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Modelling, analysis and control of linear systems using state space representations

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

Olivier Sename

State feedback control Problem formulation

Grenoble INP / GIPSA-lab

Controllability Definition Pole placement control

February 2017

Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Outline

Introduction Modelling of dynamical systems as state space representations Nonlinear models Linear models Linearisation To/from transfer functions Properties Stability State feedback control Problem formulation Controllability Definition of the state feedback control Synthesis of the state feedback control: the pole placement control Specifications Integral Control or how to ensure disturbance attenuation with a state feedback control? Observer and output feedback control Observation A preliminary property: Observability Observer design Observer-based control Introduction to optimal control Introduction to digital control

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

Introduction

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

References

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models

Some interesting books: I

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K.J. Astrom and B. Wittenmark, Computer-Controlled Systems, Information and systems sciences series. Prentice Hall, New Jersey, 3rd edition, 1997. R.C. Dorf and R.H. Bishop, Modern Control Systems, Prentice Hall, USA, 2005. G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design, Prentice Hall, New Jersey, 2001. G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall, 2005

Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

The "control design" process

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation

I

Plant study and modelling

I

Determination of sensors and actuators (measured and controlled outputs, control inputs)

To/from transfer functions

Properties Stability

State feedback control

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Performance specifications

Problem formulation

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Control design (many methods)

Definition

I

Simulation tests

Specifications

I

Implementation, tests and validation

Controllability

Pole placement control

Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

About modelling...

State space approach Olivier Sename Introduction

Identification based method I

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System excitations using PRBS (Pseudo Random Binary Signal) or sinusoïdal signals Determination of a transfer function reproducing the input/ouput system behavior

Knowledge-based method: I

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Represent the system behavior using differential and/or algebraic equations, based on physical knowledge. Formulate a nonlinear state-space model, i.e. a matrix differential equation of order 1.

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

I

I

Determine the steady-state operating point about which to linearize. Introduce deviation variables and linearize the model.

Tools: Matlab/Simulink, LMS Imagine.Lab Amesim, Catia-Dymola, ADAMS, MapleSim .....

Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Why state space equations ?

State space approach Olivier Sename Introduction Modelling Nonlinear models

I

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dynamical systems where physical equations can be derived : electrical engineering, mechanical engineering, aerospace engineering, microsystems, process plants .... include physical parameters: easy to use when parameters are changed for design. Need only for parameter identification or knowledge.

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability

I

State variables have physical meaning.

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Allow for including non linearities (state constraints, input saturation)

Definition Pole placement control

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Easy to extend to Multi-Input Multi-Output (MIMO) systems

I

Advanced control design methods are based on state space equations (reliable numerical optimisation tools)

Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties

Modelling of dynamical systems as state space representations

Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Towards state space representation

State space approach Olivier Sename Introduction Modelling Nonlinear models

What is a state space system ?

Linear models

A "matrix-form" representation of the dynamics of an N- order differential equation system into a FIRST order differential equation in a vector form of size N, which is called the state.

To/from transfer functions

Definition of a system state The state of a dynamical system is the set of variables, known as state variables, that fully describe the system and its response to any given set of inputs. Mathematically, the knowledge of the initial values of the state variables at t0 (namely xi (t0 ), i = 1, ..., n), together with the knowledge of the system inputs for time t ≥ t0 , are sufficient to predict the behavior of the future system state and output variables (for t ≥ t0 ).

Linearisation

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Example of a one-tank model

Olivier Sename Introduction Modelling

Usually the hydraulic equation is non linear and of the form

Nonlinear models Linear models Linearisation

dH = Qe − Qs S dt

To/from transfer functions

Properties

where H is the tank height, S the tank √surface, Qe the input flow, and Qs the output flow defined by Qs = a H.

Stability

State feedback control Problem formulation

Definition the state space model

Controllability Definition

The system is represented by an Ordinary Differential Equation whose solution depends on H(t0 ) and Qe . Clearly H is the system state, Qe is the input, and the system can be represented as:

Pole placement control Specifications Integral Control

Observer Observation

˙ x(t) = f (x(t), u(t)), x(0) = x0 with x = H, f

√ = − Sa x

(1)

Observability Observer design Observer-based control

+

1 Su

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Example of a pendulum

Olivier Sename

Let consider the following pendulum Introduction Modelling Nonlinear models

T

Linear models Linearisation To/from transfer functions

Properties

l

Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

θ

Observation Observability

M

Observer design Observer-based control

Introduction to optimal control

where θ is the angle, T the controlled torque, l the pendulum length, M its mass. Give the dynamical equations of motion for the pendulum angle (neglecting friction) and propose a state space model.

Introduction to digital control Conclusion

Example: Underwater Autonomous Vehicle UAV Aster x

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

Actions: axial propeller to control the velocity in Ox direction and 5 independent mobile fins : I

2 horizontals fins in the front part of the vehicle (β1 , β10 ).

I

1 vertical fin at the tail of the vehicle (δ ).

I

2 fins at the tail of the vehicle (β2 , β20 ).

Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

UAV modelling

Olivier Sename Introduction Modelling

Physical model:

Nonlinear models Linear models

M ν˙ = G(ν)ν + D(ν)ν + Γg + Γu

Linearisation

(2)

To/from transfer functions

Properties Stability

η˙ = Jc (η2 )ν

(3)

State feedback control Problem formulation

where: - M: mass matrix: real mass of the vehicle augmented by the "water-added-mass" part, - G(ν) : action of Coriolis and centrifugal forces, - D(ν): matrix of hydrodynamics damping coefficients, - Γg : gravity effort and hydrostatic forces, - Jc (η2 ): referential transform matrix, - Γu : forces and moments due to the vehicle’s actuators.

Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

UAV state definition

Olivier Sename Introduction Modelling Nonlinear models Linear models

A 12 dimensional state vector : X = η(6) I

I

T

ν(6) . T η(6): position in the inertial referential: η = η1 η2 with T T η1 = x y z and η2 = φ θ ψ . x, y and z are the positions of the vehicle , and φ , θ and ψ are respectively the roll, pitch and yaw angles. ν(6): velocity vector, in the local referential (linked to the vehicle) describing the linear and angular velocities (first derivative of the T position, considering the referential transform: ν = ν1 ν2 with T T ν1 = u v w and ν2 = p q r

Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Example: Inverted pendulum It is described by:

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Parameters:

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Example: Inverted pendulum

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models

The dynamical equations are as follows:

Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation

to be put in form (4).

Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Deinition of a NonLinear dynamical system

State space approach Olivier Sename Introduction Modelling Nonlinear models

Many dynamical systems can be represented by Ordinary Differential Equations (ODE) as ( ˙ x(t) = f ((x(t), u(t), t), x(0) = x0 (4) y (t) = g((x(t), u(t), t)

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation

where f and g are non linear functions and I

x(t) ∈ Rn

is referred to as the system state (vector of state variables),

I

u(t) ∈ Rm the control input

I

y (t) ∈ Rp the measured output

I

x0 is the initial condition.

Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Definition of linear state space representations

Olivier Sename Introduction Modelling

A continuous-time LINEAR state space system is given as : ( ˙ x(t) = Ax(t) + Bu(t), x(0) = x0 y (t) = Cx(t) + Du(t)

Nonlinear models Linear models Linearisation

(5)

To/from transfer functions

Properties Stability

I

x(t) ∈ Rn is the system state (vector of state variables),

I

u(t) ∈ Rm the control input

I y (t) ∈ Rp I I

the measured output

A, B, C and D are real matrices of appropriate dimensions x0 is the initial condition.

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability

n is the order of the state space representation. Matlab : ss(A,B,C,D) creates a SS object SYS representing a continuous-time state-space model

Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

A state space representation of a DC Motor

Olivier Sename Introduction Modelling Nonlinear models Linear models

The dynamical equations are : Ri + L

di +e = u dt

dω J = −f ω + Γm dt

Linearisation To/from transfer functions

e = Ke ω

Properties

Γm = K c i

State feedback control

Stability

System of 2 equationsof order 1 =⇒ 2 state variables. ω A possible choice x = It gives: i −f /J Kc /J 0 A= B= C= 0 −Ke /L −R/L 1/L

Problem formulation Controllability Definition Pole placement control Specifications Integral Control

1

Observer Observation Observability Observer design

Extension: measurement= motor angular position

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Example : Wind turbine

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Some important issues

State space approach Olivier Sename

I

A complete ADAMS model includes 193 DOFs to represent fully flexible tower, drive-train, and blade components ⇒ simulation model

I

Different operating conditions according to the wind speed

I

Control objectives: maximize power , enhance damping in the first drive train torsion mode, design a smooth transition different modes

I

I

A Generator torque controller to enhance drive train torsion damping in Regions 2 and 3 The control model is obtained by linearisation of a non linear electro-mechanical model: ˙ x(t) = Ax(t) + Bu(t) + Ed(t) y (t) = Cx(t) where x1 = rotor-speed x2 = drive-train torsion spring force, x3 = rotational generator speed u = generator torque, d : wind speed

Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Examples: Suspension

Olivier Sename

Let the following mass-spring-damper system.

Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design

where x1 is the relative position, M1 the system mass, k1 the spring coefficient, u the force generated by the active damper, and F1 is an external disturbance. Applying the mechanical equations it leads: M1 x¨1 = −k1 x1 + u + F1

Observer-based control

Introduction to optimal control

(6)

Introduction to digital control Conclusion

State space approach

Examples: Suspension cont.

Olivier Sename Introduction Modelling Nonlinear models Linear models

The choice x =

Linearisation

x1 x˙ 1

˙ x(t) = Ax(t) + Bu(t) + Ed(t) y (t) = Cx(t)

To/from transfer functions

gives

Properties Stability

State feedback control Problem formulation Controllability

where d = F1 , y = x1 with 0 1 0 A= , B=E = , and C = −k1 /M1 0 1/M1

Definition Pole placement control Specifications

0

1

Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Exercise

Olivier Sename Introduction Modelling

Let the following quarter car model with active suspension.

Nonlinear models Linear models Linearisation To/from transfer functions

zs and zus ) are the relative position of the chassis and of the wheel, ms (resp. mus ) the mass of the chassis (resp. of the wheel), ks (resp. kt ) the spring coefficient of the suspension (of the tire), u the active damper force, zr is the road profile.

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design

Choose some state variables and give a state space representation of this system

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

More generally

Olivier Sename Introduction

A Nth-order differential equation

Modelling

d n−1 y d ny + an−1 n−1 + . . . + a1 y˙ + a0 y = u dt n dt can be reformulated into N simultaneous first-order differential equations defining the state variables : x1 = y , , x2 = y˙ , , . . . xn =

d n−1 y , dt n−1

which gives the according state space representation. 0 0 1 0 ... 0 .. 0 0 1 0 ... . .. .. .. .. .. , B = . and . . . . . A= .. .. 0 0 . 0 1 C=

−a0 1

−a1 . . . . . . 0 ... ...

−an−1 0 .

1

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Linearisation The linearisation can be done around an equilibrium point or around a particular point defined by: ( x˙ eq (t) = f ((xeq (t), ueq (t), t), givenxeq (0) (7) yeq (t) = g((xeq (t), ueq (t), t)

Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Defining

Properties

˜ = u − ueq , y˜ = y − yeq x˜ = x − xeq , u

Stability

this leads to a linear state space representation of the system, around the equilibrium point: ( ˜ (t), x˜˙ (t) = Ax˜ (t) + B u (8) ˜ (t) y˜ (t) = C x˜ (t) + D u

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

with A = C=

∂f | , ∂ x x=xeq ,u=ueq

∂g | ∂ x x=xeq ,u=ueq

B=

and D =

∂f | , ∂ u x=xeq ,u=ueq ∂g | x=x ,u=u eq eq ∂u

Observation Observability Observer design Observer-based control

Introduction to optimal control

Usual case Usually an equilibrium point satisfies: 0 = f ((xeq (t), ueq (t), t) For the pendulum, we can choose y = θ = f = 0.

Introduction to digital control

(9)

Conclusion

Underwater Autonomous Vehicle UAV Aster x

State space approach Olivier Sename Introduction

Nonlinear model: 12 state variables; 6 control inputs. Equilibrium point chosen as [u v w p q r ] = [1 0 0 0 0 0] : all velocities are taken equal to 0, except the longitudinal velocity taken equal to 1m/s (cruising speed). Tangential linearization around the chosen equilibrium point:

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties

˜ (t) x˜˙ = Ax˜ (t) + B u y˜ = C x˜ (t) + Du(t)

where I x ˜ = δ [x u y v z w φ p θ q ψ r ]T = δ [β1 β10

β2 β20

]T

I

˜ u

I

y˜ measured output (here only the altitude z is measured)

δ 1 Qc

Matrices A, B, C and D depend on the model parameters : hydrodynamical parameters, mass of the vehicle, dimension of fins . . . Note that most of these parameters are uncertain, and that here the control design is proposed for the nominal plant case only.

Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Exercise: Inverted pendulum (2)

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models

Applying the linearisation method leads to :

Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Are state space representations equivalent to transfer functions ?

