Robust H∞ filter design for uncertain fuzzy neutral systems

Information Sciences 179 (2009) 3697–3710

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Robust H1 filter design for uncertain fuzzy neutral systems q Jun Yang a,b,*, Shouming Zhong b, Guihua Li a, Wenpin Luo c a b c

School of Computer Science, Civil Aviation Flight University of China, Guanghan, Sichuan 618307, PR China School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China College of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, PR China

a r t i c l e

i n f o

Article history: Received 29 April 2008 Received in revised form 17 June 2009 Accepted 22 June 2009

Keywords: Takagi–Sugeno (T–S) fuzzy systems Neutral systems Uncertain systems H1 filtering Lyapunov–Krasovskii functional Linear matrix inequalities (LMIs)

a b s t r a c t This paper investigates the problem of robust H1 filtering for a class of uncertain Takagi–Sugeno (T–S) fuzzy neutral systems with time-varying delay and linear fractional parametric uncertainties. The aim is to design an asymptotically stable fuzzy filter ensuring asymptotical stability and prescribed H1 performance of the filtering error system for all admissible uncertainties. Based on the Lyapunov–Krasovskii functional, the Barbalat lemma and the LMI approach, delay-dependent sufficient conditions for the solvability of this problem are proposed in terms of LMIs, and an explicit expression of the desired H1 fuzzy filter is readily given when these LMIs are feasible. Finally, the H1 filter design problem for a neutral-type nonlinear distributed network (long line with tunnel diode) is considered in the simulation example to demonstrate the effectiveness and applicability of the proposed method. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction The H1 filtering problem is to design an estimator to estimate the unknown state combination via output measurement, which guarantees that the L2 -induced gain from the external disturbance to the estimation error is less than a prescribed level [2,11,23,24,31]. In contrast with the well-known Kalman filter, one of the main advantages of the H1 filtering is that it is not necessary to know exactly the statistical properties of the external disturbance, only assuming that the external disturbance has bounded energy. This advantage renders the H1 filtering approach very appropriate to some practical applications (see, e.g. [7,16,29]). In addition, the parameter uncertainty that is the source of instability or degradation of control performance often arises in a filter system, so recently the robust H1 filtering problem has been extensively studied (see, e.g. [17,31,32,40]). The T–S fuzzy model [34] has been shown to be a powerful tool for modeling complex nonlinear system. It is well-known that, by means of the T–S fuzzy model, a nonlinear system can be represented by a weighted sum of some simple linear subsystems and then can be stabilized by a model-based fuzzy control. Therefore, it provides a good opportunity to employ the well-established theory of linear system to investigate the complex nonlinear system. Over the past two decades, many issues related to stability analysis and control synthesis of T–S fuzzy systems have been reported (see, e.g. [4,18,22,25,35,48]). On the other hand, as a source of instability, time delay is often encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems [12]. Recently, the T–S fuzzy system with time delay was introduced in [5]. During the past two decades, the study of the time-delay T–S fuzzy system has received much attention

q

This work was supported by the National Natural Science Foundation of China under Grant 60736029. * Corresponding author. Address: School of Computer Science, Civil Aviation Flight University of China, Guanghan, Sichuan 618307, PR China. E-mail address: [email protected] (J. Yang).

0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.06.024

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(see, e.g. [10,30,38,42,45,46,49]). The most conspicuous of these works is the H1 filter design for T–S fuzzy delayed system (see, e.g. [10,42,45,49]). Moreover, it is also well-known that many practical delayed processes can be modeled as general neutral systems, which contain delays both in their states and in the derivatives of their states, such as circuit analysis, computer aided design, realtime simulation of mechanical systems, power systems, chemical process simulation, optimal control [19]; in particular, some practical delayed nonlinear processes have been modeled as general nonlinear neutral systems, such as vacuum-tube oscillation [27], the car-following problem [14], distributed networks – long lines with tunnel diodes [20], the dynamics of the growth of capital stock [9], and nonlinear fluid dynamics [1]. So the stability and stabilization analysis of (nonlinear) neutral systems have recently been extensively investigated (see, e.g. [8,13,15,26,33]). As mentioned above, with the T–S fuzzy model, a nonlinear neutral system can be represented as a weighted sum of some simple linear neutral subsystems; then it provides a good chance to make use of the well-established theory of linear neutral systems to investigate the complex nonlinear neutral systems. So the T–S fuzzy neutral system was recently introduced in [41], where both the stabilization and H1 control problems were studied by the LMI approach. In [44], via a descriptor system approach introducing free matrices into the original systems, a generalized delay-dependent sufficient condition for T–S fuzzy neutral systems to achieve H1 disturbance attenuation was given. Utilizing the Lyapunov–Krasovskii functional and the LMI approach, Li and Xu [21] has investigated robust H1 control for uncertain T–S fuzzy neutral systems with both discrete and distributed delays. Based on the Lyapunov–Krasovskii functional, the descriptor system approach and the LMI approach, Yang et al. [43] presented sufficient conditions for solvability of non-fragile H1 control problem for a class of uncertain fuzzy neutral systems. However, to the best of the authors’ knowledge, the robust H1 filtering problem has not been addressed for T–S fuzzy neutral systems with time-varying delay and linear fractional parametric uncertainties, which motivates the present study. This paper aims to design an asymptotically stable fuzzy filter ensuring asymptotical stability and prescribed H1 performance of the filtering error system for all admissible uncertainties. The main contribution of this paper lies in the following aspects. First, based on the Lyapunov–Krasovskii functional, the Barbalat lemma and the LMI approach, delay-dependent sufficient conditions for solvability of the robust H1 filtering problem are obtained. It is worth mentioning that it is of great advantage to use the Barbalat lemma instead of the operator method (a very popular approach to stability analysis and control synthesis for general neutral systems; see, e.g. [12]) to deal with the H1 filter problem for T–S fuzzy neutral systems, since a strict requirement that the fuzzy weighting functions be differentiable with bounded derivatives (this requirement is inevitable when the operator method is involved) has been overcome via the Barbalat lemma. Second, as an application of these theoretical results, the H1 filtering problem for a nonlinear neutral-type distributed network (long line with tunnel diode) is investigated by the T–S approach proposed in this paper. The rest of this paper is organized as follows. The main problem is formulated in Section 2, and sufficient conditions for the solvability of the robust H1 filtering problem are derived in Section 3. A simulation example is provided in Section 4, and some concluding remarks appear in Section 5. Notations. Rn and Rnm denote, respectively, the n-dimensional Euclidean space and the set of all n  m real matrices. A > ðPÞB means that A  B is positive (semi-positive) definite. I is the identity matrix with the appropriate dimensions. A represents the sum of the A and its transpose. ‘‘*” denotes the eleA1 ðAT ) denotes the inverse (transpose) of the matrix A. ~ ments below the main diagonal of a symmetric block matrix. k  k denotes the Euclidean norm in Rn . L2 ½0; 1Þ denotes the space of square integrable functions on ½0; 1Þ and k  k2 the L2 -norm. Cð½s; 0; Rn Þ is the family of continuous functions / from the interval ½s; 0 to Rn with the norm k/ks ¼ sups6h60 k/ðhÞk.

