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peak, and on the state-derivative feedback matrix K. When feasible, LMI can be easily solved using softwares based on convex programming, for instance MATLAB. These new control designs allow new specifications, and also consider a broader class of plants than the related results available in the literature (Abdelaziz & Valášek, 2004; Duan et al., 2005; Assunção et al., 2007c). The proposed method extends the results presented in (Assunção et al., 2007c), because it can also be applied for the control of uncertain systems subject to structural failures. Examples illustrate the efficiency of these procedures.

2. Design of State-Derivative Feedback Controllers for Descriptor Systems Using a State Feedback Control Design In this section, a simple method for designing a state-derivative feedback gain using methods for state feedback control design, where the plant can be a descriptor system, is proposed. 2.1 Statement of the Problem Consider a controllable linear descriptor system described by

Ex$ (t ) = Ax(t ) + Bu (t ),

x(0) = x0 ,

(1)

where E ∈ { n×n , x(t ) ∈ { n is the state vector and u (t ) ∈ { m is the control input vector. It is × n× m are time-invariant matrices. assumed that 1 ≤ m ≤ n , and also, A ∈ { n n and B ∈ { Now, consider the state-derivative feedback control

u (t ) = − K d x$ (t ) .

(2)

Then, the problem is to obtain a state-derivative feedback gain Kd, using state feedback techniques, such that the poles of the controlled system (1), (2) are arbitrarily specified by a set { λ1 , λ 2 ,..., λ n }, where λ i ∈ } and λ i ≠ 0, i = 1, 2,..., n, such that this closed-loop systems presents a suitable performance. The motivation of this study was to investigate the possibility of designing state-derivative gains using state feedback design methods. This procedure allows the designers to use well-known methods for pole-placement using state feedback, available in the literature, for state-derivative feedback design (Chen, 1999; Valášek & Olgac, 1995a; Valášek & Olgac, 1995b). To establish the proposed results, consider the following assumptions: (A) rank [E |B] = n; (B) rank [A] = n; (C) rank [B] = m.

Remark 1. It is known (Bunse-Gerstner et al, 1992; Duan et al, 1999) that if Assumption (A) holds, then there exists Kd such that: rank[E + BKd] = n.

(3)

Assumption (B) was also considered in (Abdelaziz & Valášek, 2004) and, as will be described below, is important for the stability of the system (1), with the proposed method and the control law u = − K d x$ . Assumption (C) means that B is a full rank matrix. For Kd

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such that (3) holds, then from (2) it follows that (1) can be rewrite such as a standard linear system, given by:

Ex$ (t ) = Ax(t ) − BK d x$ (t ),

(4)

x$ (t ) = ( E + BK d ) −1 Ax(t ).

(5)

From (5) note that if rank(A) < n, then the controlled system (1), (2) given by (5) is unstable, because it presents at least one pole equal to zero. It is known that the stability problem for descriptor systems is much more complicated than for standard systems, because it is necessary to consider not only stability, but also regularity (Bunse-Gerstner et al., 1992; S. Xu & J. Lam, 2004). In this work, a descriptor system is regular if it has uniqueness in the solutions and avoid impulsive responses. In the next section, the proposed method is presented. 2.2 Design of State-Derivative Feedback Using a State Feedback Design Lemma 1 below will be very useful in the analysis of the method that solves the proposed problem. n× n , with rank(Z) = n and eigenvalues equal to λ1 , λ 2 ,..., λ n . Lemma 1. Consider a matrix Z ∈ { − −1 −1 −1 Then, the eigenvalues of Z are the following: λ1 , λ 2 ,..., λ n . 1

Proof: For each eigenvalue λ ∈ { λ1 , λ 2 ,..., λ n } of Z, there exists an eigenvector v such that

Z v = λv .

(6)

Considering that rank(Z) = n, then λ ≠ 0. Therefore, from (6),

v = Z −1λv ⇒ λ −1v = Z −1v ,

− and so λ −1 is an eigenvalue of Z 1 . Remark 2. Consider that λ = a + jb λ −1 = (a + jb)−1 =

a a2 + b2

−j

b

(7)

is an eigenvalue of Z. Then, from Lemma 1,

is also an eigenvalue of Z −1 . Therefore, note that the real parts of the

a2 + b2

λ and λ −1 present the same signal. So, if Z is Hurwitz (it has all eigenvalues with negative real −1 parts), then Z will be also Hurwitz. Now, the main result of this section will be presented. Theorem 1. Define the matrices:

An = A−1 E

and

Bn = − A−1 B

(8)

and suppose that (An, Bn) is controllable. Let Kd be a state feedback gain, such that −1 −1 −1 { λ1 , λ 2 ,..., λ n } are the poles of the closed-loop system

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x$n (t ) = An xn (t ) + Bn un (t ) ,

(9)

un (t ) = − K d xn (t ) ,

(10)

Systems, Structure and Control

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where λ i ∈ } and λ i ≠ 0, i = 1,2,...,n, are arbitrarily specified. Then, for this gain Kd, { λ1 , λ 2 ,..., λ n } are the poles of the controlled system with state-derivative feddback (1), (2) and also, the condition (3) holds. Proof: Considering that (An, Bn) is controllable, then one can find a state feedback gain Kd such that the controlled system with state feedback (9), (10), given by

x$n (t ) = ( An − Bn K d ) xn (t ) .

(11)

−1 −1 −1 −1 −1 has poles equal to λ1 , λ 2 ,..., λ n (Chen, 1999). Now, from An = A E , Bn = − A B and

λ i ≠ 0 , i = 1, 2,..., n, note that ( An − Bn K d ) −1 = [ A−1 ( E + BK d )]−1

(12)

= ( E + BK d ) −1 A

(13)

and from (11) and Lemma 1, λ1 , λ 2 ,..., λ n are the eigenvalues of ( E + BK d ) −1 A . Therefore (3) holds, the state-derivative feedback system (1) and (2) can be described by (5) and presents poles equal to λ1 , λ 2 ,..., λ n . This result is a generalization of the methods proposed in (Abdelaziz & Valášek, 2004) and (Cardim et al., 2007), because it can be applied in the control of descriptor systems (1), with det(E) = 0. 2.3 Examples The effectiveness of the proposed methods designs is demonstrated by simulation results. First Example A simple electrical circuit, can be represented by the linear descriptor system below (Nichols et al, 1992):

⎡0 1 ⎤ ⎡ x$1 (t ) ⎤ ⎡1 0 ⎤ ⎡ x1 (t ) ⎤ ⎡0 ⎤ ⎢0 0 ⎥ ⎢ x$ (t ) ⎥ = ⎢0 1 ⎥ ⎢ x (t ) ⎥ + ⎢1 ⎥ u (t ) , ⎣ ⎦⎣ 2 ⎦ ⎣ ⎦⎣ 2 ⎦ ⎣ ⎦

(14)

where x1 is the current and the x2 is the potential of a capacitor. In this system one has:

⎡0 1 ⎤ E=⎢ ⎥, ⎣0 0 ⎦

⎡1 0 ⎤ A=⎢ ⎥, ⎣0 1 ⎦

⎡0⎤ B=⎢ ⎥, ⎣1 ⎦

(15)

Consider the pole placement as design technique, using the state derivative feedback (2) with the feedback gain matrix Kd. In this example, the suitable closed-loop poles for the controlled system (2) and (14) are the following:

λ1 = −2 + 1i,

λ 2 = −2 − 1i

Note that, the system (14) with the control signal (2) satisfies the Assumptions A, B and C. From (8) one has:

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Control Designs for Linear Systems Using State-Derivative Feedback

⎡0 1 ⎤ An = ⎢ ⎥, ⎣0 0 ⎦

⎡0⎤ Bn = ⎢ ⎥ , ⎣ −1⎦

5

(16)

and (An, Bn) is controllable. From Theorem 1, the poles for the new closed-loop system with state feedback (9) and (10) with An and Bn given in (8) are the following:

λ1−1 = −0.40 − 0.20i,

λ 2 −1 = −0.40 + 0.20i.

