JO = «, I s(s + 2)' where J is the principal moment of inertia, and u is the input torque on the vehicle [7]. Consider the PD controller Design a compensator Gc(s) so that the steady-state tracking error to a ramp input is zero and the settling time (with a 2% criterion) is less than 5 seconds. Obtain the response of the closed-loop system to the input R(s) = l/.y2 and verify that the settling time FIGURE CP10.2 Single-loop feedback system with proportional controller. R(s) An aircraft pitch rate feedback control system. Gc{s) = KP + KDs. (a) Obtain a block diagram of the control system. Design a control system to meet the following specifications: Proportional controller Process K 24.2 s2+ 85 + 24.2 l Lead compensator FIGURE CP10.4 s +z s + p" Gt.(s) • Rs) Aircraft Actuator 10(5+ 1)(5 + 0.01) 10 5+ 10 2 (5 + 25 + 2)(52 + 0.025 + 0.0101) •+() 832 Chapter 10 The Design of Feedback Control Systems (1) closed-loop system bandwidth about 10 rad/s, and (2) percent overshoot less than 20% to a 10° step input. Complete the design by developing and using an interactive m-file script, (b) Verify the design by simulating the response to a 10° step input, (c) Include a closed-loop transfer function Bode plot to verify that the bandwidth requirement is satisfied. CP10.6 Consider the control system shown in Figure CP10.6. Design a lag compensator using root locus methods to meet the following specifications: (1) steadystate error less than 10% for a step input, (2) phase margin greater than 45°, and (3) settling time (with a 2% criterion) less than 5 seconds for a unit step input. (a) Design a lag compensator utilizing root locus methods to meet the design specifications. Develop a set of m-file scripts to assist in the design process. (b) Test the controller developed in part (a) by simulating the closed-loop system response to unit step input. Provide the time histories of the output y(t). (c) Compute the phase margin using the margin function. CP10.7 A lateral beam guidance system has an inner loop as shown in Figure CP10.7, where the transfer function for the coordinated aircraft is [26] G(s) = R(s FIGURE CP10.6 A unity feedback control system. Consider the PI controller Gc{s) = KP + —. 5 (a) Design a control system to meet the following specifications: (1) settling time (with a 2% criterion) to a unit step input of less than 1 second, and (2) steady-state tracking error for a unit ramp input of less than 0.1. (b) Verify the design by simulation. CP10.8 Consider again the system and the lead compensator designed in Example 10.3. The actual overshoot of the compensated system will be 46%. We want to reduce the overshoot to 32%. Using a m-file script, determine an appropriate value for the zero of Gc(s). CP10.9 Plot the frequency response of the circuit of AP10.10. CP10.10 The feedback control system shown in Figure CP10.10 has the transfer function G(s) = s + 23 -\ fc * Lag compensator Process G,(s) 5+10 s2 + 2s + 20 ** FIGURE CP10.7 A lateral beam guidance system inner loop. FIGURE CP10.10 Feedback control system with a time delay. R\s) 23 Q rah,- ~\ fc Rv) Coordinated aircraft Pi compensator Desired 7 + 6s2 The time delay is T = 0.2 s.Plot the phase margin for the system versus the gain in the range 0.1 s K <• 10. Determine the gain K that maximizes the phase margin. 23 —K) J K{s + 0.2) KB +$ + 23

Time delay

Process

e-*T

G(s)

• Yts)

4> • • Actual rate

833

Terms and Concepts

m

ANSWERS TO SKILLS CHECK True or False: (1) False; (2) False; (3) True; (4) True; (5) True Multiple Choice: (6) d; (7) b; (8) d; (9) a; (10) b; (11) c; (12) a; (13) a; (14) b; (15) b

Word Match (in order, top to bottom): a, 1, g, d, j , h, k, c, m, e, i, f, b

TERMS AND CONCEPTS Cascade compensation network A compensator network placed in cascade or series with the system process. Compensation The alteration or adjustment of a control system in order to provide a suitable performance. Compensator An additional component or circuit that is inserted into the system to compensate for a performance deficiency. Deadbeat response A system with a rapid response, minimal overshoot, and zero steady-state error for a step input. Design of a control system The arrangement or the plan of the system structure and the selection of suitable components and parameters. Integration network A network that acts, in part, like an integrator. Lag network See Phase-lag network. Lead-lag network A network with the characteristics of both a lead network and a lag network. Lead network See Phase-lead network. Phase lag compensation A widely-used compensator that possesses one zero and one pole with the pole

closer to the origin of the j-plane. This compensator reduces the steady-state tracking errors. Phase lead compensation A widely-used compensator that possesses one zero and one pole with the zero closer to the origin of the i-plane. This compensator increases the system bandwidth and improves the dynamic response. Phase-lag network A network that provides a negative phase angle and a significant attenuation over the frequency range of interest. Phase-lead network A network that provides a positive phase angle over the frequency range of interest. Thus, phase lead can be used to cause a system to have an adequate phase margin. PD controller Controller with a proportional term and a derivative term (Proportional-Derivation). PI controller Controller with a proportional term and an integral term (Proportional-Integral). Prefilter A transfer function Gp{s) that filters the input signal R(s) prior to calculating the error signal.

890

Chapter 11 The Design of State Variable Feedback Systems

Therefore, we require that 5Ka = 19290 or Ka = 3858. Furthermore, we require that 20 + 5K2Ka = 250, or K2 = 0.012. The system with the second-order model has the desired response and meets all the specifications, as shown in Table 11.2. If we add the field inductance L = 1 mH, we have a third-order model with Gi(s) =

5000

s + 1000" Using this model, which incorporates the field inductance, we test the response of the system with the feedback gains selected for the second-order model.The results are provided in Table 11.2, illustrating that the second-order model is a very good model of the system. The actual results of the third-order system meet the specifications. 11.12 SUMMARY

In this chapter, the design of control systems in the time domain was examined. The three-step design procedure for constructing state variable compensators was presented. The optimal design of a system using state variable feedback and an integral performance index was considered. Also, the s-plane design of systems utilizing state variable feedback was examined. Finally, internal model design was discussed.

El

SKILLS CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 11.45 as specified in the various problem statements.

rit)

ir^u

x(r) \
K FIGURE 11.45 Block diagram for the Skills Check. In the following True or False and Multiple Choice problems, circle the correct answer. A system is said to be controllable on the interval [r0, tf] if there exists a continuous input «(/) such that any initial state x(t0) can be transformed to any arbitrary state x{tf) in a finite interval tf - r0 > 0. The poles of a system can be arbitrarily assigned through full-state feedback if and only if the system is completely controllable and observable.

True or False

True or False

891

Skills Check 3. The problem of designing a compensator that provides asymptotic tracking of a reference input with zero steady-state error is called state-variable feedback.

True or False

4. Optimal control systems are systems whose parameters are adjusted so that the performance index reaches an extremum value.

True or False

5. Ackerman's formula is used to check observability of a system.

True or False

6. Consider the system 0 1 "0 x + u .0 - 4 . 2. y - [0 2]x

x =

The system is: a. Controllable, observable b. Not controllable, not observable c Controllable, not observable d. Not controllable, observable 7. Consider the system 10 *2(* + 2)(s2 + 2s + 5)' This system is: a. Controllable, observable b. Not controllable, not observable c. Controllable, not observable d. Not controllable, observable 8. A system has the state variable representation 1 0 0 y=[l

2

0 -3 0

0 0 x + -5

1 1 1

-l]x

Determine the associated transfer function model G(s) = 552 + 32* + 35 s + 9s2 + 23* + 15 a. G(s) = 5s2 + 32s + 35 s4 + 9s* + 23* + 15 b. G(s) = 2s2 + 16* + 22 s" + 9s2 + 23* + 15 c. G(s) = 5s + 32

Y(s) U(s)'

„, ,

3

d. G{s) = . *2 + 325 + 9 1 5~ 11.45, where -10in Figure Consider the closed-loop 12 system 0 B 1 0 I = 0 C = [3 0 0 1 0

5

-5].

892

Chapter 11 The Design of State Variable Feedback Systems Determine the state-variable feedback control gain matrix K so that the closed-loop system poles are 5 = - 3 , - 4 , and - 6 . a. K = [1 44 67] b. K = [10 44 67] c. K - [44 1 1] d. K - [1 67 44] 10. Consider the system depicted in the block diagram in Figure 11.46.

Ris)

Y(s)

FIGURE 11.46 Two-loop feedback control system. This system is: a. Controllable, observable b. Not controllable, not observable c Controllable, not observable d. Not controllable, observable 11. A system has the transfer function s +a T(s) = —:— . 4 s + 6 ? + 12s2 + 12s + 6 Determine the values of a that render the system unobservable. a. a = 1.30 or a = -1.43 b. a = 3.30 or a = 1.43 c. a = -3.30 ore = -1.43 d. a = -5.7 or a = -2.04 12. Consider the closed-loop system in Figure 11.45, where - 7 -101 1 B = C = [0 1]. A = 1 0 o Determine the state variable feedback control gain matrix K for a zero steady-state tracking error to a step input. a. K = [3 -9] b. K = [3 -6] c. K = [-3 2] d. K = [-1 4] 13. Consider the system where A =

1" - 3 01 B = 1 0.

LoJ

It is desired to place the observer poles at S]2 state-variable feedback control gain matrix L.

-

[0

1].

3 ± /3. Determine the appropriate

893

Skills Check -9" 3. "9' b. L = .3. "3" c L = 9 d. None of the above 14. A feedback system has a state-space representation "-75 0 V X+ u 1 oj y = [0 3600]x, where the feedback is u(t) - - K x + r(t). The control system design specifications are: (i) the overshoot to a step input approximately P.O. « 6 % , and (ii) the settling time Ts « 0.1 second. A state variable feedback gain matrix which satisfies the specifications is: a. K = [10 200] b. K = [6 3600] c. K = [3600 10] d. K = [100 40] 15. Consider the system 1 Y(s) = G(s)U(s) = U(s) a. L =

LJ

Determine the eigenvalues of the closed-loop system when utilizing state variable feedback, where u(t) = —2*2—2xj + r(?).We define xt = y(t) and r(t) is a reference input. a. sx = -I + /1 s2 = - 1 - / 1 b. s\ = -2 + /2 s2 = - 2 - / 2 c. Sj = — 1 + /2 s2 = -1—/2 d. si ==-1 ^2 = —1 In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Stabilizing controller b. Controllability matrix c. Stabilizable

d. Command following e. State variable feedback f. Full-state feedback control law g. Observer