2. Problem formulation In this section, a class of uncertain T–S fuzzy neutral systems is considered. For each i ¼ 1; 2; . . . ; r (r is the number of plant rules), the ith rule of the T–S fuzzy model is represented as follows: Plant Rule i: IF n1 ðtÞ is M i1 ; n2 ðtÞ is M i2 ; . . . ; np ðtÞ is M ip , THEN

_  sðtÞÞ þ Ei wðtÞ; _ xðtÞ ¼ Ai ðtÞxðtÞ þ Bi ðtÞxðt  sðtÞÞ þ C i ðtÞxðt yðtÞ ¼ Di ðtÞxðtÞ þ F i ðtÞxðt  sðtÞÞ þ Hi wðtÞ; zðtÞ ¼ Li xðtÞ; xðtÞ ¼ /ðtÞ;

t P 0;

ð1Þ

t P 0;

ð2Þ

t P 0;

ð3Þ

t 2 ½s; 0;

ð4Þ n

where n1 ðtÞ; n2 ðtÞ; . . . ; np ðtÞ are the premise variables, and each M il ðl ¼ 1; 2; . . . ; pÞ is a fuzzy set. xðtÞ 2 R is the state; yðtÞ 2 Rm is the measured output; zðtÞ 2 Rp is the signal to be estimated; wðtÞ 2 Rq is the exogenous disturbance input, which is assumed to be an arbitrary signal in L2 ½0; 1Þ; /ðtÞ 2 Cð½s; 0; Rn Þ is the initial condition, and sðtÞ is the time-varying delay satisfying

0 6 sðtÞ 6 s < 1:

ð5Þ

Moreover, Ei ; Hi and Li are known real constant matrices; Ai ðtÞ; Bi ðtÞ; C i ðtÞ; Di ðtÞ and F i ðtÞ are system matrices with appropriate dimension and admissible linear fractional parametric uncertainties, that is, these system matrices satisfy

J. Yang et al. / Information Sciences 179 (2009) 3697–3710

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½Ai ðtÞ Bi ðtÞ C i ðtÞ Di ðtÞ F i ðtÞ ¼ ½Ai þ DAi ðtÞ; Bi þ DBi ðtÞ; C i þ DC i ðtÞ; Di þ DDi ðtÞ; F i þ DF i ðtÞ ¼ ½Ai Bi C i Di F i  þ Li DðtÞ½N0i N1i N2i N3i N4i ;

DðtÞ ¼ ½I  EðtÞJ1 EðtÞ;

ð6Þ ð7Þ

T

I  JJ > 0;

ð8Þ

where Ai ; Bi ; C i ; Di ; F i ; Li ; N 0i ; N 1i ; N 2i ; N 3i ; N 4i and J are known real constant matrices with appropriate dimensions, and EðtÞ is a matrix function satisfying

EðtÞET ðtÞ 6 I:

ð9Þ

Remark 1. The time delay satisfying (5) may be either constant or time-varying, and either differentiable or nondifferentiable. Remark 2. The uncertainty DðtÞ satisfying (7)–(9) is referred to as a linear fractional parametric uncertainty. The uncertainty in (7) is well defined, since the condition (8) guarantees that I  EðtÞJ is invertible for all EðtÞ satisfying (9). This class of uncertainties has been extensively investigated in the existing literatures (see, e.g. [48] and the references therein). Note that when J ¼ 0, DðtÞ reduces to the well-known norm-bounded parametric uncertainty. Applying a center-average defuzzier, product inference and singleton fuzzifier, the dynamic fuzzy model in (1)–(3) can be represented by

_ xðtÞ ¼

r X

_  sðtÞÞ þ Ei wðtÞg; hi ðnðtÞÞfAi ðtÞxðtÞ þ Bi ðtÞxðt  sðtÞÞ þ C i ðtÞxðt

ð10Þ

hi ðnðtÞÞfDi ðtÞxðtÞ þ F i ðtÞxðt  sðtÞÞ þ Hi wðtÞg;

ð11Þ

hi ðnðtÞÞLi xðtÞ;

ð12Þ

i¼1 r X

yðtÞ ¼

i¼1

zðtÞ ¼

r X i¼1

where

Qp M il ðnl ðtÞÞ Qp hi ðnðtÞÞ ¼ Pr l¼1 ; i¼1 l¼1 M il ðnl ðtÞÞ

i ¼ 1; . . . ; r

ð13Þ

in which Mil ðnl ðtÞÞ is the grade of membership of nl ðtÞ in Mil , and nðtÞ ¼ ðn1 ðtÞ; . . . ; nr ðtÞÞ. For convenience, call hi ðnðtÞÞ ði ¼ 1; 2; . . . ; rÞ the fuzzy weighting functions. It follows from (13) that

hi ðnðtÞÞ P 0 ði ¼ 1; 2; . . . ; rÞ and

r X

hi ðnðtÞÞ ¼ 1:

ð14Þ

i¼1

For notational simplicity, hi is used to represent hi ðnðtÞÞ in the following description. Now, consider a full-order fuzzy filter as follows: Filter rule i: IF n1 ðtÞ is M i1 ; n2 ðtÞ is M i2 ; . . . ; np ðtÞ is M ip , THEN

^x_ ðtÞ ¼ Afi ^xðtÞ þ Bfi ^xðt  sðtÞÞ þ C fi ^x_ ðt  sðtÞÞ þ K fi yðtÞ; ^zðtÞ ¼ Lfi ^xðtÞ; ^xðsÞ ¼ 0;