So, one can obtain by using the command acker of MATLAB (Ogata, 2002), the feedback gain matrix Kd below: Kd = [ -0.20

-0.80 ].

(17)

Figures 1 and 2 show the simulation results of the controlled system (5) with the initial condition x(0) = [1 0]T. In this example the validity and simplicity of the proposed method can be observed. Example 2 Consider a linear descriptor MI system described by the following equations:

⎡0 0 ⎤ ⎡ x$1 (t ) ⎤ ⎡ −0.800 0.020 ⎤ ⎡ x1 (t ) ⎤ ⎡0.050 1 ⎤ u (t ) , ⎢ ⎥+ ⎢1 0 ⎥ ⎢ x$ (t ) ⎥ = ⎢ −0.020 0 ⎦⎥ ⎣ x2 (t ) ⎦ ⎣⎢ 0.001 0 ⎦⎥ ⎣ ⎦⎣ 2 ⎦ ⎣

(18)

where u(t) = [u1(t) u2(t)]T. The wanted poles for closed-loop system with the control law u ( t ) = − K d x$ ( t ) are given by:

λ1 = −2 + 1i,

λ 2 = −2 − 1i .

Observe that, the system (18) with the control signal (2) satisfies the Assumptions A, B and C. From (8) one has:

⎡ −50 0 ⎤ An = ⎢ ⎥, ⎣ −2000 0 ⎦

0 ⎤ ⎡ 0.050 Bn = ⎢ ⎥, ⎣ −0.500 −50 ⎦

(19)

and (An, Bn) is controllable. From Theorem 1, the poles for the new closed-loop system with state feedback (11), with An and Bn given in (19) are the following:

λ1−1 = −0.40 − 0.20i,

λ 2 −1 = −0.40 + 0.20i.

So, with these parameters, one can obtain with the command place of MATLAB, the feedback gain matrix Kd below:

⎡ −992.0000 4.0000 ⎤ Kd = ⎢ ⎥. ⎣ 49.9240 −0.0480 ⎦

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(20)

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Systems, Structure and Control

Figures 3 and 4 show the simulation results of the controlled system with state-derivative feedback, given by (2), (18) and (20) that can be described by (5), with the initial condition x0 = [1 0]T.

Figure 1. Transient response of the controlled system (Example 1), for x0 = [1 0]T

Figure 2. Control inputs of the controlled system (Example 1), for x0 = [1 0]T

Figure 3. Transient response of the controlled system (Example 2), for x0 = [1 0]T

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Figure 4. Control inputs of the controlled system (Example 2), for x0 = [1 0]T Example 3 In this example, is considered that the matrix E = I. So, the system (1) is in the standard space state form. The idea was to show that, for the case where det(E) ≠ 0, the proposed method is also valid. Consider the mechanical system shown in Figure 5. It is a simple model of a controlled vibration absorber, in the sense of reducing the oscillations of the masses m1 and m2. In this case, the model contains two control inputs, u1(t) and u2(t). This system is described by the following equations (Cardim et al., 2007):

y1 (t ) + b1 ( y$1 (t ) − y$ 2 (t )) + k1 y1 (t ) = u1 (t ), ⎧ m1 $$ ⎨ y2 (t ) + b1 ( y$ 2 (t ) − y$1 (t )) + k2 y2 (t ) = u2 (t ). ⎩m2 $$

(21)

The state space rorm of the mechanical system in Figure 5 is represented in equation (1) considering as state variables x(t) = [x1(t) x2(t) x3(t) x4(t)]T, where x1(t) = y1(t), x2(t) = y$1 (t), x3(t) = y2(t), x4(t) = y$ 2 (t), u(t) = [u1(t) u2(t)]T and:

⎡1 ⎢0 E=⎢ ⎢0 ⎢ ⎣0

0 1 0 0

0 0 1 0

⎡ 0 ⎢ 0⎤ ⎢ −k1 ⎢m 0 ⎥⎥ ,A= ⎢ 1 ⎥ 0 ⎢ 0 ⎥ ⎢ 1⎦ ⎢ 0 ⎢⎣

1 −b1 m1

0

0 b1 m2

0 −k2 m2

⎤ ⎡ 0 ⎥ ⎢1 ⎥ ⎢ ⎥ ⎢ m1 ⎥,B = ⎢ 1 ⎥ ⎢ 0 ⎢ −b1 ⎥ ⎥ ⎢ 0 m2 ⎥⎦ ⎣ 0 b1 m1

0

0 ⎤ ⎥ 0 ⎥ ⎥ ⎥. 0 ⎥ 1 ⎥ ⎥ m2 ⎦

(22)

For a digital simulation of the control system, assume for instance that m1 = 10kg, m2 = 30kg, k1 = 2.5kN/m, k2 = 1.5kN/m and b1 = 30Ns/m. Consider the pole placement as design technique, and the following closed-loop poles for the controlled system:

λ1 = −10,

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λ 2 = −15,

λ 3,4 = −2 ± 10i.

Systems, Structure and Control

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Figure 5. Multivariable (MI) mass-spring system with damping With these parameters and from (8), one has:

⎡0.4000 × 10−3 0 ⎤ ⎡ −0.0120 −0.0040 0.0120 ⎢ ⎢ 1.0000 ⎥ 0 0 0 ⎥ 0 ⎢ An = , Bn = ⎢⎢ ⎢ 0.0200 −0.0200 −0.0200 ⎥ 0 0 ⎢ ⎢ ⎥ 0 0 1.0000 0 ⎢ 0 ⎣ ⎦ ⎣

⎤ ⎥ ⎥ , (23) −3 ⎥ 0.6667 × 10 ⎥ ⎥⎦ 0 0 0

and (An, Bn) is controllable. From Theorem 1, the poles for the new closed-loop system with state feedback (11), with An and Bn given in (23) are the following:

λ1−1 = −0.1000,

λ −21 = −0.0667,

−1 λ3,4 = −0.0192 ± 0.0962i.

