On robust stabilization for neutral delay-differential systems with parametric uncertainties and its application

Applied Mathematics and Computation 162 (2005) 1167–1182

www.elsevier.com/locate/amc

On robust stabilization for neutral delay-differential systems with parametric uncertainties and its application Ju H. Park

a,*

, O. Kwon

b

a

b

School of Electrical Engineering and Computer Science, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, South Korea Department of Mechatronics Research, Samsung Heavy Industries Co. Ltd., Daejeon, South Korea

Abstract This paper considers the problem of robust stabilization for uncertain neutral delaydifferential systems. Based on the Lyapunov method, a delay-dependent criterion for determining the stability of systems is obtained in terms of matrix inequalities, which can be easily solved by efficient convex optimization algorithms. Then, the method presented is extended to systems with controller gain variations, that is, the design problem for the nonfragile controller is considered. A numerical example is included to illustrate the design procedure of the proposed method. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Robust stabilization; Neutral system; Lyapunov method; Nonfragile controller

1. Introduction Many physical systems have time-delay in their mathematical model [1,2]. Since the existence of delay frequently is a source of instability and performance degradation in many control systems, the stability and stabilization in such systems are of theoretical and practical importance. Therefore, these problems have received considerable attention, and many results on the problems have been reported during the three decades. Recently, more interests is focused on *

Corresponding author. E-mail address: [email protected] (J.H. Park).

0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.02.003

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Nomenclature Rn Rm
n AT P >0

n-dimensional real space set of all real m by n matrices transpose of matrix A (respectively P < 0) matrix P is symmetric positive (respectively negative) definite I identity matrix with appropriate dimension H the elements below the main diagonal of a symmetric block matrix Cn;h the Banach space of continuous functions mapping the interval ½h; 0 into Rn , with the topology of uniform convergence diagf  g block diagonal matrix kM ðAÞ maximum eigenvalue of matrix A

neutral systems, which are the general form of time-delay systems and contain delay on the derivatives of some system variables [3–10]. Using the Lyapunov method, characteristic equation approach, and state solution approach, many stability criteria have been developed in the literature; see, e.g., [3–8] and the references therein. However, only a few works consider the design problem of controllers for stabilization of neutral systems [9,10]. For example, Ma et al. [9] developed a control method for a class of neutral system with constraint on system matrices utilizing a delay-differential inequality and a transformation technique. Park [10] presents a control method for guaranteeing an adequate level of performance of neutral systems using linear matrix inequalities. The present paper considers the further investigation of robustness of the feedback stabilization of uncertain neutral delay-differential systems. The problem addressed is twofold: the first is to derive the stabilization criterion of the system, the second is to extend the result to the system with controller gain variations. For stability analysis, the Lyapunov method with a new model transformation technique is utilized. The proposed methods employs free weighting matrices to obtain less conservative stability criterion. The criterion is derived in terms of matrix inequalities, which can be easily solved by using various convex optimization algorithms [13]. A numerical example is given to show the superiority of the present result to those in the literature.

2. Problem formulation Consider the neutral control systems described by the following state equation:

J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

1169

d ½xðtÞ  A2 xðt  hÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðA1 þ DA1 ðtÞÞxðt  hÞ dt þ ðB þ DBðtÞÞuðtÞ; xðsÞ ¼ /ðsÞ; s 2 ½h; 0;

ð1Þ

where xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input, A; A1 ; A2 2 Rn
n and B 2 Rn
m are known real parameter matrices, h > 0 is a constant delay, DA; DA1 and DB are the parametric uncertainties in the system, /ðsÞ 2 Cn;h is a given continuous vector valued initial function. In this paper, it is assumed that the uncertainties are of the form DAðtÞ ¼ D1 F1 ðtÞE1 ;

DA1 ðtÞ ¼ D2 F2 ðtÞE2 ;

DBðtÞ ¼ D3 F3 ðtÞE3 ;

ð2Þ

where Di ; Ei ði ¼ 1; 2; 3Þ are known real constant matrices of appropriate dimensions, and Fi ðtÞ 2 Rki
li are unknown matrices, which satisfy FiT ðtÞFi ðtÞ 6 I

ði ¼ 1; 2; 3Þ:

Here, we introduce a definition of robust stabilizability for systems (1). Definition 1. The uncertain system (1) is said to be robustly stabilizable if there exists a feedback controller uðtÞ such that the resulting closed-loop systems is asymptotically stable for all admissible uncertainties. Then, the goal of this paper is to develop a procedure to design a memoryless state-feedback controller of the form uðtÞ ¼ KxðtÞ;

ð3Þ

where K 2 Rm
n is a gain matrix to be determined.