ð15Þ

s 2 ½s; 0;

where i ¼ 1; 2; . . . ; r; ^ xðtÞ 2 Rn and ^zðtÞ 2 Rp ; Afi ; Bfi ; C fi ; K fi and Lfi are filter gain matrices to be determined. Via the fuzzy weighting functions defined by (13), the overall fuzzy filter can be represented by

P ^x_ ðtÞ ¼ hi fAfi ^xðtÞ þ Bfi ^xðt  sðtÞÞ þ C fi ^x_ ðt  sðtÞÞ þ K fi yðtÞg; r

^zðtÞ ¼

i¼1 r P

ð16Þ hi Lfi ^xðtÞ:

i¼1

Let

xf ðtÞ ¼ ½xT ðtÞ; ^xT ðtÞT ;

zf ðtÞ ¼ zðtÞ  ^zðtÞ;

then the filtering error dynamics from systems (10)–(12) and (16) can be represented by

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x_ f ðtÞ ¼

ðRÞ :

r X r X i¼1

zf ðtÞ ¼

r X r X i¼1

hi hj fAij ðtÞxf ðtÞ þ Bij ðtÞxf ðt  sðtÞÞ þ Cij ðtÞx_ f ðt  sðtÞÞ þ Eij wðtÞg;

ð17Þ

hi hj Lij xf ðtÞ;

ð18Þ

j¼1

j¼1

where

Aij ðtÞ ¼ Aij þ DAij ðtÞ; Bij ðtÞ ¼ Bij þ DBij ðtÞ; Cij ðtÞ ¼ Cij þ DCij ðtÞ;       Ai Bi 0 DAi ðtÞ 0 0 Aij ¼ ; DAij ðtÞ ¼ ; ; Bij ¼ K fj Di Afj K fj DDi ðtÞ 0 K fj F i Bfj       Ci 0 DBi ðtÞ 0 DC i ðtÞ 0 DBij ðtÞ ¼ ; DCij ðtÞ ¼ ; Cij ¼ ; K fj DF i ðtÞ 0 0 C fj 0 0   Ei ; Lij ¼ ½ Li Lfj : Eij ¼ K fj Hi The robust H1 filtering problem addressed in this paper is formulated as follows: given the uncertain T–S fuzzy neutral system in (1)–(4) and a prescribed level of noise attenuation c > 0, design an asymptotically stable fuzzy filter in the form of (15) such that, for all admissible uncertainties, (i) when w  0, the filtering error system ðRÞ is robustly asymptotically stable; (ii) for a prescribed scalar c > 0, under zero initial conditions, ðRÞ satisfies

kzf k2 6 ckwk2

ð19Þ

for any nonzero w 2 L2 ½0; 1Þ. The filtering error system ðRÞ is said to be robustly asymptotically stable with disturbance attenuation level c if (i) and (ii) are satisfied. 3. Main results In this section, an LMI approach is developed to solve the robust H1 filtering problem for the uncertain T–S fuzzy neutral systems formulated in the previous section. Before proceeding, recall the following lemmas which will be used throughout the proofs. Lemma 1 [37]. For the fuzzy weighting functions hi ðnðtÞÞði ¼ 1; 2; . . . ; rÞ defined by (13), any vector variable fðtÞ and matrices Mij ði; j ¼ 1; 2; . . . ; rÞ with appropriate dimension, let D

VðtÞ ¼

r X r X i¼1

hi ðnðtÞÞhj ðnðtÞÞfT ðtÞMij fðtÞ;

ð20Þ

j¼1

then VðtÞ < 0; 8fðtÞ–0; if

Mii < 0;

i ¼ 1; 2; . . . ; r;

and

1 1 Mii þ ðMij þ Mji Þ < 0; r1 2

1 6 i < j 6 r:

Lemma 2 [39] (Barbalat lemma). If a function f ðtÞ is uniformly continuous and limt!1 limt!1 f ðtÞ ¼ 0.

Rt 0

f ðsÞds exists and is finite, then

Lemma 3 [6]. Suppose that matrices Mi 2 Rnm ; i ¼ 1; 2; . . . ; r, and a semi-positive-definite matrix P 2 Rnn are given, then r X i¼1

!T hi ðnðtÞÞMi

P

r X j¼1

! hj ðnðtÞÞMj

6

r X

hi ðnðtÞÞMTi PMi ;

i¼1

where hi ðnðtÞÞ; i ¼ 1; 2; . . . ; r; are defined by (13). Lemma 4 ([47,48]). Suppose that DðtÞ is given by (7)–(9), and matrices M ¼ M T ; L and N of appropriate dimensions are given. Then the following statements are equivalent:

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J. Yang et al. / Information Sciences 179 (2009) 3697–3710

(i) the inequality

M þ LDðtÞN þ NT DT ðtÞLT < 0 holds for all EðtÞ satisfying (9); (ii) for any scalar d > 0,

2

3 NT 7 dJ T 5 < 0:

M dL 6 4  dI 



dI

Theorem 1. For a prescribed scalar c > 0, the filtering error system ðRÞ is robustly asymptotically stable with disturbance attenuation level c if there exist matrices P > 0; X lk ð1 6 l 6 k 6 3Þ such that the following conditions are satisfied

2

X 11

X 12

6 X¼4 

X 22





X 13

3

7 X 23 5 P 0;

ð21Þ

X 33

Mii ðtÞ < 0; i ¼ 1; 2; . . . ; r; 1 Mij ðtÞ þ M ji ðtÞ M ii ðtÞ þ < 0; r1 2

ð22Þ 1 6 i < j 6 r;

ð23Þ

where

2 6 6 6 6 M ij ðtÞ ¼ 6 6 6 6 4

M 11 ij ðtÞ

M12 ij ðtÞ

 

sX 22  X 23

!



ATij ðtÞP

ATij ðtÞP þ PCij ðtÞ

BTij ðtÞP sX 33  2P

BTij ðtÞP P þ PCij ðtÞ







 PCij ðtÞ







3

7 7 7 7 7; 7 7 7 PEij ðtÞ 5 0 PEij ðtÞ

!



PEij ðtÞ

1 6 i 6 j 6 r;

c2 I

with !

!