So, with these parameters, one can obtain through the command place of MATLAB, the feedback gain matrix Kd below:

⎡ 178.9532 −6.4647 323.3542 19.8478 ⎤ Kd = ⎢ ⎥. ⎣ −79.637 0 −11.4321 152.3204 −26.1863 ⎦

(24)

Figures 6 and 7 show the simulation results of the controlled system (1), (2), (22), (24), that can be given by (5), with the initial condition x(0) = [0.1 0 0.1 0]T.

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Figure 6. Transient response of the controlled system (Example 3), with x(0) = [0.1 0 0.1 0]T

Figure 7. Control inputs of the controlled system (Example 3), with x(0) = [0.1 0 0.1 0]T

3. LMI-Based Control Design for State-Derivative Feedback Consider the linear time-invariant uncertain polytopic system, described as convex combinations of the polytope vertices:

x$ (t ) =

∑α A x(t ) + ∑ β B u(t ), ra

rb

i i

i =1

j

j

j =1

(25)

= A(α ) x(t ) + B( β )u (t ), and

∑α

⎫ = 1, ⎪ ⎪⎪ i =1 ⎬ rb ⎪ β j = 1, ⎪ j =1 ⎪⎭ ra

αi ≥ 0, β j ≥ 0,

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i = 1,..., ra , j = 1,..., rb ,

∑

i

(26)

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where x(t ) ∈ { n is the state vector, u (t ) ∈ { m is the input vector, ra and rb are the numbers of polytope vertices of the matrices A(α ) and B ( β ) , respectively. For i = 1, ... ,ra and j =1, ...,rb one has: Ai ∈ { n×n and B j ∈ { n×m are constant matrices and α i and β j are constant and unknown real numbers. From (8) and (25), one has:

An = A(α ) −1 E and Bn = − A(α ) −1 B ( β ) ,

(27)

Then, for the control design of the system (25) with Theorem 1, is necessary to know the real numbers α i and β j . However, in the practical problems these parameters are unknown. Therefore, Theorem 1 can not be directly applied in the control design of the system (25). For the solution of this problem, in this section sufficient Linear Matrix Inequalities (LMI) conditions for asymptotic stability of linear uncertain systems using state-derivative feedback are presented. The LMI formulation has emerged recently (Boyd et al., 1994) as an useful tool for solving a great number of practical control problems such as model reduction, design of linear, nonlinear, uncertain and delayed systems (Boyd et al., 1994; Assunção & Peres, 1999; Teixeira et al., 2001; Teixeira et al., 2002; Teixeira et al., 2003; Palhares et al., 2003; Teixeira et al., 2005; Assunção et al., 2007a; Assunção et al., 2007b; Teixeira et al., 2006). The main features of this formulation are that different kinds of design specifications and constraints that can be described by LMI, and once formulated in terms of LMI, the control problem, when it presents a solution, can be efficiently solved by convex optimization algorithms (Nesterov & Nemirovsky, 1994; Boyd et al., 1994; Gahinet et al., 1995; Sturm, 1999). The global optimum is found with polynomial convergence time (El Ghaoui & Niculescu, 2000). The state-derivative feedback has been examined with various approaches (Abdelaziz & Valášek, 2004; Kwak et al., 2002; Duan et al., 2005; Cardim et al., 2007), but neither them can be applied for uncertain systems or systems subject to structural failures (Isermann, 1997; Isermann & Ballé, 1997; Isermann, 2006). Robust state-derivative feedback LMI-based designs for linear time-invariant and time-varying systems were recently proposed in (Assunção et al., 2007c), but the results does not consider structural failures in the control design. Structural failures appear of natural form in the systems, for instance, in the following cases: physical wear of equipments, or short circuit of electronic components. Recent researches for detection of the structural failures (or faults) in systems, have been presented in LMI framework (Zhong et al., 2003; Liu et al., 2005; D. Ye & G. H. Yang, 2006; S. S. Yang & J. Chen, 2006). In this section, we will show that it is possible to extend the presented results in (Assunção et al., 2007c), for the case where there exist structural failures in the plant. A fault-tolerant design is proposed. The methods can include in the LMI-based control designs the specifications of bounds: on the decay rate, on the output peak, and on the state-derivative feedback matrix K. These design procedures allow new specifications and also, they consider a broader class of plants than the related results available in the literature. 3.1 Statement of the Problem Consider a homogeneous linear time-invariant system given by

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x$ (t ) = AN x(t )

(28)

It is known from literature that the linear system (28) is asymptotically stable if there exist a symmetric matrix P satisfying the Lyapunov conditions (Boyd et al., 1994):

⎫ ⎪ and ⎬ AN′ P + PAN < 0.⎭⎪ P > 0,

(29)

This result is useful for the design of the proposed controller. In this work, structural failure is defined as a permanent interruption of the system's ability to perform a required function under specified operating conditions (Isermann & Ballé, 1997). Systems subject to structural failures can be described by uncertain polytopic systems (25) (see Section 3.5 for details). Now, suppose that all poles of (25) are different from zero (the matrix A(α ) must have a full rank). Then, the proposed problem is defined below. Problem 1: Find a constant matrix K ∈ { 1. ( I + B( β ) K ) has a full rank; 2.

m×n

such that the following conditions hold:

the closed-loop system (25) with the state-derivative feedback control

u (t ) = − Kx$ (t ) ,

(30)

is asymptotically stable. Note that from (25) and (30) it follows that

x$ (t ) = A(α ) x (t ) − B ( β ) Kx$ (t ) or

( I + B ( β ) K ) x$ (t ) = A(α ) x(t ) . When ( I + B( β ) K ) has a full rank, the closed-loop system is well-defined and given by

x$ (t ) = ( I + B ( β ) K ) −1 A(α ) x(t ) .

(31)

This condition was also assumed in other related researches (Kwak et al., 2002; Abdelaziz & Valášek, 2004; Assunção et al., 2007c; Cardim et al., 2007). 3.2 Robust Stability Condition for State-derivative Feedback The main results of this section is presented in the next theorem, that solves Problem 1 (Assunção et al., 2007c). For the proof of this theorem, the following result will be useful. Remark 3. Recall that for any nonsymetric matrix M ( M ≠ M ′), M ∈ { n×n , if

M +M′< 0,

then M has a full rank. Theorem 2. A sufficient condition for the solution of Problem 1 is the existence of matrices

Q = Q ' and Y , where Q ∈ { n×n and Y ∈ { m×n , such that:

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Q > 0,

(32)

QAi′ + Ai Q + B j YAi′ + AiY ′B ′j < 0

(33)

where i = 1, ... ,ra and j = 1, ... ,rb. Furthermore, when (32) and (33) hold, a state-derivative feedback matrix that solves the Problem 1 is given by:

K = YQ −1

(34)

Proof: Supposing that (32) and (33) hold, then multiplying both sides of (33) by α i β j , for i = 1, ... , ra and j = 1, ... , rb and considering (26), it follows that