Occurs when the control signal for the process is a direct function of all the state variables. A system in which any initial state t0) is uniquely determined by observing the output y(t) on the interval [tOJ t/\. A system in which there exists a continuous input u(t) such that any initial state x(/„) can be driven to any arbitrary trial state x(tf) in a finite time interval tf - t0> 0. A system whose parameters are adjusted so that the performance index reaches an extremum value. An important aspect of control system design wherein a nonzero reference input is tracked. A linear system is (completely) controllable if and only if this matrix has full rank. A system in which the states that are unobservable are naturally stable. 894 Chapter 11 The Design of State Variable Feedback Systems h. Linear quadratic regulator i. Optimal control system j . Detectable k. Controllable system I. Pole placement m. Estimation error n. Kalman state-space decomposition o. Observable system p. Separation principle q. Observability matrix The difference between the actual state and the estimated state. A control law of the form u(t) - ~Kx(t) where x(t) is the state of the system assumed known at all times. A partition of the state space that illuminates the states that are controllable and unobservable, uncontrollable and unobservable, controllable and observable, and uncontrollable and observable. An optimal controller designed to minimize a quadratic performance index. A linear system is (completely) observable if and only if this matrix has full rank. A dynamic system used to estimate the state of another dynamic system given knowledge of the system inputs and measurements of the system outputs. A design methodology wherein the objective is to place the eigenvalues of the closed-loop system in desired regions of the complex plane. The principle that states that the full-state feedback law and the observer can be designed independently and when connected will function as an integrated control system in the desired manner (that is, stable). A system in which the states that are not controllable are naturally stable. A controller that stabilizes the closed-loop system. EXERCISES E l l . l The ability to balance actively is a key ingredient in the mobility of a device that hops and runs on one springy leg, as shown in Figure E l l . l [8]. The control of the attitude of the device uses a gyroscope and a feedback such that u = Kx, where -k 0 K 0 -2k and x(f) = Ax(0 + Bu(t) Servovalve Compass where 0 -1 1 0 and B = I. Two-axis gyroscope Hydraulic actuator and position/velocity sensors Determine a value for k so that the response of each hop is critically damped. E11.2 A magnetically suspended steel ball can be described by the linear equation 0 1 9 0 x + 0 1 Foot switch FIGURE E11.1 Single-leg control. Exercises 895 The state variables are x, = position and x2 = velocity, and both are measurable. Select a feedback so that the system is critically damped and the settling time (with a 2% criterion) is 4 seconds. Choose the feedback in the form u = — kiX] — k2x2 + r where r is the reference input and the gains kx and k2 are to be determined. E11.3 A system is described by the matrix equations 0 i l) L -3 J \+ -L •10 0 x + 0 -2 0]x. 0 FIGURE E11.10 State variable block diagram. 0 -3 C = [2 - 2 ] , and -l_ 0]x. x+ "o [_l_l D = [0]. Sketch a block diagram model of the system. E11.8 Consider the third-order system 0 0 -9 1 0 -3 8 0 0 I X+ 10]x + [l]w. Sketch a block diagram model of the system E11.9 Consider the second-order system y = [1 1 1 " "o" - 5 , B = _12_ 1 E11.6 A system is described by the matrix equations U(s) where -1 Determine whether the system is controllable and observable. v = [l x = Ax + Bit y = Cx + DM, l" X+ -2 0]x. -1 E11.7 Consider the system represented in state variable form V = [2 Determine whether the system is controllable and observable. El 1.5 A system is described by the matrix equations y = [l Answer: controllable and observable 0 v = [0 2]x. Determine whether the system is controllable and observable. Answer: controllable, not observable E11.4 A system is described by the matrix equations 0 -1 Determine whether the system is controllable and observable. -f x+ 1J V L*2j 0]x + [0]u. For what values of k\ and k2 is the system completely controllable? Ell.lO Consider the block diagram model in Figure El 1.10. Write the corresponding state variable model in the form x = Ax + BK y = Cx + Dit. • Y(s) 896 Chapter 11 The Design of State Variable Feedback Systems E l l . l l Consider the system shown in block diagram form in Figure E l l . l l . Obtain a state variable representation of the system. Determine if the system is controllable and observable. E11.12 Consider the single-input, single-output system is described by x(f) = Ax(0 + Bu(f) y(0 = Cx(r) where A = 0 1 " -6 -5 J ,B - V ,C = [1 0]. L6J Compute the corresponding transfer function representation of the system. If the initial conditions are zero (i.e., Xy(0) = 0 and x2(0) = 0), determine the response when u(t) is a unit step input for / > 0 , 1/(.v) FIGURE E11.11 State variable block diagram with a feedforward term. PROBLEMS PI 1.1 A first-order system is represented by the timedomain differential equation = / {x\t) + \u\t)) dt. ./n x = x + u. A feedback controller is to be designed such that u(t) = -kx, and the desired equilibrium condition is x(t) = 0 as t —> oo. The performance integral is defined as J = x2 dt, and the initial value of the state variable is x(0) = Obtain the value of k in order to make J a minimum. Is this k physically realizable? Select a practical value for the gain k and evaluate the performance index with that gain. Is the system stable without the feedback due to u(t)l PI 1.2 To account for the expenditure of energy and resources, the control signal is often included in the performance integral. Then the operation will not involve an unlimited control signal u{t). One suitable performance index, which includes the effect of the magnitude of the control signal, is (a) Repeat Problem PI 1.1 for the performance index. (b) If A = 2, obtain the value of k that minimizes the performance index. Calculate the resulting minimum value of/. P11.3 An unstable robot system is described by the vector differential equation [9] *1 _- r 2_ 1 [_-l o" *1 2 J _*2_ 4- r"(')• _ij Both state variables are measurable, and so the control signal is set as u(t) = -k(xi + x2). Following the method of Section 11.7, design gain k so that the performance index is minimized. Evaluate the minimum value of the performance index. Determine the sensitivity of the performance to a change in k. Assume that the initial conditions are x(0) Is the system stable without the feedback signals due to u(0? 897 Problems P11.4 Determine the feedback gain k of Example 11.12 that minimizes the performance index x1 x dt - / when x7(0) = [1 —1], Plot the performance index/ versus the gain k. Pll.5 Determine the feedback gain k of Example 11.13 that minimizes the performance index (x7x + u r u) dt J = when x r (0) = [1 versus the gain k. used in the feedback so that u(t) = -k{Xi - k2x2. Also, we desire to have a natural frequency w„ for this system equal to 2. Find a set of gams k\ and k2 in order to achieve an optimal system when J is given by Equation (11.63). Assume x r (0) = [1 OJ. 1]. Plot the performance index / P11.6 For the solutions of Problems PI 1.3, P11.4, and Pll.5, determine the roots of the closed-loop optimal control system. Note that the resulting closed-loop roots depend on the performance index selected. P11.7 A system has the vector differential equation as given in Equation (11.42). We want both state variables to be P11.8 For the system of Example 11.11 determine the optimum value for k2 when k] - 1 andx r (0) = [1 0]. P11.9 An interesting mechanical system with a challenging control problem is the ball and beam, shown in Figure PI 1.9(a) [ 10]. It consists of a rigid beam that is free to rotate in the plane of the paper around a center pivot, with a solid ball rolling along a groove in the top of the beam. The control problem is to position the ball at a desired point on the beam using a torque applied to the beam as a control input at the pivot. A linear model of the system with a measured value of the angle 4> and its angular velocity dfy/dt = w is available. Select a feedback scheme so that the response of the closed-loop system has an overshoot of 4% and a settling time (with a 2% criterion) of 1 second for a step input. Beam Pivot (a) Motor and amplifier o Control in pill FIGURE P11.9 Torque > Inputs lo be selected (a) Ball and beam. (b) Model of the ball and beam. (b) P11.10 The dynamics of a rocket are represented by ^0 ~0 0 0~ x+ [_l uj y = [0 l]x ~1 L°J and state variable feedback is used, where a = — IOJCI — 25.v2 + i". Determine the roots of the characteristic equation of this system and the response of the system when the initial conditions are x}(0) = 1 and x2(Q) = - 1 . Assume the reference input r(t) - 0. P l l . l l The state variable model of a plant to be controlled is x = y = [0 -5 -2 2 0 X + 0.5 0 l]x + ~[0]u. Use state variable feedback and incorporate a command input u = -Kx + ar. Select the gains K and a so that the system has a rapid response with an overshoot of approximately 1 %, a settling time (with a 2% 898 Chapter 11 The Design of State Variable Feedback Systems criterion) less than 1 second, and a zero steady-state error to a unit step input. P11.12 A DC motor has the state variable model -3 -3 0 0 0 -2 0 2 0 0 0 0 0 0 2 0.75 0 0 1 0 0" V 0 0 0 x + 0 0 0 0_ _0_ 2.75]x. [0 0 0 0 2/ (a) Determine the transfer function, G(s) = Y(s)/U(s). (b) Draw the block diagram indicating the state variables, (c) Determine whether the system is controllable, (d) Determine whether the system is observable. Pll.16 Hydraulic power actuators were used to drive the dinosaurs of the movie Jurassic Park [20]. The motions of the large monsters required high-power actuators requiring 1200 watts. One specific limb motion has dynamics represented by Determine whether this system is controllable and observable. P11.13 A feedback system has a plant transfer function K) R(s) s(s + 70)' We want the velocity error constant K„ to be 35 and the overshoot to a step to be approximately 4% so that l is 1/V2. The settling time (with a 2% criterion) desired is 0.11 second. Design an appropriate state variable feedback system for r(t) = -k^Xi — k2x2. P11.14 A process has the transfer function -10 0~ 1 °J x+ v = [0 l]x + [0]K. ~r [oJ Determine the state variable feedback gains to achieve a settling time (with a 2% criterion) of 1 second and an overshoot of about 10%. Also sketch the block diagram of the resulting system. Assume the complete state vector is available for feedback. P11.15 A telerobot system has the matrix equations [16] 1 0 0 0 -2 0 ~r 0~ 0 x+ i -3_ _0_ and y = [1 0 2]x. Ms) o o"x + " -4 1 1 y = [0 l]x + [0]u. |_oJ We want to place the closed-loop poles at s = —1 ± 3/. Determine the required state variable feedback using Ackermann's formula. Assume that the complete state vector is available for feedback. P11.17 A system has a transfer function Y{s) R(s) m s+a sA + 15.Y3 + 6852 + 106* + 80' Determine a real value of a so that the system is either uncontrollable or unobservable. P11.18 A system has a plant Y(s) U(s) -G(») 1 (s + 1)2 (a) Find the matrix differential equation to represent this system. Identify the state variables on a block diagram model, (b) Select a state variable feedback structure using u(t), and select the feedback gains so that the response y(t) of the unforced system is critically damped when the initial condition is A'I(0) = 1 and A'2(0) = 0, where *, = y(t). The repeated roots are at 5 = - v 2 . P11.19 The block diagram of a system is shown in Figure PI 1.19. Determine whether the system is controllable and observable. • Y(s) s+2 ''' FIGURE P11.19 Multiloop feedback control system. 1 o Problems 899 Pll.20 Consider the automatic ship-steering system discussed in Problems P8.ll and P9.15.The state variable form of the system differential equation is m -0.05 -10"3 1 0 -6 0 -0.15 0 0 0 1 0 0 0 x(t) + 13 0 -0.2 0.03 5(f), 0 0 where x r (/) = [v cos v 0]. The state variables are X\ = v - the transverse velocity; x2 = o>s = angular rate of ship's coordinate frame relative to response frame; x§ = y = deviation distance on an axis perpendicular to the track; x4 = 0 = deviation angle. (a) Determine whether the system is stable, (b) Feedback can be added so that 6(0 = -kxx{ P11.23 Consider again the system of Example 11.7 when we desire that the steady-state error for a step input be zero and the desired roots of the characteristic equation be s - - 2 ± y'l and .s = -10. P11.24 Consider again the system of Example 11.7 when we desire that the steady-state error for a ramp input be zero and the roots of the characteristic equation be *• = — 2 ± ;2 and s = —20. P11.25 Consider the system represented in state variable form x = Ax + Bu y = Cx + Du, where © P11.26 Consider the third-order system 0 0 8 1 0 -5 y = [2 - 4 R3 J, o<'> Rj 1 s(s + 0.4) and negative unity feedback [15]. Represent this system by a state variable signal-flow graph or block diagram and a vector differential equation, (a) Plot the response of the closed-loop system to a step input. (b) Use state variable feedback so that the overshoot is 5% and the settling time (with a 2% criterion) is 1.35 seconds, (c) Plot the response of the state variable feedback system to a step input. 0 0 4 0]x + [0]u. P11.27 Consider the second-order system FIGURE P11.21 RL circuit. G(s) = 0 1 x+ -3 Verify that the system is observable. If so, determine the observer gain matrix required to place the observer poles at 5^2 = - 1 ± j and 5'3 = —5. 1 -3 P11.22 A manipulator control system has a loop transfer function of D = [0]. Verify that the system is observable. Then design a full-state observer by placing the observer poles at *i.2 = ~ 1 - Plot the response of the estimation error e = x - x with an initial estimation error of e(0) = [l If. *. Ml'--(!) ro~ j] C = [1 - 4 ] , and hit) v(t) , B = 10 J [-.-) - £3x3. Determine whether this system is stable for suitable values of k} and k. P11.21 An RL circuit is shown in Figure PI 1.21. (a) Select the two stable variables and obtain the vector differential equation where the output is v0(t). (b) Determine whether the state variables are observable when R\/L\ — R-2JLi- (c) Find the conditions when the system has two equal roots. 'i 4 1 A = 0" l_ x+ "10" _0 J y = [1 0]x + [0]«. Determine the observer gain matrix required to place the observer poles at si2 = - 1 ± jP11.28 Consider the single-input, single-output system is described by x(t) = Ax(0 + Bfi(0 y(t) = Cx(/) where A = 0 -16 1 " -8 ,B = "o~ , C - [ 1 K 0]. 900 Chapter 11 The Design of State Variable Feedback Systems (a) Determine the value of K resulting in a zero steady-state tracking error when u(t) is a unit step input for / £ 0 . The tracking error is defined here as e(t) = tt(t) - y(t). (b) Plot the response to a unit step input and verify that the tracking error is zero for the gain K determined in part (a). P11.29 The block diagram shown in Figure PI 1.29 is an example of an interacting system. Determine a state variable representation of the system in the form x(t) = Ax(f) + Ba(r) y(t) - Cx(r) + Du(t) FIGURE P11.29 Interacting feedback system. ADVANCED PROBLEMS A P l l . l A DC motor control system has the form shown in Figure APll.l [6]. The three state variables are available for measurement; the output position is Xi(t). Select the feedback gains so that the system has a steady-state error equal to zero for a step input and a response with a percent overshoot less than 3%. API 1.2 A system has the model 3 4 0 -1~ ~3~ 0 x + 0 _0_ 0_ -1 0 1 Add state variable feedback so that the closed-loop poles are 5 = —4, —5, and - 6 . AP11.3 A system has a matrix differential equation 0 -1 0 FIGURE AP11.1 Field-controlled DC motor. 0 0 0 AP11.6 A system is represented by the differential equation dy rf2y 2 -ry 7 + y dt2 + 3dt -2 x + 0 1 0 0 -1 0 0 9.8 U(s) AP11.5 An automobile suspension system has three physical state variables, as shown in Figure API 1.5 [13]. The state variable feedback structure is shown in the figure, with Kx = 1. Select K2 and K so that the roots of the characteristic equation are three real roots lying between s - - 3 and s = - 6 . Also, select Kp so that the steady-state error for a step input is equal to zero. f What values for hx and b2 are required so that the system is controllable? AP11.4 The vector differential equation describing the inverted pendulum of Example 3.3 is dx dt Assume that all state variables are available for measurement and use state variable feedback. Place the system characteristic roots at s - -2 ± j , - 5 , and -5. 0 0 X + 1 0 0 1 0 -1 2K s + 4- + u, where y = output and u = input. (a) Develop a state variable representation and show that it is a controllable system, (b) Define the state variables as x'i = y and x2 = dy/dt - «, and determine whether the system is controllable. Note that the controllability of a system depends on the definition of the state variables. AP11.7 The Radisson Diamond uses pontoons and stabilizers to damp out the effect of waves hitting the ship, X3(.v) = //.*) Field current (lit X2(s) s+ Velocity + X,(.v) Position 901 Advanced Problems /?(.v) "> * fk \ \ \ \ \ \ \ FIGURE AP11.5 Automobile suspension system. Xj(s) 2 j-+ 4 1 5+ 2 X2(s) ^ 1 .9 + 3 x,(.v) = y K, K2 Ki ^ - ¾ Passenger • cabins Pontoon Electronical!) controlled stabilizers t _ (a) TM (f>(s) R(s) = 0 FIGURE AP11.7 (a) Radisson Diamond (courtesy of Conde-Nast Traveler, July 1993, 23). (b) Control system to reduce the effect of the disturbance. as shown in Figure API 1.7(a).The block diagram of the ship's roll control system is shown in Figure API 1.7(b). Determine the feedback gains K2 and K^ so that the characteristic roots are s = —15 and s = — 2 ± )2. Plot the roll output (j>(t) for a unit step disturbance. API 1.8 Consider again the liquid-level control system described in Problem P3.36. (a) Design a state variable controller using only h(t) as the feedback variable, so that the step response has an overshoot less than 10% and a settling time (with a 2% criterion) less than or equal to 5 seconds. (b) Design a state variable controller feedback using two state variables, level h{t) and shaft position 6(t), Roll angle (b) to satisfy the specifications of part (a), (c) Compare the results of parts (a) and (b). AP11.9 The motion control of a lightweight hospital transport vehicle can be represented by a system of two masses, as shown in Figure API 1.9, where m\ - m2 = 1 and kv = k2 = 1 [21]. (a) Determine the state vector differential equation, (b) Find the roots of the characteristic equation, (c) We wish to stabilize the system by letting u = —kx,, where u is the force on the lower mass, and x, is one of the state variables. Select an appropriate state variable .vy. (d) Choose a value for the gain k and sketch the root locus as k varies. 902 Chapter 11 The Design of State Variable Feedback Systems downward and, if slightly disturbed, will perform oscillations. If lifted to the top of its arc, the pendulum is unstable in that position. Devise a feedback compensator Gc(s) using only the velocity signal from the tachometer. T _f > r A P l l . l l Determine an internal model controller Gc(s) for the system shown in Figure A P l l . l l . We want the steady-state error to a step input to be zero. We also want the settling time (with a 2% criterion) to be less than 5 seconds. AP11.12 Repeat Advanced Problem A P l l . l l when we want the steady-state error to a ramp input to be zero and the settling time (with a 2% criterion) of the ramp response to be less than 6 seconds. Input force FIGURE AP11.9 Model of hospital vehicle. APll.lO Consider the inverted pendulum mounted to a motor, as shown in Figure API 1.10. The motor and load are assumed to have no friction damping. The pendulum to be balanced is attached to the horizontal shaft of a servomotor. The servomotor carries a tachogenerator, so that a velocity signal is available, but there is no position signal. When the motor is unpowered, the pendulum will hang vertically AP11.13 Consider the system represented in state variable form x = Ax + Bu y = Cx + D», where A = 1 -6 C = [4 2 -5 , B = _1 _ -12 - 3 ] , (ind D = [0] Verify that the system is observable and controllable. If so, design a full-state feedback law and an observer by placing the closed-loop system poles at A \ 2 —1 ± / and the observer poles at s^2 ~ _ 12AP11.14 Consider the third-order system Motor Q Tachometer 0 0 -8 A. Tachometer output FIGURE AP11.10 Motor and inverted pendulum. y = [2 1 0 -3 -9 0 1 x + -3 0 0 4 2]x + [0]«. Verify that the system is observable and controllable. Then, design a full-state feedback law and an Process G(s) R(s) O Gc(s) l O (s + 1)0? + 2) K, FIGURE AP11.11 Internal model control. - • Y(s) 903 Design Problems observer by placing the closed-loop system poles at .12 = — 1 ±7.