3. Robust stabilization Now, define an operator Dðxt Þ : Cn;h ! Rn as Z t Dðxt Þ ¼ xðtÞ þ GxðsÞ ds  A2 xðt  hÞ;

ð4Þ

th

where xt ¼ xðt þ sÞ; s 2 ½h; 0 and G 2 Rn
n is a constant matrix which will be chosen to make the system asymptotically stable. With the above operator, the transformed system is _ t Þ ¼ x_ ðtÞ þ GxðtÞ  Gxðt  hÞ  A2 x_ ðt  hÞ Dðx ¼ ðA þ DA þ GÞxðtÞ þ ðA1 þ DA1  GÞxðt  hÞ þ ðB þ DBÞKxðtÞ: ð5Þ

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J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

We will need the following well-known facts and lemmas to derive the main results. Fact 1. For given matrices D, E, F with F T F 6 I and scalar  > 0, the following inequality DFE þ ET F T DT 6 eDDT þ e1 ET E is always satisfied. Fact 2 (Schur complement). The linear matrix inequality 

ZðxÞ Y T ðxÞ

 Y ðxÞ >0 W ðxÞ

ð6Þ

is equivalent to W ðxÞ > 0 and ZðxÞ  Y ðxÞW 1 ðxÞY T ðxÞ, where ZðxÞ ¼ Z T ðxÞ, W ðxÞ ¼ W T ðxÞ and Y ðxÞ depend affinely on x. Lemma 1 [14]. For any positive-definite matrix M 2 Rn
n , scalar c > 0, vector function x : ½0; c ! Rn such that the integrations concerned are well defined, then Z

c

xðsÞ ds 0

T  Z c  Z c M xðsÞ ds 6 c xT ðsÞMxðsÞ ds: 0

ð7Þ

0

Lemma 2 [11]. For given positive scalar h, the operator Dðxt Þ is stable if there exist a positive definite matrix C and positive scalars b1 and b2 such that  T  A2 CA2  b1 C hAT2 CG b1 þ b2 < 1; < 0: ð8Þ H h2 GT CG  b2 C1

Then, we have the following theorem. Theorem 1. For given a > 1 and h, the system (1) with controller uðtÞ ¼ KxðtÞ  MX 1 xðtÞ is robustly stabilizable if there exist matrices X > 0; W > 0, positive scalars e1 ; e2 ; . . . e8 , matrices Fii P 0ði ¼ 1; . . . ; 4Þ, positive scalars b1 ; b2 , and any matrices Y ; M; Fij ði ¼ 1; . . . ; 4; i < j 6 4Þ which satisfy the following matrix inequalities:

J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

1171

2

N1 N2 A1 X  Y þ hF13 N3 0 0 0 6 1 6 H h ða  1ÞX A1 X  Y þ F23 F24 D1 D2 D3 6 6 T T H W þ hF33 XA1 þ Y þ hF34 0 0 0 6H 6 6H H H hða  1ÞX 0 0 0 6 6 6H 0 H H H e4 I 0 6 6 6H H H H H e5 I 0 6 6 6H H H H H H e6 I 6 6 6H H H H H H H 6 6 6H H H H H H H 6 6 H H H H H H 6H 6 6 H H H H H H 6H 6 6H H H H H H H 4 H