α i β j (QAi′ + Ai Q + B j YAi′ + AiY ′B′j ) < 0, ⇔

⎛ +⎜ ⎜ ⎝

rb

i

⎛ Q⎜ ⎜ ⎝

∀i, j ,

∑∑α β (QA′ + A Q + B YA′ + A Y ′B′ ) = ra

j

i

i =1 j =1

∑

⎞′ ⎛ α i Ai ⎟ + ⎜ ⎟ ⎜ i =1 ⎠ ⎝ ra

i

∑

⎛ ⎞ α i Ai ⎟ Q + ⎜ ⎟ ⎜ i =1 ⎠ ⎝ rb ⎞′ β j Bj ⎟ < 0 ⎟ j =1 ⎠ ra

∑α A ⎟⎟ Y ′ ⎜⎜ ∑ ⎞

ra

i i

i =1

⎠

⎛

⎝

j

i

∑

i

j

⎞ ⎛ β j B j ⎟Y ⎜ ⎟ ⎜ j =1 ⎠ ⎝ rb

∑

⎞′ α i Ai ⎟ ⎟ i =1 ⎠ ra

(35)

Then, from (25) one has

QA(α)′ + A(α)Q + B(β)YA(α)′ + A(α)Y ′B (β)′ < 0 . Replacing Y = KQ and Q =P −1 one obtains

P −1 A(α )′ + A(α ) P −1 + B ( β ) KP −1 A(α )′ + A(α ) P −1 K ′B( β )′ = ( I + B ( β ) K ) P −1 A(α )′ + A(α ) P −1 ( I + B ( β ) K )′ < 0

(36)

−1 From Remark 3, it follows that the matrix ( ( I + B( β ) K ) P A(α )′ has a full rank, and so the

matrices

( I + B( β ) K ) and

A(α )′ have a full rank too. Now, premultiply by

P ( I + B ( β ) K ) , posmultiply by [( I + B( β ) K )′]−1 P in both sides of (36) and replace −1

AN (α , β ) = ( I + B ( β ) K ) −1 A(α ) to obtain

A(α )′[( I + B ( β ) K )′]−1 P + P ( I + B( β ) K ) −1 A(α ) = AN (α , β )′ P + PAN (α , β ) < 0

(37)

Observe that, when the LMI (32) and (33) hold, the system (31) satisfies the Lyapunov conditions (29), considering AN (α , β ) = ( I + B ( β ) K ) −1 A(α ) . Therefore, when the LMI (32) and (33) hold the system (31) is asymptotically stable and a solution that solves Problem 1 is given by (34).

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When (32) and (33) are feasible, they can be easily solved using available softwares, such as LMISol (de Oliveira et al, 1997), that is a free software, or MATLAB (Gahinet et al, 1995; Sturm, 1999). These algorithms have polynomial time convergence. Remark 4. From the analysis presented in the proof of Theorem 2, after equation (36), note that when (32) and (33) are feasible, the matrix A(α ) , defined in (25), has a full rank. Therefore, A(α ) with a full rank is a necessary condition for the application of Theorem 2. Moreover, from (25), observe that for α i = 1 and α k = 0, i ≠ k , i, k = 1, 2,... , ra, then A(α ) = Ai . So, if A(α ) has a full rank, then

Ai , i = 1, 2,... , ra has a full rank too. Usually, only the stability of a control system is insufficient to obtain a suitable performance. In the design of control systems, the specification of the decay rate can also be very useful. 3.3 Decay Rate Conditions Consider, for instance, the controlled system (31). According to (Boyd et al., 1994), the decay rate is defined as the largest real constant γ , γ > 0 , such that

limt →∞ eγ t x(t ) = 0 holds, for all trajectories x(t ), t ≥ 0 . One can use the Lyapunov conditions (29) to impose a lower bound on the decay rate, replacing (29) by

P > 0, and AN (α , β )′ P + PAN (α , β ) < −2γ P .

(38)

where γ is a real constant (Boyd et al., 1994). Sufficient conditions for stability with decay rate for Problem 1 are presented in the next theorem (Assunção et al., 2007c). Theorem 3. The closed-loop system (31), given in Problem 1, has a decay rate greater or equal to γ if there exist a symmetric matrix Q ∈ { n×n and a matrix Y ∈ { m×n such that

Q>0

⎡QAi′ + Ai Q + B j YAi′ + AiY ′B ′j ⎢ Q + Y ′B ′j ⎢⎣

Q + B jY ⎤ ⎥<0 −Q / (2γ ) ⎥⎦

(39) (40)

where i = 1, ... , ra and j = 1, ..., rb. Furthermore, when (39) and (40) hold, then a robust statederivative feedback matrix is given by:

K = YQ −1 .

(41)

Proof: Following the same ideas of the proof of Theorem 2, multiply both sides of (40) by α i β j , for i = 1, ... , ra and j = 1, ..., rb and consider (26), to conclude that

⎡QA(α )′ + A(α )Q + B( β )YA(α )′ + A(α )Y ′B( β )′ Q + B( β )Y ⎤ <0 ⎢ −Q / (2γ ) ⎦⎥ Q + Y ′B ( β )′ ⎣

Now, using the Schur complement (Boyd et al., 1994), the equation above is equivalent to:

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QA(α )′ + A(α )Q + B ( β )YA(α )′ + A(α )Y ′B ( β )′ +(Q + B ( β )Y )2γ Q −1 ( Q + B ( β )Y )′ < 0

(42)

Replacing Y = KQ and Q = P −1 one obtains

( I + B( β ) K ) P −1 A(α )′ + A(α ) P −1 ( I + B( β ) K )′ + ( I + B ( β ) K ) P −1 (2γ P ) P −1 ( I + B ( β ) K )′ = ( I + B( β ) K ) P −1 A(α )′ + A(α ) P −1 ( I + B( β ) K )′

(43)

+ ( I + B ( β ) K )(2γ P −1 )( I + B ( β ) K )′ < 0 Premultiplying by P ( I + B ( β ) K ) −1 , posmultiplying by [( I + B ( β ) K )′]−1 P in both sides of (43) and replacing AN (α , β ) = ( I + B( β ) K ) −1 A(α ) one obtain

A(α )′[( I + B ( β ) K )′]−1 P + P( I + B( β ) K ) −1 A(α ) + 2γ P < 0 ⇔ AN (α , β )′ P + PAN (α , β ) < −2γ P,

(44)

that is equivalent to the Lyapunov condition (38). Then, when (39) and (40) hold, the system (31) satisfies the Lyapunov conditions (38), considering AN (α , β ) = ( I + B ( β ) K ) −1 A(α ) . Therefore, the system (31) is asymptotically stable with a decay rate greater or equal to γ , and a solution for the problem can be given by (41). Due to limitations imposed in the practical applications of control systems, many times it should be considered output constraints in the design. 3.4 Bounds on Output Peak Consider that the output of the system (25) is given by:

y (t ) = Cx(t ) ,

(45)

p where y (t ) ∈ { and C ∈ { p×n . Assume that the initial condition of (25) and (45) is x(0). If

the feedback system (31) and (45) is asymptotically stable, one can specify bounds on output peak as described below:

max y (t )