^3 = _ 3 and the observer poles at  u = -12 ±j2,s3 = -30. AP11.15 Consider the system depicted in Figure API 1.15. Design a full-stale observer for the system. Determine the observer gain matrix L to place the observer poles atjr li2 = -10 ± ylO. •O—•rtvi f/(.v] FIGURE AP11.15 diagram. A second-order system block DESIGN PROBLEMS CDP11.1 We wish to obtain a state variable feedback sysf- £% tem for the capstan-slide the state variable model deC VJ veloped in CDP3.1 and determine the feedback system. The step response should have an overshoot less than 2% and a settling time less than 250 ms. D P l l . l Consider the device for the magnetic levitation of a steel ball, as shown in Figures DPI 1.1(a) and (b). Obtain a design that will provide a stable response where the ball will remain within 10% of its desired position. Assume that y and dyldt are measurable. DP11.2 The control of the fuel-to-air ratio in an automobile carburetor became of prime importance in the 1980s as automakers worked to reduce exhaust-pollution emissions. Thus, auto engine designers turned to the feedback control of the fuel-to-air ratio. A sensor was placed in the exhaust stream and used as an input to a controller. The controller actually adjusts the orifice that controls the flow of fuel into the engine [3]. Select the devices and develop a linear model for the entire system. Assume that the sensor measures the actual fuel-to-air ratio with a negligible delay. With this model, determine the optimum controller when we desire a system with a zero steady-state error to a step input and an overshoot for a step command of less than 10%. DP11.3 Consider the feedback system depicted in Figure DPI 1.3. The system model is given by x(/) = Ax(f) + BK(/) y{t) = Cx(r) where A = 0 •10.5 1 11.3 j ,B = 0 [0.55_ ,C = [1 0]. Electromagnet "C^" i£s=0 CL«e^sLsource „ight 1 (a) > -20 s2 - 2000 Vertical position ol ball (b) FIGURE DP11.1 (a) The levitation of a ball using an electromagnet, (b) The model of the electromagnet and the ball. Design the compensator to meet the following specifications: 1. Tire steady-state error to a unit step input is zero. 2. The settling time Ts < 1 s and the percent overshoot is P.O. < 5%. 3. Select initial conditions for x and different initial conditions for x and simulate the response of the closed-loop system to a unit step input. 904 Chapter 11 The Design of State Variable Feedback Systems System Model -• v x = Ax + BH v = Cx r40 FIGURE DP11.3 Feedback system constructed to track a desired input r(t). Compensator (Observer + Control Law) x = (A - BK - LC)x + Lv + Mr «=-Kx- DP11.4 A high-performance helicopter has a model shown in Figure D P I 1.4. The goal is to control the pitch angle 0 of the helicopter by adjusting the rotor thrust angle 8. The equations of motion of the helicopter are d2e dd —7 = - = gd - a2 dx Gt\— + no dt dO dt dx dt where x is the translation in the horizontal direction. For a military high-performance helicopter, we find that trx = 0.415 a 2 = 1.43 o-2 = 0.0198 n = 6.27 a, = 0.0111 £ = 9.8 all in appropriate SI units. Find (a) a state variable representation of this system and (b) the transfer function representation for Body fixed axis 9(s)/S(s). (c) Use state variable feedback to achieve adequate performances for the controlled system. Desired specifications include (1) a steady-state for an input step command for dd(s), the desired pitch angle, less than 20% of the input step magnitude; (2) an overshoot for a step input command less than 20%; and (3) a settling (with a 2 % criterion) time for a step command of less than 1.5 seconds. DP11.5 The headbox process is used in the manufacture of paper to transform the pulp slurry flow into a jet of 2 cm and then spread it onto a mesh belt [22]. To achieve desirable paper quality, the pulp slurry must be distributed as evenly as possible on the belt, and the relationship between the velocity of the jet and that of the belt, called the jet/belt ratio, must be maintained. One of the main control variables is the pressure in the headbox, which in turn controls the velocity of the slurry at the jet. The total pressure in the headbox is the sum of the liquid-level pressure and the air pressure that is pumped into the headbox. Because the pressurized headbox is a highly dynamic and coupled system, manual control would be difficult to maintain and could result in degradation in the sheet properties. The state-space model of a typical headbox, linearized about a particular stationary point, is given by -0.8 -0.02 FIGURE DP11.4 Helicopter pitch angle, 0, control. + 0.02 x + 0 0.05 0.001 and y — [1 0]x. The state variables are x\ - liquid level and .r2 = pressure. The control variable is ux = pump current. (a) Design a state variable feedback system that has a characteristic equation with real roots with a magnitude greater than five, (b) Design an observer with observer poles located at least ten times farther in the left half-plane than the state variable feedback system. (c) Connect the observer and full-state feedback system and sketch the block diagram of the integrated system. 905 Design Problems DP11.6 A coupled-drive apparatus is shown in Figure DPI 1.6. The coupled drives consist of two pulleys connected via an elastic belt, which is tensioned by a third pulley mounted on springs providing an underdamped dynamic mode. One of the main pulleys, pulley A, is driven by an electric DC motor. Both pulleys A and B are fitted with tachometers that generate measurable voltages proportional to the rate of rotation of the pulley. When a voltage is applied to the DC motor, pulley A will accelerate at a rate governed by the total inertia experienced by the system. Pulley B, at the other end of the elastic belt, will also accelerate owing to the applied voltage or torque, but with a lagging effect caused by the elasticity of the belt. Integration of the velocity signals measured at each pulley will provide an angular position estimate for the pulley [23]. The second-order model of a coupled-drive is 0 -36 1 0 v+ -12 j _lj and y = x±. (a) Design a state variable feedback controller that will yield a step response with deadbeat response and a settling time (with a 2% criterion) less than 0.5 second. (b) Design an observer for the system by placing the observer poles appropriately in the left half-plane. (c) Draw the block diagram of the system including the compensator with the observer and state feedback. (d) Simulate the response to an initial state at x(0) = [1 Of and x(0) = [0 Of. DP11.7 A closed-loop feedback system is to be designed to track a reference input. The desired feedback block diagram is shown in Figure DP11.3.The system model is given by x(f) = Ax(0 + Bu(t) m = cx(/) where 0 1 A = -10_ Pulley A Pulley B FIGURE DP11.6 "o" ,B = 0 , C = [1 0 0]. 1 Design the observer and the control law to meet the following specifications: 1. The steady-state error of the closed-loop system to a unit step input is zero. 2. The gain margin G.M. ^ 6 dB. 3. The bandwidth of the closed-loop system coB > lOrad/s. 4. Select initial conditions for x and different initial conditions for x and simulate the response of the closed-loop system to a unit step input. Verify that the tracking error is zero in the steady-state. System Model N FIGURE DP11.7 Feedback system constructed to track a desired input r{f). R3 x = Ax + BH y = Cx Observer x = (A - LC)x + BH + Ly -• v 906 Chapter 11 The Design of State Variable Feedback Systems COMPUTER PROBLEMS CP11.1 Consider the system 0 -6 2 5 4 0 7 X+ 0 10 1 11_ _1 y = [\ CP11.5 A linearized model of a vertical takeoff and landing (VTOL) aircraft is [24] x = Ax + Bill] + B 2 « 2 , H, where llx. 2 Using the ctrb and obsv functions, show that the system is controllable and observable. CP11.2 Consider the system 0 -6 x= y = [l 1 -5 x + y = [l 1 x+ 2 V M, I -l]x 0 0 0 0 0 10 1 0 1 -0.5 0 0.3681 0 1 0.0188 0.0019 -0.7070 -0.4555 -4.0208 1.4200 1 0 0.4422" are S\ = - 1 and s2 = - 2 . Use state feedback u = -Kx. CP11.4 The following model has been proposed to describe the motion of a constant-velocity guided missile: 0 0.1024 0 0]x. 0 -1 y - [0 0.0271 -1.0100 and a, Determine if the system is controllable and observable. Compute the transfer function from u to y. CP11.3 Find a gain matrix K so that the closed-loop poles of the system 0 0.1 0.5 0 0.5 -0.0389 0.0482 0 0 0 0 0 0 0 0 x + 0 0 0 1 0 », 0 0 0]x. (a) Verify that the system is not controllable by analyzing the controllability matrix using the ctrb function. (b) Develop a controllable state variable model by first computing the transfer function from u to y, then cancel any common factors in the numerator and denominator polynomials of the transfer function. With the modified transfer function just obtained, use the ss function to determine a modified state variable model for the system. (c) Verify that the modified state variable model in part (b) is controllable. (d) Is the constant velocity guided missile stable? (e) Comment on the relationship between the controllability and the complexity of the state variable model (where complexity is measured by the number of state variables). 3.5446 -6.0214 ' ()_ = B2 = 0.1291 ~ -7.5922 4.4900 ()_ The state vector components are (i) .v, is the horizontal velocity (knots), (ii) x2 is the vertical velocity (knots), (iii) .¾ is the pitch rale (degrees/second), and (iv) ,v4 is the pitch angle (degrees). The input ii\ is used mainly to control the vertical motion, and u2 is used for the horizontal motion. (a) Compute the eigenvalues of the system matrix A. Is the system stable? (b) Determine the characteristic polynomial associated with A using the poly function. Compute the roots of the characteristic equation, and compare them with the eigenvalues in part (a). (c) Is the system controllable from ii] alone? What about from «2 alone? Comment on the results. CP11.6 In an effort to open up the far side of the Moon to exploration, studies have been conducted to determine the feasibility of operating a communication satellite around the translunar equilibrium point in the Earth-Sun-Moon system. The desired satellite orbit, known as a halo orbit, is shown in Figure CP11.6.The objective of the controller is to keep the satellite on a halo orbit trajectory that can be seen from the Earth so that the lines of communication are accessible at all times. The communication link is from the Earth to the satellite and then to the far side of the Moon. The linearized (and normalized) equations of motion of the satellite around the translunar equilibrium point are [25] 0 0 0 7.3809 0 0 0 0 0 u -2.1904 0 0 1 0 0 0 0 0 0 (J -2 -3.1904 0 0 1 0 2 0 0 0 0 1 0 0 0 907 Computer Problems ~(f ~o~ V 0 0 Mj 1 0 _0_ 0 0 U 0 2 1 _0_ 0 0 it*. 0 0 _1_ + + The state vector x is the satellite position and velocity, and the inputs «,, i — 1,2,3, are the engine thrust accelerations in the £, 77, and £ directions, respectively. (a) Is the translunar equilibrium point a stable location? (b) Is the system controllable from ux alone? (c) Repeat part (b) for u2. (d) Repeat part (b) for «3. (e) Suppose that we can observe the position in the 17 direction. Determine the transfer function from «2 to 77. (Hint: Let y = [0 1 0 0 0 0]x. ) (f) Compute a state-space representation of the transfer function in part (e) using the ss function. Verify that the system is controllable, (g) Using state feedback y(t) = [1 0 0]x(/). (CP11.1) Suppose that we are given three observations y(t,), i - 1,2,3, as follows: y(t{) = 1 at ?j = 0 y(t2) = -0.0256 at t2 = 2 y(t3) = -0.2522 at /3 = 4. (a) Using the three observations, develop a method to determine the initial value of the state vector x(r0) for the system in Equation CPU.l that will reproduce the three observations when simulated using the Isim function. (b) With the observations given, compute x(r0) and discuss the condition under which this problem can be solved in general, (c) Verify the result by simulating the system response to the computed initial condition. (Hint: Recall that x(f) = * A ( w „ )x(/()) for the system in Equation CP11.1.) CP11.8 A system is described by a single-input state equation with u2 = —Kx, design a controller (i.e., find K) for the system in part (f) such that the closed-loop system poles are at «1,2 ~ - 1 ± j and 53-4 = -10. A = 0 -1 0 0 and B Using the method of Section 11.7 (Equation 11.40) and a negative unity feedback, determine the optimal system when x r (0) = [1 0]. CP11.9 A first-order system is given by x = -x + it with the initial condition x(0) = .v0. We want to design a feedback controller u = -kx such that the performance index Earth (x2(t) + \u2(t)) dt J = Jo View from the earth Halo orbit of spacecraft V< .Moon, FIGURE CP11.6 CP11.7 The translunar satellite halo orbit. CPll.lO Consider the system represented in state variable form Consider the system m 0 ] 0 0 -4 2 is minimized. (a) Let A = 1. Develop a formula for / in terms of k, valid for any x(), and use an m-file to plot J'fx\ versus k. From the plot, determine the approximate value of k = kmi„ that minimizes J/XQ, (b) Verify the result in part (a) analytically, (c) Using the procedure developed in part (a), obtain a plot of &min versus A, where kmm is the gain that minimizes the performance index. 0 1 x(r), -6 x = Ax + Bu y = Cx + D«, 908 Chapter 11 The Design of State Variable Feedback Systems CP11.13 Consider the system in state variable form where 1 0 18.7 A = C = [1 10.4_ 0] and , B = [24.6 D - [0]. y = [1 Using the acker function, determine a full-state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at sl2 = - 2 and the observer poles at sx_2 = ~20 ± j4. CP11.11 Consider the third-order svstem 0 0 -4.3 v = [0 1 1 0 -1.7 0 1 -6.7 x+ 0 0 0 1 0 -1 0 0 x+ 1 -13 0 0 0 1 0]x + [0)u. Design a full-state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at i2 - -1.4 ± / 1 . 4 , ^ — —2 ± j and the observer poles s^2 = - 1 8 ± fi*s34 = ~20. Construct the state variable compensator using Figure 11.1 as a guide and simulate the closed-loop system using Simulink. Select several values of initial states and initial state estimates in the observer and display the tracking results on an x_y-graph. 0 0 0.35 0]x + [01M. (a) Using the acker function, determine a full-state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at s 12 = -1.4 ± /1.4, 4*3 = - 2 and the observer poles at S\2 = —18 ± ;5, s3 = —20. (b) Construct the state variable compensator using Figure 11.1 as a guide, (c) Simulate the closed-loop system with the state initial conditions x(0) = (1 0 0)T and initial state estimate of x(0) = (0.5 0.1 0.1)7. CP11.12 Implement the system shown in Figure CP11.12 in an m-file. Obtain the step response of the svstem. El 1 0 0 -5 0 0 0 -2 "io.r U(s) -+Q-+YU FIGURE CP11.12 Control system for Simulink implementation. ANSWERS TO SKILLS CHECK True or False: (1) True; (2) True; (3) False; (4) True; (5) False Multiple Choice: (6) c; (7) a; (8) c; (9) a; (10) a; (11) b; (12) a; (13) b; (14) b; (15) a Word Match (in order, top to bottom): e, o, k, i, d, b, j,m,f,n, h, q, g, l,p, c, a TERMS AND CONCEPTS Command following An important aspect of control system design wherein a nonzero reference input is tracked. Controllability matrix A linear system is (completely) controllable if and only if the controllability matrix Pc = [B AB A 2 B... A"_1B] has full rank, where A is an n x n matrix. For single-input, single-output linear systems, the system is controllable if and only if the determinant of the n X n controllability matrix Pc is nonzero. Controllable system A system is controllable on the interval [to, tf] if there exists a continuous input u(t) such that any initial state x(t0) can be driven to any arbitrary trial state \(tf) in a finite time interval tf - tQ > 0. Detectable A system in which the states that are unobservable are naturally stable. Estimation error The difference between the actual state and the estimated state e(r) = x(t) - x(f)- 961 Skills Check A robust control system provides stable, consistent performance as specified by the designer in spite of the wide variation of plant parameters and disturbances. It also provides a highly robust response to command inputs and a steady-state tracking error equal to zero. For systems with uncertain parameters, the need for robust systems will require the incorporation of advanced machine intelligence, as shown in Figure 12.51. Adaptive system Machine i nte lligen mired High FIGURE 12.51 Intelligence required versus uncertainty for modem control systems. m / Robust system / Moderate / Feedback system / Low Open-loop system (without feedback) Low Moderate High Uncertainty of parameters and disturbances SKILLS CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 12.52 as specified in the various problem statements. Prefdter /?(.?) G„(s) —K>-\ fc Controller Process Gc(s) G(s) i FIGURE 12.52 Block diagram for the Skills Check. In the following True or False and Multiple Choice problems, circle the correct answer. 1. A robust control system exhibits the desired performance in the presence of significant plant uncertainty. 2. For physically realizable systems, the loop gain L(s) = Gc(s)G(s) must be large for high frequencies. True or False True or False 962 Chapter 12 Robust Control Systems 3. The PID controller consists of three terms in which the output is the sum of a proportional term, an integrating term, and a differentiating term, with an adjustable gain for each term. 4. A plant model will always be an inaccurate representation of the actual physical system. 5. Control system designers seek small loop gain L(s) in order to minimize the sensitivity S(s), 6. A closed-loop feedback system has the third-order characteristic equation True or False True or False True or False q(s) - s3 + a2S2 + a-[S + OQ = 0, where the nominal values of the coefficients are a2 ~ 3, ax = 6, and a0 — 11. The uncertainty in the coefficients is such that the actual values of the coefficients can lie in the intervals 2 < a2 ^ 4, 4 < ax < 9, 6 < a0 < 17. Considering all possible combinations of coefficients in the given intervals, the system is: a. Stable for all combinations of coefficients. b. Unstable for some combinations of coefficients. c. Marginally stable for some combinations of coefficients. d. Unstable for all combinations of coefficients. In Problems 7 and 8, consider the unity feedback system in Figure 12.52, where G(s) = 2 d + 3)" 7. Assume that the prefilter is Gp(s) = l.The proportional-plus-integral (PI) controller, Gc(s), that provides optimum coefficients of the characteristic equation for ITAE (assuming ion = 12 and a step input) is which of the following: ^ « 6.9 a. Gc = 72 + — 72 b. Gc = 6.9 + — s c. Gc = 1 + s d. Gc = 14 + 10s 8. Considering the same PI controller as in Problem 7, a suitable prefilter, Gp(s), which provides optimum ITAE response to a step input is: 10.44 —— a. GD1 (s) = ' s + 12.5 12 5 b. GJs) = s + 12.5 10.44 c. GJs) p w = s + 10.44 144 d - GJs) = -— p ' s + 144 9. Consider the closed-loop system block-diagram in Figure 12.52, where G{s)= ,2* and Gp(s) = l. ^(sz + 8s) Skills Check 963 Determine which of the following PID controllers results in a closed-loop system possessing two pairs of equal roots. 22.5(5 + 1.11)2 a. u KJc\i) - 5 10.5(5' + 1.11)2 Gc(s) =• 5 2.5(5 + 2.3)2 C. Gc(s) = 5 d. None of the above 10. Consider the system in Figure 12.52 with Gp(s) = 1, G(s) = s2 + as + b' and l £ a < 3 and 1 0.1(5 + 10)2 L c Gc(s) = — 5 d. None of the above 11. Consider the system in Figure 12.52 with Gp(s) - 1 and loop transfer function m = GAs)G(s) = -JL- The sensitivity of the closed-loop system with respect to variations in the parameter K is s(s + 3) T a. S K = T b . SK - 52 52 5 c. S'TK =_ S T d. S K + 35 + K 5+ 5 + 55 + tf 2 = -= + 55 + K s{s + 5) 52 + 55 + K 12. Consider the feedback control system in Figure 12.52 with plant 1 5 + 2* A proportional-plus-integral (PI) controller and prefilter pair that results in a settling time Ts < 1.8 seconds and an ITAE step response are which of the following: G{s) = a. Gc(s) = 3.2 + — 5 b. Gc(s) = 10 + — and and GJs) = pK ' 3.2s + 13.8 Gp(s) = 5+ 1 964 Chapter 12 Robust Control Systems c Gc(s) = 1 + - and Gp(s) = — ^ ^ / v - „ * 500 _ , . 500 d. Gc.(5) = 12.5 + 5 and Gpv p (i) ' 12.55 + 500 13. Consider a unity negative feedback system with a loop transfer function (with nominal values) L(s) = Gc(s)G(s) = - - *(j - - = - - ^ + 2). Using the Routh-Hurwitz stability analysis, it can be shown that the closed-loop system is nominally stable. However, if the system has uncertain coefficients such that 0.25 < a < 3, 2 < 6 < 4, and 4 < X < 5, the closed-loop system may exhibit instability. Which of the following situations is true: a. Unstable for a = \,b = 2, and K = 4. b. Unstable for a = 2, b = 4, and K = 4.5. c. Unstable for a = 0.25, b - 3, and K = 5. d. Stable for all a, b, and /C in the given intervals. 