control design for fuzzy discrete-time singularly ...faculty.neu.edu.cn/ise/dongjiuxiang/English/Papers/Dong_J09i.pdfH1 control design for fuzzy discrete-time singularly perturbed

Information Sciences 179 (2009) 3041–3058

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Information Sciences

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H1 control design for fuzzy discrete-time singularly perturbed systemsvia slow state variables feedback: An LMI-based approach

Jiuxiang Dong, Guang-Hong Yang *

College of Information Science and Engineering, Northeastern University, Shenyang 110004, PR ChinaKey Laboratory of Integrated Automation of Process Industry (Ministry of Education), Northeastern University, Shenyang 110004, PR China

a r t i c l e i n f o

Article history:Received 15 May 2008Received in revised form 6 February 2009Accepted 16 March 2009

Keywords:Takagi–Sugeno (T–S) fuzzy systemsDiscrete-time systemsNonlinear singularly perturbed systemsH1 controlLinear matrix inequalitiesReduced-order control

0020-0255/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.ins.2009.03.012

* Corresponding author. Address: College of Info13504182968; fax: +86 24 83681939.

E-mail addresses: [email protected](G.-H. Yang).

a b s t r a c t

This paper addresses the H1 control problem via slow state variables feedback for discrete-time fuzzy singularly perturbed systems. At first, a method of evaluating the upper boundof singular perturbation parameter � with meeting a prescribed H1 performance boundrequirement is given. Subsequently, two methods for designing H1 controllers via slowstate variables feedback are presented in terms of solutions to a set of linear matrixinequalities (LMIs). In particular, one of them can be used to improve the upper boundof the singular perturbation parameter �. Finally, two numerical examples are given toillustrate the effectiveness of the proposed methods.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Slow and fast dynamic phenomena in control systems often occur due to the presence of small ‘‘parasitic” parameters,such as motor control systems, electronic circuits, magnetic-ball suspension systems, and so on. In a state space framework,such systems are commonly modeled by a mathematical description of singular perturbations, where a small parameter � isexploited to determine the degree of separation between slow and fast parts of the dynamical system. Due to the very smallsingular perturbation parameter �, the analysis and synthesis approaches for normal systems often lead to ill-conditionedresults. Therefore, a so-called reduction technique with a two-step design methodology [17] is widely adopted for overcom-ing the difficulty. Firstly, through the separate stabilization of two lower dimensional subsystems in two different timescales, a composite stabilizing controller is synthesized from separate stabilizing controllers of the two subsystems, wherethe controller could be determined without the knowledge of the small singular perturbation parameter. In the past severaldecades, many control problems of singularly perturbed systems have attracted considerable attentions, see the survey pa-per [23] and the references therein.

In control theory, a well-known convex optimization technique, i.e., linear matrix inequality (LMI) technique, has beenextensively exploited to solve control problems [3]. In contrast to Riccati approaches, linear matrix inequalities (LMIs)can be formulated as convex optimization problems that are amenable to compute solution and can be solved effectively[3]. Another good feature of LMIs is their ability of adding constraints to the parametrical optimization problem provided

. All rights reserved.

rmation Science and Engineering, Northeastern University, Shenyang 110004, PR China. Tel.: +86

, [email protected] (J. Dong), [email protected], [email protected]

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3042 J. Dong, G.-H. Yang / Information Sciences 179 (2009) 3041–3058

they are themselves linear with respect to unknowns [12]. In particular, for H1 synthesis, it has the merit of eliminating theregularity restrictions attached to the Riccati-based solutions [11]. Motivated by the merits of the LMI formulations, someLMI-based controller design approaches for singularly perturbed systems have been developed in [7–10,28] recently.

On the other hand, there has been a great deal of interest in using Takagi–Sugeno (T–S) fuzzy models to approximate non-linear systems, and many control problems of nonlinear systems have been widely studied based on T–S fuzzy systems, see[14,15,26,29,30] and the references therein. In particular, the controller design conditions for the state feedback [19], staticoutput feedback [21], and dynamic output feedback [2] cases are exploited in terms of solutions of LMIs for fuzzy singularlyperturbed systems. Moreover, the methods of designing H1 state feedback controllers with pole placement constraints aregiven in [1]. In most cases, the ‘‘fast” dynamics of singularly perturbed systems are not adequately modeled and are, therefore,neglected in order to simplify the design [24,6]. In practice the fast variables are not directly measurable sometimes, for exam-ple, the flexible variables (modeled as fast variables) of the flexible link manipulators [22]. Therefore, the study on the prob-lem of designing state feedback controllers by only using slow state variables of singularly perturbed systems is necessary.

Because most of the existing synthesis techniques for singularly perturbed systems are independent of � for avoiding toobtain ill-conditioned results, it is of great importance to find the bound of � for ensuring the stability of the closed-loopsystems. As a result, it has attracted increasing interest in the past several decades. In [5], an approach to characterizeand compute the stability bound is presented for continuous-time singularly perturbed systems. By considering criticalstability criteria with a bialternate product, systematic approaches to determine the exact stability bound of discrete-timesingularly perturbed systems are given in [13,18]. Moreover, an algorithm for finding the upper bound of the singular per-turbation parameter for D-stability is presented in [16]. However, by the authors’ knowledge, the topic of evaluating theupper bounds of singular perturbation parameters for nonlinear discrete-time singularly perturbed systems with meetingH1 performance requirements has not been studied. In this paper, the topic will be addressed by using the two lemmas thatare proposed for linear singularly perturbed systems in [8], which are given in Appendices A and B.

In this paper, a method of evaluating the upper bound of the singular perturbation parameter � for discrete-time fuzzysingularly perturbed systems with meeting a prescribed H1 performance bound requirement is given. Furthermore, twoH1 controller design methods via slow state variables feedback are presented in terms of solutions to a set of LMIs. In contrastto the conventional design methods [23], the new design methods are with twofold advantages. One is that the two designmethods are based on LMIs, which can eliminate the regularity restrictions attached to the Riccati-based solution. The otheris that one of the two methods can be used to improve the upper bound of singularly perturbed parameter � at the stage ofcontrol design, which implies that the tradeoff between the H1 performance index and the upper bound of the singularperturbation parameter � is considered in the design. Thus, the new controller design method can overcome the disadvantagethat the allowable upper bound of the singular perturbation parameter of the closed-loop system with the controllersdesigned by the existing ones is too small to be used. This paper is organized as follows. Section 2 presents system descriptionand some preliminaries. In Section 3, a sufficient condition is derived for evaluating the upper bound �� of � subject to a pre-scribed H1 performance constraint. Moreover, two new LMI-based H1 controller design methods are presented. In particular,one of them can improve the upper bound of the singular perturbation parameter � by designing controllers. The validity ofthese approaches is illustrated by two numerical examples in Section 4. Finally, Section 5 concludes the paper.

