Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems

Nonlinear Analysis 64 (2006) 2084 – 2097 www.elsevier.com/locate/na

Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems夡 Nan Xiea, b , Gong-You Tanga,∗ a College of Information Science and Engineering, Ocean University of China, Qingdao 266071, PR China b School of Computer Science and Technology, Shandong University of Technology, Zibo 255049, PR China

Received 9 November 2004; accepted 3 August 2005

Abstract This paper concerns the nonfragile guaranteed cost control problem for a class of nonlinear dynamic systems with multiple time delays and controller gain perturbations. Guaranteed cost control law is designed under two classes of perturbations, namely, additive form and multiplicative form. The problem is to design a memoryless state feedback control law such that the closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. Based on the linear matrix inequality (LMI) approach, some delay-dependent conditions for the existence of such controller are derived. A numerical example is given to illustrate the proposed method. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Uncertain systems; Nonfragile guaranteed cost control; Multiple time delays; Linear matrix inequality

1. Introduction The optimal control for nonlinear and/or time-delay systems is one of the most active subjects in control theory. For the quadratic cost functional of nonlinear and/or time-delay 夡 This work was supported by the National Natural Science Foundation of China (60574023), the Hi-Tech Research and Development Program of China (2001AA612022), and the Natural Science Foundation of Qingdao City (05-1-JC-94).

∗ Corresponding author. Tel.: +86 532 85901230; fax: +86 532 85901225.

E-mail address: [email protected] (G.-Y. Tang). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.08.005

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systems, with the exception of simplest cases, the optimal state feedback control problems do not have analytical solutions. This has motivated researchers present some approximate approaches [11,12]. The problem of designing robust controllers for systems with parameter uncertainties has drawn considerable attention in recent control system literatures. It is also desirable to design a control system which is not only stable but also guarantees an adequate level of performance. One approach to this problem is the so-called guaranteed cost control approach first introduced by Chang and Peng [2]. This approach has the advantage of providing an upper bound on a given performance index and thus the system performance degradation incurred by the uncertainties is guaranteed to be less than this bound. Based on this idea, some significant results have been proposed for the continuous-time case [10,14] and for the discrete-time case [6,15]. More recently, delay-dependent design methods for guaranteed cost control for uncertain time-delay systems have been considered for continuous-time systems in [4] and for discrete-time systems in [3]. However, the controllers yielded by using the above methods are robust with respect to uncertainties in the plant under control; their robustness with respect to the uncertainties in the controllers themselves has not been studied. Keel and Bhattacharyya [7] have shown by a number of examples that the controllers designed by using weighted H∞ ,
and l1 synthesis techniques may be very sensitive, or fragile, with respect to errors in the controller coefficients, although they are robust with respect to plant uncertainties. Recently, there have been some efforts to deal with the nonfragile controller design problem [5,13,16], and some good results have also been obtained for large-scale time-delay systems [9]. However, to our knowledge, there have been few results in the literature of an investigation for the delaydependent nonfragile guaranteed cost controller design of nonlinear systems with multiple time delays. In this paper, the problem of delay-dependent nonfragile guaranteed cost control for nonlinear dynamic systems with multiple time delays under controller gain perturbations is considered. In Section 2, the problem under consideration and some preliminaries are given. In Section 3, several delay-dependent stability criteria for the existence of the nonfragile guaranteed cost controller are derived in terms of LMIs, and their solutions provide a parameterized representation of the controller. A numerical example is given in Section 4. Finally, Section 5 concludes the paper. 2. Problem statement Consider the following nonlinear dynamic systems with multiple time-varying delays described by x(t) ˙ = Ax(t) +

2


Ai x(t − i (t)) + Bu(t)

i=1

+ f (t, x(t), x(t − 1 (t)), x(t − 2 (t))), x(t) = (t), t ∈ [−h, 0],

t > R+, (2.1)

where x(t) ∈ R n is the state vector, u(t) ∈ R m is the control vector, and f (·) : R + × R n → R n is the nonlinear uncertainty, denoted as f in the following. A, Ai , and B are known