Olivier Sename Introduction Modelling

Equivalence transfer function - state space representation

Nonlinear models

Consider a linear system given by: ˙ x(t) = Ax(t) + Bu(t), x(0) = x0 y (t) = Cx(t) + Du(t)

Linearisation

Linear models

To/from transfer functions

(10)

Properties Stability

State feedback control

Using the Laplace transform (and assuming zero initial condition x0 = 0), (10) becomes:

Problem formulation Controllability Definition Pole placement control

s.x(s) = Ax(s) + Bu(s) ⇒ (s.In − A)x(s) = Bu(s)

Specifications Integral Control

Observer

Then the transfer function matrix of system (10) is given by G(s) = C(sIn − A)−1 B + D =

N(s) D(s)

Observation Observability

(11)

Matlab: if SYS is an SS object, then tf(SYS) gives the associated transfer matrix. Equivalent to tf(N,D)

Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Conversion TF to SS

Olivier Sename Introduction Modelling Nonlinear models

There mainly three cases to be considered

Linear models Linearisation

Simple numerator 1 y = G(s) = 3 2 u s + a1 s + a2 s + a3 Numerator order less than denominator order y b s2 + b2 s + b3 N(s) = G(s) = 3 1 = 2 u D(s) s + a1 s + a2 s + a3 Numerator equal to denominator order

To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

y b s3 + b1 s2 + b2 s + b3 N(s) = G(s) = 0 3 = u D(s) s + a1 s2 + a2 s + a3

Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Canonical forms

Olivier Sename Introduction Modelling

Some specific state space representations are well-known and often usedas the so-called controllable canonical form 0 1 0 ... 0 0 0 .. 0 1 0 ... . .. .. .. .. .. . . . . A= . , B = ... and . .. 0 0 0 1 −a0 −a1 . . . . . . −an−1 1 c0 c1 ... cn−1 . C= It corresponds to the transfer function:

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

G(s) = In Matlab, use canon

c0 + c1 s + . . . + cn−1 sn−1 a0 + a1 s + . . . + an−1 sn−1 + sn

Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

How to compute the solution x(t) of a linear system?

State space approach Olivier Sename Introduction Modelling Nonlinear models

˙ Case of the autonomous equation x(t) = Ax(t)

Linear models Linearisation

It is the generalization of the scalar case: if y˙ = αy then y (t) = exp(αt)y0 . The state x(t) with initial condition x(0) = x0 is then given by x(t) = eAt x(0)

To/from transfer functions

Properties Stability

(12)

State feedback control Problem formulation Controllability

To get an explicit analytical form, this requires to compute eAt , which can be done following one of the 3 methods to compute eAt : 1. Inverse Laplace transform of (sIn

− A)−1

Definition Pole placement control Specifications Integral Control

Observer

2. Diagonalisation of A

Observation

3. Cayley-Hamilton method

Observer design

In Matlab : use expm(A*t) and not exp (if t is given).

Observability

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

How to compute the solution x(t) of a linear system ? (cont..)

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation

General case of system (10)

To/from transfer functions

The state x(t), solution of system (10), is given by

Properties Stability

x(t) =

Z t

eAt x(0) + eA(t−τ) Bu(τ)dτ | {z } |0 {z } free response forced response

(13)

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Simulation of state space systems Use lsim. Example: t = 0:0.01:5; u = sin(t); lsim(sys,u,t)

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

Properties

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Non unicity

Olivier Sename Introduction

Given a transfer function, there exists an infinity of state space representations (equivalent in terms of input-output behavior). Let ˙ x(t) = Ax(t) + Bu(t), (14) y (t) = Cx(t) + Du(t)

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

− A)−1 B + D,

the transfer matrix being G(s) = C(sIn and consider the change of variables x = Tz (T being an invertible matrix). Replacing x = Tz in the previous system gives:

State feedback control Problem formulation Controllability Definition

˙ T z(t)

=

ATz(t) + Bu(t)

(15)

y (t)

=

CTz(t) + Du(t)

(16)

Pole placement control Specifications Integral Control

Observer Observation

Hence

Observability Observer design

˙ z(t)

=

T −1 ATz(t) + T −1 Bu(t)

(17)

y (t)

=

CTz(t) + Du(t)

(18)

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach Olivier Sename

˜ = CT , the transfer function of ˜ = T −1 AT , B ˜ = T −1 B and C Defining A the previous system is:

Introduction Modelling Nonlinear models Linear models

˜ G(s)

= =

Linearisation

˜ n − A) ˜ −1 B ˜ +D C(sI C T (sIn − T

−1

AT )

(19) −1

T

−1

B +D

(20) (21)

Using In = T −1 T , we get ˜ G(s) = C T T −1 (sIn − A)−1 T T −1 B + D = G(s)

To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition

(22)

Pole placement control Specifications

Exercise: For the quarter-car model, choose: x1 = zs , x2 = z˙ s , x3 = zs − zus , x4 = z˙ s − z˙ us

Integral Control

Observer Observation Observability Observer design

and give the equivalent state space representation.

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Stability

Olivier Sename Introduction Modelling Nonlinear models

Definition

Linear models

An equilibrium point xeq is stable if, for all ρ > 0, there exists a η > 0 such that:

Linearisation To/from transfer functions

Properties

kx(0) − xeq k < η =⇒ kx(t) − xeq k < ρ, ∀t ≥ 0

Stability

State feedback control Problem formulation

Definition

Controllability

An equilibrium point xeq is asymptotically stable if it is stable and, there exists η > 0 such that:

Pole placement control

Definition

Specifications Integral Control

Observer

kx(0) − xeq k < η =⇒ x(t) → xeq , when t → ∞

Observation Observability Observer design

These notions are equivalent for linear systems (not for non linear ones).

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Stability Analysis

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation

The stability of a linear state space system is analyzed through the characteristic equation det(sIn − A) = 0. The system poles are then the eigenvalues of the matrix A. It then follows:

Proposition ˙ A system x(t) = Ax(t), with initial condition x(0) = x0 , is stable if Re(λi ) < 0, ∀i, where λi , ∀i, are the eigenvalues of A. Using Matlab, if SYS is an SS object then pole(SYS) computes the poles P of the LTI model SYS. It is equivalent to compute eig(A).

To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

The Phase Plane It consists in plotting the trajectory of the state variables (valid also for nonlinear systems). Trajectories that converge to zero are stable ! x˙ 1 (t) = x2 (t) given x1 (0) & x2 (0) x˙ 2 (t) = −5x1 (t) − 6x2 (t)

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Stability Analysis - Lyapunov

Olivier Sename Introduction

The stability of a linear state space system can be analysed through the Lyapunov theory. It is the basis of all extension of stability for non linear systems, time-delay systems, time-varying systems ...

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties

Theorem

Stability

˙ A system x(t) = Ax(t), with initial condition x(0) = x0 , is asymoptotically stable at x = 0 if and only if there exist some matrices P = P T > 0 and Q > 0 such that: T

A P + PA = −Q

State feedback control Problem formulation Controllability Definition Pole placement control

(23)

see lyap in MATLAB. Proof: The Lyapunov theory says that a linear system is stable if there exists a continuous function V (x) s.t.: V (x) > 0 with V (0) = 0 and V˙ (x) = dV dx ≤ 0. A possible Lyapunov function for the above system is : V (x) = x T Px

Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

About zeros

Olivier Sename Introduction Modelling

I

I

Nonlinear models

Roots of the transfer function numerator are called the system zeros.

Linear models Linearisation

Need to develop a similar way of defining/computing them using a state space model.

I

Zero: is a generalized frequency α for which the system can have a non-zero input u(t) = u0 eαt , but exactly zero output y (t) = 0.

I

The zeros are found by solving: A − λ In det C

To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control

B D

Specifications

=0

(24)

Integral Control

Observer Observation

In Matlab use zero

Observability Observer design

I

do remind that zeros may change the system behavior.