T M11 ij ðtÞ ¼ PAij ðtÞ þsX 11 þ X 13 þLij Lij ;

T M 12 ij ðtÞ ¼ PBij ðtÞ þ sX 12  X 13 þ X 23 :

Proof. Choose a Lyapunov–Krasovskii functional candidate for the system ðRÞ as follows:

VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ;

ð24Þ

where

V 1 ðtÞ ¼ xTf ðtÞPxf ðtÞ; 2 3T 2 xf ðhÞ X 11 Z t Z h 6 7 6 V 2 ðtÞ ¼ 4 xf ðh  sðhÞÞ 5 4  0 hsðhÞ x_ f ðsÞ  Z 0Z t V 3 ðtÞ ¼ x_ Tf ðsÞX 33 x_ f ðsÞdsdh: s

X 12 X 22 

X 13

32

xf ðhÞ

3

76 7 X 23 54 xf ðh  sðhÞÞ 5dsdh; x_ f ðsÞ X 33

tþh

The time derivative of VðtÞ along the trajectory of the system ðRÞ is given by

_ VðtÞ ¼ V_ 1 ðtÞ þ V_ 2 ðtÞ þ V_ 3 ðtÞ:

ð25Þ

Differentiating V 1 ðtÞ along ðRÞ yields

V_ 1 ðtÞ ¼ 2xTf ðtÞPx_ f ðtÞ ¼

r X r X i¼1

!

hi hj fxTf ðtÞ PAij ðtÞ xf ðtÞ þ 2xTf ðtÞP½Bij ðtÞxf ðt  sðtÞÞ þ Cij ðtÞx_ f ðt  sðtÞÞ þ Eij wðtÞg:

ð26Þ

j¼1

Taking the derivative of V 2 ðtÞ gives

V_ 2 ðtÞ ¼

Z

2 t

xf ðtÞ

3T 2

X 11

X 12

X 13

32

xf ðtÞ

3

! 76 7 X 23 54 xf ðt  sðtÞÞ 5ds 6 xTf ðtÞðsX 11 þ X 13 Þxf ðtÞ x_ f ðsÞ  X 33 Z t ! x_ Tf ðsÞX 33 x_ f ðsÞds; þ 2xTf ðtÞðsX 12  X 13 þ X T23 Þxf ðt  sðtÞÞ þ xTf ðt  sðtÞÞðsX 22  X 23 Þxf ðt  sðtÞÞ þ

6 7 6 4 xf ðt  sðtÞÞ 5 4  tsðtÞ x_ f ðsÞ 

X 22

ts

ð27Þ

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where the above inequality holds from the fact (5). V_ 3 ðtÞ is given by

V_ 3 ðtÞ ¼ sx_ Tf ðtÞX 33 x_ f ðtÞ 

Z

t ts

x_ Tf ðsÞX 33 x_ f ðsÞds:

ð28Þ

On the other hand, the following equality holds from (17):

½x_ Tf ðtÞ  x_ Tf ðt  sðtÞÞ2P

( r X r X i¼1

) hi hj ½x_ f ðtÞ þ Aij ðtÞxf ðtÞ þ Bij ðtÞxf ðt  sðtÞÞ þ Cij ðtÞx_ f ðt  sðtÞÞ þ Eij wðtÞ

¼ 0:

ð29Þ

j¼1

Using Lemma 3 twice, the following inequality holds: r X r X

zTf ðtÞzf ðtÞ ¼

i¼1

!T hi hj Lij xf ðtÞ

j¼1

r X r X

! hl hk Llk xf ðtÞ

6

r X r X i¼1

l¼1 k¼1

hi hj xTf ðtÞLTij Lij xf ðtÞ:

ð30Þ

j¼1

_ Adding the terms on the left side of (29) to VðtÞ, it follows from (25)–(28) and (30) that

_ VðtÞ þ zTf ðtÞzf ðtÞ  c2 wT ðtÞwðtÞ 6

r X r X i¼1

hi hj fT ðtÞM ij ðtÞfðtÞ;

ð31Þ

j¼1

where

fðtÞ ¼ ½xTf ðtÞ; xTf ðt  sðtÞÞ; x_ Tf ðtÞ; x_ Tf ðt  sðtÞÞ; wT ðtÞT : Taking into account (22), (23) and (31), it follows from Lemma 1 that

_ VðtÞ þ zTf ðtÞzf ðtÞ  c2 wT ðtÞwðtÞ < 0:

ð32Þ

When wðtÞ ¼ 0, (32) implies that

_ VðtÞ < 0;

8fðtÞ–0;

so

_ VðtÞ 6 ekxf ðtÞk2

e > 0. Integrating both sides of the above inequality from 0 to t gives

for a sufficiently small

VðtÞ 6 Vð0Þ  e

Z

t

kxf ðsÞk2 ds

0

and thus

Z

t

kxf ðsÞk2 ds 6

0

therefore, limt!1

Rt 0

Vð0Þ

e

;

kxf ðsÞk2 ds exists and is finite. By Lemma 2 (Barbalat Lemma), one has

limt!1 kxf ðtÞk ¼ 0: Therefore, the system (R) is robustly asymptotically stable when wðtÞ ¼ 0. When wðtÞ–0, introduce

JðTÞ ¼

Z

T

½zTf ðtÞzf ðtÞ  c2 wT ðtÞwðtÞdt;

0

where the scalar T > 0. Noting the zero initial conditions, it can be verified that for any nonzero w 2 L2 ½0; 1Þ and T > 0,

JðTÞ ¼

Z 0

T

_ ½VðtÞ þ zTf ðtÞzf ðtÞ  c2 wT ðtÞwðtÞdt  VðTÞ 6

Z 0

T

_ ½VðtÞ þ zTf ðtÞzf ðtÞ  c2 wT ðtÞwðtÞdt:

It thus follows from (32) that JðTÞ < 0 (8 T > 0), which implies that kzf k2 6 ckwk2 for any nonzero w 2 L2 ½0; 1Þ. The proof is complete.  In what follows, based on the LMI approach, we can now give the main results on the solvability of the robust H1 fuzzy filtering problem. Theorem 2. For a prescribed scalar c > 0, the robust H1 filtering problem for the uncertain T–S fuzzy neutral system in (1)–(4) is solvable if there exist scalars dij > 0 ð1 6 i 6 j 6 rÞ and matrices Z > 0; Y > 0; Xlk ð1 6 l 6 k 6 3Þ; Aj ; Bj ; Cj ; Kj ; Lj ðj ¼ 1; 2; . . . ; rÞ, with