Notation: Rn denotes the set which consists of real n-vectors (n� 1 matrices). For a symmetric block matrix, (*) is used forthe blocks induced by symmetry, for example,

M11 � �M21 M22 �M31 M32 M33

264375 ¼ M11 MT

21 MT31

M21 M22 MT32

M31 M32 M33

264375

The superscript T stands for matrix transposition and the notation M�T denotes the transpose of the inverse matrix of M.

2. System description and some preliminaries

A class of nonlinear singularly perturbed systems under consideration are described by the following fuzzy system model:

Plant Rule i :

IF v1ðkÞ is Mi1 and v2ðkÞ is Mi2; . . . ; vpðkÞ is Mip; THEN

x1ðkþ 1Þ ¼ Ai11x1ðkÞ þ �Ai

12x2ðkÞ þ Biw1wðkÞ þ Bi

u1uðkÞx2ðkþ 1Þ ¼ Ai

21x1ðkÞ þ �Ai22x2ðkÞ þ Bi

w2wðkÞ þ Biu2uðkÞ

zðkÞ ¼ Ciz1x1ðkÞ þ �Ci

z2x2ðkÞ þ DizwwðkÞ þ Di

zuuðkÞ

ð1Þ

where i ¼ 1;2; . . . ; r, r is the number of IF–THEN rules. Mil, 1 6 i 6 r, 1 6 l 6 p are fuzzy sets. v iðkÞ are the premise variables,x1ðkÞ 2 Rn1 and x2ðkÞ 2 Rn2 are respectively the slow and fast state vectors, uðkÞ 2 Rnu is the control input, wðkÞ 2 Rnw is thedisturbance, zðkÞ 2 Rnz is the controlled output, the matrices Ai

11, Ai12, Ai

21, Ai22, Bi

w1, Biw2, Bi

u1, Biu2, Ci

z1, Ciz2, Di

zw and Dizu are of

appropriate dimensions. � > 0 is a singular perturbation parameter, which determines the degree of separation betweenthe ‘‘slow” and ‘‘fast” modes of the system [1].

J. Dong, G.-H. Yang / Information Sciences 179 (2009) 3041–3058 3043

Denote

-iðvðkÞÞ ¼Yp

j¼1

lijðv jðkÞÞ

lijðv jðkÞÞ is the grade of membership of v jðkÞ in Mij, where it is assumed that
Xr
i¼1

-iðvðkÞÞ > 0; -iðvðkÞÞP 0 i ¼ 1;2; . . . ; r

Let aiðvðkÞÞ ¼-iðvðkÞÞPri¼1-iðvðkÞÞ

, then

0 6 aiðvðkÞÞ 6 1; andXr

i¼1

aiðvðkÞÞ ¼ 1 ð2Þ

aiðvðkÞÞ, i ¼ 1; . . . ; r are said to be normalized membership functions. Then, the T–S fuzzy model of (1) is inferred as follows:

x1ðkþ 1Þ ¼Xr

i¼1

aiðvðkÞÞ Ai11x1ðkÞ þ �Ai


u1uðkÞ� �

x2ðkþ 1Þ ¼Xr

i¼1

aiðvðkÞÞ Ai21x1ðkÞ þ �Ai


u2uðkÞ� �

zðkÞ ¼Xr

i¼1

aiðvðkÞÞ Ciz1x1ðkÞ þ �Ci

z2x2ðkÞ þ DizwwðkÞ þ Di

zuuðkÞ� �

ð3Þ

which can be rewritten as follows:

xðkþ 1Þ ¼ AðaðkÞÞE�xðkÞ þ BwðaðkÞÞwðkÞ þ BuðaðkÞÞuðkÞzðkÞ ¼ CzðaðkÞÞE�xðkÞ þ DzwðaðkÞÞwðkÞ þ DzuðaðkÞÞuðkÞ

ð4Þ

where

AðaðkÞÞ ¼Xr

i¼1

aiðvðkÞÞAi; BwðaðkÞÞ ¼

Xr

i¼1

aiðvðkÞÞBiw;

BuðaðkÞÞ ¼Xr

i¼1

aiðvðkÞÞBiu; CzðaðkÞÞ ¼

Xr

i¼1

aiðvðkÞÞCiz;

DzwðaðkÞÞ ¼Xr

i¼1

aiðvðkÞÞDizw; DzuðaðkÞÞ ¼

Xr

i¼1

aiðvðkÞÞDizu

xðkÞ ¼x1ðkÞx2ðkÞ

� �E� ¼

In1�n1 00 �In2�n2

� �Ai ¼

Ai11 Ai

12

Ai21 Ai

22

" #Bi

w ¼Bi

w1

Biw2

" #

Biu ¼

Biu1

Biu2

" #Ci

z ¼ ½Ciz1Ci

z2�

ð5Þ

In this paper, the concept of parallel distributed compensation (PDC) is used to design fuzzy controllers, i.e., the designedfuzzy controller shares the same fuzzy sets with the fuzzy model in the premise parts. The more details can be found in[25]. For the fuzzy model (1), the following slow state feedback controller is adopted, where Ki

a, 1 6 i 6 r are the parametersto be designed.

Control Rule i :

IF v1ðkÞ is Mi1 and v2ðkÞ is Mi2; . . . ;vpðkÞ is Mip

THEN uðkÞ ¼ Kiax1ðkÞ ð6Þ

Because the control rules are the same as the plant rules, the fuzzy controller can be obtained as follows:

uðkÞ ¼Xr

i¼1

aiðvðkÞÞKiax1ðkÞ ¼

Xr

i¼1

aiðvðkÞÞKi x1ðkÞx2ðkÞ

� �ð7Þ

where

Ki ¼ Kia 0

� �ð8Þ


Combining (7) and (4), then the resulting closed-loop system is given as follows:

xðkþ 1Þ ¼ AðaðkÞÞE� þ BuðaðkÞÞKðaðkÞÞð ÞxðkÞ þ BwðaðkÞÞwðkÞzðkÞ ¼ CzðaðkÞÞE� þ DzuðaðkÞÞKðaðkÞÞð ÞxðkÞ þ DzwðaðkÞÞwðkÞ