1 14. Consider the feedback control system in Figure 12.52 with GJs) = 1 and G(s) = —r. /r The nominal value of J = 5, but it is known to change with time. It is thus necessary to design controller with sufficient phase margin to retain stability as J changes. A suitable PID controller such that the phase margin is greater than P.M. > 40° and bandwidth (ob < 20 rad/s is which of the following: 50(52 + 105 + 26) 'a. Gc(5) = - ^ s 5(52 + 25 + 2) b. Gc(s) = 60(52 + 205 + 200) 'c. Gc(s) = - * d. None of the above 15. A feedback control system has the nominal characteristic equation q{s) = s3 + a2s2 + «i5 + o0 = 53 + 352 + 25 + 3 = 0. The process varies such that 2 < a2 s 4, 1 < « t < 3, 1 < a0 - 5. Considering all possible combinations of coefficients a2, ah and a0 in the given intervals, the system is: a. Stable for all combinations of coefficients. b. Unstable for some combinations of coefficients. c. Marginally stable for some combinations of coefficients. d. Unstable for all combinations of coefficients. In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Root sensitivity b. Additive perturbation A system that exhibits the desired performance in the presence of significant plant uncertainty. A controller with three terms in which the output is the sum of a proportional term, an integrating term, and a differentiating term, with an adjustable gain for each term. 965 Exercises c. Complementary sensitivity function d. Robust control system e. System sensitivity f. Multiplicative perturbation g. PID controller h. Robust stability criterion i. Prefilter j . Sensitivity function k. Internal model principle A transfer function that filters the input signal R(s) prior to the calculation of the error signal. A system perturbation model expressed in the additive form Ga(s) = G(s) + A(s) where G(s) is the nominal plant, A(s) is the perturbation that is bounded in magnitude, and Ga(s) is the family of perturbed plants. The function C(s) = Gc(s)G(s)[l + Gc(s)G(s)yl that satisfies the relationship C(s) + S(s) = 1, where S(s) is the sensitivity function. The principle that states that if Gc(s)G(s) contains the input R(s), then the output y(t) will track the input asymptotically (in the steady state) and the tracking is robust. A system perturbation model expressed in the multiplicative form Gm(s) = G(s)[l + M(s)] where G(s) is the nominal plant, M(s) is the perturbation that is bounded in magnitude, and G„,(s) is the family of perturbed plants. A test for robustness with respect to multiplicative perturbations. A measure of the sensitivity of the roots (that is, the poles and zeros) of the system to changes in a parameter. The function S(s) = [1 + Gc(s)G(s)]~l that satisfies the relationship C(s) + S(s) = 1, where C(s) is the complementary sensitivity function. A measure of the system sensitivity to changes in a parameter. EXERCISES E12.1 Consider a system of the form shown in Figure 12.1, where G(s) = (s + 3)" Using the ITAE performance method for a step input, determine the required Gc(s), Assume a>„ = 30 for Table 5.6. Determine the step response with and without a prefilter Gp(s). E12.2 For the ITAE design obtained in Exercise E12.1, determine the response due to a disturbance Td(s) = 1/s, E12.3 A closed-loop unity feedback system has the loop transfer function •bs Answer: Sj, s2 + bs + 25 E12.4 A PID controller is used in the system in Figure 12.1, where G(s) 1 (s + 20)(5 + 36)' The gain KD of the controller (Equation (12.33)) is limited to 200. Select a set of compensator zeros so that the pair of closed-loop roots is approximately equal to the zeros. Find the step response for the approximation in Equation (12.35) and the actual response, and compare them. E12.5 A system has a process function L(s) = Gc(s)G(s) 25 s(s + bY where b is normally equal to 8. Determine Si and plot |r(;'a>)| and \S(ja))\ on a Bode plot. G(s) K s(s + 3)( s + 10) 966 Chapter 12 Robust Control Systems of C = 0.6. Determine the step response of the system. Predict the effect of a change in K of ±50% on the percent overshoot. Estimate the step response of the worst-case system. with K = 10 and negative unity feedback with a PD compensator Gc(s) = Kp+ KDs. The objective is to design Gc(s) so that the overshoot to a step is less than 5% and the settling time (with a 2% criterion) is less than 3 sec. Find a suitable Gc(s). What is the effect of varying the process gain 5 :£ K < 20 on the percent overshoot and settling time? E12.6 Consider the control system shown in Figure E12.6 when G{s) - \/{s + 5) 2 , and select a PID controller so that the settling time (with a 2% criterion) is less than 1.5 second for an ITAE step response. Plot y(t) for a step input r(t) with and without a prefilter. Determine and plot y(t) for a step disturbance. Discuss the effectiveness of the system. E12.10 A system has the form shown in Figure El 2.6 with G(*) = where K = 1. Design a PI controller so that the dominant roots have a damping ratio £ — 0.70. Determine the step response of the system. Predict the effect of a change in K of ±50% on the percent overshoot. Estimate the step response of the worst-case system. E12.ll Consider the closed-loop system represented in state variable form X == Answer: One possible controller is where 0 -4 E12.7 For the control system of Figure E12.6 with G(s) = \/(s + 4) 2 , select a PID controller to achieve a settling time (with a 2% criterion) of less than 1.0 second for an ITAE step response. Plot y(t) for a step input r(t) with and without a prefilter. Determine and plot y(t) for a step disturbance. Discuss the effectiveness of the system. E12.8 Repeat Exercise 12.6, striving to achieve a minimum settling time while adding the constraint that \u(t)\ < 80 for t > 0 for a unit step input, r(t) = l , l > 0 . Answer: Gc(s) = Ax + Br y = Cx + D r , 0.5s2 + 52 As + 216 Gc(s) K s(s + 3)(5 + 6)' , C = [5 0], and D = [0]. B = The nominal value of k -2. However, the value of k can vary in the range 0.1 ^ fe S 4, Plot the percent overshoot to a unit step input as k varies from 0.1 to 4. E12.12 Consider the second-order system 3600 + 80s x = E12.9 A system has the form shown in Figure E12.6 with G{s) 1 ~ -k 0 -a y = [1 K s(s + 7)(s + 9)' 1 Tc, , x + u -b] [_c2_ 0]x + [0]«. The parameters a, b, C\, and c2 are unknown a priori. Under what conditions is the system completely controllable? Select valid values of a, b, c l7 and c2 to ensure controllability and plot the step response. where K = 1. Design a PD controller such that the dominant closed-loop poles possess a damping ratio Disturbance Controller FIGURE E12.6 System with controller. R(s) Desired input Gc(s) Gp(s) Si Uis \n > JU > as) . Yis) Output Problems 967 PROBLEMS P12.1 Consider the unmanned underwater vehicle (UUV) problem presented in DP4.7. The control system is shown in Figure P12.1, where R(s) - 0, the desired roll angle, and TA[s) = 1/s. We select Gc(s) = K(s + 2), where K = 4. (a) Plot 20 log|r| and 20 \og\STK\ on a Bode diagram, (b) Evaluate \Sl\ at cog, (Oftj2, and <*B;*(T(S) = Gc(s) = K(s - 1) (s + 0.02)' (a) Find the range of K for a stable system, (b) Select a gain so that the steady-state error of the system is less than 0.1 for a step input command, (c) Find y(t) for the gain of part (b). (d) Find y(t) when K varies ±15% from the gain of part (b). Y(s)/R(s)). P12.2 Consider the mobile, remote-controlled video camera P12.4 An automatically guided vehicle is shown in Figure system presented in DP4.8.The control system is shown P12.4(a) and its control system is shown in Figure P12.4(b). in Figure P12.2, where TX - 20 ms and TI - 2 ms. The goal is to track the guide wire accurately, to be insensitive to changes in the gain K^, and to reduce the effect (a) Select K so that Mpto = 1.84. (b) Plot 20 log|r| of the disturbance [15, 22]. The gain K] is normally and 201og|s£| on one Bode diagram, (c) Evaluate equal to 1 and T\ - 1/25 second. |S£[ at coB, a)B/2, and coB^. (d) Let R(s) = 0 and de(a) Select a compensator Gc{s) so that the percent termine the effect of T(l(s) = 1/s for the gain K of overshoot to a step input is less than or equal to part (a) by plotting y(/). 10%, the settling time (with a 2% criterion) is less P12.3 Magnetic levitation (maglev) trains may replace airthan 100 ms, and the velocity constant Kv for a planes on routes shorter than 200 miles. The maglev ramp input is 100. train developed by a German firm uses electromagnet(b) For the compensator selected in part (a), deteric attraction to propel and levitate heavy vehicles, carmine the sensitivity of the system to small rying up to 400 passengers at 300-mph speeds. But the changes in Kx by determining SrKi or Sj^. j-inch gap between car and track is difficult to maintain (c) If # i changes to 2 while Gc(s) of part (a) remains [7,12,17]. unchanged, find the step response of the system The block diagram of the air-gap control system and compare selected performance figures with is shown in Figure P12.3.The compensator is those obtained in part (a). TAs) FIGURE P12.1 Gc{s) — • R(s) = 0 s L Control of an underwater vehicle [13]. -k H Roll angle S TAs) K FIGURE P12.2 I 1 rts + I s(r2s+ 1) •** Vis) Remote-controlled TV camera. Vehicle and levitation coil Controller FIGURE P12.3 Maglev train control. Desired gap + ">) H^J fc Gr(s) Coil current s-4 (s + If Y(s) Air gap 968 Chapter 12 Robust Control Systems Battery bay Component bin ^ Steerable wheel ^ Transponder antenna I | v&y I J Steerable wheel ( i n \ (¾ AGV Transponder „ antenna rasss \^7 L_ » ^ ^ C~ ^ \ Idler ^-—^ wheels Top View Side V i e w In floor / transponder "tag" (a) Actuator and wheels Controller R(s) Guide wire signal -^ Error *W" •i>u • Ct(j) FIGURE P12.4 Automatically guided vehicle. *l s ( w + 1) (b) (d) Determine the effect of T(l(s) = \/s y(t) when R(s) = 0. positioning station, waiting station, wrapping station, and so forth. We will focus on the positioning station shown in Figure P12.5(a). The positioning station is the first station that sees a paper roll. This station is responsible for receiving and weighing the roll, measuring its diameter and width, determining the desired by plotting P12.5 A roll-wrapping machine (RWM) receives, wraps, and labels large paper rolls produced in a paper mill [9, 16]. The RWM consists of several major stations: Diameter measuring arm . If Paper roll Width measuring arm Laser Positioning arm W aw — • Q — • FIGURE P12.5 Roll-wrapping machine control. Y(s) • Direction of travel Gc(s) s(s + p) (b) 1-*" K*> Problems 969 wrap for the roll, positioning it for downstream processing, and finally ejecting it from the station. Functionally, the RWM can be categorized as a complex operation because each functional step (e.g., measuring the width) involves a large number of field device actions and relies upon a number of accompanying sensors. The control system for accurately positioning the width-measuring arm is shown in Figure P12.5(b).The negative pole p of the positioning arm is normally equal to 2, but it is subject to change because of loading and misalignment of the machine, (a) For p = 2, design a compensator so that the complex roots are s = -2 ± ; 2 V 3 . (b) Plot y(t) for a step input R(s) = \/s. (c) Plot y(t) for a disturbance Td(s) = 1/s, with R(s) — 0. (d) Repeat parts (b) and (c) when p changes to 1 and Gc(s) remains as designed in part (a). Compare the results for the two values of the negative pole p. P12.6 The function of a steel plate mill is to roll reheated slabs into plates of scheduled thickness and dimension [5, 10). The final products are of rectangular plane view shapes having a width of up to 3300 mm and a thickness of 180 mm. A schematic layout of the mill is shown in Figure P12.6(a). The mill has two major rolling stands, denoted No. 1 and No. 2. These are equipped with large rolls (up to 508 m m in diameter), which are driven by high-power electric motors (up to 4470 kW). Roll gaps and forces are maintained by large hydraulic cylinders. Typical operation of the mill can be described as follows. Slabs coming from the reheating furnace initially go through the No. 1 stand, whose function is to reduce the slabs to the required width. The slabs proceed through the No. 2 stand, where finishing passes are carried out to produce the required slab thickness. Finally, they go through the hot plate leveller, which gives each plate a smooth finish. O n e of the key systems controls the thickness of the plates by adjusting the rolls. The block diagram of this control system is shown in Figure P12.6(b). The plant is represented by G(s) 1 s(s2 + 4s + 5)' The controller is a P1D with two equal real zeros, (a) Select the PID zeros and the gains so that the closed-loop system has two pairs of equal roots, (b) For the design of part (a), obtain the step response without a prefilter (Gp(s) = 1). (c) Repeat part (b) for an appropriate prefilter. (d) For the system, determine the effect of a unit step disturbance by evaluating y(t) with r(t) = 0. P12.7 A motor and load with negligible friction and a voltage-to-current amplifier Ka is used in the feedback control system, shown in Figure P12.7. A designer selects a PID controller is Gc(s) = KP + — + KDs, where KP = 5, Kt = 500, and KD = 0.0475. (a) Determine the appropriate value of Ku so that the phase margin of the system is 30°. (b) For the gain Ka, plot the root locus of the system and determine the roots of the system for the Ka of part (a), (c) Determine the maximum value of y(t) when T{i(s) = 1/s and R(s) = 0 for the Ka of part (a), (d) Determine the response to a step input r(t), with and without a prefilter. P12.8 A unity feedback system has a nominal characteristic equation q(s) = s 3 + 3s2 + 3s + 6 = 0. Furnace Hot plate leveller No. 2 stand No. 1 stand (a) R{s) Desired thickness FIGURE P12.6 Steel-rolling mill control. Y(s) Thickness (b) 970 Chapter 12 Robust Control Systems '/;,i.v) PID controller V(s) R(s) FIGURE P12.7 Hs) Gr(s) Ka 10 >'(.V) PID controller for the motor and load system. have a range of 620 miles and could be used for missions of up to six months. Boeing Company engineers first analyzed the Apollo-era Lunar Roving Vehicle, then designed the new vehicle, incorporating improvements in radiation and thermal protection, shock and vibration control, and lubrication and sealants. The steering control of the moon buggy is shown in Figure P12.9(b).The objective of the control design is to achieve a step response to a steering command with The coefficients vary as follows: 2 < a2 ^ 4, 1 s a, as 4, 4 < a0 <• 5. Determine whether the system is stable for these uncertain coefficients. P12.9 Future astronauts may drive on the Moon in a pressurized vehicle, shown in Figure P12.9(a), that would 7"(/(s) Disturbun FIGURE P12.9 (a) A moon vehicle. (b) Steering control for the moon vehicle. R(s) Steering commands Gc(s) Gp(s) U(s) ^ L (b) b- Dynamics 0.2* + 1 rt.s) Angle ol steering 971 Advanced Problems zero steady-state error, an overshoot less than 20%, and a peak time less than 0.3 second with a \u(t) | s 50. It is also necessary to determine the effect of a step disturbance Td(s) = \/s when R(s) = 0, in order to ensure the reduction of moon surface effects. Using (a) a PI controller and (b) a PID controller, design an acceptable controller. Record the results for each design in a table. Compare the performance of each design. Use a prefilter Gp(s) if necessary. P12.10 A plant has a transfer function G(s) = % in DP5.7. The control of x may be achieved with a DC motor and position feedback of the form shown in Figure P12.ll, with the DC motor and load represented bv where 1 £ K < 3 and 1 <; p < 3. Normally K = 2.5 and /7 = 2. Design an ITAE system with a PID controller so that the peak time response to a step input is less than 3 seconds for the worst-case performance. P12.12 Consider the closed-loop second-order system We want to use a negative unity feedback with a PID controller and a prefilter. The goal is to achieve a peak time of 1 second with ITAE-type performance. Predict the system overshoot and settling time (with a 2% criterion) for a step input. P12.ll Consider the three dimensional cam shown in Figure P12.ll [18]. This problem was first introduced K s(s + p)(s + 4)' G(s) = 0 i 3~ -id J v + y = [2 0]x + [0]«. Compute the sensitivity of the closed-loop system to variations in the parameter K. R(s) FIGURE P12.11 An x-axis control system. ADVANCED PROBLEMS AP12.1 To minimize vibrational effects, a telescope is magnetically levitated. This method also eliminates friction in the azimuth magnetic drive system. The photodetectors for the sensing system require electrical connections. The system block diagram is shown in Figure AP12.1. Design a PID controller so that the velocity error constant is Kv = 100 and the maximum overshoot for a step input is less than 10%. AP12.2 One promising solution to traffic gridlock is a magnetic levitation (maglev) system. Vehicles are suspended on a guideway above the highway and guided by magnetic forces instead of relying on wheels or aerodynamic forces. Magnets provide the propulsion for the vehicles [7,12,17]. Ideally, maglev can offer the environmental and safety advantages of a high-speed train, the speed and low friction of an airplane, and the convenience of an automobile. All these shared attributes notwithstanding, the maglev system is truly a new mode of travel and will enhance the other modes of travel by relieving congestion and providing connections among them. Maglev travel would be fast, operating at 150 to 300 miles per hour. The tilt control of a maglev vehicle is illustrated in Figures AP12.2(a) and (b). The dynamics of the plant G(s) are subject to variation so that the poles will lie within the boxes shown in Figure AP12.2(c), and 1 < K < 2. The objective is to achieve a robust system with a step response possessing an overshoot less than 10%, as well as a settling time (with a 2% criterion) less Process dynamics FIGURE API 2.1 Magnetically levitated telescope position control system. «(.*) to _ i I Gc(s) l 50s2 + l - • Y(s) 972 Chapter 12 Robust Control Systems Superconducting coils Windings for propulsion and suspension Support pier (a) TAs) R{s) Desired tilt angle G(s) Vis) A Gr(s) Gn(s) k K s2 + as + b Tilt angle (b) Region for poles of -/2 G(s) -n I -3 -2 I I --yi --72 (c) FIGURE AP12.2 (a) and (b) Tilt control for a maglev vehicle, (c) Plant dynamics. Advanced Problems than 2 seconds when \u(t)\ ^ 100. Obtain a design with a PI. PD, and PID controller and compare the results. Use a prefilter Gp(s) if necessary. AP12.3 Antiskid braking systems present a challenging control problem, since brake/automotive system parameter variations can vary significantly (e.g., due to the brake-pad coefficient of friction changes or road slope variations) and environmental influences (e.g., due to adverse road conditions). The objective of the antiskid system is to regulate wheel slip to maximize the coefficient of friction between the tire and road for any given road surface [8]. As we expect, the braking coefficient of friction is greatest for dry asphalt, slightly reduced for wet asphalt, and greatly reduced for ice. One simplified model of the braking system is represented by a plant transfer function G(s) with a system as shown in Figure 12.16 with w U(s) (s + a)(s + b)' where normally a = 1 and b - 4. (a) Using a PID controller, design a very robust system where, for a step input, the overshoot is less than 4% and the settling time (with a 2% criterion) is 1 second or less. The steady-state error must be less than 1% for a step. We expect a and b to vary by ±50%. (b) Design a system to yield the specifications of part (a) using an ITAE performance index. Predict the overshoot and settling time for this design. AP12.4 A robot has been designed to aid in hip-replacement surgery. The device, called RoBoDoc, is used to precisely orient and mill the femoral cavity for acceptance of the prosthetic hip implant. Clearly, we want a very robust surgical tool control, because there is no opportunity to redrill a bone [21,27].The control system will be as shown in Figure 12.1 with G(s) = -= , s + as + b where 1 < a < 2, and 4 s b < 12. Select a PID controller so that the system is very robust. Use the s-plane root locus method. Select the appropriate Gp(s) and plot the response to a step input. AP12.5 Consider the system of Figure 12.1 with G(s) = sJs-TWy where K\ = 1 under normal conditions. Design a PID controller to achieve a phase margin of 50°. The controller is 973 K(s2 + 20s + b) Gc{s) = - * '- s with complex zeros. Determine the effect of a change of ±25% in Ki by developing a tabular record of the phase margin as K\ varies. AP12.6 Consider the system of Figure 12.1 with GW = *TW where Kt = 1.5 and T « 0.001 second, which may be neglected. (Check this later in the design process.) Select a PID controller so that the settling time (with a 2% criterion) for a step input is less than 1 second and the overshoot is less than 10%. Also, the effect of a disturbance at the output must be reduced to less than 5% of the magnitude of the disturbance. Select &>„, and use the ITAE design method. AP12.7 Consider the system of Figure 12.1 with <*) = i. The goal is to select a PI controller using the ITAE design criterion while constraining the control signal as |w(0l — 1 for a unit step input. Determine the appropriate PI controller and the settling time (with a 2% criterion) for a step input. Use a prefilter. AP12.8 A machine tool control system is shown in Figure AP12.8. The transfer function of the power amplifier, prime mover, moving carriage, and tool bit is Kf s(s + l)(s + 4)(s + 5)' The goal is to have an overshoot less than 25% for a step input while achieving a peak time less than 3 seconds. Determine a suitable controller using (a) PD control, (b) PI control, and (c) PID control, (d) Then select the best controller. AP12.9 Consider a system with the structure shown in Figure 12.1 with r + las + av where 1 < a < 3 and 2 == K < 4. Use a PID controller and design the controller for the worst-case condition. We desire that the settling time (with a 2% criterion) be less than 0.8 second with an ITAE performance. AP12.10 A system of the form shown in Figure 12.1 has . G(s) = s +r -, (s + p)(s + q) 974 Chapter 12 Robust Control Systems Lead Screw External energy Differential amplifier Power amplifier J l|| Tool bit Gear box Rotating cam WlMMilU Moving tool carriage ^ \ t Prime mover Gc(s) ,,--,— WUUUMUIMUIMI o Position feedback -AAAq FIGURE AP12.8 A machine tool control system. where 3 < /; -& 5,0 ^ q £ 1, and 1 < /• < 2. We will use a compensator Gc(s) = K(s + zi)(s + z2) (s + p{)(s + p2) ' with all real poles and zeros. Select an appropriate compensator to achieve robust performance. G(s) = (s + 2)(5 + 4)(s + 6) We want to attain a steady-state