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J. Yang et al. / Information Sciences 179 (2009) 3697–3710

" Xlk ¼

X11 lk

X12 lk

X21 lk

X22 lk

# ð1 6 l 6 k 6 3Þ;

;

such that the following LMIs hold:

2

X11 6 X¼4  

X12 X22 

X13

3

7 X23 5 P 0; X33

ð33Þ

Z  Y > 0;

ð34Þ

Nii < 0;

ð35Þ

i ¼ 1; 2; . . . ; r;

and

2 6 6 6 6 6 6 6 6 6 6 Nij ðrÞ ¼ 6 6 6 6 6 6 6 6 6 4

N11 N12 N13 N14 ij ðrÞ ij ðrÞ ij ðrÞ ij ðrÞ

N15 ij ðrÞ

N22 N23 N24 ij ðrÞ ij ðrÞ ij ðrÞ



N16 N17 ij ðrÞ ij ðrÞ

0



N33 N34 ij ðrÞ ij ðrÞ







44 ij ðrÞ









45 ij ðrÞ r 2  r1 I

    

    

    

    

    



N

N19 Nij1;10 ðrÞ ij ðrÞ

N26 ij ðrÞ

0

0

0

0

0

N37 ij ðrÞ

0

0

0

47 ij ðrÞ

N35 ij ðrÞ N

N18 ij ðrÞ

46 ij ðrÞ

0

0

0

0

0

0

0

0

dij I    

N67 ij

0 0 ðr  1ÞI  

0 0 0 2I 

0 0 0 0 2I

N

c

N

dij I   

3 7 7 7 7 7 7 7 7 7 7 7 < 0; 7 7 7 7 7 7 7 7 5

1 6 i < j 6 r;

ð36Þ where

2 6 6 6 6 6 6 6 6 6 6 Nij ¼ 6 6 6 6 6 6 6 6 6 4

N11 N12 N13 N14 ij ij ij ij 

N15 ij

N16 ij

0

N18 ij

N19 ij

0

N26 ij

0

0

0

N33 N34 ij ij

N35 ij

0

0

N38 ij

N39 ij

N44 ij

N45 ij

0

N47 ij

N48 ij

N49 ij

0

N22 N23 N24 ij ij ij











2

 

 

 

 

c I 

0 dij I

0 0

N68 ij

0 0

   

   

   

   

   

   

dij I   

0 dij I  

dij J T 0 dij I 

Nij1;10

3

7 0 7 7 7 0 7 7 7 0 7 7 0 7 7; 0 7 7 7 0 7 7 7 0 7 7 0 5

I

with

2

N

11 ij

N12 ij N14 ij N13 ij N16 ij N19 ij N23 ij

3 ! YAi þ ATi X þ DTi KTj þ ATj 5 þ sX11 þ X13 ;  !  ZAi þ Kj Di   YBi YBi þ sX12  X13 þ X32 ; ¼ ZBi þ Kj F i þ Bj ZBi þ Kj F i " # ðATi Z þ DTi Kj T þ ATj Þ þ YC i ATi Y þ YC i ; ¼ ATi Y þ Z T C i þ Cj ðATi Z þ Di T KTj Þ þ ZC i " T #   Ai Y ATi Z þ DTi KTj þ ATj YEi 15 45 ; ¼ N ¼ ¼ N35 ij ij ¼ Nij ; T T ZEi þ Kj Hi Ai Y Ai Z þ DTi KTj " #   YLi 0 NT0i NT3i 18 48 ; ¼ dij N ¼ ¼ N38 ij ij ¼ Nij ; ZLi Kj Li NT0i NT3i " #   ! LTi  LTj YLi 1;10 39 49 ; N22 ¼ Nij ¼ Nij ; Nij ¼ ¼ ij ¼ sX22  X23 ; T ZLi Li " T # " # Bi Y BTi Z þ F Ti KTj þ BTj NT1i NT4i 24 26 ¼ Nij ; Nij ¼ dij ; ¼ BTi Y BTi Z þ F Ti KTj NT1i NT4i ¼4

!

YAi

1 6 i 6 j 6 r;

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N33 ij ¼

 2

4 N44 ij ¼

2Y

2Y



2Z !



þ sX33 ;

N34 ij ¼

 YC i

ðYC i þ C Ti Z þ CTj Þ



 ZC i

!



3 5;

Y þ YC i

Y þ YC i

Y þ ZC i þ Cj Z þ ZC i " # NT2i ; N47 ¼ d ij ij NT2i

 ;

T T N68 ij ¼ diagfdij J ; dij J g;

and

mn

mn

Nij þ Nji 1 Nmn ; ð1 6 i < j 6 r; m ¼ 1; . . . ; 4; n ¼ 1; . . . ; 5Þ; ii þ r 1 2 " T # N0i NT3i 0 NT0j N T3j 0 N16 ðrÞ ¼ d ; ij ij NT0i NT3i 0 NT0j N T3j 0 " #  1  1 1 þ 12 YLi 0 þ 12 YLi 12 YLj 0 YLj 17 47 r1 r1 2 1  1  ¼ N37 Nij ðrÞ ¼  1 1 ij ðrÞ ¼ Nij ðrÞ; þ 2 ZLi Ki þ 12 Kj Li þ 12 ZLi 12 ZLj 12 Ki Lj 12 ZLj r1 r1 r1 " # " T # Li  LTj LTi  LTi 18 19 ; Nij ðrÞ ¼ ; Nij ðrÞ ¼ LTi LTi " T # " T # N1i NT4i 0 N T1j NT4j 0 Lj  LTi 26 ; ; N1;10 ðrÞ ¼ N ðrÞ ¼ d ij ij ij LTj NT1i NT4i 0 N T1j NT4j 0 " # 0 0 NT2i 0 0 NT2j T T 46 ; N67 Nij ðrÞ ¼ dij ij ¼ diagfdij J ; . . . ; dij J g: 0 0 NT2i 0 0 NT2j Nmn ij ðrÞ ¼

Furthermore, the H1 fuzzy filter is given in the form of (15), with the following parameters:

½Afj ; Bfj ; C fj  ¼ S1 ½Aj ; Bj ; Cj Y 1 W T ;

K fj ¼ S1 Kj ;

Lfj ¼ Lj Y 1 W T ;

j ¼ 1; 2; . . . ; r;

ð37Þ

where S and W are any nonsingular matrices satisfying

SW T ¼ I  ZY 1 :