ð9Þ

where

KðaðkÞÞ ¼Xr

i¼1

aiðvðkÞÞ Kia 0

� �ð10Þ

The replacement of �x2 by n2 in system (9) will result in the equivalent system

nðkþ 1Þ ¼ E� AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞnðkÞ þ E�BwðaðkÞÞwðkÞzðkÞ ¼ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞnðkÞ þ DzwðaðkÞÞwðkÞ

ð11Þ

where

nðkÞ ¼x1ðkÞn2ðkÞ

� �
The H1 norm in [20] for nonlinear discrete-time systems is applicable for nonlinear discrete-time singularly perturbed sys-tems. The definition is given as follows:
Definition 1 [20]. Given a real number c > 0, it is said that the H1 norm of the closed-loop system (11) is less than or equalto c (i.e., the exogenous signals are locally attenuated by c) if there exists a neighborhood U of x ¼ 0 such that for everypositive integer N and for every w 2 l2ð½0;NÞ;Rnw Þ for which the state trajectory of the closed-loop system (11) startingxð0Þ ¼ 0 remains in U for all k 2 ½0;N�, the response z 2 l2ð½0;N�;Rnz Þ of (11) satisfies
XN
i¼0

kzkk26 c2

XN

i¼0

kwkk2; for all N

In this paper, the following problems will be addressed.

2.1. Evaluation of the upper bound of � with meeting stability and H1 performance bound requirement

Let c > 0 be a given constant and the gains Ki be given. Find an �� > 0 as big as possible such that the system (1) with (7) isasymptotically stable and its H1-norm is less than or equal to c for any � 2 ð0; ��.

2.2. H1 controller designs without the consideration of improving the upper bound of �

Let c > 0 be a given constant. Find gains Ki ði ¼ 1; . . . ; rÞ, and there exists a positive scalar ��, such that the system (1) with(7) is asymptotically stable and its H1-norm is less than or equal to c for any � 2 ð0; ��.

2.3. H1 controller design with the consideration of improving the upper bound of �

Let c > 0 be a given constant and �� > 0 be a prescribed upper bound of the singular perturbation parameter �. Find gainsKi

a ði ¼ 1; . . . ; rÞ and an �� > 0 such that the system (1) with (7) is asymptotically stable and its H1-norm is less than or equalto c for any � 2 ð0; ��.

The following lemmas will be used in this sequel.

Lemma 2. If there exists a symmetric positive-definite matrix PðaðkÞÞ such that the following LMIs hold,

U11ðk; kþ 1Þ �U21ðk; kþ 1Þ U22ðk; kþ 1Þ

� �< 0; f or � 2 ð0; �� ð12Þ

where AðaðkÞÞ, BwðaðkÞÞ, BuðaðkÞÞ, CzðaðkÞÞ, DzwðaðkÞÞ and DzuðaðkÞÞ are the same as in (5), KðaðkÞÞ is the same as in (10), and

U11ðk; kþ 1Þ ¼ �PðaðkÞÞ þ AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞT

� E�Pðaðkþ 1ÞÞE� AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð Þ

þ 1c

CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞT

� CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞU21ðk; kþ 1Þ ¼ BT

wðaðkÞÞE�Pðaðkþ 1ÞÞE� AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð Þ

þ 1c

DTzwðaðkÞÞ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð Þ

U22ðk; kþ 1Þ ¼ BTwðaðkÞÞE�Pðaðkþ 1ÞÞE�BwðaðkÞÞ þ

1c

DTzwðaðkÞÞDzwðaðkÞÞ � cI

ð13Þ


Then for each singular perturbation parameter � 2 ð0; ��, the closed-loop system (11) is asymptotically stable and its H1 norm isless than or equal to c.

Proof. See Appendix A. h

Lemma 3 [8]. For a given positive scalar ��, if the following conditions are satisfied,

a P 0 ð14Þa��2 þ b�� þ c < 0 ð15Þc < 0 ð16Þ

where a, b and c are constants, then

a�2 þ b�þ c < 0; f or � 2 ½0; �� ð17Þ

Proof. See Appendix B. h

Lemma 4 [8]. For a given positive scalar �� and matrices T1, T2, T3, if following conditions are satisfied,

T1 P 0 ð18Þ��2T1 þ ��T2 þ T3 < 0 ð19ÞT3 < 0 ð20Þ

then

�2T1 þ �T2 þ T3 < 0; f or � 2 ½0; �� ð21Þ

Proof. See Appendix C. h

3. Main results

In this section, a method of evaluating the upper bound of singularly perturbed parameter � subject to the stability of theclosed-loop system with meeting H1 performance bound requirements is presented. Moreover, two sufficient conditions fordesigning H1 controllers are given. In particular, one of them can improve the upper bound of the singular perturbationparameter � by designing controllers.

3.1. Computation of stability bound of � subject to an H1 performance bound constraint

Firstly, the following preliminary lemma is needed.

Lemma 5. If there exists a symmetric matrix

QðaðkÞÞ ¼ Q 11ðaðkÞÞ QT21ðaðkÞÞ

Q 21ðaðkÞÞ Q22ðaðkÞÞ

" #;

such that the following inequalities hold, " #
�2
0 00 Q 22ðaðkÞÞ

� ��

0 0 � �0 0 0 �0 0 0 0

2666664

3777775þ �0 Q T

21ðaðkÞÞQ21ðaðkÞÞ 0

� � �

0 0 � �0 0 0 �0 0 0 0

26666664

37777775

þ

W11ðk; kþ 1Þ � � �0 �cI � �

W31ðk; kþ 1Þ BwðaðkÞÞ �Qðaðkþ 1ÞÞÞ �W41ðk; kþ 1Þ DzwðaðkÞÞ 0 �cI

2666437775 < 0; f or � 2 ð0; �� ð22Þ

where � �
W11ðkÞ ¼
Q11ðaðkÞÞ 00 0

� SðaðkÞ;aðkþ 1ÞÞ � STðaðkÞ;aðkþ 1ÞÞ

W31ðkÞ ¼ AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞSðaðkÞ;aðkþ 1ÞÞW41ðkÞ ¼ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞSðaðkÞ;aðkþ 1ÞÞ

ð23Þ

then for each singular perturbation parameter � 2 ð0; ��, the system (9) is asymptotically stable and its H1 norm is less than orequal to c.


Proof. See Appendix D. h

Based on Lemma 5, a method of evaluating the upper bound of the singular perturbation parameter � with a prescribedH1 performance bound constraint is given in the following theorem.