error for a step input. Select a compensator Gc(s) and gain K< using the pseudo-QFT method, and determine the performance of the system when all the poles of G(s) change by - 5 0 % . Describe the robust nature of the system. AP12.11 A system of the form shown in Figure 12.44 has a plant DESIGN PROBLEMS CDP12.1 Design a P I D controller for the capstan-slide f k> system of Figure CDP4.1. The percent overshoot "V, should be less than 3 % and the settling time should be (with a 2 % criterion) less than 250 ms for a step input r{t). Determine the response to a disturbance for the designed system. DP12.1 A position control system for a large turntable is shown in Figure DP12.1(a), and the block diagram of the system is shown in Figure DP12.1(b) [11,14]. This system uses a large torque motor with K„, = 15. The objective is to reduce the steady-state effect of a step change in the load disturbance to 5 % of the magnitude of the step disturbance while maintaining a fast response to a step input command R(s), with less than 5 % overshoot. Select K] and the compensator when (a) Gc(s) = K and (b) Ge(s) = KP + KQS (a P D compensator). Plot the step response for the disturbance and the input for both compensators. Determine whether a prefilter is required to meet the overshoot requirement. DP12.2 Consider the closed-loop system depicted in Figure D P I 2.2. The process has a parameter K that is nominally K = \. Design a controller that results in a percent overshoot P.O. < 10% for a unit step input for all K in the range 0.1 < K < 2. DP12.3 Many university and government laboratories have constructed robot hands capable of grasping and manipulating objects. But teaching the artificial devices to perform even simple tasks required formidable computer programming. Now, however, the Dexterous Hand Master (DHM) can be worn over a human hand to record the side-to-side and bending motions of finger joints. Each joint is fitted with a sensor that changes its signal depending on position. The signals from all the sensors are translated into computer data and used to operate robot hands [1]. The D H M is shown in parts (a) and (b) of Figure DP12.3. The joint angle control system is shown in part (c). The normal value of Km is 1.0. The goal is to design a PID controller so that the steady-state error 975 Torque motor Position sensor Position signal (a) T,U) R HQ—• >'(.v) FIGURE DP12.1 Turntable control. FIGURE DP12.2 A unity feedback system with a process with varying parameter K. (b) R(s) Controller Process Gc(s) K s(s + 5) \. for a ramp input is zero. Also, the settling time (with a 2% criterion) must be less than 3 seconds for the ramp input. We want the controller to be Gc(s) = KD(s2 + 65 + 18) (a) Select KD and obtain the ramp response. Plot the root locus as KD varies, (b) If K„, changes to one-half of its normal value and Gc(s) remains as designed in part (a), obtain the ramp response of the system. Compare the results of parts (a) and (b) and discuss the robustness of the system. DP12.4 Objects smaller than the wavelengths of visible light are a staple of contemporary science and technology. Biologists study single molecules of protein or DNA; materials scientists examine atomic-scale flaws in crystals; microelectronics engineers lay out circuit patterns only a few tenths of atoms thick. Until recently, this minute world could be seen only by cumbersome, often destructive methods, such as • Y(.s) electron microscopy and X-ray diffraction. It lay beyond the reach of any instrument as simple and direct as the familiar light microscope. New microscopes, typified by the scanning tunneling microscope (STM), are now available [3]. The precision of position control required is in the order of nanometers. The STM relies on piezoelectric sensors that change size when an electric voltage across the material is changed. The "aperture" in the STM is a tiny tungsten probe, its tip ground so fine that it may consist of only a single atom and measure just 0.2 nanometer in width. Piezoelectric controls maneuver the tip to within a nanometer or two of the surface of a conducting specimen—so close that the electron clouds of the atom at the probe tip and of the nearest atom of the specimen overlap. A feedback mechanism senses the variations in tunneling current and varies the voltage applied to a third, z-axis, control. The z-axis piezoelectric moves the probe vertically to stabilize the current and to maintain a constant gap between the 976 Chapter 12 Robust Control Systems (b) (a.i Motor and joint Controller n.v) K Ms) > FIGURE DP12.3 Dexterous Hand Master. 17,640 s(s2 + 59 As + 1764)' and the controller is chosen to have two real, unequal zeros so that we have Gc(s) Kj(TlS + l)(T2S + 1) (a) Use the ITAE design method to determine Gc(s). (b) Determine the step response of the system with and without a prefilter Gp(s). (c) Determine the response of the system to a disturbance when Td(s) = l/s. (d) Using the prefilter and Gc(s) of parts (a) and (b), determine the actual response when the process changes to G(s) = s(s + 5)0 + 10) angle (0 microscope's tip and the surface. The control system is shown in Figure DP12.4(a), and the block diagram is shown in Figure DP12.4(b).The process is G(s) m Gt.(s) 16,000 s(s2 + 4Qs + 1600)' DP12.5 The system described in DP12.4 is to be designed using the frequency response techniques described in Section 12.6 with Gc(s) = K,(T,S + l)(r 2 5 + 1) Select the coefficients of Gc(s) so that the phase margin is approximately 45°. Obtain the step response of the system with and without a prefilter Gp(s). DP12.6 The use of control theory to provide insight into neurophysiology has a long history. As early as the beginning of the last century, many investigators described a muscle control phenomenon caused by the feedback action of muscle spindles and by sensors based on a combination of muscle length and rate of change of muscle length. This analysis of muscle regulation has been based on the theory of single-input, single-output control systems. An example is a proposal that the stretch reflex is an experimental observation of a motor control strategy, namely, control of individual muscle length by the spindles. Others later proposed the regulation of individual muscle stiffness (by sensors of both length and force) as the motor control strategy [30]. One model of the human standing-balance mechanism is shown in Figure DPI2.6. Consider the case of a paraplegic who has lost control of his standing mechanism. We propose to add an artificial 977 Design Problems c-axis control I Reference signal voltage ^ Probe (a) W Process Controller G((.v) G„(s) Desired gap i FIGURE DP12.4 Microscope control. -k^ Y(s) Gap G(s) (b) Muscle dynamics and nervous system Artificial controller R(s) Gp(s) leg ansle I tnJ _ 1 G((s) K Nerves 2 s + as + b J Y(s) -» Leg angle FIGURE DP12.6 Artificial control of standing and leg articulation. controller to enable the person to stand and move his legs, (a) Design a controller when the normal values of the parameters are K - 10, a = 12, and b = 100, in order to achieve a step response with percent overshoot less than 10%, steady-state error less than 5%, and a settling time (with a 2% criterion) less than 2 seconds. Try a controller with proportional gain, PI, PD, and PID. (b) When the person is fatigued, the parameters may change to K = 15, a - 8, and b = 144. Examine the performance of this system with the controllers of part (a). Prepare a table contrasting the results of parts (a) and (b). DP12.7 The goal is to design an elevator control system so that the elevator will move from floor to floor 1 Biosensors J rapidly and stop accurately at the selected floor (Figure DPI2.7). The elevator will contain from one to three occupants. However, the weight of the elevator should be greater than the weight of the occupants; you may assume that the elevator weighs 1000 pounds and each occupant weighs 150 pounds. Design a system to accurately control the elevator to within one centimeter. Assume that the large DC motor is field-controlled. Also, assume that the time constant of the motor and load is one second, the time constant of the power amplifier driving the motor is onehalf second, and the time constant of the field is negligible. We seek an overshoot less than 6% and a settling time (with a 2% criterion) less than 4 seconds. 978 Chapter 12 Robust Control Systems mo, w(o Drum v(0 Motor Command from computer FIGURE DP12.7 Elevator position control. FIGURE DP12.8 Feedback control system for an electric ventricular assist device. Controller R(s) Desired flow rate Gc(s) Motor, pump, and blood sac Y(s) G(s) - e~sT L DP12.8 A model of the feedback control system is shown in Figure DP12.8 for an electric ventricular assist device. This problem was introduced in AP9.11. The motor, pump, and blood sac can be modeled by a time delay with T = 1 s. The goal is to achieve a step response with less than 5% steady-state error and less than 10% overshoot. Furthermore, to prolong the batteries, the voltage is limited to 30 V [26]. Design a controller using (a) Gc(s) = K/s, (b) a PI controller, and (c) a PID controller. In each case, also design the pre-filter Gp(s). Compare the results for the three controllers by recording in a table the percent overshoot, peak time, settling time (with 2% criterion) and the maximum value of v(t). DP12.9 One arm of a space robot is shown in Figure DP12.9(a). The block diagram for the control of the arm is shown in Figure DP12.9(b). The transfer function of the motor and arm is G(s) = Vis) Motor voltage 1 s(s + 10) Blood flow rate (a) If Gc(s) = K, determine the gain necessary for an overshoot of 4.5%, and plot the step response, (b) Design a proportional plus derivative (PD) controller using the ITAE method and w„ = 10. Determine the required prefilter Gp(s). (c) Design a PI controller and a prefilter using the ITAE method, (d) Design a PID controller and a prefilter using the ITAE method with co„ - 10. (e) Determine the effect of a unit step disturbance for each design. Record the maximum value of )'(/) and the final value of y(r) for the disturbance input, (f) Determine the overshoot, peak time, and settling time (with a 2% criterion) step R(s) for each design above, (g) The process is subject to variation due to load changes. Find the magnitude of the sensitivity at a) = 5, \SG(fi)\, where T = Gc(s)G(s) 1 + Gc(s)G(s)' (h) Based on the results of parts (e), (f), and (g), select the best controller. 979 Design Problems (a) W * Y(s) R(s) FIGURE DP12.9 Space robot control. (b) DP12.10 A photovoltaic system is mounted on a space station in order to develop the power for the station. The photovoltaic panels should follow the Sun with good accuracy in order to maximize the energy from the panels. The system uses a DC motor, so that the transfer function of the panel mount and the motor is G(s) = l s(s + 19)' We will select a controller Gc(s) assuming that an optical sensor is available to accurately track the sun's position, and thus H(s) = 1. The goal is to design Gc(s) so that (1) the percent overshoot to a step is less than 7% and (2) the steadystate error to a ramp input is less than or equal to 1%. Determine the best phase-lead controller. Examine the robustness of the system to a 10% variation in the motor time constant. DP12.11 Electromagnetic suspension systems for aircushioned trains are known as magnetic levitation (maglev) trains. One maglev train uses a superconducting magnet system [17]. It uses superconducting coils, and the levitation distance x(t) is inherently unstable. The model of the levitation is G(s) = X(s) K v(s) (*T, -i-1)(*2 - o>iy where V(s) is the coil voltage; TX is the magnet time constant; and u}\ is the natural frequency. The system uses a position sensor with a negligible time constant. A train traveling at 250 km/hr would have T± - 0.75  and (x>] = 75 rad/s. Determine a controller that can maintain steady, accurate levitation when disturbances occur along the railway. Use the system model of Figure 12.1. DP12.12 Consider again the Mars rover problem described in DP6.2.The system uses a PID controller, and a robust system is desired. The specifications are (1) maximum overshoot equal to 18%, (2) settling time (with a 2% criterion) less than 2 seconds, (3) rise time equal to or greater than 0.20 to limit the power requirements, (4) phase margin greater than 65°, (5) gain margin greater than 8 dB, (6) maximum root sensitivity (magnitude of real and imaginary parts) less than 1. Select the best value of the gain K. DP12.13 A benchmark problem consists of the massspring system shown in Figure DP12.13, which represents a flexible structure. Let m\ = m2 — 1 and 0.5 :£ k < 2.0 [29]. It is possible to measure X\ and x2 and use a controller prior to u{t). Obtain the system description, choose a control structure, and design a robust system. Determine the response of the system to a unit step disturbance. Assume that the output x2(t) is the variable to be controlled. 980 Chapter 12 Robust Control Systems x? - •- i u m VWW- \ Input FIGURE DP12.13 Two-mass cart system. (o) I (o) „ d m2 Disturbance (o) (o) EF|ii| COMPUTER PROBLEMS CP12.1 A closed-loop feedback system is shown in responses for 0.5 < p < 20, with the gain K as deterFigure CP12.1. Use an m-file to obtain a plot of | 5 ^ | mined above. Plot the settling time as a function of p. versus u>. Plot \T(s)\ versus w, where T(s) is the CP12.3 Consider the control system in Figure CP12.3, closed-loop transfer function. where The value of / is known to change slowly with time, although, for design purposes, the nominal value is chosen to be J = 25. (a) Design a PID compensator (denoted by Gc(s)) to achieve a phase margin greater than 45° and a bandwidth less than 4 rad/s. (b) Using the PID controller designed in part (a), develop an m-file script to generate a plot of the phase margin as J varies from 10 to 40. At what J is the closed-loop system unstable. + Y(s) FIGURE CP12.1 Closed-loop feedback system with gain K. CP12.2 An aircraft aileron can be modeled as a firstorder system G(s) R(s) s + p where p depends on the aircraft. Obtain a family of step responses for the aileron system in the feedback configuration shown in Figure CP12.2. The nominal value of p = 10. Compute a reasonable value of K so that the step response (with p = 10) has Ts < 0.1s. Then, use an m-file to obtain the step Aileron P s+p K R(s) k FIGURE CP12.2 aircraft aileron. • Y(s) Closed-loop control system for the Controller FIGURE CP12.4 A feedback control system with uncertain parameter b. + R(s) * rJ^ i k _ K FIGURE CP12.3 A feedback control system with compensation. CP12.4 Consider the feedback control system in Figure CP12.4. The exact value of parameter b is unknown; however, for design purposes, the nominal value is taken to be h = 4. The value of a - 8 is known very precisely. (a) Design the proportional controller K so that the closed-loop system response to a unit step input has a settling time (with a 2% criterion) less than 5 seconds and an overshoot of less than 10%. Use the nominal value of b in the design. (b) Investigate the effects of variations in the parameter b on the closed-loop system unit step response. Plant l s + bs + a •*- Y(s) 981 Computer Problems Let b = 0,1,4, and 40, and co-plot the step response associated with each value of b. In all cases, use the proportional controller from part (a). Discuss the results. CP12.7 A unity negative feedback loop has the loop transfer function a(s - 0.5) Gc(s)G(s) = ^ T T T T s + 2s + \ We know from the underlying physics of the problem that the parameter a can vary only between 0 < a < 1. Develop an m-file script to generate the following plots: (a) The steady-state tracking error due to a negative unit step input (i.e., R(s) = —1/s ) versus the parameter a. (b) The maximum percent initial undershoot (or overshoot) versus parameter a. (c) The gain margin versus the parameter a. (d) Based on the results in parts (a)-(c), comment on the robustness of the system to changes in parameter a in terms of steady-state errors, stability, and transient time response. CP12.5 A model of a flexible structure is given by C(5) = (1 + ka>,?)s2 + 2£a)ns + w„2 .V2(A'2 + 2£cj)ts + a>„2) where a>„ is the natural frequency of the flexible mode, and £ is the corresponding damping ratio. In general, it is difficult to know the structural damping precisely, while the natural frequency can be predicted more accurately using well-established modeling techniques. Assume the nominal values of ca„ - 2 rad/s, f - 0.005, and* = 0.1. (a) Design a lead compensator to meet the following specifications: (1) a closed-loop system response to a unit step input with a settling time (with a 2% criterion) less than 200 seconds and (2) an overshoot of less than 50%. (b) With the controller from part (a), investigate the closed-loop system unit step response with £ = 0, 0.005, 0.1, and 1. Co-plot the various unit step responses and discuss the results. (c) From a control system point of view, is it preferable to have the actual flexible structure damping less than or greater than the design value? Explain. CP12.6 The industrial process shown in Figure CP12.6 is known to have a time delay in the loop. In practice, it is often the case that the magnitude of system time delays cannot be precisely determined. The magnitude of the time delay may change in an unpredictable manner depending on the process environment. A robust control system should be able to operate satisfactorily in the presence of the system time delays. (a) Develop an m-file script to compute and plot the phase margin for the industrial process in Figure CP12.6 when the time delay, T, varies between 0 and 5 seconds. Use the pade function with a second-order approximation to approximate the time delay. Plot the phase margin as a function of the time delay. (b) Determine the maximum time delay allowable for system stability. Use the plot generated in part (a) to compute the maximum time delay approximately. Controller FIGURE CP12.6 An industrial controlled process with a time delay in the loop. fi{s) »nj 1 w K CP12.8 The Gamma-Ray imaging Device (GRID) is a NASA experiment to be flown on a long-duration, high-altitude balloon during the coming solar maximum. The GRID on a balloon is an instrument that will qualitatively improve hard X-ray imaging and carry out the first gamma-ray imaging for the study of solar high-energy phenomena in the next phase of peak solar activity. From its long-duration balloon platform, GRID will observe numerous hard X-ray bursts, coronal hard X-ray sources, "superhot" thermal events, and microflares [2]. Figure CP12.8(a) depicts the GRID payload attached to the balloon. The major components of the GRID experiment consist of a 5.2meter canister and mounting gondola, a high-altitude balloon, and a cable connecting the gondola and balloon. The instrument-sun pointing requirements of the experiment are 0.1 degree pointing accuracy and 0.2 arcsecond per 4 ms pointing stability. An optical sun sensor provides a measure of the sun-instrument angle and is modeled as a first-order system with a DC gain and a pole at s = -500. A torque motor actuates the canister/gondola assembly. The azimuth angle control system is shown in Figure CP12.8(b). Tlie PID controller is selected by the design team so that Gc(s) Delay • e~r> KD(s2 + as + b) Plant • 1 s2 +105 + 2 • Yls) 982 Chapter 12 Robust Control Systems Tether to balloon GRID payload (a) Prefilter Controller Motor Balloon and canister-gondola dynamics GJs) Gc(s) 1 ,v + 2 1 (s + 4)(s + 10) tf(.v) Yls) Azimuth unfile FIGURE CP12.8 (b) The GRID device. Develop a simulation to study the control system performance. Use a step response to confirm the percent overshoot meets the specification. where a and b are to be selected. A prefilter is used as shown in Figure CP12.8(b). Determine the value of KD, a, and b so that the dominant roots have a £ of 0.8 and the overshoot to a step input is less than 3%. m ANSWERS TO SKILLS CHECK True or False: (1) True; (2) False; (3) True; (4) True; (5) False Multiple Choice: (6) b; (7) b; (8) c; (9) d; (10) a; (11) d; (12) a; (13) c; (14) a; (15) b Word Match (in order, top to bottom): d, g, i, b, c, k, f,h,a,j,e TERMS AND CONCEPTS Additive perturbation A system perturbation model expressed in the additive form Ga(s) = G(s) + A(s), where G(s) is the nominal process function, A (s) is the perturbation that is bounded in magnitude, and Gn(s) is the family of perturbed process functions. Complementary sensitivity function The function Gc{s)G(s) T(s) = — . —: . that satisfies the relationship 1 + Gc(s)G(s) S(s) + T(s) = 1, where S(s) is the sensitivity function. The function T(s) is the closed-loop transfer function. Internal model principle The principle that states that if Gc(s)G(s) contains the input R(s), then the output y(t) will track R(s) asymptotically (in the steady-state) and the tracking is robust. Multiplicative perturbation A system perturbation model expressed in the multiplicative form G,„(s) = G(s)(l + M(s)), where G(s) is the nominal process function, M(s) is the perturbation that is bounded in magnitude, and G,„(s) is the family of perturbed process functions. PID controller A controller with three terms in which the output is the sum of a proportional term, an integrating 983 Terms and Concepts term, and a differentiating term, with an adjustable gain for each term. Prefilter A transfer function Gp(s) that filters the input signal R(s) prior to the calculation of the error signal. Process controller See PID controller. Robust control system A system that exhibits the desired performance in the presence of significant plant uncertainty. Robust stability criterion A test for robustness with respect to multiplicative perturbations in which stability I is guaranteed if \M(i(o)\ < 1 + _,,. x , for all where M(s) is the multiplicative perturbation. Root sensitivity A measure of the sensitivity of the roots (i.e., the poles and zeros) of the system to changes in a parameter defined by S^ = ——, where a is the da/a parameter and /', is the root. Sensitivity function The function S(s) = [1 4- Gc(s)G(s)]~l that satisfies the relationship S(s) + T(s) = 1, where T(s) is the complementary sensitivity function. System sensitivity A measure of the system sensitivity to dT/T T changes in a parameter defined by S„ = ——, where da/a a is the parameter and 7"is the system transfer function. 