Example: find the zero of :

s+3 s2 +5s+2

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Objective of any control system

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties

shape the response of the system to a given reference and get (or keep) a stable system in closed-loop, with desired performances, while minimising the effects of disturbances and measurement noises, and avoiding actuators saturation, this despite of modelling uncertainties, parameter changes or change of operating point.

Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Formal objective of any control system

State space approach Olivier Sename Introduction Modelling Nonlinear models

Nominal stability (NS): The system is stable with the nominal model (no model uncertainty) Nominal Performance (NP): The system satisfies the performance specifications with the nominal model (no model uncertainty) Robust stability (RS): The system is stable for all perturbed plants about the nominal model, up to the worst-case model uncertainty (including the real plant)

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Robust performance (RP): The system satisfies the performance specifications for all perturbed plants about the nominal model, up to the worst-case model uncertainty (including the real plant).

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

About Feedback control

State space approach Olivier Sename Introduction Modelling Nonlinear models

How to design a controller using a state space representation ? Tow cases are possible : I

Static controllers (output or state feedback)

I

Dynamic controllers (output feedback or observer-based)

Linear models Linearisation To/from transfer functions

Properties Stability

What for ?

State feedback control Problem formulation Controllability

I

Closed-loop stability (of state or output variables)

I

disturbance rejection

I

Model tracking

I

Input/Output decoupling

Observation

I

Other performance criteria : H2 optimal, H∞ robust...

Observer design

Definition Pole placement control Specifications Integral Control

Observer Observability

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Why state feedback and not output feedback?

State space approach Olivier Sename Introduction Modelling

Example: G(s) =

y (s) u(s)

Nonlinear models

=

1 s2 −s

1. Give the controllable canonical form considering x1 = y , x2 = y˙ . 2. Case of output feedback= Proportional control : u = −Kp y I

I

Compute the closed-loop transfer function and check that the closed-loop poles are given by the roots of the characteristic polynomial PBF (s) = s2 − s + L. Can the closed-loop system be stabilized ?

3. Case of state feedback : consider the control law u = −x1 + 3x2 + yref I I I I

Compute the state space representation of the closed-loop system. What are the poles of the closed-loop systems? Is it stable? If yes why this second control solves the problem?

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

A preliminary property analysis: Controllability

State space approach Olivier Sename

Controllability refers to the ability of controlling a state-space model using state feedback.

Introduction Modelling Nonlinear models

Definition Given two states x0 and x1 , the system (10) is controllable if there exist t1 > 0 and a piecewise-continuous control input u(t), t ∈ [0, t1 ], such that x(t) takes the values x0 for t = 0 and x1 for t = t1 .

Proposition The controllability matrix is defined by C = [B, A.B, . . . , An−1 .B]. Then system (10) is controllable if and only if rank (C ) = n. If the system is single-input single output (SISO), it is equivalent to det(C ) 6= 0. Using Matlab, if SYS is an SS object then crtb(SYS) returns the controllability matrix of the state-space model SYS with realization (A,B,C,D). This is equivalent to ctrb(sys.a,sys.b)

Exercices Test the controllability of the previous examples: DC motor, suspension, inverted pendulum.

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Definition of the state feedback control

Olivier Sename Introduction Modelling Nonlinear models

A state feedback controller for a continuous-time system is: u(t) = −Fx(t)

Linear models Linearisation

(25)

To/from transfer functions

Properties

where F is a m × n real matrix. When the system is SISO, it corresponds to : u(t) = −f1 x1 − f2 x2 − . . . − fn xn with F = [f1 , f2 , . . . , fn ]. When the system is MIMO we have x1 u1 f . . . f 11 1n u2 x . .. 2 . = .. . . .. .. fm1 . . . fmn um xn

Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

State feedback (2)

Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation

Using state feedback controllers (25), we get in closed-loop (for simplicity D = 0) ˙ x(t) = (A − BF )x(t), y (t) = Cx(t)

To/from transfer functions

Properties Stability

(26)

State feedback control Problem formulation Controllability

The stability (and dynamics) of the closed-loop system is then given by the eigenvalues of A − BF . Then the solution y (t) = C exp(A−BF )t x0 converges asymptotically to zero!

Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Complete state feedback control for reference tracking

Olivier Sename Introduction Modelling Nonlinear models

Objective: y should track some reference signal r , i.e.

Linear models Linearisation

y (t) −−→ r (t)

To/from transfer functions

t→∞

Properties

Then the state feedback control must be of the form: u(t)

=

−Fx(t)+Gr (t)

Stability

(27)

State feedback control Problem formulation Controllability

G is a m × p real matrix. Then the closed-loop transfer matrix is : GCL (s) = C(sIn − A + BF )

−1

Definition Pole placement control Specifications

BG

(28)

Integral Control

Observer Observation

G is chosen to ensure a unitary steady-state gain as: G = [C(−A + BF )

−1

B]

−1

Observability Observer design

(29)

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Synthesis of the state feedback control: the pole placement control

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models

Problem definition Given a linear system (5), does there exist a state feedback control law (25) such that the closed-loop poles are in predefined locations (denoted γi , i = 1, ..., n ) in the complex plane ?

Proposition Let a linear system given by A, B, and let γi , i = 1, ..., n , a set of complex elements (i.e. the desired poles of the closed-loop system). There exists a state feedback control u = −Fx such that the poles of the closed-loop system are γi , i = 1, ..., n if and only if the pair (A, B) is controllable. In Matlab, use F=acker(A,B,P) or F=place(A,B,P) where P = [γ1 , . . . , γn ] is the set of desired closed-loop poles.

Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Pole placement control: the case of controllable canonical forms

State space approach Olivier Sename Introduction

In this first case, we assume that the system state space representation is of the form: 0 0 1 0 ... 0 0 .. 0 1 0 ... . .. .. .. .. .. , B = . and . . . . . A= .. .. 0 0 . 0 1 −a0 −a1 . . . . . . −an−1 c0 c1 ... cn−1 . C= Let F = [ f1 f2 . . . fn ] Then 0 1 0 0 0 1 .. .. .. . . . A − BF = . .. 0 −a0 − f1 −a1 − f2 . . .

1

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability

Definition Pole placement control Specifications Integral Control

Observer

... 0 .. .

0 ... .. .

0 ...

1 −an−1 − fn

Observation Observability

Observer design Observer-based control

(30)

Introduction to optimal control Introduction to digital control Conclusion

the case of controllable canonical forms (cont..)

State space approach Olivier Sename Introduction Modelling

Remind that the desired closed-loop polynomial writes: (s − γ1 )(s − γ2 )...(s − γn ) or equivalently sn + αn−1 sn−1 + . . . + α1 s + α0 . The choice:

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

fi = −ai−1 + αi−1 , i = 1, .., n is a problem solution and ensures that the poles of A − BF are {γi }, i = 1, n

State feedback control Problem formulation Controllability Definition Pole placement control Specifications

Remark: the case of controllable canonical forms is very important since , when we consider a general state space representation, it is first necessary to use a change of basis to make the system under canonical form, which will simplify a lot the computation of the state feedback control gain F (see next slide).

Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Pole placement control: general procedure

Olivier Sename

Procedure for the general case:

Introduction

1. Check controllability of (A, B) 2. Calculate C = [B, AB, . . . , An−1 B]. q1 Note C −1 = ... . Define T = qn

State space approach

Modelling Nonlinear models

qn qn A .. . qn An−1

−1

¯ = T −1 AT and B ¯ = T −1 B (which are under the controllable 3. Note A canonical form)

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control

4. Choose the desired closed-loop poles and define the desired closed-loop characteristic polynomial: sn + αn−1 sn−1 + . . . + α1 s + α0 5. Calculate the state feedback u = −F¯ x with:

Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

¯f = −a + α , i = 1, .., n i i−1 i−1 6. Calculate (for the original system): u = −Fx, with F = F¯ T −1

Introduction to optimal control Introduction to digital control Conclusion

How to specificy the desired closed-loop performances?

State space approach Olivier Sename

The required closed-loop performances should be chosen in the following zone

Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control

which ensures a damping greater than ξ = sin φ . −γ implies that the real part of the CL poles are sufficiently negatives.

Conclusion

State space approach

Specifications (2)

Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation

Some useful rules for selection the desired pole/zero locations (for a second order system):

To/from transfer functions

Properties Stability

1.8 ωn

I

Rise time : tr '

I

Seetling time : ts '

I

Overshoot Mp = exp(−πξ /sqrt(1 − ξ 2 )): ξ = 0.3 ⇔ Mp = 35%, ξ = 0.5 ⇔ Mp = 16%, ξ = 0.7 ⇔ Mp = 5%.

4.6 ξ ωn

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Specifications(3) Some rules do exist to shape the transient response. The ITAE (Integral of Time multiplying the Absolute value of the Error), defined as: ITAE =

Z ∞

Olivier Sename Introduction Modelling

t|e(t)|dt

0

Nonlinear models Linear models Linearisation To/from transfer

functions can be used to specify a dynamic response with relatively small Properties overshoot and relatively little oscillation (there exist other methods to do Stability so). The optimum coefficients for the ITAE criteria are given below (see State feedback Dorf & Bishop 2005). control Problem formulation Order Characteristic polynomials dk (s) Controllability Definition 1 d1 = [s + ωn ] Pole placement control 2 d2 = [s2 + 1.4ωn s + ωn2 ] Specifications Integral Control 3 d3 = [s3 + 1.75ωn s2 + 2.15ωn2 s + ωn3 ] Observer 4 3 2 2 3 4 4 d4 = [s + 2.1ωn s + 3.4ωn s + 2.7ωn s + ωn ] Observation Observability 5 d5 = [s5 + 2.8ωn s4 + 5ωn2 s3 + 5.5ωn3 s2 + 3.4ωn4 s + ωn5 ] Observer design 6 d6 = [s6 + 3.25ωn s5 + 6.6ωn2 s4 + 8.6ωn3 s3 + 7.45ωn4 s2 + 3.95ωn5 s +Observer-based ωn6 ] control

and the corresponding transfer function is of the form: Hk (s) =

ωnk dk (s)

, ∀k = 1, ..., 6

Introduction to optimal control Introduction to digital control Conclusion

Specifications(4)

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Integral Control or how to ensure disturbance attenuation with a state feedback control?

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models

Preliminary remrak: a state feedback controller may not allow to reject the effects of disturbances (particularly of input disturbances). Let us consider the system: ( ˙ x(t) = Ax(t) + Bu(t)+Ed(t), x(0) = x0 (31) y (t) = Cx(t)

Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition

where d is the disturbance. The objective is to keep y following a reference signal r even in the presence of d, i.e y −→ 1 r − t→∞ y I −→ 0 d − t→∞ I

Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Formulation of the Integral Control

Olivier Sename Introduction Modelling

A very useful method consists in adding an integral term to ensure a unitary static closed-loop gain . The method consists in extending the system by adding a new state variable:

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

˙ z(t) = r (t) − y (t)

State feedback control

x . z Then the new state space representation is given as:

Problem formulation

We need to define the extended state vector

Controllability Definition Pole placement control Specifications Integral Control

˙ x(t) ˙ z(t)

y (t)

= =

A −C C

0 x 0 z x 0 z

+

0 1

u(t) +

B 0

r (t) +

E 0

Observer

d(t)

Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Synthesis of the Integral Control Let us define:

Olivier Sename

Ae =

A −C

0 0

, Be =

B 0

, Ce =

C

0

Introduction Modelling Nonlinear models

The new state feedback control is now of the form x u(t) = −Fe = −Fx(t)−Hz(t) z with Fe = [F H]. Then the synthesis of the control law u(t) requires: I the verification of the extended system controllability I the specification of the derised closed-loop performances, i.e. a set Pe of n + 1 desired closed-loop poles has to be chosen, I the computation of the full state feedback Fe using Fe=acker(Ae,Be,Pe) We then get the closed-loop system ˙ x(t) A − BF BH x 0 E = + r (t) + d(t) ˙ z(t) −C 0 z 1 0 x C 0 y (t) = z

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Integral control scheme

State space approach Olivier Sename Introduction Modelling

The complete structure has the following form:

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

When an observer is to be used, the control action simply becomes: u(t) = −F xˆ (t) − Hz(t)

Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties

Observer and output feedback control

Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Introduction

Olivier Sename

A first insight

Introduction

To implement a state feedback control, the measurement of all the state variables is necessary. If this is not available, we will use a state estimation through a so-called Observer.

Observation or Estimation The estimation theory is based on the famous Kalman contribution to filtering problems (1960), and accounts for noise induced problems. The observation theory has been developed for Linear Systems by Luenberger (1971), and doe snot consider the noise effects.

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control

Other interest of observation/estimation In practice the use of sensors is often limited for several reasons: feasibility, cost, reliability, maintenance ... An observer is a key issue to estimate unknown variables (then non measured variables) and to propose a so-called virtual sensor. Objective: Develop a dynamical system whose state xˆ (t) satisfies: I

(x(t) − xˆ (t)) −−→ 0 t→∞

I

(x(t) − xˆ (t)) → 0 as fast as possible

Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

A first approach to estimation from input data: Open loop observers Let consider

State space approach Olivier Sename Introduction

(

˙ x(t) = Ax(t) + Bu(t), x(0) = x0 y (t) = Cx(t)

Given that we know the plant matrices and the inputs, we can just perform a simulation that runs in parallel with the system ( xˆ˙ (t) = Ax(t) + Bu(t), given xˆ (0)

Modelling

(32)

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control

(33)

yˆ (t) = C xˆ (t)

Problem formulation Controllability Definition Pole placement control

Therefore, if we would have xˆ (0) = x(0), then xˆ (t) = x(t), ∀t ≥ 0.

Specifications Integral Control

BUT

Observer Observation

I I

I

x(0) is UNKNOWN so we cannot choose xˆ (0) = x(0), the estimation error (e = x − xˆ ) dynamics is determined by A, i.e ˙ satisfies e(t) = Ae(t) (could be unstable AND cannot be modified) the effects of disturbance and noise cannot be attenuated (leads to estimation biais)

NEED FOR A FEEDBACK FROM MEASURED OUTPUTS TO CORRECT THE ESTIMATION !

Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

A preliminary property: Observability Observability refers to the ability to estimate a state variable (often not measured !!).

Definition

Olivier Sename Introduction Modelling

A linear system (5) is completely observable if, given the control and the output over the interval t0 ≤ t ≤ T , one can determine any initial state x(t0 ). It is equivalent to characterize the non-observability as : A state x(t) is not observable if the corresponding output vanishes, i.e. if the following holds: y (t) = y˙ (t) = y¨ (t) = . . . = 0

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation

Where does observability come from ?

Controllability

Compare the transfer function of the two different systems*

Pole placement control

Definition

Specifications Integral Control

x˙

=

−x + u

y

=

2x

Observer Observation Observability Observer design Observer-based control

and x˙

=

y

=

−1 0

0 −2

x+

1 1

Introduction to optimal control

u

Introduction to digital control Conclusion

2

0

x

State space approach

Observability cont.

Olivier Sename Introduction

Proposition

Modelling

The observability matrix is defined by O =

C CA .. . CAn−1

Nonlinear models Linear models

. Then system

(10) is observable if and only if rank (O) = n. If the system is single-input single output (SISO), it is equivalent to det(O) 6= 0. Using Matlab, if SYS is an SS object then obsv(SYS) returns the observability matrix of the state-space model SYS with realization (A,B,C,D). This is equivalent to OBSV(sys.a,sys.c).

Exercices Test the observability of the previous examples: DC motor, suspension, inverted pendulum. Analysis of different cases, according to the considered number of sensors.

Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Closed-loop Observer: estimation from input AND output data

State space approach Olivier Sename Introduction

Objective: since y is KNOWN (measured) and is function of the state variables, use an on line comparison of the measured system output y and the estimated output yˆ . Observer description:

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

xˆ˙ (t) xˆ0 ∈ R n to be defined

=

Axˆ (t) + Bu(t) + L(y (t) − C xˆ (t))

(34)

State feedback control Problem formulation Controllability

where xˆ (t) ∈ Rn is the estimated state of x(t) and L is the n × p constant observer gain matrix to be designed.

Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Analysis of the observer properties

Olivier Sename Introduction

Th estimated error, e(t) := x(t) − xˆ (t), satisfies:

Modelling Nonlinear models

˙ e(t)

=

(A − LC)e(t)

(35)

Linear models Linearisation To/from transfer functions

If L is designed such that A − LC is stable, then xˆ (t) converges asymptotically towards x(t).

Properties

Proposition

State feedback control

(34) is an observer for system (5) if and only if the pair (C,A) is observable, i.e.

Stability

Problem formulation Controllability Definition Pole placement control Specifications

rank (O) = n

Integral Control

Observer

where O =

C CA .. . CAn−1

Observation Observability

.

Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Observer design The observer design is restricted to find L such that A − LC is stable (so that (x(t) − xˆ (t)) −−→ 0) and has some desired eigenvalues (so that t→∞

(x(t) − xˆ (t)) → 0 as fast as possible). This is still a pole placement problem.

Specifications Usually the observer poles are chosen around 5 to 10 times higher than the closed-loop system, so that the state estimation is good as early as possible. This is quite important to avoid that the observer makes the closed-loop system slower.

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition

Design method

Pole placement control Specifications Integral Control

I

I

I

In order to use the acker Matlab function, we will use the duality property between observability and controllability, i.e. : (C, A) observable ⇔ (AT , C T ) controllable. Then there exists LT such that the eigenvalues of AT − C T LT can be randomly chosen. As (A − LC)T = AT − C T LT then L exists such that A − LC is stable. Matlab : use L=acker(A’,C’,Po)’ where Po is the set of desired observer poles.

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Observer-based control: integration of the observer into the control loop

State space approach Olivier Sename Introduction

When an observer is built, we will use as control law:

Modelling Nonlinear models

u(t) = −F xˆ (t) + Gr (t)

(36)

Linear models Linearisation

But: can we ensure that the scheme observer+state feedback works (i.e is stable and meets the requirements)? We then need to study the stability of the complete closed-loop system, using the extended state:

To/from transfer functions

Properties Stability

State feedback control Problem formulation

xe (t) =

x(t)

e(t)

T

Controllability Definition Pole placement control

The closed-loop system with observer (34) and control (36) is: A − BF BF BG x˙ e (t) = xe (t) + r (t) 0 A − LC 0 y

=

[C 0]xe (t)

The closed-loop system from r to y is then: y = C(sIn − A + BF )BG r

Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Separation principle

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation

Therefore, the characteristic polynomial of the extended system is: det(sIn − A + BF ) × det(sIn − A + LC) If the observer and the control are designed separately then the closed-loop system with the dynamic measurement feedback is stable, given that the controlled and observer systems are stable. The whole set of closed-loop poles then include those of the state-feedback controlled system (i.e. the eigenvalues of A − BF ), plus those of the observer (i.e. the eigenvalues of A − LC). This corresponds to the so-called separation principle.

To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

Introduction to optimal control

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Introduction

Olivier Sename Introduction

The objective of an optimal control is to minimize a cost function which penalizes simultaneously the state and input behaviors, of the form R∞ L(x, y )dt, i.e to reach a tradeoff between the transient response and 0 the control effort. This objective is defined through the following criteria always considered in the quadratic form: J=

Z ∞

(x T Qx + u T Ru)dt

0

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition

In that form: I

x T Qx is the state cost,

I

u T Ru is the control cost,

I

Q and R are respectively the state and cost penalties.

Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

It can be proved that the state feedback control that minimizes J in closed-loop (given Q and R) is obtained solving an Algebraic Riccati Equation (ARE)

Introduction to optimal control Introduction to digital control Conclusion

Linear-Quadratic Regulator (LQR) design

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models

LQR problem solution

Linearisation

˙ Given a linear system x(t) = Ax(t) + Bu(t), with (A, B) stabilizable, and given positive definite matrices Q = Q T > 0 and R = R T > 0, if there exists P = P T > 0 s.t: AT P + PA − PBR −1 B T P + Q = 0

To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition

then the state feedback control u = −Kx such that:

Pole placement control Specifications

K =R

−1 T

B P

Integral Control

Observer Observation

minimizes the quadractic criteria J (for given Q and R).

Observability Observer design

This problem is handled in Matlab through the lqr command.