H

H

H

0

0

0

0

0

0

0

0

0

0

0

0

e2 XE2T e5 XE2T e8 XE2T 0

0

0

D1

D2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

e2 I

0

0

0

0

H

e5 I

0

0

0

H

H

e8 I

0

0

H

H

H

e7 I

0

H

H

H

H

e8 I

H

H

H

H

H

N4

H

H

H

3

7 07 7 7 07 7 07 7 7 07 7 7 07 7 7 07 7 < 0; 7 07 7 7 07 7 7 07 7 7 07 7 07 5

ð9Þ

N5

X þ F22 6 0;

ð10Þ

X þ F44 6 0;

ð11Þ

b1 þ b2 < 1;

ð12Þ

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J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

2

b1 X 0 4 H b2 X H H 2 F11 F12 F13 6 H F22 F23 6 4 H H F33 H H H

3 XAT2 hY T 5 < 0; X 3 F14 F24 7 7 P 0; F34 5 F44

ð13Þ

ð14Þ

where N1 ¼ AX þ XAT þ Y þ Y T þ BM þ M T BT þ W þ hF11 ; N2 ¼ XAT þ Y T þ M T BT þ F12 ; N3 ¼  XAT  Y T  M T BT þ hF14 ; N4 ¼ ½ ahY T

ahXAT2 D1 D2 D3 e1 XE1T e4 XE1T e7 XE1T e3 M T E3T e6 M T E3T ; N5 ¼ diagfahX ; ahX ; e1 I; e2 I; e3 I; e1 I; e4 I; e7 I; e3 I; e6 Ig: ð15Þ

Proof. Consider a legitimate Lyapunov function candidate [1] as V ¼ V1 þ V2 þ V3 þ V4 þ V5 ;

ð16Þ

where V1 ¼ DT ðxt ÞP Dðxt Þ; Z t Z t V2 ¼ a xT ðuÞGT PGxðuÞ du ds; th s Z t V3 ¼ xT ðsÞQxðsÞ ds; th Z t V4 ¼ ah xT ðsÞAT2 PA2 xðsÞ ds; th 2 3 F11 F12 F13 F14 Z tZ s 6H F F23 F24 7 22 6 7 ZTP 6 V5 ¼ 7P Z du ds: 4 H H F33 F34 5 0 sh H

H

H

F44

where a > 1, Q > 0, P > 0, P ¼ diagfP ; P ; P ; P g and 2 3 xðsÞ 6 GxðuÞ 7 7 Z¼6 4 xðs  hÞ 5: A2 xðs  hÞ

ð17Þ ð18Þ ð19Þ ð20Þ

ð21Þ

J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

1173

Taking the time-derivative of V leads to _ tÞ V_1 ¼ 2DT ðxt ÞP Dðx

T Z t GxðsÞds  A2 xðt  hÞ P fðA þ DAðtÞ þ GÞxðtÞ ¼ 2 xðtÞ þ th

þ ðA1 þ DA1 ðtÞ  GÞxðt  hÞ þ ðB þ DBðtÞÞKxðtÞg T

¼ xT ðtÞ½P ðA þ G þ BKÞ þ ðA þ G þ BKÞ P xðtÞ Z t T GxðsÞds P ðA þ G þ BKÞxðtÞ þ 2xT ðtÞP ðA1  GÞxðt  hÞ þ 2 th

þ2

Z

T

t

GxðsÞds

P ðA1  GÞxðt  hÞ  2xT ðt  hÞAT2 P ðA þ G þ BKÞxðtÞ

th

 2xT ðt  hÞAT2 P ðA1  GÞxðt  hÞ þ 2xT ðtÞP DAðtÞxðtÞ þ 2xT ðtÞP DA1 ðtÞxðt  hÞ þ 2xT ðtÞP DBðtÞKxðtÞ Z t T Z t T GxðsÞds P DAðtÞxðtÞ þ 2 GxðsÞds P DA1 ðtÞxðt  hÞ þ2 th

þ2

Z

th

T

t

GxðsÞds

P DBðtÞKxðtÞ  2xT ðt  hÞAT2 DAðtÞxðtÞ

th

 2xT ðt  hÞAT2 DA1 ðtÞxðt  hÞ  2xT ðt  hÞAT2 DBðtÞKxðtÞ;