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real constant matrices of appropriate dimensions; i (t) are the time-varying bounded delays satisfying 0
i (t)
hi < ∞, ˙ i (t)
di < 1, h = max(hi ), d = max(di ), i = 1, 2,

(2.2)

i

i

and  is a given continuous vector-valued initial function on [−h, 0]. Assumption 2.1. The nonlinear uncertainty f satisfies f T f
[x T (t) x T (t − 1 (t)) x T (t − 2 (t))]H [x T (t) x T (t − 1 (t)) x T (t − 2 (t))]T , (2.3) where H is a known constant matrix satisfying H = Block − diag{H0T H0 , H1T H1 , H2T H2 } > 0.

(2.4)

Associated with this system is the cost function  ∞ J= (x T (t)Qx(t) + uT (t)Ru(t)) dt,

(2.5)

0

where Q ∈ R n×n and R ∈ R m×m are given positive-definite matrices. The objective of this paper is to develop a procedure to design a memoryless state feedback control law u(t) = (K +
K)x(t),

(2.6)

such that the resulting closed-loop system x(t) ˙ = (A + B(K +
K))x(t) + x(t) = (t),

2


Ai x(t − i (t)) + f,

t ∈ R+,

i=1

t ∈ [−h, 0],

(2.7)

is asymptotically stable and cost function (2.5) satisfies J
J ∗ , where J ∗ is some specified constant. In the controller (2.6), K is the nominal controller gain, and
K represents the gain perturbations. In this paper, the following two classes of perturbations are considered: (a)
K is of the additive form
K = M1 F1 (t)N1 ,

F1T (t)F1 (t)
I ,

(2.8)

where M1 and N1 are known constant matrices, and F1 (t) is the uncertain parameter matrix. (b)
K is of the multiplicative form
K = M2 F2 (t)N2 K,

F2T (t)F2 (t)
I ,

(2.9)

where M2 and N2 are known constant matrices, and F2 (t) is the uncertain parameter matrix.

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Remark 2.1. The controller gain perturbations can result from the actuator degradations, as well as from the requirements for re-adjustment of controller gains during the controller implementation stage [16]. These perturbations in the controller gains are modeled here as uncertain gains that are dependent on the uncertain parameters. The models of additive uncertainties (2.8) and multiplicative uncertainties (2.9) are used to describe the controller gain variations in [13] and [7], respectively. Definition 2.1. If there exists a control law u∗ (t) and a positive scalar J ∗ such that for all admissible uncertainties, the closed-loop system (2.7) is asymptotically stable and J
J ∗ , then J ∗ is said to be a guaranteed cost and u∗ (t) in (2.6) is said to be a nonfragile guaranteed cost control law for the uncertain system (2.1) and cost function (2.5). Before proceeding to the main results, the following useful lemmas are needed in the proof. Lemma 2.1 (Schur complement). Given the constant matrices 1 , 2 , 3 of appropriate dimensions, where 1 = T1 and 2 = T2 > 0, then 1 + T3 −1 2 3 < 0 if and only if   1 T3 <0 3 −2

 or

−2 T3

 3 < 0. 1

(2.10)

Lemma 2.2 (Barmish [1]). Given matrices Y, H, E of appropriate dimensions and with Y symmetric, then for all F satisfying F T F
I and Y + HFE + E T F T H T < 0, if and only if there exists  > 0 such that Y + HHT + −1 E T E < 0. 3. Main results In this section, we will consider the nonfragile guaranteed cost control under gain perturbations of the form (2.8) and (2.9). We will first present a sufficient condition for the existence of state feedback guaranteed cost control law, and then give a parameterized representation of the guaranteed cost control law in terms of the feasible solutions to a certain LMI. Rewrite system (2.7) in an equivalent form    t 2 2

x(t) ˙ = A + B(K +
K) + Ai x(t) − Ai x(s) ˙ ds + f, t ∈ R + , x(t) = (t),

t ∈ [−h, 0]

i=1

i=1

t−i (t)

(3.1a)

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or  x(t) ˙ =

A + B(K +
K) + − +

2
i=1 2


 Ai

t−i (t)

 Ai x(t) + f

i=1



t

2


(A + B(K +
K))x(s) 

Ai x(s − i (s)) + f

t ∈ R+,

ds,

i=1

x(t) = (t),

t ∈ [−h, 0].