Theorem 6. For a given positive scalar ��, if there exist matrices Q i ¼ ðQiÞT , Sij, 1 6 i; j 6 r

Q i ¼Q i

11 ðQ i21Þ

T

Q i21 Q i

22

" #; Sij ¼

S11 0Sij

21 Sij22

� �
satisfying the following LMIs
Miil < 0; 1 6 i; l 6 r ð24aÞ1

r � 1Miil þ

12ðMijl þMjilÞ < 0; 1 6 i – j 6 r; 1 6 l 6 r ð24bÞ

Kiil < 0; 1 6 i; l 6 r ð25aÞ1

r � 1Kiil þ

12ðKijl þKjilÞ < 0; 1 6 i – j 6 r; 1 6 l 6 r ð25bÞ

where

Mijl ¼

Q i11 0

0 0

" #� Sil � ðSilÞT � � �

0 �cI � �

Ai þ BiuKj

� �Sjl Bi

w �Ql �

Ciz þ Di

zuKj� �

Sjl Dizw 0 �cI

266666666664

377777777775

Kijl ¼ ��2

0 0

0 Q i22

" #0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

266666664

377777775þ ��

0 ðQ i21Þ

T

Q i21 0

" #0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

2666666664

3777777775

þ

Q i11 0

0 0

" #� Sil � ðSilÞT � � �

0 �cI � �

Ai þ BiuKj

� �Sjl Bi

w �Ql �

Ciz þ Di

zuKj� �

Sjl Dizw 0 �cI

266666666664

377777777775
then for each singular perturbation parameter � 2 ð0; ��, the system (9) is asymptotically stable and its H1 norm is less than orequal to c.
Proof. See Appendix E. h

Remark 7. Theorem 6 presents a method of estimating the upper bound of singularly perturbed parameter � subject to thestability of the closed-loop system (9) while satisfying an H1 performance bound requirement. An upper bound of � can beobtained by solving the following optimization problem:

Minimize �� subject to ð24Þ and ð25Þ

which can be effectively solved by using the LMI Control Toolbox [11].

3.2. H1 controller design

In this subsection, two LMI-based methods of designing H1 controllers are given. The two methods are with the merit ofeliminating the regularity restrictions attached to the Riccati-based solutions. In particular, one of them can improve theupper bound of � by designing controllers.


3.2.1. Design without the consideration of improving the upper bound of �In the following, an LMI-based design method without considering the improvement of the upper bound of the singular

perturbation parameter � is given.

Theorem 8. If there exist matrices Qi ¼ ðQiÞT , Sil, Li, 1 6 i; l 6 r, with

Q i ¼Q i

11 ðQ i21Þ

T

Q i21 Q i

22

" #; Sil ¼

S11 0Sil

21 Sil22

� �; Li ¼ Li

a 0� �

ð26Þ

satisfying the following LMIs,

Caiil < 0; 1 6 i; l 6 r ð27aÞ1

r � 1Caiil þ

12ðCaijl þ CajilÞ < 0; 1 6 i – j 6 r; 1 6 l 6 r ð27bÞ

where

Caijl ¼

Q i11 0

0 0

" #� Sil � ðSilÞT � � �

0 �cI � �AiSjl þ Bi

uLj Biw �Q l �

CizSjl þ Di

zuLj Dizw 0 �cI

266666664

377777775 < 0; f or 1 6 i; j; l 6 r

then there exists a sufficient small �� > 0 such that for � 2 ð0; ��, the closed-loop system (9) with

Kia ¼ Li

aS�111 ; 1 6 i 6 r ð28Þ

is asymptotically stable and its H1 norm is less than or equal to c.

Proof. The proof is easily obtained from Theorem 6 and omitted. h

Remark 9. Theorem 8 presents a sufficient condition for designing H1 controllers for discrete-time fuzzy singularly per-turbed systems. The method is based on LMIs, and with the merit of eliminating the regularity restrictions attached tothe Riccati-based solutions. Example 12 in Section 4 will illustrate the effectiveness of the method. However, in the method,the issue of improving the upper bound �� of the singular perturbation parameter � is not addressed. As a result, the obtainedcontroller might give a very small stability bound so that the resulting closed-loop system is unstable for a practical singularperturbation parameter �, see Example 14 in Section 4. In order to overcome the difficulty, another new method with theconsideration of improving the upper bound of the singular perturbation parameter � while satisfying H1 performance con-straints will be proposed in the next subsection.

3.2.2. Design with the consideration of improving the upper bound of �In this subsection, a new LMI-based H1 controller design method with the consideration of improving the upper bound of

the singular perturbation parameter � is given as follows:

Theorem 10. For a given positive scalar ��, if there exist matrices Q i ¼ ðQiÞT , Sil, Li, 1 6 i; l 6 r, with

Q i ¼Q i

11 ðQ i21Þ

T

Q i21 Q i

22

" #; Sil ¼

S11 0Sil

21 Sil22

� �; Li ¼ Li

a 0� �

satisfying (27b) and the following LMIs,

Cbiil < 0; 1 6 i; l 6 r ð29aÞ1

r � 1Cbiil þ

12ðCbijl þ CbjilÞ < 0; 1 6 i – j 6 r; 1 6 l 6 r ð29bÞ

where

Cbijl ¼ ��2

0 00 Q i

22

� �0 0 0

0 0 0 00 0 0 00 0 0 0

2666664

3777775þ ��0 ðQi

21ÞT

Q i21 0

" #0 0 0

0 0 0 00 0 0 00 0 0 0

26666664

37777775þQ i

11 00 0

" #� Sil � ðSilÞT � � �


uLj Biw �Q l �

CizSjl þ Di

zuLj Dizw 0 �cI

266666664

377777775
Then for � 2 ð0; ��, the closed-loop system (9) with (28) is asymptotically stable and its H1 norm is less than or equal to c.


Proof. The proof is easily obtained from Theorem 6 and omitted. h

Remark 11. In this section, Theorem 6 provides an LMI-based condition for estimating the upper bound of singularly per-turbed parameter � of discrete-time singularly perturbed systems subject to H1 performance bound constraints. In contrastto the existing techniques for estimating stability bounds, the new method can be used to estimate the bound of singularlyperturbed parameter � subject to H1 performance bound constraints. Moreover, two controller design methods are respec-tively given by Theorems 8 and 10. The two methods can eliminate the regularity restrictions attached the existing Riccati-based solution [23], the fact is shown by Example 12. In particular, the method given by Theorem 10 can be used to improvethe upper bound of singularly perturbed parameter � by designing controllers, which is illustrated by Example 14.

4. Example

In Example 12, a discrete-time fuzzy singularly perturbed system is obtained by discretizing a tunnel diode circuit, whichis borrowed from [2]. Because the example does not satisfy the regularity property, the conventional Riccati-based method isnot applicable. The new LMI-based method given by Theorem 8 will be applied to the example for illustrating itseffectiveness.