V_2 ¼ ahxT ðtÞGT PGxðtÞ  a 6 ahxT ðtÞGT PGxðtÞ 

Z

Z

ð22Þ

t

xT ðsÞGT PGxðsÞ ds th

t

xT ðsÞGT PGxðsÞ ds th

 h1 ða  1Þ

Z

t

GxðsÞ ds

T  Z P

th

t

 GxðsÞ ds ;

ð23Þ

th

V_3 ¼ xT ðtÞQxðtÞ  xT ðt  hÞQxðt  hÞ;

ð24Þ

V_4 ¼ ahxT ðtÞAT2 PA2 xðtÞ  ahxT ðt  hÞAT2 PA2 xðt  hÞ; ¼ ahxT ðtÞAT2 PA2 xðtÞ  ða  1ÞhxT ðt  hÞAT2 PA2 xðt  hÞ  hxT ðt  hÞAT2 PA2 xðt  hÞ;

ð25Þ

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J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

V_5 ¼ hxT ðtÞPF11 PxðtÞ þ 2xT ðtÞPF12 P Z T þ 2hx ðtÞPF14 PA2 xðt  hÞ þ þ2 þ2

Z Z

T

t

GxðsÞ ds þ 2hxT ðtÞPF13 Pxðt  hÞ

th t

xT ðsÞGT PF22 PGxðsÞ ds

PF23 Pxðt  hÞ

GxðsÞ ds th

t

th

T

t

Z

GxðsÞ ds

PF24 PA2 xðt  hÞ þ hxT ðt  hÞPF33 Pxðt  hÞ

th

þ 2hxT ðt  hÞPF34 PA2 xðt  hÞ þ hxT ðt  hÞAT2 PF44 PA2 xðt  hÞ;

ð26Þ

where Lemma 1 is utilized in (23). Using Fact 1, we get a bound of some terms which include uncertainties as 2xT ðtÞP DAðtÞxðtÞ ¼ 2xT ðtÞPD1 F1 ðtÞE1 xðtÞ T T T T 6 e1 1 x ðtÞPD1 D1 PxðtÞ þ e1 x ðtÞE1 E1 xðtÞ;

ð27Þ

T T T T 2xT ðtÞP DA1 ðtÞxðt  hÞ 6 e1 2 x ðtÞPD2 D2 PxðtÞ þ e2 x ðt  hÞE2 E2 xðt  hÞ;

ð28Þ T T T T T 2xT ðtÞP DBðtÞKxðtÞ 6 e1 3 x ðtÞPD3 D3 PxðtÞxðtÞ þ e3 x ðtÞK E3 E3 KxðtÞ;

Z

T

t

2

GxðsÞds

P DAðtÞxðtÞ6e1 4

Z

GxðsÞds

th þ e4 xT ðtÞE1T E1 xðtÞ;

Z

6 e1 5

Z

th



t

GxðsÞds th

ð30Þ

T

t

GxðsÞ ds

PD2 DT2 P

th

Z



t

GxðsÞ ds th

T

þ e5 x ðt  Z

Z

P DA1 ðtÞxðt  hÞ

GxðsÞ ds th

hÞE2T E2 xðt

 hÞ:

ð31Þ

T

t

2

PD1 DT1 P

T

t

2

T

t

ð29Þ

GxðsÞ ds th

6 e1 6

Z

P DBðtÞKxðtÞ T

t

GxðsÞ ds th

PD3 DT3 P

Z

t



GxðsÞ ds þ e6 xT ðtÞKE3T E3 KxðtÞ;

th

ð32Þ T T T 2xT ðt  hÞAT2 P DAðtÞxðtÞ 6 e1 7 x ðt  hÞA2 PD1 D1 PA2 xðt  hÞ

þ e7 xT ðtÞE1T E1 xðtÞ;

ð33Þ

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1175

T T T 2xT ðt  hÞAT2 P DA1 ðtÞxðt  hÞ 6 e1 8 x ðt  hÞA2 PD2 D2 Pxðt  hÞ

þ e8 xT ðt  hÞE2T E2 xðt  hÞ;

ð34Þ

T T T 2xT ðt  hÞAT2 P DBðtÞKxðtÞ 6 e1 9 x ðt  hÞA2 PD3 D3 Pxðt  hÞ

þ e9 xT ðtÞK T E3T E3 KxðtÞ:

ð35Þ

Substituting (27)–(35) into (22) and using (23)–(26) gives that the timederivative of V has new upper bound as follows: 2

3T 2 3 xðtÞ xðtÞ R R t t 6 6 7 GxðsÞ ds 7 6 7 6 th GxðsÞ ds 7 V_ 6 6 th 7 X6 7 4 xðt  hÞ 5 4 xðt  hÞ 5 A2 xðt  hÞ A2 xðt  hÞ Z t T T x ðsÞG ðP þ PF22 P ÞGxðsÞ ds þ hxT ðt  hÞAT2 ðP þ PF44 P ÞA2 xðt  hÞ; þ th

ð36Þ where 2

R1 6H X¼6 4H H

R5 R2 H H

P ðA1  GÞ þ hPF13 P P ðA1  GÞ þ PF23 P R3 H

3 R6 7 PF24 P 7; T ðA1  GÞ P þ hPF34 P 5 R4

R1 ¼ P ðA þ G þ BKÞ þ ðA þ G þ BKÞT P þ ahGT PG þ ahAT2 PA2 þ Q T T 1 T 1 T þ hPF11 P þ e1 1 PD1 D1 P þ e1 E1 E1 þ e2 PD2 D2 P þ e3 PD3 D3 P

þ e3 K T E3T E3 K þ e4 E1T E1 þ e6 K T E3T E3 K þ e7 E1T E1 ; T 1 T 1 T R2 ¼  h1 ða  1ÞP þ e1 4 PD1 D1 P þ e5 PD2 D2 P þ e6 PD3 D3 P

R3 ¼  Q þ hPF33 P þ e2 E2T E2 þ e5 E2T E2 þ e8 E2T E2 T 1 T R4 ¼  hða  1ÞP þ e1 7 PD1 D1 P þ e8 PD2 D2 P ;

R5 ¼ ðA þ G þ BKÞT P þ PF12 P ; R6 ¼  ðA þ G þ BKÞT P þ hPF14 P : ð37Þ Hence, if X < 0, then a positive scalar c exists which satisfies 2 V_ < ckxðtÞk :

ð38Þ

Let X ¼ P 1 ;

W ¼ XQX ;

Y ¼ GX ;

M ¼ KX :

ð39Þ

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J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

By pre- and post-multiplying the inequalities P þ PF22 P 6 0, P þ PF44 P 6 0 and X < 0 by X , X , and diagfX ; X ; X ; X g, respectively, the resulting inequalities are equivalent to (10) and (11) and the following inequality: 2

C1 6H X1 ¼ 6 4H H

C5 C2 H H

A1 X  Y þ hF13 A1 X  Y þ F23 C3 H

3 XAT  Y T  M T BT þ hF14 7 F24 7 < 0; T T 5 XA1 þ Y þ hF34 C4 ð40Þ

where C1 ¼ AX þ Y þ BM þ XA þ Y T þ M T BT þ W þ hF11 þ ahY T X 1 Y T 1 T 1 T T þ ahXAT2 X 1 A2 X þ e1 1 D1 D1 þ e2 D2 D2 þ e3 D3 D3 þ e1 XE1 E1 X

þ e4 XE1T E1 X þ e7 XE1T E1 X þ e3 M T E3T E3 M þ e6 M T E3T E3 M T 1 T 1 T C2 ¼  h1 ða  1ÞX þ e1 4 D1 D1 þ e5 D2 D2 þ e6 D3 D3

C3 ¼  W þ hF33 þ e2 XE2T E2 X þ e5 XE2T E2 X þ e8 XE2T E2 X T 1 T C4 ¼  hða  1ÞX þ e1 7 D1 D1 þ e8 D2 D2 ;

C5 ¼ XAT þ Y T þ M T BT þ F12 : ð41Þ The inequality (40) is equivalent to the inequality (9) by Fact 2. Also, the inequality (13) is equivalent to 