2

= max y ′(t ) y (t ) < ξ0

(46)

for t ≥ 0 , where ξ0 is a known positive constant. From (Boyd et al., 1994), (46) is satisfied when the following LMI hold:

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x(0)′⎤ ⎡ 1 > 0, ⎢ x(0) Q ⎥⎦ ⎣ ⎡ Q QC ′⎤ > 0, ⎢ 2 ⎥ ⎣⎢CQ ξ0 I ⎥⎦

(47)

(48)

Control Designs for Linear Systems Using State-Derivative Feedback

15

and the LMI that guarantee stability (Theorem 2), given by (32) and (33), or stability and decay rate (Theorem 3), given by (39) and (40). In some cases, the entries of the state-derivative feedback matrix K must be bounded. In (Assunção et al., 2007c) is presented an optimization procedure to obtain bounds on the state-derivative feedback matrix K, that can help the practical implementation of the controllers. The result is the following: Theorem 4. Given a constant µ0 > 0 , then the specification of bounds on the state-derivative feedback matrix K can be described by finding the minimum value of β , β > 0 , such that

KK ′ < β I / µ02 . The optimal value of β can be obtained by the solution of the following optimization problem: min β s.t.

⎡β I Y ⎤ ⎢Y′ I ⎥ > 0 , ⎣ ⎦

(49)

Q > µ0 I ,

(50)

(Set of LMI), where the Set of LMI can be equal to (33), or (40), with or without the LMI (47) and (48). Proof: See (Assunção et al., 2007c) for more details. In the next section, a numerical example illustrates the efficiency of the proposed methods for solution of Problem 1. 3.5 Example The presented methods are applied in the design of controllers for an uncertain mechanical system subject to structural failures. For the designs and simulations, the software MATLAB was used. Active Suspension Systems Consider the active suspension of a car seat given in (E. Reithmeier and G. Leitmann, 2003; Assunção et al., 2007c) with other kind of control inputs, shown in Figure 8. The model consists of a car mass Mc and a driver-plus-seat mass ms. Vertical vibrations caused by a street may be partially attenuated by shock absorbers (stiffness k1 and damping b1). Nonetheless, the driver may still be subjected to undesirable vibrations. These vibrations, again, can be reduced by appropriately mounted car seat suspension elements (stiffness k2 and damping b2). Damping of vibration of the masses Mc and ms can be increased by changing the control inputs u1(t) and u2(t). The dynamical system can be described by

⎡ 0 ⎢ $ ⎡ x1 (t ) ⎤ ⎢ 0 ⎢ x$ (t ) ⎥ ⎢ ⎢ 2 ⎥ = ⎢ − k1 − k2 ⎢ x$3 (t ) ⎥ ⎢ M c ⎢ ⎥ ⎣ x$4 (t ) ⎦ ⎢⎢ k2 ⎣⎢ ms

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0 0 k2 Mc

1 0 −b1 − b2 Mc

−k2 ms

b2 ms

⎤ ⎡ 0 ⎥ x (t ) ⎢ ⎥⎡ 1 ⎤ ⎢ 0 ⎥ ⎢ x2 (t ) ⎥ ⎢ 1 ⎥+⎢ ⎥⎢ ⎢ ⎥ Mc x t ( ) ⎥ 3 ⎢ ⎥ ⎢⎢ ⎥ −b2 ⎣ x4 (t ) ⎦ ⎥ ⎢ 0 ms ⎦⎥ ⎣⎢

0 1 b2 Mc

0 ⎤ 0 ⎥⎥ −1 ⎥ ⎥ u (t ) , Mc ⎥ 1 ⎥ ⎥ ms ⎦⎥

(51)

Systems, Structure and Control

16

⎡ x1 (t ) ⎤ ⎢ ⎥ ⎡ y1 (t ) ⎤ ⎡1 0 0 0 ⎤ ⎢ x2 (t ) ⎥ . ⎢ y (t ) ⎥ = ⎢ ⎥ ⎣ 2 ⎦ ⎣ 0 1 0 0 ⎦ ⎢ x3 (t ) ⎥ ⎢ ⎥ ⎣ x4 (t ) ⎦

(52)

The state vector is defined by x(t ) = [ x1 (t ) x2 (t ) x$1 (t ) x$2 (t )]T . As in (E. Reithmeier and G. Leitmann, 2003), for feedback only the accelerations signals $$ x1 (t ) and $$ x2 (t ) are available (that are measured by accelerometer sensors). The velocities

x$1 (t ) and x$2 (t ) are estimated from their measured time derivatives. Therefore the accelerations and velocities signals are available (derivative of states), and so one can use the proposed method to solve the problem. Consider that the driver weight can assume values between 50kg and 100kg. Then the system in Figure 8 has an uncertain constant parameter ms such that, 70kg ≤ ms ≤ 120kg. Additionally, suppose that can also happen a fail in the damper of the seat suspension (in other words, the damper can break after some time). The fault can be described by a polytopic uncertain system, where the system parameters without failure correspond to a vertice of the polytopic, and with failures, the parameters are in another vertice. Then, one can obtain the polytopic plant given in (25) and (26), composed by the polytopic sets due the failures and the uncertain plant parameters.

Figure 8. Active suspension of a car seat

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Control Designs for Linear Systems Using State-Derivative Feedback

17

The damper of the seat suspension b2 can be considered as an uncertain parameter such that: b2 = 5 x 102Ns/m while the damper is working and b2 = 0 when the damper is broken. Hence, and supposing Mc = 1500kg (mass of the car), k1 = 4 x 104N/m (stiffness), k2 = 5 x 103N/m (stiffness) and b1 = 4 x 103Ns/m (damping), the plant (51) and (52) can be described by equations (25), (26) and (45), and the matrices Ai and Bj, where ra = 4, rb, = 2, are given by:

0 1 0 ⎤ 0 1 0 ⎤ ⎡ 0 ⎡ 0 ⎢ 0 ⎥ ⎢ 0 0 1 ⎥ 0 0 0 1 ⎥⎥ , , A2 = ⎢ A1 = ⎢ ⎢ −30 ⎢ −30 3.33 0.33 ⎥ 3.33 0.33 ⎥ −3 −3 ⎢ ⎥ ⎢ ⎥ ⎣71.43 −71.43 7.143 −7.143⎦ ⎣ 41.67 −41.67 4.167 −4.167 ⎦

while the damper is working (in this case b2 = 5 x 102 Ns/m, ms = 70kg in A1 and ms = 120kg in A2),

0 1 ⎡ 0 ⎢ 0 0 0 A3 = ⎢ ⎢ −30 3.33 −2.67 ⎢ − 71.43 71.43 0 ⎣

0⎤ 0 1 ⎡ 0 ⎥ ⎢ 1⎥ 0 0 0 , A4 = ⎢ ⎥ ⎢ −30 0 3.33 −2.67 ⎥ ⎢ − 0⎦ 41.67 41.67 0 ⎣

0⎤ 1 ⎥⎥ , 0⎥ ⎥ 0⎦

when the damper is broken (in this case b2 = 0, ms = 70kg in A3 and ms = 120kg in A4) and