1024 FIGURE 13.44 Feedback control system with a digital controller. Note that G(z) = Z[G0(s)Gp(s)}. Chapter 13 R(z) O Digital Control Systems Eiz) D(z) G(z) •• Y(Z) we have 1 - e-sT s s(s + 20) sT We note that for s = 20 and 7 = 1 ms, e~ is equal to 0.98. Then we see that the pole at 5 = -20 in Equation (13.71) has an insignificant effect. Therefore, we could approximate GQ(s)GJs) = 0.25 GJs) Then we need G(z) = Z 1 - = (1 - z"1)(0.25)Z = (1 - z"1)(0.25) 0.257 1 Tz (z ~ 1)2 0.25 X 10 -3 1 We need to select the digital controller D(z) so that the desired response is achieved for a step input. If we set D{z) = K, then we have D(z)G(z) = £(0.25 X 10"3) z - 1 The root locus for this system is shown in Figure 13.45. When K = 4000, D(z)G(z) = 1 z - 1' Therefore, the closed-loop transfer function is T(z) = D(z)G(z) 1 + D(z)G(z) z We expect a rapid response for the system. The percent overshoot to a step input is 0%, and the settling time is 2 ms. 1025 Skills Check Unit circle Root when K = 4000 FIGURE 13,45 Root locus. 13.13 SUMMARY The use of a digital computer as the compensation device for a closed-loop control system has grown during the past two decades as the price and reliability of computers have improved dramatically. A computer can be used to complete many calculations during the sampling interval T and to provide an output signal that is used to drive an actuator of a process. Computer control is used today for chemical processes, aircraft control, machine tools, and many common processes. The z-transform can be used to analyze the stability and response of a sampled system and to design appropriate systems incorporating a computer. Computer control systems have become increasingly common as low-cost computers have become readily available. m SKILLS CHECK In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 13.46 as specified in the various problem statements. R(s) -\ iO e(/) _/ Zero-order hold Process G0(s) Op(s) 1 FIGURE 13.46 Block diagram for the Skills Check. 1026 Chapter 13 Digital Control Systems In the following True or False and Multiple Choice problems, circle the correct answer. 1. A digital control system uses digital signals and a digital computer to control a process. 2. The sampled signal is available only with limited precision. 3. Root locus methods are not applicable to digital control system design and analysis. 4. A sampled system is stable if all the poles of the closed-loop transfer function lie outside the unit circle of the z-plane. 5. The /-transform is a conformal mapping from the s-plane to the z -plane by the relation z = esT. 6. Consider the function in the s-domain True or False True or False True or False True or False True or False s(s + 2)0 + 6)' Let The the sampling time. Then, in the z-domain the function Y(s) is 5 z 5 z , 5 z v/ N 2T + a. Y(z) = --—r !== 6 z - 1 - 74 z - e~ b c -y(z)^ri"^ Y(z)-l 6z - 1 7 z - e~6T + ^ 12 z - e~27 S 7 Z w 6 z - 1 1 - e~2T 6 1 - e~6T 7. The impulse response of a system is given by Y(z) z3 - 25z2 + 0.6z' Determine the values of y(nT) at the first four sampling instants. a. y(0) = 1, y(T) = 27, y(2T) = 647, y(37) = 660.05 b. y(0) = 0, y(T) = 27, y(2T) = 47, y(3T) - 60.05 c y(0) = l,y(T) = 27, y(27) = 674.4, y(3T) - 16845.8 d. y(0) = l,y(7) = 647,y(27) = 47,y(3T) = 27 8. Consider a sampled-data system with the closed-loop system transfer function T(z) = K This system is: a. Stable for all finite K. b. Stable for -0.5 < K < oo. c. Unstable for all finite K. d. Unstable for -0.5 < K < oo. Z2 + 0.2z - 0.5' 1027 Skills Check 9. The characteristic equation of a sampled system is q{z) = z2 + (2K - 1.75)2 + 2.5 = 0, where K > 0. The range of K for a stable system is: a. 0 < K < 2.63 b. K > 2.63 c. The system is stable for all K > 0. d. The system is unstable for all K > 0. 10. Consider the unity feedback system in Figure 13.46, where IS G (S) = " ,(0.2, + 1) with the sampling time T = 0.4 second. The maximum value for K for a stable closedloop system is which of the following: a. K = 7.25 b. K = 10.5 c Closed-loop system is stable for all finite K d. Closed-loop system is unstable for all K > 0 In Problems 11 and 12, consider the sampled data system in Figure 13.46 where p w 225 s + 225 2 11. The closed-loop transfer function T(z) of this system with sampling at T = 1 second is _, , 1.76« + 1.76 Z2 + 3.279z + 2.76 a. T(z) = z + 1.76 z1 + 2.76 b. T{z) = 1.76z + 1.76 Z2 + 1.519z + 1 c. T(z) = z 12. The unit step response of the closed-loop system is: 1.76z + 1.76 a. Y(z) = 2 z + 3.279z + 2.76 1.76z + 1.76 b. Y{z) = 3 z + 2.279z2 - 0.5194* - 2.76 1.76z2 + 1.76z c. Y(z) = 2 z + 2.279z2 - 0.5194z - 2.76 1.76Z2 + 1.76* , „, , d. Y(z) = z 2.279z2 - 0.5194* - 2.76 1028 Chapter 13 Digital Control Systems In Problems 13 and 14, consider the sampled data system with a zero-order hold where 20 Gp(s) = s(s + 9)' 13, The closed-loop transfer function T(z.) of this system using a sampling period of T = 0.5 second is which of the following: _, . 1.76* + 1.76 a. T(z) = 2 z + 2.76 0.87^ + 0.23 b. T(z) r - 0.14z + 0.24 0.87z + 0.23 c. T(z) z2 - 1.0U + 0.011 0.92z + 0.46 d. T(z) LOU 14. The range of the sampling period T for which the closed-loop system is stable is: a. T < 1.12 b. The system is stable for all T > 0. c. 1.12 < T < 10 d. T < 4.23 15. Consider a continuous-time system with the closed-loop transfer function T(s) s2 + 4s + 8 Using a zero-order hold on the inputs and a sampling period of T = 0.02 second, determine which of the following is the equivalent discrete-time closed-loop transfer function representation: 0.019z - 0.019 a. T(z) = z2 + 2.76 0.87z + 0.23 b. T(z) = 2 z - 0A4z + 0.24 0.019z - 0.019 c T(z) = 2 z - l.9z + 0.9 0.043^ - 0.02 d. T(z) = 1 Z + 1.9231 In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided. a. Precision b. Digital computer compensator c. s-plane d. Backward difference rule A system where part of the system acts on sampled data (sampled variables). The stable condition exists when all the poles of the closed-loop transfer function T(z) are within the unit circle on the z-plane. The plane with the vertical axis equal to the imaginary part of z and the horizontal axis equal to the real part of z. A control system using digital signals and a digital computer to control a process. 1029 Exercises e. Minicomputer f. Sampled-data system g. Sampled data h. Digital control system i. Microcomputer j . Forward rectangular integration k. Stability of a sampled-data system 1. Amplitude quantization error m. PID controller n. z -transform o. Sampling period p. Zero-order hold Data obtained for the system variables only at discrete intervals. The period when all the numbers leave or enter the computer. A conformal mapping from the s-plane to the z -plane by the relation z = e*r. The sampled signal available only with a limited precision. A system that uses a digital computer as the compensator element. A computational method of approximating the time derivative of a function. A computational method of approximating the integration of a function. A small personal computer (PC) based on a microprocessor. A stand-alone computer with size and performance between a microcomputer and a large mainframe. A controller with three terms in which the output is the sum of a proportional term, an integral term, and a differentiating term. The degree of exactness or discrimination with which a quantity is stated. A mathematical model of a sample and data hold operation. EXERCISES E13.1 State whether the following signals are discrete or continuous: (a) Elevation contours on a map. (b) Temperature in a room. (c) Digital clock display. (d) The score of a basketball game. (e) The output of a loudspeaker. E13.2 (a) Find the values y(kT) when Y(z) r - 3z + 2 for k = 0 to 4. (b) Obtain a closed form of solution for y(kT) as a function of k. Answer: v(0) = 0, y(T) = 1, y(2T) = 3, y(3T) = 7, y(4T) = 15 E13.3 A system has a response y{kT) = kT for k Find Y(z) for this response. Tz Answer: Y(z) = r (z - 1)2 E13.4 We have a function 5 s(s + 2)(5 + 10)* Y(s) Using a partial fraction expansion of Y(s) and Table 13.1, find Y(z) when T - 0 . 1 s . E13.5 The space shuttle, with its robotic arm, is shown in Figure E13.5(a). An astronaut controls the robotic arm and gripper by using a window and the TV cameras [9]. Discuss the use of digital control for this system and sketch a block diagram for the system, including a computer for display generation and control. 1030 Chapter 13 Digital Control Systems TV camera and lights "\Robot arm TV camera i TV camera (a) FIGURE E13.5 (a) Space shuttle and robotic arm. (b) Astronaut control of the arm. (b) E13.6 Computer control of a robot to spraypaint an automobile is shown by the system in Figure E13.6 [l].The system is of the type shown in Figure 13.24, where E13J Find the response for the first four sampling instants for Y{z) = Then find y(Q),y(l), and we want a phase margin of 45°. A compensator for this system was obtained in Section 10.8. Obtain the D(z) required when T = 0.001 s. z3 + 2z2 + 1 3 z - l.5z2 + Q.5z y(2), and y(3). Line conveyor iiiHiHitiitiiHiiiiiiiimnnmK /-*>. Line encoder Screw Tabic encoder FIGURE E13.6 Automobile spraypaint system. Computer Input 1031 Problems E13.8 Determine whether the closed-loop system with T(z) is stable when sine wave input with the same frequency as the natural frequency of the system. E13.12 T(z) = Find the ^-transform of r + Q.2z - 1.0 X(s) = when the sampling period is 1 second. (a) Determine y(kT) for k = 0 to 3 when E13.13 z + 1 Y(z) = 1 (b) Determine the closed form solution for y(kT) as a function of k. E13.10 A system has G{z) as described by Equation (13.34) with T = 0.01 s and r = 0.008 s. (a) Find K so that the overshoot is less than 40%. (b) Determine the steady-state error in response to a unit ramp input. (c) Determine K to minimize the integral squared error. E13.ll A system has a process transfer function Gp(s) = 100 2 s + 100' (a) Determine G(z) for Gp(s) preceded by a zero-order hold with T = 0.05 s. (b) Determine whether the digital system is stable, (c) Plot the impulse response of G(z) for the first 15 samples, (d) Plot the response for a FIGURE E13.15 An open-loop sampled-data system with sampling time Zero-order hold r*(t) tit) GQ(s) T= 1 r=is. FIGURE E13.16 An open-loop sampled-data system with sampling time T = 0.5 s. + 1 s + 5s + 6 Answer: unstable E13.9 " 2 Zero-order hold r*(t) >•(/) 0.5 Got*) The characteristic equation of a sampled system is z2 + (K - 4)z + 0.8 = 0. Find the range of K so that the system is stable. Answer: 2.2 < K < 5.8 E13.14 A unity feedback system, as shown in Figure 13.18, has a plant K GJs) s(s + 3)' with T = 0.5. Determine whether the system is stable when K = 5. Determine the maximum value of K for stability. E 13.15 Consider the open-loop sampled-data system shown in Figure E13.15. Determine the transfer function G(z) when the sampling time is T = 1 s. E13.16 Consider the open-loop sampled-data system shown in Figure E13.16. Determine the transfer function G(z) and when the sampling time T - 0.5 s. GM (s + 1 )(s + 4) •*- >'(/) Gp(s) 3 5(5 + 2) • * v(0 PROBLEMS P13.1 The input to a sampler is r(t) = sin(wf), where co = I/IT. Plot the input to the sampler and the output r*{t) for the first 2 seconds when T - 0.25 s. P13.2 The input to a sampler is r{t) = sin(art), where (o = l/7r. The output of the sampler enters a zeroorder hold, as shown in Figure 13.7. Plot the output of the hold circuit p(t) for the first 2 seconds when T = 0.25 s. P13.3 A unit ramp r(t) — *, t > 0, is used as an input to a process where G(s) = l/(s + 1), as shown in Figure P13.3. Determine the output y(kT) for the first four sampling instants. 1032 Chapter 13 Digital Control Systems + yU) FIGURE P13.3 Sampling system. j 3 fS£| -f*\ P13.4 A closed-loop system has a hold circuit and process as shown in Figure 13.18. Determine G(z) when T = 1 and TV camera Gp(s) s +2 P13.5 For the system in Problem P13.4, let r(r) be a unit step input and calculate the response of the system by synthetic division. P13.6 For the output of the system in Problem P13.4, find the initial and final values of the output directly from Y(z). P13.7 A closed-loop system is shown in Figure 13.18. This system represents the pitch control of an aircraft. The process transfer function is Gp(s) = K/[s(0.5s + 1)]. Select a gain K and sampling period T so that the overshoot is limited to 0.3 for a unit step input and the steady-state error for a unit ramp input is less than 1.0. P13.8 Consider the computer-compensated system shown in Figure 13.24 when T — \ and KGJs) = K s(s + 10)' (a) Let D(z) = K and determine the transfer function G(z)D(z). (b) Determine the characteristic equation of the closed-loop system, (c) Calculate the maximum value of K for a stable system. (d) Determine K such that the overshoot is less than 30%. (e) Calculate the closed-loop transfer function T(z) for K of part (d) and plot the step response. (f) Determine the location of the closed-loop roots and the overshoot if K is one-half of the value determined in part (c). (g) Plot the step response for the K of part (f). P13.ll z - 0.3678 z +r Select within the range 1 < K < 2 and 0 < r < 1. Determine the response of the compensated system and compare it with the uncompensated system. P13.9 A suspended, mobile, remote-controlled system to bring three-dimensional mobility to professional NFL football is shown in Figure P13.9. The camera can be moved over the field as well as up and down. The motor control on each pulley is represented by Figure 13.18 with r M - 1Q (b) (c) (d) p{S) s(s + l)(s/10 + 1)" We wish to achieve a phase margin of 45° using Gc(s). Select a suitable crossover frequency and sampling period to obtain D(z). Use the Gc(s)-to-D(z) conversion method. P13.10 Consider a system as shown in Figure 13.15 with a zero-order hold, a process Gp(s) = and T = 0.1 s. 1 s(s + 10)' Motor and pulley FIGURE P13.9 Mobile camera for football field. Select the parameters K and r oiD{z) when D(z) = ~~ (e) (a) For the system described in Problem P13.10, design a lag compensator Gc(s) using the methods of Chapter 10 to achieve an overshoot less than 30% and a steady-state error less than 0.01 for a ramp input. Assume a continuous nonsampled system with Gp(s). Determine a suitable D(z) to satisfy the requirements of part (a) with a sampling period T = 0.1 s. Assume a zero-order hold and sampler, and use the Gc(s)-to-D(z) conversion method. Plot the step response of the system with the continuous-time compensator Gc(s) of part (a) and of the digital system with the D(z) of part (b). Compare the results. Repeat part (b) for T = 0.01 s and then repeat part (c). Plot the ramp response for D(z) with T = 0.1 s and compare it with the continuous-system response. P 13.12 The transfer function of a plant and a zero-order hold (Figure 13.18) is G(Z) = K{z + 0.5) z(z - 1) * (a) Plot the root locus, (b) Determine the range of gain K for a stable system. 1033 Advanced Problems P13.13 The space station orientation controller described in Exercise E7.6 is implemented with a sampler and hold and has the transfer function (Figure 13.18) P13.16 A closed-loop system as shown in Figure 13.18 has G K(z2 + 1.1206* - 0.0364) G(z) = -=3 : . z - 1.7358Z2 + 0.87Hz - 0.1353 (a) Plot the root locus, (b) Determine the value of K so that two of the roots of the characteristic equation are equal, (c) Determine all the roots of the characteristic equation for the gain of part (b). P13.14 A sampled-data system with a sampling period T = 0.05s (Figure 13.18) is K{z* + 10.36Hz2 + 9.758z + 0.8353) Z) 4 ~ z - 3.7123z3 + 5.1644? - 3.195z + 0.7408' (a) Plot the root locus, (b) Determine K when the two real poles break away from the real axis, (c) Calculate the maximum K for stability. P13.15 A closed-loop system with a sampler and hold, as shown in Figure 13.18, has a process transfer function Calculate and plot y{kT) for 0 < T < 0.6 when T - 0.1 s. The input signal is a unit step. ^ ) = ^ r e - calculate and plot y{kT) for 0 ^ k < 8 when T = 1 s and the input is a unit step. P13.17 A closed-loop system, as shown in Figure 13.18, has is G {S) > = , ( , + 0.75) and T = 1 s. Plot the root locus for K ^ 0, and determine the gain K that results in the two roots of the characteristic equation on the z-circle (at the stability limit). P13.18 A unity feedback system, as shown in Figure 13.18, has If the system is continuous (T = 0), then K = 1 yields a step response with an overshoot of 16% and a settling time (with a 2% criterion) of 8 seconds. Plot the response for 0 < T ^ 1.2, varying T by increments of 0.2 when K ~ 1. Complete a table recording overshoot and settling time versus T. ADVANCED PROBLEMS AP13.1 A closed-loop system, as shown in Figure 13.18, has a process K{\ + as) j2 , Gp(s) = where a is adjustable to achieve a suitable response. Plot the root locus when a = 10. Determine the range of K for stability when T = 1 s. AP13.2 A manufacturer uses an adhesive to form a seam along the edge of the material, as shown in Figure AP13.2. It is critical that the glue be applied evenly to avoid flaws; however, the speed at which the material passes beneath the dispensing head is not constant. The glue needs to be dispensed at a rate proportional to the varying speed of the material. The controller adjusts the valve that dispenses the glue [12]. The system can be represented by the block diagram shown in Figure 13.15, where Gp(s) = 2/(0.03, + 1 ) with a zero-order hold G()(s). Use a controller FIGURE AP13.2 A glue control system. 1034 Chapter 13 Digital Control Systems that represents an integral controller. Determine G(z)D(z) for T = 30 ms, and plot the root locus. Select an appropriate gain K and plot the step response. AP13.3 A system of the form shown in Figure 13.15 has D(z) = k and Gp(s) = 12 s(s + 12) When T = 0.05, find a suitable K for a rapid step response with an overshoot less than 10%. AP13.4 GJs) ms) AP13.5 Consider the closed-loop sampled-data system shown in Figure AP13.5. Determine the acceptable range of the parameter K for closed-loop stability. G0(.v) T =().1 10 s + f Determine the range of sampling period T for which the system is stable. Select a sampling period Tso that the system is stable and provides a rapid response. Zero-order hold FIGURE AP13.5 A closed-loop sampled-data system with sampling time T = 0.1 s. A system of the form shown in Figure 13.18 has GJs) sis + 3) •*> V(s) DESIGN PROBLEMS CDP13.1 Design a digital controller for the system using f ^\ the second-order model of the motor-capstan-slide as described in CDP2.1 and CDP4.1. Use a sampling period of T = 1 ms and select a suitable D(z) for the system shown in Figure 13.15. Determine the response of the designed system to a step input r(t). DP13.1 A temperature system, as shown in Figure 13.15. has a process transfer function 0.8 c o(s) - 1 , + 1 and a sampling period T of 0.5 second. (a) Using D(z) = K, select a gain K so that the system is stable, (b) The system may be slow and overdamped, and thus we seek to design a lead network using the method of Section 10.5. Determine a suitable controller Gc(s) and then calculate D(z). (c) Verify the design obtained in part (b) by plotting the step response of the system for the selected D(z)DP13.2 A disk drive read-write head-positioning system has a system as shown in Figure 13.15 [1 l].The process transfer function is Q GJs) = 2 s + 0.855 + 788 Accurate control using a digital compensator is required. Let T = 10 ms and design a compensator, D(z), using (a) the Gc(s)-to-D(z) conversion method and (b) the root locus method. DP13.3 Vehicle traction control, which includes antiskid braking and antispin acceleration, can enhance vehicle performance and handling. The objective of this control is to maximize tire traction by preventing the wheels from locking during braking and from spinning during acceleration. Wheel slip, the difference between the vehicle speed and the wheel speed (normalized by the vehicle speed for braking and the wheel speed for acceleration), is chosen as the controlled variable for most of the traction-control algorithm because of its strong influence on the tractive force between the tire and the road [17]. A model for one wheel is shown in Figure DPI3.3 when v is the wheel slip. The goal is to minimize the slip when a disturbance occurs due to road conditions. Design a controller D(z) so that the £ of the system is 1 / V 2 , and determine the resulting K. Assume T = 0.1 s. Plot the resulting step response, and find the overshoot and settling time (with a 2 % criterion). 1035 Design Problems Disturbance Controller * U) = 0 Desired FIGURE DP13.3 Vehicle fraction control system. l+ _+/~v *~