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

Introduction to digital control

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Toward digital control

State space approach Olivier Sename

Digital control

Introduction

Usually controllers are implemented in a digital computer as:

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

This requires the use of the discrete theory. m (Sampling theory + Z-Transform) m

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Definition of the Z-Transform

Olivier Sename Introduction Modelling

Mathematical definition

Nonlinear models

Because the output of the ideal sampler, with values x(kTe ), we have: x ∗ (t) =

x ∗ (t),

is a series of impulses

Linear models Linearisation To/from transfer functions

∞

Properties

∑ x(kTe )δ (t − kTe )

k =0

Stability

State feedback control Problem formulation

by using the Laplace transform,

Controllability Definition

∗

L [x (t)] =

∞

∑ x(kTe )e

−ksTe

Pole placement control Specifications Integral Control

k =0

Observer

Noting z = esTe , we can derive the so called Z-Transform

Observation Observability Observer design

∞

X (z) = Z [x(k )] =

∑ x(k )z

k =0

−k

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Properties

Olivier Sename Introduction Modelling

Definition

Nonlinear models Linear models

∞

X (z) = Z [x(k )] =

∑ x(k )z

−k

k =0

Linearisation To/from transfer functions

Properties Stability

Properties

State feedback control

Z [αx(k ) + β y (k )] Z [x(k − n)] Z [kx(k )]

= = =

Z [x(k ) ∗ y (k )]

=

lim x(k )

=

k →∞

αX (z) + β Y (z) z

−n

Z [x(k )] d −z Z [x(k )] dz X (z).Y (z) lim (z − 1)X (z)

1→z −1

The z −1 can be interpreted as a pure delay operator.

Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Zero order holder

State space approach Olivier Sename Introduction

Sampler and Zero order holder A sampler is a switch that close every Te seconds. A Zero order holder holds the signal x for Te seconds to get h as: h(t + kTe ) = x(kTe ), 0 ≤ t < Te

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Zero order holder (cont’d)

Olivier Sename Introduction Modelling Nonlinear models Linear models

Model of the Zero order holder

Linearisation

The transfer function of the zero-order holder is given by: GBOZ (s)

= =

1 e−sTe − s s 1 − e−sTe s

To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Influence of the D/A and A/D Note that the precision is also limited by the available precision of the converters (either A/D or D/A). This error is also called the amplitude quantization error.

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Representation of the discrete linear systems

State space approach Olivier Sename

The discrete output of a system can be expressed as:

Introduction Modelling

∞

y (k ) =

∑ h(k − n)u(n)

n=0

hence, applying the Z-transform leads to

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

Y (z) = Z [h(k )]U(z) = H(z)U(z)

State feedback control Problem formulation Controllability

Y b + b1 z + · · · + bm z m = H(z) = 0 a0 + a1 z + · · · + an z n U where n (≥ m) is the order of the system Corresponding difference equation:

Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design

y (k )

= −

1 b u(k − n) + b1 u(k − n + 1) + · · · + bm u(k − n + m) an 0 a0 y (k − n) − a2 y (k − n + 1) − · · · − an−1 y (k − 1)

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Some useful transformations

Olivier Sename Introduction Modelling Nonlinear models Linear models Linearisation

x(t) δ (t) δ (t − kTe ) u(t) t e−at 1 − e−at sin(ωt) cos(ωt)

X (s) 1 e−ksTe

X (z) 1 z −k

1 s 1 s2 1 s+a 1 s(s+a) ω s2 +ω 2 s s2 +ω 2

z z−1 zTe (z−1)2 z z−e−aTe z(1−e−aTe ) (z−1)(z−e−aTe ) zsin(ωTe ) z 2 −2zcos(ωTe )+1 z(z−cos(ωTe )) z 2 −2zcos(ωTe )+1

To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Poles, Zeros and Stability

Olivier Sename Introduction

Equivalence {s} ↔ {z}

Modelling

The equivalence between the Laplace domain and the Z domain is obtained by the following transformation: z =e

sTe

Nonlinear models Linear models Linearisation To/from transfer functions

Properties

Two poles with a imaginary part witch differs of 2π/Te give the same pole in Z.

Stability

State feedback control Problem formulation

Stability domain

Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Approximations for discretization

State space approach Olivier Sename Introduction Modelling

Forward difference (Rectangle inferior)

Nonlinear models Linear models Linearisation To/from transfer functions

s=

z −1 Te

Properties Stability

State feedback control Problem formulation Controllability Definition

Backward difference (Rectangle superior)

Pole placement control Specifications Integral Control

Observer Observation

z −1 s= zTe

Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Approximations for discretization (cont’d) Trapezoidal difference (Tustin)

State space approach Olivier Sename Introduction Modelling Nonlinear models

2 z −1 s= Te z + 1

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Systems definition A discrete-time state space system is as follows: ( x((k + 1)h) = Ad x(kh) + Bd u(kh), x(0) = x0 y (kh) = Cd x(kh) + Dd u(kh)

Olivier Sename Introduction Modelling

(37)

Nonlinear models Linear models Linearisation

where h is the sampling period. Matlab : ss(Ad ,Bd ,Cd ,Dd ,h) creates a SS object SYS representing a discrete-time state-space model From a discretization step (c2d) we have:

To/from transfer functions

Properties Stability

State feedback control Problem formulation

Ad = exp(Ah), Bd = (

Z h

Controllability Definition

exp(Aτ)dτ)B

Pole placement control

0

Specifications Integral Control

For discrete-time systems, x((k + 1)h) = Ad x(kh) + Bd u(kh), x(0) = x0 y (kh) = Cd x(kh) + Dd u(kh)

Observer Observation

(38)

Observer design Observer-based control

Introduction to optimal control

the discrete transfer function is given by G(z) = Cd (zIn − Ad )−1 Bd + Dd

Observability

(39)

Introduction to digital control Conclusion

where z is the shift operator, i.e. zx(kh) = x((k + 1)h)

State space approach

Solution of state space equations - discrete case

Olivier Sename Introduction Modelling

The state xk , solution of system xk +1 = Ad xk with initial condition x0 , is given by

Nonlinear models Linear models Linearisation To/from transfer functions

x1

=

Ad x0

(40)

x2

=

A2d x0

(41)

xn

=

And x0

(42)

Properties Stability

State feedback control Problem formulation Controllability

The state xk , solution of system (37), is given by

Definition Pole placement control

x1

=

Ad x 0 + Bd u 0

(43)

x2

=

A2d x0 + Ad Bd u0 + Bd u1

(44)

=

And x0 +

xn

n−1

∑

i=0

Specifications Integral Control

Observer Observation Observability

An−1−i Bd ui d

(45)

Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space analysis (discrete-time systems)

State space approach Olivier Sename Introduction

Stability A system (state space representation) is stable iff all the eigenvalues of the matrix F are inside the unit circle.

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Controllability definition

Properties

Definition

State feedback control

Given two states x0 and x1 , the system (37) is controllable if there exist K1 > 0 and a sequence of control samples u0 , u1 , . . . , uK1 , such that xk takes the values x0 for k = 0 and x1 for k = K1 .

Stability

Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observability definition

Observer Observation Observability

Definition The system (37) is said to be completely observable if every initial state x(0) can be determined from the observation of y (k ) over a finite number of sampling periods.

Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

State space analysis (2)

Olivier Sename Introduction Modelling

Controllability

Nonlinear models

The system is controllable iff

Linearisation

Linear models

Cd (A

d ,Bd

= rg[Bd Ad Bd . . . An−1 d Bd ] = n )

To/from transfer functions

Properties Stability

State feedback control

Observability

Problem formulation

The system is observable iff

Definition

Controllability

Pole placement control

T O(Ad ,Cd ) = rg[Cd Cd Ad . . . Cd An−1 d ]

=n

Specifications Integral Control

Observer Observation

Duality Observability of (Cd , Ad ) ⇔ Controllability of (ATd , CdT ). Controllability of (Ad , Bd ) ⇔ Observability of (BdT , ATd ).

Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

About sampling period

State space approach Olivier Sename Introduction

Influence of the sampling period on the time response

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability

Impose a maximal time response to a discrete system is equivalent to place the poles inside a circle defined by the upper bound of the bound given by this time response. The closer to zero the poles are , the faster the system is.

Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Frequency analysis

State space approach Olivier Sename

As in the continuous time, the Bode diagram can also be used. Example with sampling Time Te = 1s ⇔ fe = 1Hz ⇔ we = 2π):

Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

Note that, in our case, the Bode is cut at the pulse w = π. see SYSD = c2d(SYSC,Ts,METHOD) in MATLAB.

Observation Observability Observer design Observer-based control

Sampling ↔ Limitations Recall the Shannon theorem which imposes the sampling frequency at least 2 times higher than the system maximum frequency. Related to the anti-aliasing filter . . .

Introduction to optimal control Introduction to digital control Conclusion

Frequency analysis

State space approach Olivier Sename

As in the continuous time, the Bode diagram can also be used. Example with sampling Time Te = 1s ⇔ fe = 1Hz ⇔ we = 2π):

Introduction Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer

Note that, in our case, the Bode is cut at the pulse w = π. see SYSD = c2d(SYSC,Ts,METHOD) in MATLAB.

Observation Observability Observer design Observer-based control

Sampling ↔ Limitations Recall the Shannon theorem which imposes the sampling frequency at least 2 times higher than the system maximum frequency. Related to the anti-aliasing filter . . .

Introduction to optimal control Introduction to digital control Conclusion

About sampling period and robustness

State space approach Olivier Sename Introduction

Influence of the sampling period on the poles In theory, smaller the sampling period Te is, closer the discrete system is from the continuous one.

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation

But reducing the sampling time modify poles location . . . Poles and zeros become closer to the limit of the unit circle ⇒ can introduce instability (decrease robustness). ⇒ Sampling influences stability and robustness ⇒ Over sampling increase noise sensitivity

Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Stability

Olivier Sename

Recall

Introduction

A linear continuous feedback control system is stable if all poles of the closed-loop transfer function T (s) lie in the left half s-plane. The Z-plane is related to the S-plane by z = e−sTe = e(σ +jω)Te . Hence |z| = e

σ Te

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

and ∠z = ωTe

Properties Stability

State feedback control

Jury criteria The denominator polynomial (den(z) = a0 z n + a1 z n−1 + · · · + an = 0) has all its roots inside the unit circle if all the first coefficients of the odd row are positive.

Problem formulation Controllability Definition Pole placement control Specifications Integral Control

1 2 3 2 .. . 2n + 1

a0 an b0 bn−1 .. . s0

a1 an−1 b1 bn−2

a2 an−2 b2 bn−3

... ... ... ...

an−k ak bn−1 b0

... ...

an a0

b0

=

b1

=

bk

=

ck

=

an a0 − an a0

Observer Observation Observability

an Observer design a1 − an−1 Observer-based control a0Introduction to anoptimal control ak − an−k Introduction to a0digital control b Conclusion bk − bn−1−k n−1 b0

How to get a discrete controller

State space approach Olivier Sename Introduction Modelling

First way I

Obtain a discrete-time plant model (by discretization)

I

Design a discrete-time controller

I

Derive the difference equation

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation

Second way

Controllability Definition

I

Design a continuous-time controller

I

Converse the continuous-time controller to discrete time (c2d)

I

Derive the difference equation

Pole placement control Specifications Integral Control

Observer Observation Observability Observer design

Now the question is how to implement the computed controller on a real-time (embedded) system, and what are the precautions to take before?

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Discretisation

Olivier Sename Introduction Modelling

The idea behind discretisation of a controller is to translate it from continuous-time to discrete-time, i.e. A/D + algorithm + D/A ≈ G(s)

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

To obtain this, few methods exists that approach the Laplace operator (see lecture 1-2).

State feedback control Problem formulation Controllability

Recall

Definition Pole placement control Specifications

s

=

s

=

s

=

z −1 Te z −1 zTe 2 z −1 Te z + 1

Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Implementation characteristics

State space approach Olivier Sename Introduction Modelling

Anti-aliasing

Nonlinear models

Practically it is smart to use a constant high sampling frequency with an analog filter matching this frequency. Then, after the A/D converter, the signal is down-sampled to the frequency used by the controller. Remember that the pre-filter introduce phase shift.

Linearisation

Linear models

To/from transfer functions

Properties Stability

State feedback control

Sampling frequency choice

Problem formulation

The sampling time for discrete-time control are based on the desired speed of the closed loop system. A rule of thumb is that one should sample 4 − 10 times per rise time Tr of the closed loop system.

Definition

Nsample =

Controllability

Tr ≈ 4 − 10 Te

where Te is the sampling period, and Nsample the number of samples.

Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Delay

State space approach Olivier Sename Introduction Modelling

Problematic Sampled theory assume presence of clock that synchronizes all measurements and control signal. Hence in a computer based control there always is delays (control delay, computational delay, I/O latency).

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

Origins There are several reasons for delay apparition

State feedback control Problem formulation Controllability

I

Execution time (code)

I

Preemption from higher order process

I

Interrupt

I

Communication delay

Observation

I

Data dependencies

Observer design

Definition Pole placement control Specifications Integral Control

Hence the control delay is not constant. The delay introduce a phase shift ⇒ Instability!

Observer Observability

Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Delay (cont’d)

Olivier Sename

Admissible delay (Bode)

Introduction Modelling

I

I

Measure the phase margin: PM = 180 + ϕw0 [ˇr], where ϕw0 is the phase at the crossover frequency w0 , i.e. |G(jw0 )| = 1 Then the delay margin is DM =

PMπ 180w0 [s]

Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

Exercise: compute delay margin for these 3 cases

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

State space approach

Quantification

Olivier Sename

Effects

Introduction

I

Non linear phenomena

I

Limit cycles

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Example (stable for K<2)

Properties Stability

State feedback control

H(z) =

0.25 (z − 1)(z − 0.5)

Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Quantification (cont’d)

State space approach Olivier Sename Introduction

Results

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Quantification (cont’d)

State space approach Olivier Sename Introduction

Results

Modelling Nonlinear models Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

Conclusion

State space approach Olivier Sename Introduction Modelling Nonlinear models

I

I

I

A state space approach for continuous-time and discrete-time MIMO systems A first insight in optimal control... that can be extended towards predictive control (over a finite horizon) The state space approach is also considered in Robust control, in order to I I

I

I I

design H∞ controllers provide a robustness analysis in the presence of parameter uncertainties prove the stability of a closed-loop system in the presence of non linearities (as state or input constraints) design non linear controllers (feedback linearisation...) solve an optimisation problem using efficient numerical tools as Linear Matrix Inequalities

Linear models Linearisation To/from transfer functions

Properties Stability

State feedback control Problem formulation Controllability Definition Pole placement control Specifications Integral Control

Observer Observation Observability Observer design Observer-based control

Introduction to optimal control Introduction to digital control Conclusion

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