0 ⎡ ⎢ 0 B1 = ⎢⎢ 6.67 × 10−4 ⎢ ⎢⎣ 0

0 ⎤ ⎡ ⎥ ⎢ 0 0 ⎥ ⎢ −4 ⎥ , B2 = ⎢ −6.67 ×10 6.67 ×10−4 ⎥ ⎢ ⎢⎣ 1.43 × 10−2 ⎥⎦ 0 0

⎤ ⎥ 0 ⎥, −6.67 × 10−4 ⎥ ⎥ 8.33 × 10−3 ⎥⎦ 0

because the input matrix B ( β ) depends only on the uncertain parameter ms (in this case ms = 70kg in B1 and ms = 120kg in B2). Specifying an output peak bound ξ0 = 300, an initial condition x(0) = [0.1 0.3 0 0]T and using the MATLAB (Gahinet et al, 1995) to solve the LMI (32) and (33) from Theorem 2, with (47) and (48), the feasible solution was:

⎡ 2.4006 × 104 ⎢ ⎢ 2.2812 × 104 Q=⎢ 4 ⎢ −4.1099 × 10 ⎢ 4 ⎣ −2.6578 × 10

⎡ −7.9749 ×106 Y =⎢ ⎢⎣ 1.7401× 106

2.2812 × 104

−4.1099 × 104

2.3265 × 104

−2.1628 × 104

−2.1628 × 104

5.29 × 105

−2.9019 × 104

8.3897 × 104

−3.0334 ×107

−4.4436 × 106

2.2947 × 106

−8.0344 × 106

From (34), we obtain the state-derivative feedback matrix below:

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−2.6578 × 104 ⎤ ⎥ −2.9019 × 104 ⎥ ⎥, 8.3897 × 104 ⎥ ⎥ 1.8199 × 105 ⎦

6.5815 ×108 ⎤ ⎥. −1.616 × 107 ⎥⎦

Systems, Structure and Control

18

⎡ 2.894 × 103 K =⎢ ⎣⎢ −498.14

−442.06 4.3902 × 103 ⎤ ⎥. −75.996 ⎦⎥ 471.29 −22.567 923.6

(53)

The locations in the s-plane of the eigenvalues λ i , for the eight vertices (Ai, Bj), i = 1, 2, 3, 4 and j = 1, 2, of the robust controlled system, are plotted in Figure 9. There exist four eigenvalues for each vertice. Consider that driver weight is 70kg, and so ms = 90kg. Using the designed controller (53) and the initial condition x(0) defined above, the controlled system was simulated. The transient response and the control inputs (30), of the controlled system, while the damper is working are presented in Figures 10 and 11. Now suppose that happen a fail in the damper of the seat suspension b2 after 1s (in other words, b2 = 5 x 102Ns/m if t ≤ 1s and b2 = 0 if t > 1s). Then, the transient response and the control inputs (30), of the controlled system, are displayed in Figures 12 and 13. The required condition max y ′(t ) y (t ) < ξ0 = 300 was satisfied.

Figure 9. The eigenvalues in the eight vertices of the controlled uncertain system

Figure 10. Transient response of the system with the damper working

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Control Designs for Linear Systems Using State-Derivative Feedback

Figure 11. Control inputs of the controlled system with the damper working

Figure 12. Transient response of the system with a fail in the damper b2 after 1s

Figure 13. Control inputs of the controlled system with a fail in the damper b2 after 1s

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Systems, Structure and Control

20

Observe in Figures 10 and 12, that the happening of a fail in the damper b2 does not change the settling time of the controlled system, and had little influence in the control inputs. Furthermore, as discussed before, considering ms = 90kg and the controller (53), the matrix ( I + B( β ) K ) has a full rank (det ( I + B( β ) K ) = 0.85868 ≠ 0). There exist problems where only the stability of the controlled system is insufficient to obtain a suitable performance. Specifying a lower bound for the decay rate equal γ = 3, to obtain a fast transient response, Theorem 3 is solved with (47) and (48) ( ξ0 = 300). The solution obtained with the software MATLAB was:

⎡ 3.9195 × 103 ⎢ ⎢ 3.1064 × 103 Q=⎢ 4 ⎢ −2.6316 × 10 ⎢ 4 ⎣ −1.6730 × 10

3.1064 ×103

−2.6316 × 104

3.6868 × 103

−1.3671× 104

−1.3671×104

5.3775 × 105

−1.8038 ×104

1.0319 × 105

⎡ 4.3933 × 107 2.8021× 107 Y =⎢ 6 6 ⎣⎢1.3888 × 10 1.8426 × 10

−7.9356 ×108 −9.1885 × 106

From (41), we obtain the state-derivative feedback matrix below:

⎡ −621 3.8664 × 103 K =⎢ 365.55 ⎢⎣ −313.58

−1.452 × 103 −8.79

−1.6730 × 104 ⎤ ⎥ −1.8038 × 104 ⎥ ⎥, 1.0319 × 105 ⎥ ⎥ 1.9587 × 105 ⎦

−1.6408 ×108 ⎤ ⎥. −1.69 × 107 ⎦⎥ 230.33 ⎤ ⎥ −74.77 ⎥⎦

(54)

The locations in the s-plane of the eigenvalues λ i , for the eight vertices (Ai, Bj), i = 1, 2, 3, 4 and j = 1, 2, of the robust controlled system, are plotted in Figure 14. There exist four eigenvalues for each vertice.

Figure 14. The eigenvalues in the eight vertices of the controlled uncertain system

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Control Designs for Linear Systems Using State-Derivative Feedback

21

From Figure 14, one has that all eigenvalues of the vertices have real part lower than −γ = −3 . Therefore, the controlled uncertain system has a decay rate greater or equal to γ . Again, considering that ms = 90kg and using the designed controller (54) the matrix ( I + B( β ) K ) has a full rank (det ( I + B( β ) K ) = 0.026272). For the initial condition x(0) defined above, the controlled system was simulated. The transient response and the control inputs (30) of the controlled system are presented in Figures 15, 16, 17 and 18, respectively.

Figure 15. Transient response of the system with the damper working Observe that, the settling time in Figures 15 and 17 are smaller than the settling time in Figures 10 and 12, where only stability was required and also, max y ′(t ) y (t ) is equal to 0.31623 < ξ0 = 300 . Then, the specifications were satisfied by the designed controller (54). Moreover, the happening of a fail in the damper b2 does not significantly change the settling time (Figures 15 and 17) of the controlled system. In spite of the change in the control inputs from Figures 16 and 18, the fail in the damper does not changed the maximum absolute value of the control signal (u(t) = 1.1161 x 105N).