z+ 1 (z- 1)(2-0 5)

. Yiz) Slip

slip

temperatures. The heat produced by the heaters in the barrel, together with the heat released from the friction between the raw polymer and the surfaces of the barrel and the screw, eventually causes the melting of the polymer, which is then pushed by the screw out from the die, to be processed further for various purposes. The output variables are the outflow from the die and the polymer temperature. The main controlling variable is the screw speed, since the response of the process to it is rapid. The control system for the output polymer temperature is shown in Figure DP13.5. Select a gain K and a sampling period T to obtain a step overshoot of 10% while reducing the steady-state error for a ramp input.

DP13.4 A machine-tool system has the form shown in Figure 13.28 with [10] CM

O.J s(s + 0.1)'

The sampling rate is chosen as T = I s . We desire the step response to have an overshoot of 16% or less and a settling time (with a 2 % criterion) of 12 seconds or less. Also, the error to a unit ramp input, r(t) = r, must be less than or equal to 1. Design a D(z) to achieve these specifications. DP13.5 Plastic extrusion is a well-established method widely used in the polymer processing industry [12]. Such extruders typically consist of a large barrel divided into several temperature zones, with a hopper at one end and a die at the other. Polymer is fed into the barrel in raw and solid form from the hopper and is pushed forward by a powerful screw. Simultaneously, it is gradually heated while passing through the various temperature zones set in gradually increasing

DP13.6 A sampled-data system closed-loop block diagram is shown in Figure DPI 3.6. Design D{z) to such that the closed-loop system response to a unit step response has a percent overshoot P.O. s 12% and a settling time T% < 20 s.

Polymer

Heated barrel

Die ^j.

^

^

J^j,

5UT« y_^_

(a)

m

Temperature setting FIGURE DP13.5 Control system for an extruder.

(b)

Actual temperature

1036

Chapter 13

Digital Control Systems

FIGURE DP13.6 A closed-loop sampled-data system with sampling time 7 = 1s.

R(s)

COMPUTER PROBLEMS CP13.1 Develop an m-file to plot the unit step response of the system G(Z)

CP13.3 The closed-loop transfer function of a sampleddata system is given by

0.2U5z + 0.1609 Z2 - 0.75z + 0.125'

T(z) =

Verify graphically that the steady-state value of the output is 1. CP13.2 Convert the following continuous-time transfer functions to sampled-data systems using the c2d function. Assume a sample period of 1 second and a zeroorder hold GQ(S).

Y(z) _ 1.7(z + 0.46) R(z) ~ z2 + z + 0.5'

(a) Compute the unit step response of the system using the step function, (b) Determine the continuous-time transfer function equivalent of T(z) using the d2c function and assume a sampling period of T = 0.1 s. (c) Compute the unit step response of the continuous (nonsampled) system using the step function, and compare the plot with part (a).

(a) Gp(s) = -

s (b) Gp{s) =

CP13.4 Plot the root locus for the system

s2 + 2

G(z)D(z) = K V - z + 0.45'

s +4 (c) Gn(s) = s +3 (d) Gp(s) =

Find the range of K for stability. CP13.5 Consider the feedback system in Figure CP13.5. Obtain the root locus and determine the range of K for stability.

1 s(s + 8)

Controller R(z) FIGURE CP13.5

z-0.2 z-0.8

Process

£±1

*

Y(z)

2-1

Control system with a digital controller.

CP13.6 Consider the sampled data system with the loop transfer function r + 3z + 3.75 G(z)D(z) = K 2 z - 0.2z - 1.9' (a) Plot the root locus using the rlocus function. (b) From the root locus, determine the range of K for stability. Use the rlocfind function.

CP13.7 An industrial grinding process is given by the transfer function [15]

The objective is to use a digital computer to improve the performance, where the transfer function of the computer is represented by £>(<:). The design specifications are (1) phase margin greater than 45°, and (2) settling time (with a 2% criterion) less than 1 second.

1037

Terms and Concepts (a) Design a controller

(c) Simulate the continuous-time, closed-loop system with a unit step input, (d) Simulate the sampled-data, closed-loop system with a unit step input. (e) Compare the results in parts (c) and (d) and comment.

to meet the design specifications, (b) Assuming a sampling time of T = 0.02 s, convert Gc(s) to D(z).

m

ANSWERS TO SKILLS CHECK True or False: (1) True; (2) True; (3) False; (4) False; (5) True Multiple Choice: (6) a; (7) c; (8) a; (9) d; (10) a; (11) a; (12) c; (13) b; (14) a; (15) c

Word Match (in order, top to bottom): f, k, c, h, g, o, n, 1, b, d, j , i, e, m, a, p

TERMS AND CONCEPTS Amplitude quantization error The sampled signal available only with a limited precision. The error between the actual signal and the sampled signal.

term, with an adjustable gain for each term, given by K2Ts z - 1

Backward difference rule A computational method of approximating the time derivative of a function given x(kT) - x((k - l)T) by x(kT) w —— Y » w h e r e ' = kT^ Tis the sample time, and k = 1,2, Digital computer compensator A system that uses a digital computer as the compensator element.

Pi Precision The degree of exactness or discrimination with which a quantity is stated. cfl Sampled data Data obtained for the system variables only at discrete intervals. Data obtained once every sampling period. Sa Sampled-data system A system where part of the system acts on sampled data (sampled variables). Sa Sampling period The period when all the numbers leave or enter the computer. The period for which the sampled variable is held constant. Stability of a sampled-data system The stable condition St exists when all the poles of the closed-loop transfer function T(z) are within the unit circle on the z-plane. z-plane The plane with the vertical axis equal to the z-\ imaginary part of z and the horizontal axis equal to the real part of z. A conformal mapping from the i-plane to Z2-transform H the z-plane by the relation z = esT. A transform from the .?-domain to the z-domain. Z ( Zero-order hold A mathematical model of a sample and data hold operation whose input-output transfer func1 - e'sT tion is represented by Gn{s) = .

Digital control system A control system using digital signals and a digital computer to control a process. Forward rectangular integration A computational method of approximating the integration of a function given by x(kT) « x((k - 1)3*) + Tx((k - 1)T), where r = kT, Tis the sample time, and k = 1,2, — Microcomputer A small personal computer (PC) based on a microprocessor. Minicomputer A stand-alone computer with size and performance between a microcomputer and a large mainframe. The term is not commonly used today, and computers in this class are now often known as midrange servers. P1D controller A controller with three terms in which the output is the sum of a proportional term, an integrating term, and a differentiating

APPENDIX

A

MATLAB Basics

A.1 INTRODUCTION MATLAB is an interactive program for scientific and engineering calculations. The MATLAB family of programs includes the base program plus a variety of toolboxes, a collection of special files called m-files that extend the functionality of the base program [1-8]. Together, the base program plus the Control System Toolbox provide the capability to use MATLAB for control system design and analysis. Whenever MATLAB is referenced in this book, it means the base program plus the Control System Toolbox. Most of the statements, functions, and commands are computer-platform-independent. Regardless of what particular computer system you use, your interaction with MATLAB is basically the same. This appendix concentrates on this computer platform-independent interaction. A typical session will utilize a variety of objects that allow you to interact with the program: (1) statements and variables, (2) matrices, (3) graphics, and (4) scripts. MATLAB interprets and acts on input in the form of one or more of these objects. The goal in this appendix is to introduce each of the four objects in preparation for our ultimate goal of using MATLAB for control system design and analysis. The manner in which MATLAB interacts with a specific computer system is computer-platform-dependent. Examples of computer-dependent functions include installation, the file structure, hard-copy generation of the graphics, the invoking and exiting of a session, and memory allocation. Questions related to platformdependent issues are not addressed here. This does not mean that they are not important, but rather that there are better sources of information such as the MATLAB User's Guide or the local resident expert. The remainder of this appendix consists of four sections corresponding to the four objects already listed. In the first section, we present the basics of statements and variables. Following that is the subject of matrices. The third section presents an introduction to graphics, and the fourth section is a discussion on the important topic of scripts and m-files. All the figures in this appendix can be constructed using the m-files found at the MCS website.

A.2 STATEMENTS AND VARIABLES Statements have the form shown in Figure A.l. MATLAB uses the assignment so that equals ("-") implies the assignment of the expression to the variable. The command

1038

Section A.2

1039

Statements and Variables

Command prompt

»variab!e=expression FIGURE A.1 MATLAB statement form.

» A = [ 1 2; 4 6] < ret > A= 1 2 4 6

Carriage return

FIGURE A.2 Entering and displaying a matrix A.

prompt is two right arrows," » ."A typical statement is shown in Figure A.2, where we are entering a 2 x 2 matrix to which we attach the variable name A. The statement is executed after the carriage return (or enter key) is pressed. The carriage return is not explicitly denoted in the remaining examples in this appendix. The matrix A is automatically displayed after the statement is executed following the carriage return. If the statement is followed by a semicolon (;), the output matrix A is suppressed, as seen in Figure A.3.The assignment of the variable A has been carried out even though the output is suppressed by the semicolon. It is often the case that your MATLAB sessions will include intermediate calculations for which the output is of little interest. Use the semicolon whenever you have a need to reduce the amount of output. Output management has the added benefit of increasing the execution speed of the calculations since displaying screen output takes time. The usual mathematical operators can be used in expressions. The common operators are shown in Table A.LThe order of the arithmetic operations can be altered by using parentheses. The example in Figure A.4 illustrates that MATLAB can be used in a "calculator" mode. When the variable name and "=" are omitted from an expression, the result is assigned to the generic variable cms. MATLAB has available most of the trigonometric and elementary math functions of a common scientific calculator. Type help elfun at the command prompt to view a complete list of available trigonometric and elementary math functions; the more common ones are summarized in Table A.2. »A=[1 2;4 6]; <—

»

»A=[1 2;4 6] «— A= 1 2 4 6

suppresses L Semicolon the output. No semicolon displays the output.

FIGURE A.3 Using semicolons to suppress the output.

»12.4/6.9 FIGURE A.4 Using the calculator mode.

ans = 1.7971

Table A.1 Mathematical Operators

+ * /A

1040

Appendix A

MATLAB Basics

Table A.2 Common Mathematical Functions sin(x) sinh(x) asin(x) asinh(x) cos(x) cosh(x) acos(x) acosh(x) tan(x) tanh(x) atan(x) atan2(y,x) atanh(x) sec(x) sech(x) asec(x) asech(x) csc(x) csch(x) acsc(x) acsch(x) cot(x) coth(x) acot(x)

Sine Hyperbolic sine Inverse sine Inverse hyperbolic sine Cosine Hyperbolic cosine Inverse cosine Inverse hyperbolic cosine Tangent Hyperbolic tangent Inverse tangent Four quadrant inverse tangent Inverse hyperbolic tangent Secant Hyperbolic secant Inverse secant Inverse hyperbolic secant Cosecant Hyperbolic cosecant Inverse cosecant Inverse hyperbolic cosecant Cotangent Hyperbolic cotangent Inverse cotangent

acoth(x) exp(x) log(x) loglO(x) log2(x) pow2(x) sqrt(x) nextpow2(x) abs(x) angle(x) complex(x,y) conj(x) imag(x) real(x) unwrap(x) isreal(x) cplxpair(x) fix(x) floor(x) ceil(x) round(x) mod(x,y) rem(x,y)

Inverse hyperbolic cotangent Exponential Natural logarithm Common (base 10) logarithm Base 2 logarithm and dissect floating point number Base 2 power and scale floating point number Square root Next higher power of 2 Absolute value Phase angle Construct complex data from real and imaginary parts Complex conjugate Complex imaginary part Complex real part Unwrap phase angle True for real array Sort numbers into complex conjugate pairs Round towards zero Round towards minus infinity Round towards plus infinity Round towards nearest integer Modulus (signed remainder after division) Remainder after division

Variable names begin with a letter and are followed by any number of letters and numbers (including underscores). Keep the name length to N characters, since MATLAB remembers only the first N characters, where N = namelengthmax. It is a good practice to use variable names that describe the quantity they represent. For example, we might use the variable name vel to represent the quantity aircraft velocity. Generally, we do not use extremely long variable names even though they may be legal MATLAB names. Since MATLAB is case sensitive, the variables M and m are not the same. By case, we mean upper- and lowercase, as illustrated in Figure A.5. The variables M and m are recognized as different quantities. MATLAB has several predefined variables, including pi, Inf, NaN, i, and/.Three examples are shown in Figure A.6. NaN stands for Not-a-Number and results from undefined operations. Inf represents +oo, and pi represents IT. The variable i = V — 1 is used to represent complex numbers. The variable j = v - 1 can be used for complex arithmetic by those who prefer it over i. These predefined variables can be inadvertently overwritten. Of course, they can also be purposely overwritten in order to free the variable name for other uses. For instance, you might want to use i as an integer and reserve ; for complex arithmetic. Be safe and leave these predefined variables alone, as

FIGURE A.5 Variables are case sensitive.

» M = [ 1 21; » m = [ 3 5 7];

Section A.2

1041

Statements and Variables

»who »z=3+4*i

z= 3.0000 + 4.0000i

A

M

ans

m

z

FIGURE A.7 Using the who function to display variables.

»lnf ans = Inf »0/0 FIGURE A.6 Three predefined variables i, Inf, and NaN.

ans = NaN

there are plenty of alternative names that can be used. Predefined variables can be reset to their default values by using clear name (e.g., clear/?/)The matrix A and the variable ans, in Figures A.3 and A.4, respectively, are stored in the workspace. Variables in the workspace are automatically saved for later use in your session. The who function gives a list of the variables in the workspace, as shown in Figure A.7. MATLAB has a host of built-in functions. Refer to the MATLAB User's Guide for a complete list or use the MATLAB help browser. Each function will be described as the need arises. The whos function lists the variables in the workspace and gives additional information regarding variable dimension, type, and memory allocation. Figure A.8 gives an example of the whos function. The memory allocation information given by the whos function can be interpreted as follows: Each element of the 2 x 2 matrix A requires 8 bytes of memory for a total of 32 bytes, the l x l variable ans requires 8 bytes, and so forth. All the variables in the workspace use a total of 96 bytes. Variables can be removed from the workspace with the clear function. Using the function clear, by itself, removes all items (variables and functions) from the workspace; clear variables removes all variables from the workspace; clear name] name!... removes the variables runnel, name2, and so forth. The procedure for removing the matrix A from the workspace is shown in Figure A.9.

»whos

FIGURE A.8 Using the whos function to display variables.

Name

Size

Bytes

Class

A

2x2

32

double

M

1x2

16

double

ans

1x1

8

double

m

1x3

24

double

z

1x1

16

double

Attributes

complex

1042

Appendix A

MATLAB Basics

»clear A »who FIGURE A.9 Removing the matrix A from the workspace.

ans

m

Computations in MATLAB are performed in double precision. However, the screen output can be displayed in several formats. The default output format contains four digits past the decimal point for nonintegers. This can be changed by using the format function shown in Figure A. 10. Once a particular format has been specified, it remains in effect until altered by a different format input. The output format does not affect internal MATLAB computations. On the other hand, the number of digits displayed does not necessarily reflect the number of significant digits of the number. This is problem-dependent, and only the user can know the true accuracy of the numbers input and displayed by MATLAB. Other display formats (not shown in Figure A.10) include format long g (best of fixed or floating point format with 14 digits after the decimal point), format short g (same as format long g but with 4 digits after the decimal point), format hex (hexadecimal format), format bank (fixed format for dollars and cents), format rat (ratio of small integers) and format (same as format short). Since MATLAB is case sensitive, the functions who and WHO are not the same functions. The first function, who, is a built-in function, and typing who lists the variables in the workspace. On the other hand, typing the uppercase WHO results in the error message shown in Figure A.ll. Case sensitivity applies to all functions.

»pl ans = 3.1416 «-

4-digit scaled fixed point

»format long; pi ans = 3.141592653589793-«-

15-digit scaled fixed point

»forrnat short e; pi ans = 3.1416e+00 M—

FIGURE A.10 Output format control illustrates the four forms of output.

»format long e; pi . ans= T 3.141592653589793e+000

4-digit scaled floating point

15-digit scaled floating point

Section A.3

1043

Matrices

»WHO ??? Undefined function or variable 'WHO'.

»Who FIGURE A.11 Function names are case sensitive.

??? Undefined function or variable 'Who'.

A.3 MATRICES MATLAB is short for matrix laboratory. Although we will not emphasize the matrix routines underlying our calculations, we will learn how to use the interactive capability to assist us in the control system design and analysis. We begin by introducing the basic concepts associated with manipulating matrices and vectors. The basic computational unit is the matrix. Vectors and scalars can be viewed as special cases of matrices. A typical matrix expression is enclosed in square brackets, [ • ]. The column elements are separated by blanks or commas, and the rows are separated by semicolons or carriage returns. Suppose we want to input the matrix 1 A = log(-l) _asin(0.8)

-4/ sin(7r/2) acos(0.8)

V2 cos(7r/3) exp(0.8)

One way to input A is shown in Figure A. 12. The input style in Figure A. 12 is not unique. Matrices can be input across multiple lines by using a carriage return following the semicolon or in place of the semicolon. This practice is useful for entering large matrices. Different combinations of spaces and commas can be used to separate the columns, and different combinations of semicolons and carriage returns can be used to separate the rows, as illustrated in Figure A. 12.

» A = [ 1 , -4*j, sqrt(2); < log(-1) sin(pi/2) cos(pi/3) +— asin(0.5), acos(0.8) exp(0.8)] A= 1.0000 0 + 3.1416i 0.5236

FIGURE A.12 Complex and real matrix input with automatic dimension and type adjustment.

» A = [ 1 2;4 5] 4A= 1 2 4 5

3 X 3 complex matrix

ll

Carriage return

0 - 4.0000i 1.4142 1.0000 0.5000 0.6435 2.2255

2 X 2 real matrix

Appendix A

MATLAB Basics

No dimension statements or type statements are necessary when using matrices; memory is allocated automatically. Notice in the example in Figure A. 12 that the size of the matrix A is automatically adjusted when the input matrix is redefined. Also notice that the matrix elements can contain trigonometric and elementary math functions, as well as complex numbers. The important basic matrix operations are addition and subtraction, multiplication, transpose, powers, and the so-called array operations, which are element-toelement operations. The mathematical operators given in Table A.l apply to matrices. We will not discuss matrix division, but be aware that MATLAB has a leftand right-matrix division capability. Matrix operations require that the matrix dimensions be compatible. For matrix addition and subtraction, this means that the matrices must have the same dimensions. If A is an n X m matrix and B is a p x r matrix, then A ± B is permitted only if n = p and m = r. Matrix multiplication, given by A * B, is permitted only if m = p. Matrix-vector multiplication is a special case of matrix multiplication. Suppose b is a vector of length/?. Multiplication of the vector b by the matrix A, where A is an n X m matrix, is allowed if m = p. Thus, y = A * b is the n x 1 vector solution of A * b. Examples of three basic matrix-vector operations are given in Figure A. 13. The matrix transpose is formed with the apostrophe ('). We can use the matrix transpose and multiplication operation to create a vector inner product in the following manner. Suppose w and v are / « x l vectors. Then the inner product (also known as the dot product) is given by w' * v. The inner product of two vectors is a scalar. The outer product of two vectors can similarly be computed as w * v'. The outer product of two m X 1 vectors is an m X m matrix of rank 1. Examples of inner and outer products are given in Figure A. 14. The basic matrix operations can be modified for element-by-element operations by preceding the operator with a period. The modified matrix operations are known

»A=[1 3; 5 9]; B^=14-7; 10 01; Matrix addition ans = 5 -4 15 9

»b=[1 ;5]; Matrix multiplication ans = 16 50

FIGURE A.13 Three basic matrix operations: addition, multiplication, and

Matrix transpose ans = 1 5 3 9

Section A.3

1045

Matrices

»x=[5;pi;. sin(pi/2)]; y==[exp(-0.5); -13 piA2]; Inner product ans = -27.9384

Outer product ans = 3.0327 -65.0000 1.9055 -40.8407 0.6065 -13.0000

Table A.3 Mathematical Array Operators

49.3480 31.0063 9.8696

+ .* ./ .A

FIGURE A.14

Inner and outer products of two vectors.

as array operations. The commonly used array operators are given in Table A.3. Matrix addition and subtraction are already element-by-element operations and do not require the additional period preceding the operator. However, array multiplication, division, and power do require the preceding dot, as shown in Table A.3. Consider A and B as 2 x 2 matrices given by «12 «21

B

«22

*n

bn

*21

*22

Then, using the array multiplication operator, we have A.*B =

«11*11

«12*12

«21*21

«22*22

The elements of A.* B are the products of the corresponding elements of A and B. A numerical example of two array operations is given in Figure A.15.