AT2 X 1 A2  b1 P H

hAT2 PG 2 T h G PG  b2 P

 <0

ð42Þ

by pre- and post-multiplying the inequality (13) by diagfX 1 ; X 1 ; X 1 g. Therefore, from Lemma 2, if the inequality (12) and (13) hold, then operator Dðxt Þ is stable. The inequality (14) means that V5 is non-negative. According Theorem 9.8.1 in [1], we conclude that if matrix inequalities (9)–(14) holds, then, system (1) is asymptotically stable. This completes our proof. h Remark 1. In this paper, we use the operator Dðxt Þ to transform the original system. Note that if the free parameter G in Dðxt Þ is A1 , then the transfor-

J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

1177

mation is the neutral model transformation one [1]. Since the operator Dðxt Þ has free weighting matrix, it is less conservative than the results obtained by using the neutral model transformation. Remark 2. The solutions of Theorem 1 can be obtained by solving the generalized eigenvalue problem with respect to solution variables, which is a quasiconvex optimization problem. In this paper, in order to solve the problem, we utilize MatlabÕs LMI Control Toolbox [15] which implements interiorpoint algorithms. These algorithms are significantly faster than classical convex optimization algorithms [13].

Remark 3. The method presented in the paper can be easily extended to uncertain time-delay system with A2 ¼ 0 or neutral systems with multiple time delays.

4. Extension to nonfragile controller In the field of control for uncertain systems, it is generally known that feedback systems designed for robustness with regard to plant parameters may require very accurate controllers. Recently, it is shown that relatively small perturbations in controller parameters could even destabilize the closed-loop system [12]. Therefore, it is necessary that any controller should be able to tolerate some level of controller gain variations. This has motivated the study of nonfragile controller problems. Since gain perturbations may arise when implementing the controller given in (3), the actual controller will be of the form uðtÞ ¼ ðK þ DKðtÞÞxðtÞ;

ð43Þ

where DK is the additive controller gain variations of the form: DKðtÞ ¼ D4 F4 ðtÞE4 ; where D4 and E4 are known matrices, and F4 ðtÞ is an unknown perturbation matrix satisfying F4 ðtÞT F4 ðtÞ 6 I:

1178

J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

In this case, using Fact 1, the matrix X given in (36) 20 1 PBDKðtÞþDK T ðtÞBT P 6B C 6 B þe3 K T E3T E3 DKðtÞ C 6B C 6 B þe DK T ðtÞET E K C 3 6B C 3 3 6B C 6 B þe3 DK T ðtÞET E3 DKðtÞ C DK T ðtÞBT P 3 6B C 6B C 6 B þe6 K T E3T E3 DKðtÞ C C B X ¼ Xþ 6 6B C 6 @ þe6 DK T ðtÞE3T E3 K A 6 6 þe DK T ðtÞET E DKðtÞ 6 6 3 3 6 6 H 0 6 6 H H 4 H 20

T T e1 9 PBD4 D4 B P

H

is modified as 3 7 7 7 7 7 7 T T 7 0 DK ðtÞB P 7 7 7 7 7 7 7 7 7 7 7 0 0 7 7 0 0 5 0

H

1

C 6B 6 B þe9 E4T E4 þ e3 ðE4T E4 C C 6B 6 B þK T ET E D DT ET E KÞ C C 6B 3 3 4 4 3 3 C 6B 6 B þe3 bE4T E4 þ e6 ðE4T E4 C 0 0 C B 6 C 6B 6 B þK T E3T E3 D4 DT4 E3T E3 KÞ C C B 6Xþ 6 6 B þe bET E þ e1 ET E C 6@ 6 4 4 10 4 4 A 6 T 6 þe1 6 11 E4 E4 6 6 H e10 PBD4 DT4 BT P 0 6 6 H H 0 4 H  XþX2 ;

H

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

0

0 0

H e11 PBD4 DT4 BT P ð44Þ

where b ¼ kM ðDT4 E3T E3 D4 Þ and e9 , e10 and e11 are positive scalars. Then, the fact that ðX þ X2 Þ < 0 is equivalent to X ðX þ X2 ÞX < 0, and we have X ðX þ X2 ÞX ¼ X1 þ X X2 X < 0;