Moreover, Example 14 is given for better illustrating the effectiveness of the method given by Theorem 10 (i.e., the H1controller design method with the consideration of improving the upper bound of singularly perturbed parameter �).

Example 12. Consider a tunnel diode circuit (Fig. 1), where the diode current is iDðtÞ and the diode voltage vDðtÞ, and theysatisfy that

iDðtÞ ¼ �0:2vDðtÞ � 0:05v3DðtÞ

Moreover, C, R and L denote the capacitor, the resistance and the inductance, respectively. vCðtÞ, vRðtÞ and vLðtÞ are thecapacitor voltage, the resistance voltage and the inductor voltage, respectively. iCðtÞ, iRðtÞ and iLðtÞ are the capacitor current,the resistance current and the inductor current, respectively. uðtÞ is the input voltage, wðtÞ is the disturbance. ConsideringFig. 1 and applying the Kirchoff voltage and current law, we have that

CdðvCðtÞÞ

dt¼ icðtÞ; RiLðtÞ ¼ vRðtÞ

LdðiLðtÞÞ

dt¼ vLðtÞ; iCðtÞ ¼ iLðtÞ � iDðtÞ

vLðtÞ ¼ uðtÞ � vRðtÞ � vCðtÞ þwðtÞ

ð30Þ

Let x1ðtÞ ¼ vCðtÞ, x2ðtÞ ¼ iLðtÞ. Combining them and (30), then it follows that

C _x1ðtÞ ¼ 0:2x1ðtÞ þ 0:05x31ðtÞ þ x2ðtÞ

L _x2ðtÞ ¼ �x1ðtÞ � Rx2ðtÞ þ uðtÞ þwðtÞzðtÞ ¼ x1ðtÞ þ 0:1wðtÞ

ð31Þ

where zðtÞ is the controlled output. Assume that the parameters of the circuit are C ¼ 100 mF, L ¼ 1 mH and R = 20 X. Withthese parameters, (31) can be rewritten as follows:

Fig. 1. The tunnel diode circuit in Example 12.


_x1ðtÞ ¼ 2x1ðtÞ þ 0:5x31ðtÞ þ 10x2ðtÞ

� _x2ðtÞ ¼ �0:1x1ðtÞ � 2x2ðtÞ þ 0:1uðtÞ þ 0:1wðtÞzðtÞ ¼ x1ðtÞ þ 0:1wðtÞ

ð32Þ

where xðtÞ ¼ ½xT1ðtÞ; xT

2ðtÞ�T and � ¼ 10�4. Assume that jxðtÞj 6 3 and model the nonlinear system (32) by the sector nonlinear-

ity approach in [14], then the following T–S fuzzy model can be obtained:

E� _xðtÞ ¼X2

i¼1

aiðtÞ AicxðtÞ þ Bi

cwwðtÞ þ BicuuðtÞ

� �zðtÞ ¼

X2

i¼1

aiðtÞ CiczxðtÞ þ Di

czwwðtÞ� �

where

A1c ¼

2 10�0:1 �2

� �; A2

c ¼6:9 10�0:1 �2

� �; B1

cw ¼ B2cw ¼

00:1

� �B1

cu ¼ B2cu ¼

00:1

� �; C1

cz ¼ C2cz ¼ ½1 0�; D1

czw ¼ D2czw ¼ 0:1; E� ¼

1 00 �

� �
and aiðtÞ is the normalized time-varying fuzzy weighting function for each rule i ¼ 1;2, satisfies a1ðtÞ ¼ 1� x2
1ðtÞ9 ,

a2ðtÞ ¼ 1� a1ðtÞ.

From (32), it can be seen that x2ðtÞ, i.e., the inductor current iLðtÞ, changes very fast and exhibits a large range of variation.Therefore, it is very difficult to measure x2ðtÞ. On the other hand, the slow variable x1ðtÞ changes slowly and exhibits a smallrange of variation, it can be easily measured. Therefore, the slow state feedback controller is needed for this example.

Inhere, we discretize the model with a sampling period T ¼ 0:3 s and a zero-order holder, then the following discrete-time singularly perturbed model is obtained:

xðkþ 1Þ ¼X2

i¼1

aiðkÞ AidE�xðkÞ þ Bi

dwwðkÞ þ BiduuðkÞ

� �zðkÞ ¼

X2

i¼1

aiðkÞ CidzxðkÞ þ Di

dzwwðkÞ� �

where

A1d ¼

1:5683 7:8415�0:0784 �0:3921

� �; A2

d ¼6:8210 34:1039�0:3410 �1:7051

� �; B1

dw ¼0:18940:0405

� �B2

dw ¼0:45470:0273

� �; B1

du ¼0:18940:0405

� �; B2

du ¼0:45470:0273

� �; C1

dz ¼ C2dz ¼ ½1 0�;

D1dzw ¼ D2

dzw ¼ 0:1

Due to DdzwðaðkÞÞ– 0, the regularity property is not satisfied, the conventional Riccati-based methods [23] are not applicable.Applying Theorem 8 (the H1 controller design method without the consideration of improving the upper bound of the sin-gular perturbation parameter �), then the following results are obtained

Q 1 ¼0:4220 0:03080:0308 995:8465

� �; Q 2 ¼

0:4220 0:02930:0293 994:0820

� �;

L1a ¼ �4:4462; L2

a ¼ �6:5378;

K1a ¼ �10:5348; K2

a ¼ �15:4908; copt ¼ 0:51639

With the obtained controller gains, the upper bound of the singular perturbation parameter � of the tunnel diode circuit sys-tem are estimated as �� ¼ 0:1772 by using Theorem 6 under the H1 performance constraint c ¼ 0:5614. It is bigger than thepractical parameter � ¼ 10�4. Therefore, the designed controller can be used and some simulation results are shown in Figs.

2–4 with an initial state xð0Þ ¼ ½0:51�T , and wðkÞ ¼ 1; 5 6 k 6 100; others:

�From Figs. 2, and 3, it can be seen that the resulting closed-loop system is asymptotically stable and with a good H1 per-

formance, which further shows the effectiveness of the proposed condition in Theorem 8.

Remark 13. Note that Theorem 8 is applied for designing an H1 controller in Example 12 and the upper bound of singularlyperturbed parameter � for the resulting closed-loop system is more than the practical �, hence the controller is applicable.Since the controller design condition in Theorem 8 is without considering the tradeoff of the H1 performance bound c andthe stability bound ��, the obtained controller by Theorem 8 might be not applicable for some singularly perturbed systems.For solving the problem, Theorem 10 can be applied as an alternative, which is shown in Example 14.