(3.1b)

The following Lyapunov–Krasovskii functional is applied: V (t) = V1 (t) + V2 (t) + V3 (t),

(3.2a)

V1 (t) = x T (t)P x(t),

(3.2b)

where

V2 (t) =

2 


2 
i=1



t

−i (t) t+

i=1

V3 (t) =

0

t t−i (t)

x˙ T (s)Ri x(s) ˙ ds d,

(3.2c)

x T ()Si x() d.

(3.2d)

Then, the following theorem gives the delay-dependent nonfragile guaranteed cost control for systems (2.1) and (2.5). Theorem 3.1. u(t) = (K +
K)x(t) is a nonfragile guaranteed cost control law if there exist positive-definite matrices P , Si , Ri , matrices Yi , Zi and a scalar 1 > 0 such that the following matrix inequalities hold: ⎡

11

⎢ ⎢ ∗ ⎢ ⎢ =⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗

12 22

2

i=1

∗

13 AT1 hi Ri A2 33

2

i=1 2

i=1

∗

∗

2

i=1

14 AT1 hi Ri AT2 hi Ri

hi Ri − −1 1 I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎦

(3.3a)

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and 

Zi ∗

Yi Ri

 0,

i = 1, 2,

(3.3b)

where (∗) denotes the symmetric element of a matrix, and 11 =

2


[A + B(K +
K)]T hi Ri [A + B(K +
K)]

i=1

+ [A + B(K +
K)]T P + P [A + B(K +
K)] +

2


(hi Zi + Yi + YiT + Si )

i=1 T T + −1 1 H0 H0 + Q + (K +
K) R(K +
K),

12 =

2


[A + B(K +
K)]T hi Ri A1 + P A1 − Y1 ,

i=1

13 =

2


[A + B(K +
K)]T hi Ri A2 + P A2 − Y2 ,

i=1

14 =

2


[A + B(K +
K)]T hi Ri + P ,

i=1 T 22 = − S1 (1 − d1 ) + −1 1 H1 H1 +

2


AT1 hi Ri A1 ,

i=1 T 33 = − S2 (1 − d2 ) + −1 1 H2 H2 +

2


AT2 hi Ri A2 .

(3.3c)

i=1

Moreover, the cost function (2.5) satisfies the following bound: J
V (0) = x (0)P x(0) + T

2 
i=1

∗

=J .

0



−i (t) 

0

x˙ (s)Ri x(s) ˙ ds d + T

2 
i=1

0 −i (t)

x T ()Si x() d (3.4)

Proof. Taking u(t) = (K +
K)x(t) in system (2.1), the resulting closed-loop system is given by (2.7).

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Differentiating V1 with respect to t gives V˙1 (t) = x˙ T (t)P x(t) + x T (t)P x(t) ˙    2 2

Ai x(t)− Ai = A+B(K+
K)+ i=1

 + x (t)P

A + B(K +
K) +

T

−

2


 Ai

t−i (t)

i=1

= x (t) T

⎧ ⎨ ⎩

t

i=1

x(s) ˙ ds+f

P x(t)

Ai x(t)

x(s) ˙ ds + f

A + B(K +
K) +

2


 x T (t)P Ai

T Ai

i=1

×P + P A + B(K +
K) + 2




t−i (t)

T

i=1





−2

i=1 2


t

2
i=1

t t−i (t)

⎫ ⎬ Ai

⎭

x(t)

x(s) ˙ ds + x T (t)Pf + f T P x(t).