ð45Þ

where X ¼ diagfX ; X ; X ; X g. Then we have the following theorem. Theorem 2. Consider system (1) with given a > 1 and h. the system with controller gain variation (43) is robustly stabilizable if there exist matrices X > 0, W > 0, positive scalars e1 ; e2 ; . . . e10 , matrices Fii P 0ði ¼ 1; . . . ; 4Þ, positive scalars b1 ; b2 , and any matrices Y ; M; Fij ði ¼ 1; . . . ; 4; i < j 6 4Þ which satisfy the following matrix inequalities:

J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

2

N1 6 6H 6 6H 6 6 6H 6 6 6H 6 6H 6 6 6H 6 6 6H 6 6H 4

N2

A1 X  Y þ hF13

h ða  1ÞX

A1 X  Y þ F23

1

H

0

N3 F24 XAT1

1179

T

H

W þ hF33

H

H

hða  1ÞX

0

H

H

H

N7

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

0

0

0

0

0

0

0

0

e2 XE2T

e5 XE2T

e8 XE2T

0

0

0

0

N8

0

0

0

0

e2 I

0

0

0

H

e5 I

0

0

H

H

e8 I

0

H

H

H

N9

H

H

H

H

N10

þ Y þ hF34

N6 0

3

7 0 7 7 0 7 7 7 0 7 7 7 0 7 7<0 0 7 7 7 0 7 7 7 0 7 7 0 7 5

N11

matrix inequalities ð10Þ–ð14Þ;

ð46Þ

where N1 , N2 and N3 are the same with (15), and N6 ¼ ½D1 D2 D3 e10 BD4 ; N7 ¼ diagfe4 I;e5 I;e6 I;e10 Ig; N8 ¼ ½D1 D2 e11 BD4 ; N9 ¼ diagfe7 I;e8 I;e11 Ig; N10 ¼ ½ahY T ahXAT2 D1 D2 D3 e1 XE1T e4 XE1T e7 XE1T e3 M T E3T e6 M T E3T BD4 e9 E4T e3 XE4T e3 M T E3T E3 D4 e3 bE4T e6 XE4T e6 M T E3T E3 D4 e6 bE4T XE4T XE4T ; N11 ¼ diagfahX ;ahX ;e1 I;e2 I;e3 I;e1 I;e4 I;e7 I;e3 I;e6 I;e9 I; e9 I;e3 I;e3 I;e3 bI;e6 I;e6 I;e6 bI;e10 I;e11 Ig: ð47Þ

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Proof. The inequality in (45) is equivalent to (46) by Fact 2. The rest of proof is similar to that of Theorem 1. This completes the proof. h Remark 4. The method presented in this paper can be easily applied to multiplicative controller gain variation of the form DKðtÞ ¼ DF ðtÞEK; where D and E are known matrices and F ðtÞ is an unknown matrix satisfying F T ðtÞF ðtÞ 6 I.

5. Numerical example We demonstrate the applicability of the proposed approach by a example. Consider the following neutral systems (1) with       0:9 0:2 1:1 0:2 0:2 0 A¼ ; A1 ¼ ; A2 ¼ ; 0 0:9 0:1 1:1 0:2 0:1   0 B¼ ; D1 ¼ D2 ¼ I; D3 ¼ 0:1I; 1 E1 ¼ E2 ¼ 0:2I;

E3 ¼ 1;

FiT ðtÞFi ðtÞ 6 I

for i ¼ 1; 2; 3:

For the case of h ¼ 0:5 and a ¼ 2:6, by solving the inequalities of Theorem 1, we found the solutions as     12:9506 14:4042 8:2874 8:7788 ; W ¼ ; X ¼ 14:4042 22:1891 8:7788 12:9182   1:0491 2:3915 Y ¼ ; M ¼ ½ 29:0232 49:2531 ; 1:1702 6:1155     14:1799 15:6437 9:1104 9:4473 F11 ¼ ; F12 ¼ ; 15:6437 24:2074 10:6990 15:6443     10:5045 9:1388 12:3706 12:4273 F13 ¼ ; F14 ¼ ; 13:6963 17:1465 14:9856 21:2286     12:6713 14:0754 6:9249 6:2398 F22 ¼ ; F23 ¼ ; 14:0754 21:7868 8:8066 11:3512     4:8038 5:0393 8:4743 8:9845 F24 ¼ ; F33 ¼ ; 5:6044 8:2670 8:9845 13:1880     9:5492 11:3467 12:8363 14:2733 F34 ¼ ; F44 ¼ ; 9:4074 15:6106 14:2733 22:0351