ð38Þ

Proof. Let

    Li 0 0 e e ðtÞ ¼ DðtÞ L ij ¼ ; H ¼ ½I ; 1 6 i 6 j 6 r; D 0 K fj Li DðtÞ 0       Li e 0i ¼ N0i ; N e 1i ¼ N 1i ; e N Li ¼ ; i ¼ 1; 2; . . . ; r; N3i N4i 0

0 ;

then it can be verified that

e ðtÞ½N ~ 0i H; N e 1i H; ½DAij ðtÞ; DBij ðtÞ ¼ e L ij D DCij ðtÞ ¼ eL i DðtÞðN2i HÞ; i ¼ 1; 2; . . . ; r:

1 6 i 6 j 6 r;

Hence (22) in Theorem 1 can be rewritten as

2

X11

6 6 6  6 Mii ðtÞ ¼ 6  6 6 4   2

ATii P

ATij P þ PCii



BTii P sX 33  2P

 

 

BTii P P þ PCii !  PCii 

X12 !

sX 22  X 23

PEii

3

7 7 7 7 7 7 7 PEii 5 0 PEii

c2 I 3 e e P L ii P Li 6 0 " #" # 0 7 7 e 6 e 0i H N e 1i H 0 N 0 0 0 7 D ðtÞ 6 þ 6 Pe L ii Pe Li 7 7 0 6 0 0 0 N2i H 0 DðtÞ 4 P e L ii P e Li 5 0 0 3 2 T ~T H N 0i 0 7" 6 T ~T #" # 0 7 eT 6 H N1i e L Tii P 0 e L Tii P e L Tii P 0 0 7 D ðtÞ 6 þ6 0 < 0; 0 7 7 6 0 DT ðtÞ eL Ti P 0 eL Ti P eL Ti P 0 4 0 HT N T2i 5 0 0

i ¼ 1; 2; . . . ; r;

ð39Þ

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J. Yang et al. / Information Sciences 179 (2009) 3697–3710

where !

!

X12 ¼ PBii þ sX 12  X 13 þ X T23 :

X11 ¼ PAii þsX 11 þ X 13 þLTii Lii ; Let

  EðtÞ 0 e EðtÞ ¼ ; 0 EðtÞ

eJ ¼



J

0

0

J

 ;

then

"

# # #" #!1 "   "e e ðtÞ e eJ 0 I 0 EðtÞ 0 EðtÞ 0 D 0 ¼ ;  0 I DðtÞ 0 0 EðtÞ 0 EðtÞ 0 J " #" #T " #" #T ~ eJ 0 eJ 0 e EðtÞ 0 EðtÞ 0 < I; 6 I: 0 EðtÞ 0 J 0 J 0 EðtÞ

ð40Þ

ð41Þ

Observing (40) and (41), and using Lemma 4 and the Schur complement, one can see that (39) holds if and only if, for scalars dii > 0; i ¼ 1; 2; . . . ; r, the following inequality holds:

2 6 6 6 6 6 6 6 6 6 6 6 6 M ii ¼ 6 6 6 6 6 6 6 6 6 6 6 6 4

M 11 ii

M12 ii

ATii P

M14 ii

PEii

eT dii HT N 0i

0

Pe L ii

Pe Li



M22 ii

BTii P

BTii P

0

~T dii HT N 1i

0

0

0





M 33 ii

M 34 ii

PEii

0

0

Pe L ii

Pe Li







M 44 ii

PEii

0

dii HT N T2i

P e L ii

P e Li









c2 I

0

0

0

0











dii I

0

dii~J T

0













dii I

0

dii J T















dii I

0

















dii I



















LTii

3

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

i ¼ 1; 2; . . . ; r;

ð42Þ

I

where !

!

M 11 ii ¼ PAii þsX 11 þ X 13 ; !

M 22 ii ¼ sX 22  X 23 ;

T M 12 ii ¼ PBii þ sX 12  X 13 þ X 23 ;

M 33 ii ¼ sX 33  2P;

M34 ii ¼ P þ PCii ;

T M14 ii ¼ Aii P þ PCii ; !

M 44 ii ¼  PCii :

On the other hand, (34) implies that I  ZY 1 is nonsingular. Therefore, there always exist nonsingular matrices S and W such that (38) holds. Now, let

"

P1 ¼

Y 1

I

WT

0

# ;



P2 ¼

I

Z



0 ST

and

P ¼ P2 P1 1 ; then

P¼



Z

S

T



S

 ;

where

 ¼ W 1 Y 1 ðZ  YÞY 1 W T : Observe that

Z  S 1 ST ¼ Y > 0; thus it follows from the Schur complement that P > 0. Let

e ¼ diagfY 1 ; Ig; Y

e P1 ; Y e P1 ; Y e P1 g; e ¼ diagf Y P 1 1 1

e TXP e: X¼P

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J. Yang et al. / Information Sciences 179 (2009) 3697–3710

Pre- and post-multiply LMI (35) by T e T e T e e diagfPT 1 Y ; P1 Y ; P1 Y ; P1 Y ; I; I; I; I; I; Ig

and its transpose, respectively; then, observing (37) and the above-introduced matrix variables P and X yielding (42), thus (22) follows. Similarly, LMI (36) implies (23). Therefore, by Theorem 1, the desired results follow immediately. The proof is complete. h Remark 3. To prove the negative-definiteness of the fuzzy summations (20), Proposition 2 in [28] provides a set of sufficient conditions that are progressively less conservative than those proposed in Lemma 1, when the number of plant rules is r P 4 and the complexity parameter is n P 4. However, using Proposition 2 is a sort of a ‘‘not-too-practical” way of obtaining less conservative conditions because the computational cost grows wildly with the complexity parameters there. Therefore, at the cost of dramatically high computational complexity, less conservative conditions can be derived via Proposition 2 than those presented in Theorem 2 by means of Lemma 1. Remark 4. Based on the LMI approach, Theorem 2 establishes delay-dependent sufficient conditions for the solvability of the robust H1 filtering problem for a class of uncertain T–S fuzzy neutral systems with time-varying delay and linear fractional parametric uncertainties. The desired H1 fuzzy filter is readily constructed by solving the LMIs in (33)–(36), which can be implemented by standard numerical algorithm [3]without requiring any tuning of parameters. Remark 5. For the uncertain delayed T–S fuzzy system in (1)–(4) without neutral terms, the robust H1 filtering problem has been investigated in [42]; so, in view of this, the results in this paper to some extent extend the corresponding results in [42].