0 5 10 15 20 25 30−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

x(k)

x1x2

Fig. 2. Trajectories of xðkÞ in Example 12.

0 5 10 15 20 25 30−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time (sec)

z(k)

, w(t)

zw

Fig. 3. Trajectories of zðkÞ and wðkÞ in Example 12.

Fig. 4. Membership functions aiðx1ðkÞÞ for Example 14.



Example 14. Consider a numerical example, which is described by (4) with

Table 1Upper b

c ¼ 1:5c ¼ 2c ¼ 2:5

A1 ¼

0:6 1 �2 10 2 0 �3

0:1 0 0:3 12:5 0:1 0:2 0:4

2666437775; A2 ¼

0:7 1 �3 0:80 1 0 �2

0:1 0 0:3 18 0:5 0:1 0:6

2666437775

B1w ¼

00:20

0:2

2666437775; B2

w ¼

00:30

0:1

2666437775; B1

u ¼

0:510

0:8

2666437775; B2

u ¼

0:310

0:4

2666437775

C1z ¼ C2

z ¼1 0 0 00 1 0 0

� �; D1

zw ¼01

� �; D2

zw ¼0

1:2

� �; D1

zu ¼ D2zu ¼

00:1

� �

E� ¼

1 0 0 00 1 0 00 0 � 00 0 0 �

2666437775

where � ¼ 0:05 and the membership functions aiðx1ðkÞÞ, i ¼ 1;2 are given in Fig. 4.

Applying Theorem 8 to the example (the H1 controller design method without the consideration of improving the upperbound of the singular perturbation parameter �), we can obtain the following controller gains and the optimal H1 perfor-mance bound.

K1a ¼ ½�0:1544 � 2:1685�; K2

a ¼ ½�0:0764 � 1:9145�copt ¼ 1:3334

ð33Þ

ounds of �.

Theorem 8 Theorem 10

0.0014 0.05270.0049 0.05560.0073 0.0576

0 5 10 15 20 25 30−3

−2

−1

0

1

2

3x 1015

Time (sec)

x(k)

x1x2

Fig. 5. Trajectories of xðkÞ via the gain (33) in Example 14.


Moreover, by using Theorem 10 with �� ¼ 0:05 (the H1 controller design method with the consideration of improving theupper bound of the singular perturbation parameter �), we can obtain

K1a ¼ ½�0:2233 � 2:1834�; K2

a ¼ ½�0:7358 � 1:8368�copt ¼ 1:4284

ð34Þ

Then the allowable upper bound of the singular perturbation parameter � of the closed-loop system with the controller gain(33) or (34) can be estimated by using Theorem 6. The obtained results are shown in Table 1.

From Table 1, it can be seen that,for the larger H1 performance indices,the larger upper bounds of the singular perturba-tion parameter � are achieved by using Theorem 10, which shows that the method of Theorem 10 is effective for improvingthe upper bound of the singular perturbation parameter �.

0 5 10 15 20 25 30−6

−4

−2

0

2

4

6

8

Time (sec)

x(k)

x1x2

Fig. 6. Trajectories of xðkÞ via the gain (34) in Example 14.

0 5 10 15 20 25 300

5

10

15

20

25

30

Time (sec)

The

ratio

1.3572

Fig. 7.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPk

i¼0zT ðiÞzðiÞ=Pk

i¼0wT ðiÞwðiÞq

with the gain (34) in Example 14.


Note that � ¼ 0:05, then for the case of a1ðkÞ ¼ 0, a2ðkÞ ¼ 1, the eigenvalues of

AðaðkÞÞE� þ BuðaðkÞÞKðaðkÞÞ ¼ A2E� þ B2uK2 ¼ A2E� þ B2

u½K2a0�

with the controller gain (33) designed by using Theorem 8 are �1.1190, 0.6411, 0.4496, �0.1641, which implies that theresulting closed-loop system is unstable. Therefore,the controller cannot be applied. On the other hand, Table 1 showsthe achieved upper bounds of � of the closed-loop system with the controller gains (34) under different H1 performancerequirements. The upper bounds are more than � ¼ 0:05. The fact shows that Theorem 10 can be used to effectively improvethe upper bound of the singular perturbation parameter � at the stage of controller design.

What it follows, some simulation results will be given in order to further validate the effectiveness of Theorem 10. As-sume that the initial state xð0Þ ¼ ½�2 8 0 0�T and the disturbance

wðkÞ ¼2; 3 6 k 6 100; others

�
Figs. 5 and 6 show the trajectories of the state xðkÞ of the closed-loop system with the controller gains (33) and (34), respec-
tively. Fig. 7 shows the ratio offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPk

i¼0zTðiÞzðiÞ=Pk

i¼0wTðiÞwðiÞq

in the simulation.From Fig. 5, it can be seen that the closed-loop system with the controller gain (33), which is obtained by Theorem 8, is

unstable. It can be seen that from Figs. 6 and 7 that the controller gain (34), which is obtained by Theorem 10, can guaranteethe stability and meet the H1 performance requirement. These simulation results further show the advantage of the methodgiven by Theorem 10, i.e., Theorem 10 can be used to effectively improve the upper bound of the singular perturbationparameter � at the stage of controller design.

5. Conclusion

In this paper, the H1 control problem via slow state variables feedback for discrete-time fuzzy singularly perturbed systemshas been investigated. Two LMI-based methods for designing H1 controllers via slow state variables feedback are presented,and one of them can be used to improve the upper bound of the singular perturbation parameter �, which can overcome thedisadvantage in the conventional design methods where the designed controller might not be used because the resulting allow-able upper bound of the singular perturbation parameter is too small. Moreover, a method of evaluating the upper bound of asingular perturbation parameter �with meeting a prescribed H1 performance bound requirement is given in terms of solutionsto a set of LMIs. The effectiveness of the proposed methods has been illustrated by the numerical examples.

Acknowledgements

This study was supported in part by the Funds for Creative Research Groups of China (No. 60821063), the State Key Pro-gram of National Natural Science of China (Grant No. 60534010), National 973 Program of China (Grant No. 2009CB320604),the Funds of National Science of China (Grant No. 60674021), the 111 Project (B08015) and the Funds of PhD program ofMOE, China (Grant No. 20060145019).