J.H. Park, O. Kwon / Appl. Math. Comput. 162 (2005) 1167–1182

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Table 1 Control gains in case of certain h h¼1 a ¼ 9:7 K ¼ ½ 5:7582 4:6512  a ¼ 4:2 K ¼ ½ 1:9419 3:3200  a ¼ 2:6 K ¼ ½ 1:0198 2:8281 

h¼2

h¼3

K ¼ ½ 10:1083 5:1341  K ¼ ½ 2:3362 3:2078  Infeasible

K ¼ ½ 14:277 5:2132  Infeasible Infeasible

e1 ¼ 105
6:9474;

e2 ¼ 105
7:0083;

e3 ¼ 106
3:5622;

e4 ¼ 105
7:2417;

e5 ¼ 105
7:2578;

e6 ¼ 106
5:6565;

e7 ¼ 105
7:1855;

e1 8 ¼ 105
7:5637;

b1 ¼ 1=3;

b2 ¼ 1=3:

Therefore, the free parameter G of operator Dðzt Þ and gain matrix K of controller are as follows:   0:1398 0:1985 G¼ ; K ¼ ½ 0:8194 2:7516 : 0:7777 0:7804 For other h and a, the control gains for above system are given in Table 1. 6. Conclusions In this article, the problem of finding a state-feedback controller that asymptotically stabilizes a class of neutral delay-differential systems with uncertainties has been investigated. A delay-dependent criterion expressed in terms of matrix inequalities has been derived using a Lyapunov functional. Moreover, the design method is extended to the systems with controller gain variations. References [1] J. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, SpringerVerlag, New York, 1993. [2] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992. [3] G.D. Hu, G.D. Hu, Some simple stability criteria of neutral delay-differential systems, Applied Mathematics and Computation 80 (1996) 257–271. [4] J.H. Park, S. Won, A note on stability of neutral delay-differential systems, Journal of The Franklin Institute 336 (1999) 543–548. [5] J.H. Park, S. Won, Asymptotic stability of neutral systems with multiple delays, Journal of Optimization Theory and Applications 103 (1999) 187–200. [6] J.H. Park, Stability criterion for neutral differential systems with mixed multiple time-varying delay arguments, Mathematics and Computers in Simulation 59 (2002) 401–412. [7] J.H. Park, Simple criterion for asymptotic stability of interval neutral delay differential systems, Applied Mathematics Letters 16 (2003) 1063–1068.

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[8] K.K. Fan, J.D. Chen, C.H. Lien, J.G. Hsieh, Delay-dependent stability criterion for neutral time-delay systems via linear matrix inequality approach, Journal of Mathematical Analysis and Applications 273 (2002) 580–589. [9] W.B. Ma, N. Adachi, T. Amemiya, Delay-independent stabilization of uncertain linear systems of neutral type, Journal of Optimization Theory and Application 84 (1995) 393–405. [10] J.H. Park, Guaranteed cost stabilization of neutral differential systems with parametric uncertainty, Journal of Computational and Applied Mathematics 151 (2003) 371–382. [11] D. Yue, S. Won, O. Kwon, Delay-dependent stability of neutral systems with time delay: an LMI approach, IEE Proceedings––Control Theory and Applications 150 (2003) 23–27. [12] P. Dorato, Nonfragile controller design: an overview, in: Proceedings of American Control Conference, Philadelphia, Pennsylvania, 1998, pp. 2829–2831. [13] B. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, 1994. [14] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of 39th IEEE CDC, Sydney, Australia, 2000, pp. 2805–2810. [15] P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI Control Toolbox UserÕs Guide, The Mathworks, Natick, MA, 1995.