4. Simulation example In this section, as an application of the theoretical results, the H1 filtering problem for a distributed network (long line with tunnel diode) is considered. The tunnel diode is a two-electrode device on the basis of semiconducting crystals having a very narrow potential barrier hindering the motion of electrons. Such diodes are wildly used in high-frequency amplifiers of electric oscillations and in other devices. The dynamic of the distributed network (long line with tunnel diode) is represented by the following nonlinear neutral functional differential equation (see, e.g. [18]).

_  K xðt _  hÞ ¼ ðaðtÞ  1ÞxðtÞ  ð1 þ aðtÞÞKxðt  hÞ þ ½xðtÞ  Kxðt  hÞ2 þ ewðtÞ; xðtÞ

ð43Þ

with 1

h ¼ 2lCC l ;

Z¼

pffiffiffiffiffiffiffiffiffiffi LC 1 ;

K ¼ ðZ  R0 ÞðZ þ R0 Þ1 ;

a ¼ aZ; aðtÞ ¼ a þ DaðtÞ;

where l is the length of the conductor, L and C are the inductivity and capacity of the conductor per unit length, respectively, R0 is the resistance at the input, C l is the capacity at the output, a > 0 is a scalar, DaðtÞ is the perturbation of the coefficient a, and e is the coefficient of the exogenous disturbance input wðtÞ. In what follows, applying the idea of ‘‘using local sector nonlinearity in T–S fuzzy model construction”[36], the nonlinear system (43) will be represented by the T–S fuzzy system. For simplicity, it is assumed that xðtÞ; xðt  hÞ 2 ½1; 1. Of course, any ranges for xðtÞ and xðt  hÞ can be assumed to construct the T–S fuzzy model. Define nðtÞ ¼ xðtÞ  Kxðt  hÞ; then it follows from (43) that

_  K xðt _  hÞ ¼ ðaðtÞ  1ÞxðtÞ  ð1 þ aðtÞÞKxðt  hÞ þ nðtÞ½xðtÞ  Kxðt  hÞ þ ewðtÞ: xðtÞ Next, calculate the maximum and minimum values of nðtÞ under xðtÞ; xðt  hÞ 2 ½1; 1:

max

xðtÞ2½1;1; xðthÞ2½1;1

nðtÞ ¼ 1 þ K;

min

xðtÞ2½1;1; xðthÞ2½1;1

nðtÞ ¼ ð1 þ KÞ:

From the maximum and minimum values, nðtÞ can be represented by

nðtÞ ¼ M 1 ðnðtÞÞ  ð1 þ KÞ þ M 2 ðnðtÞÞ  ð1  KÞ; where

M1 ðnðtÞÞ þ M 2 ðnðtÞÞ ¼ 1: Therefore, the membership functions are given by

M1 ðnðtÞÞ ¼

nðtÞ 1 þ ; 2ð1 þ KÞ 2

which are shown in Fig. 1.

M 2 ðnðtÞÞ ¼

1 nðtÞ  ; 2 2ð1 þ KÞ

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J. Yang et al. / Information Sciences 179 (2009) 3697–3710

With the membership functions M 1 ðnðtÞÞ and M 2 ðnðtÞÞ, the nonlinear neutral system (43) with (2) and (3) can be represented by the following T–S fuzzy model. Plant Rule 1: IF nðtÞ is M1 , THEN

_ _  hÞ þ E1 wðtÞ; xðtÞ ¼ A1 ðtÞxðtÞ þ B1 ðtÞxðt  hÞ þ C 1 ðtÞxðt yðtÞ ¼ D1 ðtÞxðtÞ þ F 1 ðtÞxðt  hÞ þ H1 wðtÞ;

ð44Þ

zðtÞ ¼ L1 xðtÞ; Plant Rule 2: IF nðtÞ is M2 , THEN

_  hÞ þ E2 wðtÞ; _ xðtÞ ¼ A2 ðtÞxðtÞ þ B2 ðtÞxðt  hÞ þ C 2 ðtÞxðt yðtÞ ¼ D2 ðtÞxðtÞ þ F 2 ðtÞxðt  hÞ þ H2 wðtÞ;

ð45Þ

zðtÞ ¼ L2 xðtÞ; where

A1 ðtÞ ¼ aðtÞ þ K; B1 ðtÞ ¼ ðaðtÞ þ K þ 2ÞK; C 1 ðtÞ ¼ K; E1 ¼ e; A2 ðtÞ ¼ aðtÞ  K  2; B2 ðtÞ ¼ ðK  aðtÞÞK; C 2 ðtÞ ¼ K; E1 ¼ e: Assume aðtÞ ¼ a þ a sinðtÞ , then 5

A1 ðtÞ ¼ ða þ KÞ þ 0:2aDðtÞ;

B1 ðtÞ ¼ ða þ K þ 2ÞK  0:2aK DðtÞ;

A2 ðtÞ ¼ ða  K  2Þ þ 0:2aDðtÞ;

B2 ðtÞ ¼ ðK  aÞK  0:2aK DðtÞ;

Fig. 1. Membership functions M1 ðnðtÞÞ and M 2 ðnðtÞÞ.

0.4

0.2

States

0

−0.2

−0.4

−0.6

−0.8

0

5

10

15

20

25

Time (s)

Fig. 2. State responses xðtÞðÞ and ^ xðtÞð  Þ.