Figure 16. Control inputs of the controlled system with the damper working

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Systems, Structure and Control

Figure 17. Transient response of the system with a fail in the damper b2 after 0.3s

Figure 18. Control inputs of the controlled system with a fail in the damper b2 after 0.3s Note that some absolute values of the entries of (53) and (54) are great values and it could be a trouble for the practical implementation of the controller. For the reduction of this problem in the implementation of the controller, the specification of bounds on the state-derivative feedback matrix K can be done using the optimization procedure stated in Theorem 4, with µ0 = 0.1. The optimal values, obtained with the software MATLAB, for Theorem 4 considering: (33) for stability, or (40) for stability with bound on the decay rate ( γ = 3), and (47) and (48) ( ξ0 = 300) are displayed in Table 1. Considering that ms = 90kg and the initial condition x(0) defined above, the transient response and the control inputs obtained by Theorem 4 considering (33) or (40), are displayed in Figures 19, 20, 21 and 22 respectively.

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Control Designs for Linear Systems Using State-Derivative Feedback

Theorem 4 with (33)

⎡ 1.2265 ⎢ 1.5357 Q=⎢ -1.667 ⎢ ⎣-5.8859

Y =

K =

⎡ 17.423 ⎢⎣-25.896

1.5357

Theorem 4 with (40) -5.8859 ⎤

0.6289 -5.1654 ⎥ -1.667

⎥ 0.6289 27.177 30.007 ⎥ -5.1654 30.007 67.502 ⎦ 2.5422

19.928 -13.793 20.088 -2.8711

23

⎤ 0.69624 ⎥⎦ 12.407

⎡ 39.536 -6.5518 -2.7229 4.3402 ⎤ ⎢⎣-276.41 173.56 -17.953 -2.829 ⎥⎦

⎡ 0.16831 ⎢ 0.088439 Q=⎢ −0.52166 ⎢ ⎣ 0.25122

Y =

0.088439 −0.52166 −0.25122 ⎤ 0.56992

−0.07813

−0.07813

5.1595

−2.3703

−2.9849

−2.3703 ⎥

⎥ ⎥ 43.238 ⎦

−2.9849

⎡ 918.06 749.73 -3.3745×10 3 204.86 ⎤ ⎢ 3⎥ -102.46 -3.5475×10 ⎦ ⎣30.057 468.97

K =

⎡ 4.7321×10 3 859.72 -121.49 70.976 ⎤ ⎢ -559.07 ⎥ 664.62 -98.521 -55.661⎦ ⎣

Table 1. The solutions with Theorem 4

Figure 19. Transient response of the system with a fail in the damper b2 after 1s, obtained with Theorem 4 and (33)

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Systems, Structure and Control

Figure 20. Control inputs of the controlled system with a fail in the damper b2 after 1s

Figure 21. Transient response of the system with a fail in the damper b2 after 0.3s, obtained with Theorem 4 and (40)

Figure 22. Control inputs of the controlled system with a fail in the damper b2 after 0.3s

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Control Designs for Linear Systems Using State-Derivative Feedback

25

The matrix norm of the controller (53) obtained with Theorem 2 is equal to K = 5.3628xl03 and the maximum absolute value of the control signal is u(t) = 6.0356 x 104N, while that the matrix norm of the same controller obtained with Theorem 4 considering (33) is equal to

K = 328.96 and the maximum absolute value of the control signal is u(t) = 68.111N. Then, Theorem 4 was able to stabilize the controlled system with a smaller state-derivative feedback matrix gain. The similar form, the maximum absolute value of the control signal u(t) from (54), obtained with Theorem 3 is u(t) = 1.1161 x 105N, and of the same controller obtained with Theorem 4 considering (40) is u(t) = 2.0362 x 103N. This example shows that the proposed methods are simple to use and it is easy to specify the constraints in the design.

4. Conclusions In this chapter two new control designs using state-derivative feedback for linear systems were presented. Firstly, considering linear descriptor plants, a simple method for designing a state-derivative feedback gain (Kd) using methods for state feedback control design was proposed. The descriptor linear systems must be time-invariant, Single-Input (SI) or Multiple-Input (MI) system. The procedure allows that the designers use the well-known state feedback design methods to directly design state-derivative feedback control systems. This method extends the results described in (Cardim et al, 2007) and (Abdelaziz & Valášek, 2004) to a more general class of control systems, where the plant can be a descriptor system. As the first design can not be directly applied for uncertain systems, then a design considering sufficient stability conditions based on LMI for state-derivative feedback, that provide an extension of the methods presented in (Assunção et al., 2007c) were presented. The designers can include in the LMI-based control design, the specification of the decay rate and bounds on output peak and on state-derivative feedback gains. The plant can be subject to structural failures. So, in this case, one has a fault-tolerant design. Furthermore, the new design methods allow a broader class of plants and performance specifications, than the related results available in the literature, for instance in (E. Reithmeier and G. Leitmann, 2003; Abdelaziz & Valášek, 2004; Duan et al., 2005; Assunção et al., 2007c; Cardim et al., 2007). The presented method offers LMI-based designs for state-derivative feedback that, when feasible, can be efficiently solved by convex programming techniques. In Sections 2.3 and 3.5, the validity and simplicity of the new control designs can be observed with some numerical examples.

5. Acknowledgments The authors acknowledge the financial support by FAPESP, CAPES and CNPq, from Brazil.

6. References A. Bunse-Gerstner, R. Byers, V. M. & Nichols, N. (1999), Feedback design for regularizing descriptor systems, in Linear Algebra and its Applications, pp. 119-151. Abdelaziz, T. H. S. & Valášek, M. (2005), Direct Algorithm for Pole Placement by StateDerivative Feedback for Multi-Input Linear Systems - Nonsingular Case, Kybernetika 41(5), 637-660.