(3.5)

Since, by Moon et al. [8], for any a ∈ R p , b ∈ R q , N ∈ R p×q , R ∈ R q×q , Y ∈ Z ∈ R p×p , the following holds:  T    Z Y −N a a , (3.6a) −2a T Nb
Y T − NT b R b

R p×q ,

where



Z YT

 Y 0. R

(3.6b)

Taking N = Ni = P Ai , R = Ri , Z = Zi , Y = Yi , a = x(t) and b = x(s), ˙ we obtain  t − 2x T (t)P Ai x(s) ˙ ds t−i (t)  t = −2 x T (t)P Ai x(s) ˙ ds t−i (t)     t x(t) Zi Yi − P A i T T
ds [x (t) x˙ (s)] x(s) ˙ YiT − ATi P Ri t−i (t)  t
x T (t)(YiT − ATi P + Yi − P Ai )x(t) + x˙ T (s)Ri x(s) ˙ ds t−i (t)

− x T (t − i (t))(YiT − ATi P )x(t) − x T (t)(Yi − P Ai ) × x(t − i (t)) + hi x T (t)Zi x(t).

(3.7)

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Substituting (3.7) into (3.5) and using (2.3), we have  T V˙1 (t)
x (t) (A + B(K +
K))T P + P (A + B(K +
K)) +YiT

+ Yi +

2


 hi Zi x(t) + x T (t)Pf + f T P x(t)

i=1

− −

2


x T (t)(Yi − P Ai )x(t − i (t))

i=1 2


x T (t − i (t))(YiT − ATi P )x(t)

i=1

+

2 


t t−i (t)

i=1

T x˙ T (s)Ri x(s) ˙ ds − −1 1 f f

T T T + −1 1 [x (t) x (t − 1 (t)) x (t − 2 (t))]

× H [x T (t) x T (t − 1 (t)) x T (t − 2 (t))]T . Differentiating V2 and V3 with respect to t gives 2  0
˙ ˙ + )] d [x˙ T (t)Ri x(t) ˙ − x˙ T (t + )Ri x(t V2 (t) = =


i=1 −i (t) 2 
i=1 2


hi x˙ T (t)Ri x(t) ˙ −

i=1






i (t)x˙ T (t)Ri x(t) ˙ −

2
i=1

t−i (t)

2  t
i=1



t

t−i (t)

 x˙ T (s)Ri x(s) ˙ ds x˙ T (s)Ri x(s) ˙ ds

hi (A + B(K +
K))x(t) +

2 
i=1

V˙3 (t) =


2


Ai x(t − i (t)) + f

2


 Ai x(t − i (t)) + f

i=1 t t−i (t)

x˙ T (s)Ri x(s) ˙ ds,

x T (t)Si x(t) −

2


i=1

i=1

2


2


i=1

T

i=1



× Ri (A + B(K +
K))x(t) + −

2


x T (t)Si x(t) −

i=1

(1 − ˙ i )x T (t − i (t))Si x(t − i (t)) (1 − di )x T (t − i (t))Si x(t − i (t)).

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From the above inequalities, we can obtain V˙ (t)
V˙1 (t)+V˙2 (t)+V˙3 (t)
T (t) (t)−x T (t)[Q + (K+
K)T R(K +
K)]x(t), (3.8a) where (t) = col{x(t) x(t − 1 (t)) x(t − 2 (t)) f }.

(3.8b)

Then, the matrix inequality (3.3) implies that V˙ (t) < − x T (t)[Q + (K +
K)T R(K +
K)]x(t) < 0.

(3.9)

Noting that Q > 0 and R > 0, this implies that system (2.7) is asymptotically stable by the Lyapunov stability theory. Moreover, from (3.9) we have x T (t)[Q + (K +
K)T R(K +
K)]x(t) < − V˙ ,

(3.10)

by integrating both sides of (3.10) from 0 to Tf , we obtain  Tf x T (t)[Q + (K +
K)T R(K +
K)]x(t) dt < V (0) − V (Tf ) 0

= x T (0)P x(0) − x T (Tf )P x(Tf ) + + −

2  0
i=1 −i (t) 2  Tf
i=1

x T ()Si x() d −

Tf −i (t)

2 


0



0

i=1 −i (t)  2
 0  Tf i=1

−i (t) Tf +

x T ()Si x() d.

x˙ T (s)Ri x(s) ˙ ds d x˙ T (s)Ri x(s) ˙ ds d (3.11)

As the closed-loop systems (2.7) is asymptotically stable when Tf → ∞, we obtain x T (Tf )P x(Tf ) → 0,  Tf 2  0
x˙ T (s)Ri x(s) ˙ ds d → 0, i=1 −i (t) Tf + 2  Tf
T i=1

Tf −i (t)

x ()Si x() d → 0.