Array multiplication -6 14 30

FIGURE A. 15 Array operations.

»A.A2 ans= 1 4 9

4

Array raised to a power

1046

Appendix A

MATLM3 Basics

.

Starting value

I

I

Final value

)<=[xi:dx:xf]

t Increment

FIGURE A.16 The colon notation.

Before proceeding to the important topic of graphics, we need to introduce the notion of subscripting using colon notation. The colon notation, shown in Figure A.16, allows us to generate a row vector containing the numbers from a given starting value, Xj, to a final value, xf, with a specified increment, dx. We can easily generate vectors using the colon notation, and as we shall soon see, this is quite useful for developing x-y plots. Suppose our objective is to generate a plot of v = x sin(x) versus x for x = 0, 0.1, 0.2,..., 1.0. Our first step is to generate a table of x-y data. We can generate a vector containing the values of x at which the values of y(x) are desired using the colon notation, as illustrated in Figure A.17. Given the desired x vector, the vector y(x) is computed using the multiplication array operation. Creating a plot of y = x sin(.x:) versus x is a simple step once the table of x-y data is generated. A.4 GRAPHICS Graphics plays an important role in both the design and analysis of control systems. An important component of an interactive control system design and analysis tool is an effective graphical capability. A complete solution to the control system design and analysis will eventually require a detailed look at a multitude of data types in many formats. The objective of this section is to acquaint the reader with the basic

»x=[0:0.' :1]';y=x.*sin(x); » [ x y] J

Starting value Final value Increment

I ans =

FIGURE A.17 Generating vectors using the colon notation.

0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000

0 0.0100 0.0397 0.0887 0.1558 0.2397 0.3388 0.4510 0.5739 0.7050 0.8415

U1 x=[0:0.1:1]'

1047

Section A.4 Graphics Table A.4

Plot Formats

plot(x,y) semilogx(x,y) semilogy(x,y) loglog(x,y)

Plots the vector x versus the vector y. Plots the vector x versus the vector y. The *-axis is log1();the y-axis is linear. Plots the vector x versus the vector y. The .v-axis is linear; the y-axis is logjo. Plots the vector x versus the vector y. Creates a plot with logl0scales on both axes.

x-y plotting capability of MATLAB. More advanced graphics topics are addressed in the chapter sections on MATLAB. MATLAB uses a graph display to present plots. The graph display is activated automatically when a plot is generated using any function that generates a plot (e.g., the plot function). The plot function opens a graph display, called a FIGURE window. You can also create a new figure window with the figure function. Multiple figure windows can exist in a single MATLAB session; the function figure (n) makes n the current figure. The plot in the graph display is cleared by the elf function at the command prompt. The shg function brings the current figure window forward. There are two basic groups of graphics functions. The first group, shown in Table A.4, specifies the type of plot. The list of available plot types includes the x-y plot, semilog plots, and log plots. The second group of functions, shown in Table A.5, allows us to customize the plots by adding titles, axis labels, and text to the plots and to change the scales and display multiple plots in subwindows. The standard x-y plot is created using the plot function. The x-y data in Figure A. 17 are plotted using the plot function, as shown in Figure A.18. The axis scales and line types are automatically chosen. The axes are labeled with the xlabel and ylabel functions; the title is applied with the title function. The legend function puts a legend on the current figure. A grid can be placed on the plot by using the grid on function. A basic x-y plot is generated with the combination of functions plot, legend, xlabel, ylabel, title, and grid on. Multiple lines can be placed on the graph by using the plot function with multiple arguments, as shown in Figure A.19. The default line types can also be altered. The available line types are shown in Table A.6.The line types will be automatically

Table A.5

Functions for Customized Plots

title('text') legend (stringl, sthng2,...) xlabel('text') ylabel('text') text(p1 ,p2, 'text') subplot grid on grid Off grid

Puts vtexf at the top of the plot Puts a legend on current plot using specified strings as labels Labels the x-axis with 'text' Labels the y-axis with 'text1 Adds 'text' to location (pl,p2), where (pl,p2) is in units from the current plot Subdivides the graphics window Adds grid lines to the current figure Removes grid lines from the current figure Toggles the grid state

1048

Appendix A

MATI-AB Basics

Table A.6 Commands for Line Types for Customized Plots

Solid line Dashed line Dotted line Dashdot line

: _-.

chosen unless specified by the user. The use of the text function and the changing of line types are illustrated in Figure A. 19. The other graphics functions—loglog, semilogx, and semilogy—are used in a fashion similar to that of plot. To obtain an x-y plot where the x-axis is a linear scale and the y-axis is a log10 scale, you would use the semilogy function in place of the plot function. The customizing features listed in Table A.5 can also be utilized with the loglog, semilogx, and semilogy functions. The graph display can be subdivided into smaller subwindows. The function subplot(m,n,p) subdivides the graph display into an m X n grid of smaller subwindows. The integer p specifies the window, numbered left to right, top to bottom, as illustrated in Figure A.20, where the graphics window is subdivided into four subwindows.

»x=[0:0.1:1]'; »y=x.*sin(x); »plot(x,y) »title('Plot of x sin(x) vs x') »xlabel('x') »ylabel('y') » g r i d on

» »

x=[0:0.1:1]'; y1=x.*sin(x); y2=sin(x);

Dashed line for yl Dashed-dot line for y2

»text(0.1,0.85,'y_1 = x sin(x) —') »text(0.1,0.80,'y_2 = sin(x) A_.\_') » xlabel('x'), ylabel('y_1 and y_2'), grid on

(a)

(a) 0.9 y, = xsin(x) — y2 = sin(x) _ . _

0.8

1

...^v-

r_U :/"'/

0.7 0.6

Text indicating lines

0.3 0.2 0.1 0 0 (b) FIGURE A.18 (a) MATLAB commands, (b) A basic x-y plot of x sin(x) versus x.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b)

FIGURE A.19 (a) MATLAB commands, (b) A basic x-y plot with multiple lines.

I

Section A.5

Scripts

1049

FIGURE A.20 Using subplot to create a 2 x 2 partition of the graph display.

A.5 SCRIPTS Up to this point, all of our interaction with MATLAB has been at the command prompt. We entered statements and functions at the command prompt, and MATLAB interpreted our input and took the appropriate action. This is the preferable mode of operation whenever the work sessions are short and nonrepetitive. However, the real power of MATLAB for control system design and analysis derives from its ability to execute a long sequence of commands stored in a file. These files are called m-files, since the filename has the form filename.m. A script is one type of m-file. The Control System Toolbox is a collection of m-files designed specifically for control applications. In addition to the preexisting m-files delivered with MATLAB and the toolboxes, we can develop our own scripts for our applications. Scripts are ordinary ASCII text files and are created using a text editor. A script is a sequence of ordinary statements and functions used at the command prompt level. A script is invoked at the command prompt level by typing in the filename or by using the pull-down menu. Scripts can also invoke other scripts, When the script is invoked, MATLAB executes the statements and functions in the file without waiting for input at the command prompt. The script operates on variables in the workspace. Suppose we want to plot the function y(t) = sin at, where a is a variable that we want to vary. Using a text editor, we write a script that we call plotdata.m, as

1050

Appendix A

MATLAB Basics

»alpha=50; »plotdata plotdata.m

' r

% This is a script to plot the function y=sin(alpha*t).

% % The value of alpha must exist in the workspace prior % to invoking the script.

% t=[0:0.01:1]; y=sin(alpha*t); plot(t.y) xlabel('Time (sec)') ylabel('y(t) = sin(\alpha t)') grid on

FIGURE A.21 A simple script to plot the function y(f) = sin at.

shown in Figure A.21, then input a value of a at the command prompt, placing a in the workspace. Then we execute the script by typing in plotdata at the command prompt; the script plotdata.m will use the most recent value of a in the workspace. After executing the script, we can enter another value of a at the command prompt and execute the script again. Your scripts should be well documented with comments, which begin with a %. Put a header in the script; make sure the header includes several descriptive comments regarding the function of the script, and then use the help function to display the header comments and describe the script to the user, as illustrated in Figure A.22. Use plotdata.m to develop an interactive capability with a as a variable, as shown in Figure A.23. At the command prompt, input a value of a = 10 followed by the script filename, which in this case is plotdata. The graph of y(t) = sin at is automatically generated. You can now go back to the command prompt, enter a value of a = 50, and run the script again to obtain the updated plot. A limited subset of TeX1 characters are available to allow you to annotate plots with symbols and mathematical characters. Table A.7 shows the available symbols. Figure A.21 illustrates the use of '\alpha' to generate the a character in the y-axis label. The 'V character preceeds all TeX sequences. Also, you can modify the characters with the following modifiers: • Q Q Q • • l

\bf—bold font \it—italics font \rm—normal font \fontname—specify the name of the font family to use \fontsize—specify the font size \color—specify color for succeeding characters

TeX is a trademark of the American Mathematical Society.

Section A.5

1051

Scripts

» h e l p plotdata This is a script to plot the function y=sin(alpha*t). FIGURE A.22 Using the help function.

The value of alpha must exist in the workspace prior to invoking the script.

Command prompt

»alpha=10; plotdata

Graph display

Script filename

»alpha=50; plotdata

FIGURE A.23 An interactive session using a script to plot the function y(t) = sin at.

Graph display

Subscripts and superscripts arc obtained with "_" and "A", respectively. For example, ylabel('y_l and y_2') generates the y-axis label shown in Figure A. 19. The graphics capability of MATLAB extends beyond the introductory material presented here. A table of MATLAB functions used in this book is provided in Table A.8.

1052

Appendix A

MATLAB Basics

Table A.7 Character Sequence

TeX Symbols and Mathematics Characters Symbol

Character Sequence

Symbol

Character Sequence

I

Symbol

\alpha

a

\upsilon

V

Vsim

~

\beta

0

\phi

\leq

^

\gamma

y

\chi

X

\infty

00

\delta

8

\psi

*

\clubsuit

*

\epsilon

e

\omega

(0

\diamondsuit

\zeta

£

\Gamma

r

\heartsuit

V

\eta

\Delta

A

A

\theta

•n e

VTheta

e

Meftrightarrow

<-»

wartheta

<&

XLambda

A

Meftarrow

<—

Mota

i

\Xi

0

\uparrow

T

\kappa

K

\Pi

n

Vrightarrow

->

\lambda

X

\Sigma

2

\downarrow

I

\mu

M<

\Upsilon

Y

\circ

o

Vnu

V

\Phi

O

\pm

+

\xi

e

\Psi

*

\geq

^:

\pi

IT

\Omega

i)

\propto

oc

\rho

P

\forall

V

\partial

d

\sigma

CT

\exists

3

\bullet

Warsigma

i

\ni

3

\div

*

Mau

T

\cong

S3

\neq

5*

\equiv

S3

\approx

»

\aleph

K

Mm

3

\Re

Dt

\wp

P

\otimes

®

\oplus

\oslash

0

\cap

n

\cup

u

\supseteq

D

\supset

D

\subseteq

c

\subset

C

\in

3

\o

o

Vint

1

Section A.5 Scripts Table A.8

MATLAB Functions

Function Name abs acos ans asin atan atan2 axis bode c2d clear elf con] conv cos ctrb diary d2c eig end exp expm eye feedback for format grid on help hold on i imag impulse inf ] legend linspace load log Iog10 log log logspace Isim margin max mesh meshgrid min mi n real

1053

Function Description Computes the absolute value Computes the arccosine Variable created for expressions Computes the arcsine Computes the arctangent (2 quadrant) Computes the arctangent (4 quadrant) Specifies the manual axis scaling on plots Generates Bode frequency response plots Converts a continuous-time state variable system representation to a discrete-time system representation Clears the workspace Clears the graph window Computes the complex conjugate Multiplies two polynomials (convolution) Computes the cosine Computes the controllability matrix Saves the session in a disk file Converts a discrete-time state variable system representation to a continuous-time system representation Computes the eigenvalues and eigenvectors Terminates control structures Computes the exponential with base e Computes the matrix exponential with base e Generates an identity matrix Computes the feedback interconnection of two systems Generates a loop Sets the output display format Adds a grid to the current graph Prints a list of HELP topics Holds the current graph on the screen Computes the imaginary part of a complex number Computes the unit impulse response of a system Represents infinity

V^T Puts a legend on the current plot Generates linearly spaced vectors Loads variables saved in a file Computes the natural logarithm Computes the logarithm base 10 Generates log-log plots Generates logarithmically spaced vectors Computes the time response of a system to an arbitrary input and initial conditions Computes the gain margin, phase margin, and associated crossover frequencies from frequency response data Determines the maximum value Creates three-dimensional mesh surfaces Generates arrays for use with the mesh function Determines the minimum value Transfer function pole-zero cancellation Table A.8 continued

1054 Table A.8

Appendix A

Continued

Function Name NaN ngrid nichols num2str nyquist obsv ones pade parallel plot pole poly polyval printsys pzmap rank real residue rlocfind rlocus roots semilogx semilogy series shg sin sqrt ss step subplot tan text title tf who whos xlabel ylabel zero zeros

MATLAB Basics

Function Description Representation for Not-a-Number Draws grid lines on a Nichols chart Computes a Nichols frequency response plot Converts numbers to strings Calculates the Nyquist frequency response Computes the observability matrix Generates a matrix of integers where all the integers are 1 Computes an nth-order Pade approximation to a time delay Computes a parallel system connection Generates a linear plot Computes the poles of a system Computes a polynomial from roots Evaluates a polynomial Prints state variable and transfer function representations of linear systems in a pretty form Plots the pole-zero map of a linear system Calculates the rank of a matrix Computes the real part of a complex number Computes a partial fraction expansion Finds the gain associated with a given set of roots on a root locus plot Computes the root locus Determines the roots of a polynomial Generates an x-y plot using semilog scales with the .v-axis logU) and the y-axis linear Generates an x-y plot using semilog scales with the y-axis logio and the .v-axis linear Computes a series system connection Shows graph window Computes the sine Computes the square root Creates a state-space model object Calculates the unit step response of a system Splits the graph window into subwindows Computes the tangent Adds text to the current graph Adds a title to the current graph Creates a transfer function model object Lists the variables currently in memory Lists the current variables and sizes Adds a label to the x-axis of the current graph Adds a label to the y-axis of the current graph Computes the zeros of a system Generates a matrix of zeros

Section A.5

1055

Scripts

MATLAB BASICS: PROBLEMS A.l

Consider the two matrices A=

B=

4 6/ ~6j _1T

A.4 Develop a MATLAB script to plot the function

-13TT~ 16

Using MATLAB, compute the following: (a) (c) (e) (g)

A +B A2 B~x A2 + fl2 - Afl

4 4 y(x) = — cos o»x H cos 3wx.

2TT

10 + Vlj_

(b) AB (d) A' (f) B'A'

A.2 Consider the following set of linear algebraic equations: 5.x + 6y + lOz = 4, - 3 x + Uz = 10, -ly + 21z = 0. Determine the values of x, y, and z so that the set of algebraic equations is satisfied. {Hint: Write the equations in matrix vector form.) A.3 Generate a plot of y(x) - e_0-5ir sin OJX, where co = lOrad/s, andO s x :£ 10.Utilize the colon notation to generate the x vector in increments of 0.1.

where co is a variable input at the command prompt. Label the x-axis with time (sec) and the y-axis with y(x) = (A/IT) * cos(wx) + (4/9TT) * cos(3o>x). Include

a descriptive header in the script, and verify that the help function will display the header. Choose to = 1,3,10 rad/s and test the script. A.5 Consider the function y(x) - 10 + 5e~xcos((ox + 0.5). Develop a script to co-plot y(x) for the three values of co = 1,3,10 rad/s with 0 s j < 5 seconds.The final plot should have the following attributes: Title x-axis label 3^-axis label Line type Grid

y(x) = 10 + 5 exp(-x) * cos(wx + 0.5) Time (sec)

y(x) co = 1: solid line co - 3: dashed line co = 10: dotted line grid on

Design Process Establish the control goals i'

Identify the variables to be controlled

\

(1) Establishment of goals, variables to be controlled. and specifications.

ir

Write the specifications V

Establish the system configuration

(2) System definition and modeling.

i Obtain a model of the process, the actuator, and the sensor i

'

Describe a controller and select key parameters to be adjusted i

(3) Control system design, simulation, and analysis.

'

Optimize the parameters and analyze the performance

\ If the performance does not meet the specifications, then iterate the configuration.

If the performance meets the specifications, thenfinalizethe design.

EXAMPLES Q Insulin delivery control system (Section 1.6) Q Fluid flow modeling (Section 2.13) • Space station orientation modeling (Section 3.7) Q Blood pressure control during anesthesia (Section 4.4) • Attitude control of an airplane (Section 5.11) • Robot-controlled motorcycle (Section 6.11) • Automobile velocity control (Section 7.16) Q Control of one leg of a six-legged robot (Section 8.8) Q Hot ingot robot control (Section 9.12) Q Milling machine control system (Section 10.15) • Diesel electric locomotive control (Section 11.16) • Digital audio tape controller (Section 12.16) • Fly-by-wire aircraft control surface (Section 13.11)

31 94 193 259 346 406 505 588 681 790 876 943 1011

Selected Tables and Formulas for Design A second-order closed-loop system

+ Y(s)

s{s + 2gai„)

UNIT STEP RESPONSE

CLOSED-LOOP MAGNITUDE PLOT

Overshoot

Rise time •

Pealc

ftme

Settling time

Settling time (to within 2% of the final value)

Q Maximum magnitude (£ < 0.7) M

P.=

2£Vl - ?

Percent overshoot MD = 1 + e-fr'VW 2

Q

I

and

P.O. =

Time-to-peak

HKk-**"^ •

Resonant frequency (£ ^ 0.7)

wr = w n Vi - 2£2

Rise time (time to rise from 10% to 90% of final value)

Bandwidth (0.3 < { < 0.8) ft>5 = ( - U 9 6 £ + 1.85)wn

2.16£ + 0.60 Try =

(0.3 ss £ =s 0.8)

w.

PID Controller:

^i (s + zi)(s + z2) Gc(s) = KP + KDs + — = — s s

TABLE PAGE 5.5 Summary of Steady-State Errors 325 5.6 The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Step Input 335 5.7 The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Ramp Input 339 7.6 Effect of Increasing the PID Gain, Kpi KD, and K, on the Step Response 483 7.7 Ziegler-Nichols PID Timing Using Ultimate Gain, K0, and Oscillation Period, Py 488 7.8 Ziegler-Nichols PID Tuning Using Reaction Curve Characterized by Time Delay, Td, and Reaction Rate, R 490 782 10.2 Coefficients and Response Measures of a Deadbeat System 10.7 A Summary of the Characteristics of Phase-Lead and Phase-Lag Compensation Networks 805 *

## Modern Control Systems

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