Appendix A. Proof of Lemma 2

Proof. Consider the system (11) (which is equivalent to the system (9)), and let PðaðkÞÞ > 0. Choose Lyapunov function

VðkÞ ¼ cnTðkÞPðaðkÞÞnðkÞ

then

Vðkþ 1Þ � VðkÞ þ zTðkÞzðkÞ � c2wTðkÞwðkÞ¼ c AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞnðkÞ þ BwðaðkÞÞwðkÞð ÞT E�Pðaðkþ 1ÞÞE� � AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞnðkÞðþ BwðaðkÞÞwðkÞÞ � cnTðkÞPðaðkÞÞnðkÞ þ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞnðkÞð þ DzwðaðkÞÞwðkÞÞT � CzðaðkÞÞððþ DzuðaðkÞÞKðaðkÞÞÞnðkÞ þ DzwðaðkÞÞwðkÞÞ � c2wTðkÞwðkÞ

¼ cnðkÞwðkÞ

� �TU11ðk; kþ 1Þ UT

21ðk; kþ 1ÞU21ðk; kþ 1Þ U22ðk; kþ 1Þ

" #nðkÞwðkÞ

� �

where /11ðk; kþ 1Þ, /21ðk; kþ 1Þ, /22ðk; kþ 1Þ are the same as in (13).From (12) and the above equality, then it follows that

Vðkþ 1Þ � VðkÞ þ zTðkÞzðkÞ � c2wTðkÞwðkÞ 6 0; for � 2 ð0; ��


From the above inequality, we have that the system (11) with � 2 ð0; �� is asymptotically stable in the disturbance-free case.For the initial condition xð0Þ ¼ 0 and every positive integer N, sum the above inequality from k ¼ 0 to N, then we can obtain

VðNÞ � Vð0Þ þX1k¼0

zTðkÞzðkÞ � c2X1k¼0

wTðkÞwðkÞ ¼ VðNÞ þX1k¼0

zTðkÞzðkÞ � c2X1k¼0

wTðkÞwðkÞ 6 0; for � 2 ð0; ��

which implies that
XN
k¼0

zTðkÞzðkÞ 6 c2XN

k¼0

wTðkÞwðkÞ; for � 2 ð0; ��

Combining it and Definition 1, it follows that the H1 norm of the closed-loop system (11) is less than or equal to c. h

Appendix B. Proof of Lemma 3

Proof. Consider the following two cases:

(i): If a ¼ 0, then from (15) and (16), (17) obviously holds.(ii): If a > 0, we consider the following quadratic function of �,

yð�Þ ¼ a�2 þ b�þ c ð35Þ

Since a > 0, yð�Þ is convex function of � [4]. From (15) and (16), it follows that yð��Þ < 0 and yð0Þ < 0, which further impliesthat yð�Þ < 0 for � 2 ½0; ��, i.e., when a > 0, (17) holds. Thus, the proof is complete. h

Appendix C. Proof of Lemma 4

Proof. For all nonzero vector xðkÞ, pre- and post-multiplying (18)–(20) by xTðkÞ and its transpose, then we have

xTðkÞT1xðkÞP 0 ð36Þ

��2xTðkÞT1xðkÞ þ ��xTðkÞT2xðkÞ þ xTðkÞT3xðkÞ < 0 ð37Þ

xTðkÞT3xðkÞ < 0 ð38Þ

Denote axk¼ xTðkÞT1xðkÞ, bxk

¼ xTðkÞT2xðkÞ, cxk¼ xTðkÞT3xðkÞ. Substituting axk

, bxkand cxk

into (36)–(38), then it follows that

axkP 0 ð39Þ

axk��2 þ bxk

�� þ cxk< 0 ð40Þ

cxk< 0 ð41Þ

From (39)–(41) and applying Lemma 3, we can obtain axk�2 þ bxk

�þ cxk< 0 for � 2 ½0; ��, i.e., for all nonzero vector xðkÞ,

�2xTðkÞT1xðkÞ þ �xTðkÞT2xðkÞ þ xTðkÞT3xðkÞ < 0; for � 2 ½0; ��

which implies that matrix inequality (21) holds. Thus, the proof is complete. h

Appendix D. Proof of Lemma 5

Proof. (22) can be rewritten as follows,

W11ðkÞ � � �0 �cI � �

W31ðkÞ BwðaðkÞÞ �Qðaðkþ 1ÞÞ �W41ðkÞ DzwðaðkÞÞ 0 �cI

2666437775 < 0; for � 2 ð0; ��: ð42Þ

where W31ðkÞ, W41ðkÞ are the same as in (23) and

W11ðkÞ ¼ E�QðaðkÞÞE� � SðaðkÞ;aðkþ 1ÞÞ � STðaðkÞ;aðkþ 1ÞÞ

where E� is same as in (5).


From (42), we can obtain that W11ðkÞ < 0. Considering the block (3,3) of (42), then yields QðaðkÞÞ > 0. Combiningit and W11ðkÞ < 0, then we have SðaðkÞ;aðkþ 1ÞÞ þ STðaðkÞ;aðkþ 1ÞÞ > 0. Pre- and post-multiply (42) by

S�TðaðkÞ;aðkþ 1ÞÞ 0 0 00 I 0 00 0 I 00 0 0 I

26643775 and

S�1ðaðkÞ;aðkþ 1ÞÞ 0 0 00 I 0 00 0 I 00 0 0 I

26643775, respectively. Then it follows that

eW11ðkÞ � � �0 �cI � �

AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞ BwðaðkÞÞ �Qðaðkþ 1ÞÞ �CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞ DzwðaðkÞÞ 0 �cI

266664377775 < 0; for � 2 ð0; �� ð43Þ

where
eW11ðkÞ ¼ S�TðaðkÞ;aðkþ 1ÞÞE�QðaðkÞÞE�S�1ðaðkÞ;aðkþ 1ÞÞ � S�1ðaðkÞ;aðkþ 1ÞÞ � S�TðaðkÞ;aðkþ 1ÞÞ
Let P�1ðaðkÞÞ ¼ E�QðaðkÞÞE�, therefore,

PðaðkÞÞ > 0 ð44Þ
then
S�TðaðkÞ;aðkþ 1ÞÞ � PðaðkÞÞ� �

P�1ðaðkÞÞ S�1ðaðkÞ;aðkþ 1ÞÞ � PðaðkÞÞ� �

P 0

which implies that

S�TðaðkÞ;aðkþ 1ÞÞP�1ðaðkÞÞS�1ðaðkÞ;aðkþ 1ÞÞ � S�1ðaðkÞ;aðkþ 1ÞÞ � S�TðaðkÞ;aðkþ 1ÞÞP �PðaðkÞÞ

Combining it with (43), we can obtain

�PðaðkÞÞ � � �0 �cI � �

AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞ BwðaðkÞÞ �E�1� P�1ðaðkþ 1ÞÞE�1

� �CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞ DzwðaðkÞÞ 0 �cI

2666437775 < 0; for � 2 ð0; �� ð45Þ

Applying Schur complement lemma to (45), then we have

�PðaðkÞÞ � �0 �cI �


�

264375

þ 1c

CTz ðaðkÞÞ þ KTðaðkÞÞDT

zuðaðkÞÞDT

zwðaðkÞÞ0

264375� CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞ DzwðaðkÞÞ 0½ �

¼� 11ðkÞ � �� 21ðkÞ � 22ðkÞ �


�

2643750; for � 2 ð0; ��

where

� 11ðkÞ ¼ �PðaðkÞÞ þ 1c

CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞT

� CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð Þ

� 21ðkÞ ¼1c

DTzwðaðkÞÞ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð Þ

� 22ðkÞ ¼ �cI þ 1c

DTzwðaðkÞÞDzwðaðkÞÞ

Applying Schur complement lemma to the above inequality, again, then it follows that

� 11ðkÞ � T21ðkÞ

� 21ðkÞ � 22ðkÞ

� �þ ATðaðkÞÞ þ KTðaðkÞÞBT

uðaðkÞÞBT

wðaðkÞÞ

� �E�Pðaðkþ 1ÞÞE� � AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞ BwðaðkÞÞ½ �

¼ U11ðk; kþ 1Þ UT21ðk; kþ 1Þ

U21ðk; kþ 1Þ U22ðk; kþ 1Þ

� �< 0; for � 2 ð0; �� ð46Þ

where U11ðk; kþ 1Þ, U21ðk; kþ 1Þ and U22ðk; kþ 1Þ are the same as in (13).


Then applying Lemma 2 to (46), we have that, for each singular perturbation parameter � 2 ð0; ��, the system (11) isasymptotically stable and its H1 norm is less than c. Then the conclusion follows. h

Appendix E. Proof of Theorem 6

Proof. Considering the block (3,3) of (24a) and from (24a), it follows that

Qi > 0 1 6 i 6 r

which also implies that

Qi22 > 0 1 6 i 6 r ð47Þ

Let

T1i ¼

0 00 Q i

22

� �0 0 0

0 0 0 00 0 0 00 0 0 0

2666664

3777775

T2i ¼

0 ðQ i21Þ

T

Q i21 0

" #0 0 0

0 0 0 00 0 0 00 0 0 0

26666664

37777775T3il ¼ Miil

From (47), it follows that
T1i P 0 ð48aÞ
From (25a), we have

��2T1i þ ��T2i þ T3il < 0 ð48bÞ
From (24a), we have
T3il < 0 ð48cÞ

Applying Lemma 4 to (48a), then yields

�2T1i þ �T2i þ T3il < 0; for � 2 ½0; ��; 1 6 i; l 6 r

i.e.,

Kiil < 0; for � 2 ½0; ��; 1 6 i; l 6 r ð49aÞ

where

Kiil ¼ �2

0 00 Q i

22

� �0 0 0

0 0 0 00 0 0 00 0 0 0

2666664

3777775þ �0 ðQ i

21ÞT

Q i21 0

" #0 0 0

0 0 0 00 0 0 00 0 0 0

26666664

37777775þQ i

11 00 0

" #� Sil � ðSilÞT � � �

0 �cI � �Ai þ Bi

uKi� �

Sil Biw �Q l �

Ciz þ Di

zuKi� �

Sil Dizw 0 �cI

26666666664

37777777775
Similarly, from (47), (24b) and (25b), we can also obtain
1r � 1

Kiil þ12ðKijl þKjilÞ < 0; for � 2 ½0; ��; 1 6 i – j 6 r; 1 6 l 6 r ð49bÞ

where

Kijl ¼ �2

0 00 Q i

22

� �0 0 0

0 0 0 00 0 0 00 0 0 0

2666664

3777775þ �0 ðQ i

21ÞT

Q i21 0

" #0 0 0

0 0 0 00 0 0 00 0 0 0

26666664

37777775þQ i

11 00 0

" #� Sil � ðSilÞT � � �


uKjSjl Biw �Q l �

CizSjl þ Di

zuKjSjl Dizw 0 �cI

266666664

377777775


Applying the parameterized linear matrix inequality (PLMI) technique in [27] to (49a), then yields
Xr
i¼1

Xr

j¼1

aiðkÞajðkÞKijl < 0; 1 6 l 6 r

Multiplying the above inequality by alðkþ 1Þ and summing them from l ¼ 1 to r, then we can obtain
Xr
i¼1

Xr

j¼1

Xr

l¼1

aiðkÞajðkÞalðkþ 1ÞKijl < 0 ð50Þ

Note that

Xr

j¼1

Xr

l¼1

ajðkÞalðkþ 1ÞKjSjl ¼Xr

j¼1

Xr

l¼1

ajðkÞalðkþ 1Þ KjaS11 0Sjl

21 Sjl22

" #¼

Prj¼1

ajðkÞKjaS11 0

Prj¼1

Prl¼1

ajðkÞalðkþ 1ÞSjl21

Prj¼1

Prl¼1


2666437775

¼Prj¼1

ajðkÞKja 0

� ��

S11 0Prj¼1

Prl¼1


Prj¼1

Prl¼1


24 35¼ KðaðkÞÞ S11 0

S21ðaðkÞ;aðkþ 1ÞÞ S22ðk; kþ 1Þ

� �¼ KðaðkÞÞSðaðkÞ;aðkþ 1ÞÞ

where KðaðkÞÞ is the same as in (10) and

SðaðkÞ;aðkþ 1ÞÞ ¼S11 0Pr

j¼1

Prl¼1


Prj¼1

Prl¼1


24 35
Then Xr
i¼1

Xr

j¼1

Xr

l¼1

aiðkÞajðkÞalðkþ 1ÞBiuKjSjl ¼ BuðaðkÞÞKðaðkÞÞSðaðkÞ;aðkþ 1ÞÞ

Xr

i¼1

Xr

j¼1

Xr

l¼1

aiðkÞajðkÞalðkþ 1ÞDizuKjSjl ¼ DzuðaðkÞÞKðaðkÞÞSðaðkÞ;aðkþ 1ÞÞ

Therefore, (50) implies that (22) holds. Combining it and Lemma 5, it follows that for the singular perturbation parameter� 2 ð0; ��, the system (9) is asymptotically stable and its H1 norm is less than c. Then the conclusion follows. h

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control design for fuzzy discrete-time singularly ...faculty.neu.edu.cn/ise/dongjiuxiang/English/Papers/Dong_J09i.pdfH1 control design for fuzzy discrete-time singularly perturbed