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Abdelaziz, T. H. S. & Valášek, M. (2004), Pole-placement for SISO Linear Systems by StateDerivative Feedback, IEE Proceedings-Control Theory Applications 151(4), 377-385. Assunção, E., Andrea, C. Q. & Teixeira, M. C. M. (2007a), 2 and ∞, -optimal control for the tracking problem with zero variation, IET Control Theory Applications 1(3), 682688. Assunção, E., Marchesi, H. E, Teixeira, M. C. M. & Peres, P. L. D. (2007b), Global Optimization for the H∞-Norm Model Reduction Problem, International Journal of Systems Science 38(2), 125-138. Assunção, E. & Peres, P. L. D. (1999), A global optimization approach for the 2-norm model reduction problem, in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, USA, pp. 1857-1862. Assunção, E., Teixeira, M. C. M., Faria, F. A., da Silva, N. A. P. & Cardim, R. (2007c), Robust State-Derivative Feedback LMI-Based Designs for Multivariable Linear Systems, International Journal of Control 80(8), 1260-1270. Boyd, S., El Ghaoui, L., Feron, E. & Balakrishnan, V. (1994), Linear Matrix Inequalities in Systems and Control Theory, 2nd edn, SLAM Studies in Applied Mathematics, USA. http://www.stanford.edu/ boyd/lmibook/lmibook.pdf. Bunse-Gerstner, A., Nichols, N. & Mehrmann, V. (1992), Regularization of Descriptor Systems by Derivative and Proportional State Feedback, in SIAM J. Matrix Anal Appl., pp. 46-67. Cardim, R., Teixeira, M. C. M., Assunção, E. & Covacic, M. R. (2007), Design of StateDerivative Feedback Controllers Using a State Feedback Control Design, in 3rd IFAC Symposium on System, Structure and Control, Vol. 1, Iguassu Falls, PR, Brazil, pp. Article 135-6 pages. Chen, C. T. (1999), Linear System Theory and Design, 2nd edn, Oxford University Press, New York. D. Ye & G. H. Yang (2006), Adaptive fault-tolerant tracking control against actuator faults with application to flight control, Control Systems Technology, IEEE Transactions on 14(6), 1088-1096. de Oliveira, M. C., Farias, D. P. & Geromel, J. C. (1997), LMISol, User's guide, UNICAMP, Campinas-SP, Brazil, http://www.dt.fee.unicamp.br/~mauricio/software.html. Duan, G. R., Irwin, G. W. & Liu, G. P. (1999), Robust Stabilization of Descriptor Linear Systems via Proportional-plus-derivative State Feedback, in Proceedings of the 1999 American Control Conference, San Diego, CA, USA, pp. 1304-1308. Duan, G. R. & Zhang, X. (2003), Regularizability of Linear Descriptor Systems via Output plus Partial State Derivative Feedback, Asian Journal of Control 5(3), 334-340. Duan, Y. E, Ni, Y Q. & Ko, J. M. (2005), State-Derivative feedback control of cable vibration using semiactive magnetorheological dampers, Computer-Aided Civil and Infrastructure Engineering 20(6), 431-449. E. Reithmeier and G. Leitmann (2003), Robust vibration control of dynamical systems based on the derivative of the state, Archive of Applied Mechanics 72(11-12), 856864. El Ghaoui & Niculescu, S. (2000), Advances in Linear Matrix Inequalities Methods in Control, SIAM Advances in Design and Control, USA. Gahinet, P., Nemirovski, A., Laub, A. J. & Chilali, M. (1995), LMI control toolbox - For use with Matlab, The Math Works Inc.

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Isermann, R. (1997), Supervision, fault-detection and fault-diagnosis methods - an introduction, Control Engineering Practice 5(5), 639-652. Isermann, R. (2006), Fault-Diagnosis systems: An introduction from fault detection to fault tolerance, Springer, Berlin. Isermann, R. & Ballé, P. (1997), Trends in the application of model-based fault detection and diagnosis of technical processes, Control Engineering Practice 5(5), 709-719. Kwak, S. K., Washington, G. & Yedavalli, R. K. (2002), Acceleration-Based vibration control of distributed parameter systems using the "reciprocal state-space framework", Journal of Sound and Vibration 251(3), 543-557. Liu, J., Wang, J. L. & Yang, G. H. (2005), An LM1 approach to minimum sensitivity analysis with application to fault detection, Automatica 41(11), 1995-2004. Nesterov, Y. & Nemirovsky, A. (1994), Interior-Point Polynomial Algorithms in Convex Programming, SLAM Studies in Applied Mathematics, USA. Nichols, N., Bunse-Gerstner, A. & Mehrmann, V. (1992), Regularization of descriptor systems by derivative and proportional state feedback, in SIAM J. Matrix Anal. Appl., pp. 46-67. Ogata, K. (2002), Modern Control Engineering, 4th edn, Prentice-Hall, New Jersey. Palhares, R. M., Hell, M. B., Duraes, L. M., Ribeiro Neto, J. L., Teixeira, M. C. M. & Assunção, E. (2003), Robust H∞ Filtering for a Class of State-delayed Nonlinear Systems in an LMI Setting, International Journal Of Computer Research 12(1), 115122. S. S. Yang & J. Chen (2006), Sensor faults compensation for MLMO fault-tolerant control systems, Transactions of the Institute of Measurement and Control 28(2), 187-205. S. Xu & J. Lam (2004), Robust Stability and Stabilization of Discrete Singular Systems: An Equivalent Characterization, IEEE Transactions on Automatic Control 49(4), 568574. Sturm, J. (1999), Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software 11-12, 625-653. http://citeseer.ist.psu.edu/sturm99using.html. Teixeira, M. C. M., Assunção, E. & Avellar, R. G. (2003), On relaxed LMI-based designs for fuzzy regulators and fuzzy observers, IEEE Transactions on Fuzzy Systems 11(5), 613-623. Teixeira, M. C. M., Assunção, E. & Palhares, R. M. (2005), Discussion on: H∞ Output Feedback Control Design for Uncertain Fuzzy Systems with Multiple Time Scales: An LMI Approach, European Journal of Control 11(2), 167-169. Teixeira, M. C. M, Assunção, E. & Pietrobom, H. C. (2001), On Relaxed LMI-Based Design Fuzzy, in Proceedings of the 6th European Control Conference, Porto, Portugal, pp. 120-125. Teixeira, M. C. M., Covacic, M., Assunção, E. & Lordelo, A. D. (2002), Design of SPR Systems and Output Variable Structure Controllers Based on LMI, in 7th IEEE International Workshop on Variable Structure Systems, Vol. 1, Sarajevo, Bosnia, pp. 133-144. Teixeira, M. C. M., Covacic, M. R. & Assunção, E. (2006), Design of SPR Systems with Dynamic Compensators and Output Variable Structure Control, in International Workshop on Variable Structure Systems, Vol. 1, Alghero-Italy, pp. 328-333.

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Systems, Structure and Control

Valášek, M. & Olgac, N. (1995a), An Efficient Pole-Placement Technique for Linear TimeVariant SISO System, 1EE Proceedings-Control Theory and Applications 142(5), 451458. Valášek, M. & Olgac, N. (1995b), Efficient Eigenvalue Assignments for General Linear MIMO Systems, Automatica 31(11), 1605-1617. Zhong, M., Ding, S. X., Lam, J. & Wang, H. (2003),'An LMI approach to design robust fault detection filter for uncertain LTI systems, Automatica 39(3), 543-550.

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Systems Structure and Control Edited by Petr Husek

ISBN 978-953-7619-05-3 Hard cover, 248 pages Publisher InTech

Published online 01, August, 2008

Published in print edition August, 2008 The title of the book System, Structure and Control encompasses broad field of theory and applications of many different control approaches applied on different classes of dynamic systems. Output and state feedback control include among others robust control, optimal control or intelligent control methods such as fuzzy or neural network approach, dynamic systems are e.g. linear or nonlinear with or without time delay, fixed or uncertain, onedimensional or multidimensional. The applications cover all branches of human activities including any kind of industry, economics, biology, social sciences etc.

How to reference

In order to correctly reference this scholarly work, feel free to copy and paste the following: Rodrigo Cardim, Marcelo C. M. Teixeira, Edvaldo Assuncao and Flavio A. Faria (2008). Control Designs for Linear Systems Using State-Derivative Feedback, Systems Structure and Control, Petr Husek (Ed.), ISBN: 978-953-7619-05-3, InTech, Available from: http://www.intechopen.com/books/systems_structure_and_control/control_designs_for_linear_systems_using_ state-derivative_feedback

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