Hence, we obtain  ∞ (x T (t)Qx(t) + uT (t)Ru(t)) dt
V (0) = J ∗ .

(3.12)

0

This completes the proof.



In the sequel, we will show that criterion (3.3) for the existence of guaranteed cost controller is equivalent to the feasibility of an LMI.

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Theorem 3.2. For system (2.1) with the controller gain perturbation (2.8) and the cost function (2.5), if there exist scalars 1 > 0, 2 > 0, matrices L, V , Ui , and with L, Ui positivedefinite such that the following LMI is satisfied: ⎡

A1 U1 A2 U2 1 I GT GT V 0 0 (A1 U1 )T (A1 U1 )T 0 ⎢ ∗ (d1 − 1)U1 ⎢ ⎢∗ 0 (A2 U2 )T (A2 U2 )T 0 ∗ (d2 − 1)U2 ⎢ ⎢∗  I  I 0 ∗ ∗ −1 I 1 1 ⎢ −1 ⎢∗ ∗ ∗ ∗ −h−1 0 0 1 R1 ⎢ ⎢∗ −1 0 ∗ ∗ ∗ ∗ −h−1 ⎢ 2 R2 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ −R −1 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎣∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (N1 L)T (H0 L)T 0 0 L L 0 0 (H1 U1 )T 0 0 0 0 0 0 0 0 (H2 U2 )T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 2 I ∗ − 1 I 0 0 0 0 0 0 0 ∗ ∗ −1 I ∗ ∗ ∗ − 1 I 0 0 −1 0 ∗ ∗ ∗ ∗ −h−1 1 Z1 ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

−1 −h−1 2 Z2 ∗ ∗ ∗

2 BM1 0 0 0 2 BM1

2 BM1 2 M 1 − 2 I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

⎤

L 0 0 0 0 0 0 0 0 0 0 0 0

L 0 0 0 0 0 0 0 0 0 0 0 0

L 0 0 0 0 0 0 0 0 0 0 0 0

0 −U1 ∗ ∗

0 0 −U2 ∗

0 0 0 −Q−1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.13) where  = G + GT ,

G = AL + BV , L = P −1 ,

V = KP −1 ,

Ui = Si−1 ,

i = 1, 2,

then the nonfragile guaranteed cost control law for (2.1) and (2.5) is given by u(t) = V L−1 x(t)

(3.14)

and the corresponding closed-loop value of the cost function satisfies J
J ∗ , in which J ∗ is given in (3.4). Proof. Letting Yi ≡ 0 and Zi > 0, Ri > 0 in (3.3b), substituting the representation of matrix
K = M1 F1 (t)N1 into (3.3a), in light of Lemma 2.2. and using the Schur complement,

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pre- and post-multiplying both sides of the obtained inequality by T and , where ⎫ ⎧ ⎬ ⎨ (3.15)

= Block − diag P −1 , S1−1 , S2−1 , 1 I, I, I, I, 2 I, I, . . . , I ,   ⎭ ⎩ 9

we can obtain (3.13). This completes the proof.