30

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J. Yang et al. / Information Sciences 179 (2009) 3697–3710

where DðtÞ ¼ sinðtÞ. For the convenience of the simulation, set the above parameters to

l ¼ 50 m; L ¼ 200 lH; C ¼ 3600 lF; C l ¼ 540000 lF; R0 ¼ 160 X; e ¼ 0:5;

D1 ðtÞ ¼ 0:6;

F 1 ðtÞ ¼ 0:4;

D2 ðtÞ ¼ 0:5;

F 2 ðtÞ ¼ 0:3;

H2 ¼ 0:2;

a ¼ 0:0025; L1 ¼ 1;

H1 ¼ 0:3; L2 ¼ 1:4:

In this example, the H1 performance level is specified to be c ¼ 0:8. Solving the LMIs in (33)–(36), the feasible solutions are obtained as follows:



 ; 0:3053 2:3934   0:3700 0:2748 ¼ ; 0:2748 2:7766

X11 ¼ X22

0:0828

0:3053

A1 ¼ 0:5893;

B1 ¼ 2:1055;

B2 ¼ 1:5002;

C2 ¼ 0:3438;

 X12 ¼ X23

0:3434

0:4658



; 0:2420 2:9302   0:3779 0:1114 ¼ ; 0:0866 1:5939

C1 ¼ 0:4135;

 X13 ¼

L2 ¼ 1:2722;

0:0530 1:5879  0:0839 0:0161 ¼ ; 0:0161 0:6586

 ;



X33

L1 ¼ 0:8666;

K1 ¼ 2:8736;

K2 ¼ 2:9178;

0:1393

0:3522

X ¼ 1:9908;

A2 ¼ 3:0182; Y ¼ 0:6525:

Therefore, by Theorem 2, the robust H1 filtering problem for system (43) is solvable. In order to construct the desired H1 fuzzy filter, choose

S ¼ 1:4321;

W ¼ 1:4321;

it can be verified that S and W satisfy the equality (38); thus by (37) the fuzzy filter gain matrices are given as follows:

Af 1 ¼ 0:4403;

Bf 1 ¼ 1:5733;

C f 1 ¼ 0:3090;

K f 1 ¼ 2:0065;

Lf 1 ¼ 0:9274;

Af 2 ¼ 0:3652;

Bf 2 ¼ 1:1210;

C f 2 ¼ 0:2569;

K f 2 ¼ 2:0374;

Lf 2 ¼ 1:3614:

0.5 0.45 0.4 0.35

y(t)

0.3 0.25 0.2 0.15 0.1 0.05 0

0

5

10

15

Time (s)

20

25

30

Fig. 3. Measured output yðtÞ.

0

zf(t)

−0.05

−0.1

−0.15

−0.2

−0.25 0

5

10

15

Time (s)

20

Fig. 4. Response of the error signal zf ðtÞ.

25

30

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J. Yang et al. / Information Sciences 179 (2009) 3697–3710 1

0.95

0.9

0.85

0.8

0

2

4

Time (s)

6

8

10

Fig. 5. H1 performance level c.

The simulation results of state responses of the plant and filter are given in Fig. 2, where the initial condition is zero, and the 1 exogenous disturbance input is wðtÞ ¼ 1þt 2 ðt P 0Þ, which belongs to L2 ½0; 1Þ; Figs. 3, 4 show the measured output yðtÞ and response of the error signal zf ðtÞ, respectively; the H1 performance level c is given in Fig. 5. From the simulation results, it follows that the designed H1 fuzzy filter meets the specified requirements. 5. Conclusion In this paper, the problem of robust H1 filtering for a class of uncertain T–S fuzzy neutral systems with time-varying delay and linear fractional parametric uncertainties has been investigated. By means of the Lyapunov–Krasovskii functional, the Barbalat lemma and the LMI approach, delay-dependent sufficient conditions for the solvability of the robust H1 filtering problem have been established, and the desired H1 fuzzy filter is readily constructed by solving a set of LMIs via standard numerical algorithm without requiring any tuning of parameters. Finally, the H1 filter design problem for the nonlinear distributed network (long line with tunnel diode) is considered in a simulation example to demonstrate the effectiveness and applicability of the proposed method. Acknowledgements The authors would like to thank the Editor in Chief and anonymous reviewers for their helpful and insightful comments for further improving the quality of this work. References [1] S.R. Barone, A new approach to some nonlinear fluid dynamics problems, Phys. Lett. A 70 (3) (1979) 260–262. [2] M. Basin, E. Sanchez, R. Martinez-Zuniga, Optimal linear filtering for systems with multiple state and observation delays, Int. J. Innov. Comput. Inform. Contr. 3 (5) (2007) 1309–1320. [3] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA, 1994. [4] S.G. Cao, N.W. Rees, G. Feng, Analysis and design of a class of continuous time fuzzy control systems, Int. J. Contr. 64 (1996) 1069–1087. [5] Y.Y. Cao, P.M. Frank, Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach, IEEE Trans. Fuzzy Syst. 8 (2000) 200–211. [6] Y.Y. Cao, Z.L. Lin, Y. Shamash, Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation, Syst. Cont. Lett. 46 (2) (2002) 137–151. [7] B.S. Chen, C.L. Tsai, Y.F. Chen, Mixed H2 =H1 filtering design in multirate transmultiplexer systems: LMI approach, IEEE Trans. Signal Process. 49 (11) (2001) 2693–2701. [8] W.H. Chen, W.X. Zheng, Delay-dependent robust stabilization for uncertain neutral systems with distributed delays, Automatica 43 (1) (2007) 95–104. [9] E.N. Chukwu, Mathematical controllability theory of capital growth of nations, Appl. Math. Comput. 52 (2-3) (1992) 317–344. [10] C. Gong, B. Su, Robust H1 filtering of convex polyhedral uncertain tim-delay fuzzy systems, Int. J. Innov. Comput. Inform. Contr. 4 (4) (2008) 793–802. [11] M.J. Grimble, A.E. Sayed, Solution of the H1 optimal linear filtering problem for discrete-time systems, IEEE Trans. Acoust. Speech Signal Process. 38 (7) (1990) 1092–1104. [12] J.K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977. [13] Q.L. Han, A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays, Automatica 40 (10) (2004) 1791–1796. [14] J. Harband, The existence of monotonic solutions of a nonlinear car-following equation, J. Math. Anal. Appl. 57 (2) (1977) 257–272. [15] Y. He, M. Wu, J.H. She, G.P. Liu, Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Syst. Contr. Lett. 51 (1) (2004) 57–65. [16] J.C. Hung, B.S. Chen, Genetic algorithm approach to speech fixed-order mixed H2 =H1 optimal deconvolution filter designs, IEEE Trans. Signal Process. 48 (12) (2000) 3451–3461. [17] S.H. Jin, J.B. Park, Robust H1 filtering for polytopic uncertain systems via convex optimisation, IEE Proc. Contr. Theory Appl. 148 (2001) 55–59. [18] E. Kim, H. Lee, New approaches to relaxed quadratic stability condition of fuzzy control systems, IEEE Trans. Fuzzy Syst. 8 (5) (2000) 523–534. [19] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Netherlands, 1992.

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