Theorem 3.2 presents a method of designing a state feedback nonfragile guaranteed cost controller. The following theorem presents a method of selecting a controller minimizing the upper bound of the guaranteed cost (3.4). Theorem 3.3. Consider systems (2.1) with performance index (2.5), if the following optimization problem: min

L,,11 ,12 ,21 ,22

(i) (3.13), (ii) 

{ + tr(11 ) + tr(12 ) + tr(21 ) + tr(22 )}

 x T (0) < 0, −L

− x(0)

(iii) 

−1i C

 CT < 0, −Ri−1

i = 1, 2,

−2i D

 DT < 0, −Si−1

i = 1, 2,

(iv) 

(3.16)

has a solution set (L, , 11 , 12 , 21 , 22 ), then controller (3.14) is an optimal nonfragile guaranteed cost control law which ensures the minimization of the guaranteed cost (3.4) for the uncertain dynamic system (3.1), where 

0



−i (t) 

0

 x(s) ˙ x˙ (s) ds d = C C, T

T

0 −i (t)

x()x T () d = D T D.

(3.17)

Proof. By Theorem 3.2,(i) in (3.16) is clear. It follows Lemma 2.1 that (ii)–(iv) in (3.16) are equivalent to x T (0)L−1 x(0) < ,

C T Ri C < 1i ,

D T Si D < 2i ,

i = 1, 2,

(3.18)

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2095

respectively. Furthermore, 



0

0

−i (t) 

and



 x˙ (s)Ri x(s) ˙ ds d = T

−i (t)

0

tr(x˙ T (s)Ri x(s)) ˙ ds d

−i (t)  = tr(C T Ri C) < tr(1i )



0



0

x T ()Si x() d =

0 −i (t)

(3.19)

tr(x T ()Si x()) d = tr(D T Si D) < tr(2i ). (3.20)

Hence, it follows from (3.4) that ∗

J = x (0)P x(0) + T

2 
i=1

0



−i (t) 

0

x˙ (s)Ri x(s) ˙ ds d + T


 + tr(11 ) + tr(12 ) + tr(21 ) + tr(22 ).

2 
i=1

0 −i (t)

x T ()Si x() d (3.21)

Thus, the minimization of (3.21) implies the minimization of the guaranteed cost for system (3.1). The optimality of the solutions of the optimization problem (3.16) follows from the convexity of the objective function and of the constraints, which ensures that a global optimum, when it exists, is reachable. This completes the proof.  Remark 3.1. When the controller gain perturbation
K is of the multiplicative form (2.9), the criterion for the nonfragile guaranteed cost control of the systems is identical to the LMI (3.13), except that BM1 , N1 L are changed as BM2 , N2 V , respectively. The proof is omitted.

4. Simulations Consider the nonlinear dynamic systems with multiple time-varying delays in (2.1)–(2.5) with         0 1 0.3 0.2 0.2 0.1 0 , A1 = A= , A2 = , B= , −1 1 0.2 0.3 0.1 0.2 1 

  T 0.1 0 −0.1 , M1,2 = 0.2, N1,2 = , h1 = h2 = h = 0.3, H0 = H1 = H2 = 0 0.1 0.2 d1 = d2 = d = 0.2, x(t) = [e−0.5t e0.5t ]T , −h
t
0. The cost function associated with this system is given in (2.5) with Q = I , R = 0.5I . (a) When the gain perturbation
K is of the additive form (2.8), using the LMI toolbox, we obtain the guaranteed cost controller u(t) = [−2.3986 − 4.4593]x(t), and the least upper bound of the corresponding closed-loop cost function J ∗ = 26.5491.

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N. Xie, G.-Y. Tang / Nonlinear Analysis 64 (2006) 2084 – 2097

(b) When the gain perturbation
K is of the multiplicative form (2.9), using the LMI toolbox, we obtain the guaranteed cost controller u(t) = [−2.9764 − 5.7231]x(t), and the least upper bound of the corresponding closed-loop cost function J ∗ = 31.8539.

5. Conclusions In this paper, based on the Lyapunov method, we have presented a design method to the nonfragile guaranteed cost control via memoryless state feedback controller for nonlinear dynamic systems with multiple time-varying delays and controller gain perturbations in an LMI framework. The parameterized representation of a set of the controller, which guaranteed not only the robust stability of the closed-loop system but also the cost function bound constraint, has been provided in terms of the feasible solutions to the LMI. Furthermore, a convex optimization problem has been introduced to select the optimal nonfragile guaranteed cost controller. Finally, a numerical example is given for illustration of the controller design.

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