Applied Mathematics and Computation 229 (2014) 327–339

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Observer design of discrete-time impulsive switched nonlinearsystems with time-varying delays

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.12.053

⇑ Corresponding author.E-mail addresses: xiangzr@mail.njust.edu.cn, xiangzr@tom.com (Z. Xiang).

Xia Li, Zhengrong Xiang ⇑School of Automation, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China

a r t i c l e i n f o

Keywords:Impulsive systemsSwitched systemsTime-varying delaysObserver designAverage dwell time

a b s t r a c t

This paper investigates the problem of observer design for discrete-time impulsiveswitched nonlinear systems with time-varying delays. Firstly, based on the average dwelltime approach and the Lyapunov–Krasovskii functional technique, a delay-dependentexponential stability criterion for the discrete-time impulsive switched nonlinear systemswith time-varying delays is derived in terms of a set of linear matrix inequalities (LMIs).Then, sufficient conditions for the existence of an observer that guarantees the exponentialstability of the corresponding error system are proposed. Finally, two numerical examplesare given to illustrate the effectiveness of the proposed method.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Switched systems are a class of important hybrid systems consisting of subsystems and a switching law which defines aspecific subsystem being activated during a certain interval of time [1]. Due to the theoretical development as well aspractical applications, analysis and synthesis of switched systems have recently gained considerable attention, and a largenumber of results in this field have appeared (see, e.g., [2–6]). However, in the real world, some switched systems exhibitimpulsive dynamical behavior due to sudden changes in the states of these systems at certain instants of switching, this kindof systems are usually called impulsive switched systems [7]. During the last decade, impulsive switched systems havereceived considerable attention. Some results on stability and stabilization of such systems have been achieved (see, e.g.,[8–10]). For instance, stability analysis of systems with impulse effects was investigated in [8]. Analysis and design of impul-sive control systems were considered in [10].

In many practical systems, there inevitably exists time delay which plays a crucial role in destroying the system perfor-mance. Although, in recent years, many stability conditions of time-delay systems have been obtained (see, e.g., [11–16]), itis worth mentioning that till now, there is only little effort putting on impulsive switched systems with time delays [17–27].For example, some stability criteria for impulsive switched systems with constant time delays were developed in [17–26]. Adelay-dependent stability criterion for a class of impulsive switched discrete systems with time-varying delays was estab-lished by using the Lyapunov–Krasovskii functional technique in [27].

On the other hand, it is necessary to design state observers for the systems due to the fact that the states of the systemsare not all measurable in practice. Some significant results on this topic have been obtained in [28–33]. For example, theasymptotic stability property of the proposed switching observer was discussed and an LMI-based algorithm was givenfor a class of impulsive switched systems in [33]. However, to the best of our knowledge, the problem of observer designfor discrete-time impulsive switched systems has not been fully investigated, especially for impulsive switched nonlinearsystems with time-varying delays, which motivates us for this study.

328 X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339

In this paper, we are interested in investigating the observer design problem for impulsive switched nonlinear systemswith time-varying delays. The main contributions of this paper are as follows: (1) based on the dwell time approach, weestablish a delay-dependent exponential stability criterion of the underlying system, which is different from those proposedin [33]; and (2) an observer design scheme for the underlying system is proposed.

The remainder of the paper is organized as follows. In Section 2, the problem formulation and some necessary lemmas arepresented. In Section 3, the main results are obtained. Section 4 gives two numerical examples to illustrate the effectivenessof the proposed approach. Finally, concluding remarks are given in Section 5.

Notations. Throughout this paper, the superscript ‘‘T‘’ denotes the transpose, and the notation X P YðX > YÞmeans that matrix X � Y ispositive semi-definite (positive definite, respectively). k � k denotes the Euclidean norm. I represents the identity matrix. diagfbigdenotes a diagonal matrix with the diagonal elements bi,i = 1, 2, . . ., n. X�1 denotes the inverse of X. The asterisk ⁄ in a matrix is used todenote the term that is induced by symmetry. Zþ0 denotes the set of all nonnegative integers. The set of all positive integers isrepresented by Z+.

2. Problem formulation and preliminaries

Consider the following discrete-time impulsive switched nonlinear systems with time-varying delays:

xðkþ 1Þ ¼ ArðkÞxðkÞ þ AdrðkÞxðk� dðkÞÞ þ BrðkÞuðkÞ þ DfrðkÞfrðkÞðxðkÞÞ; k–kb � 1; b 2 Zþ; ð1aÞ

xðkþ 1Þ ¼ Erðkþ1ÞrðkÞxðkÞ; k ¼ kb � 1; b 2 Zþ; ð1bÞ

yðkÞ ¼ CrðkÞxðkÞ; ð1cÞ

xðk0 þ hÞ ¼ /ðhÞ; h 2 ½�d2;0�; ð1dÞ

where x(k) 2 Rn is the state vector, uðkÞ 2 Rm is the control input, yðkÞ 2 Rp is the output, /ðhÞ is a discrete vector-valued initialfunction defined on the interval ½�d2;0�. d(k) is the discrete time-varying delay which is assumed to satisfy d1 6 dðkÞ 6 d2. d1

and d2 are constant positive scalars representing the minimum and maximum delays, respectively. k0 is the initial time. Thefunction rðkÞ : Zþ0 ! N :¼ f1; . . . ;Ng is the switching signal with N being the number of subsystems. For each i 2 N,fiðxðkÞÞ 2 Rn is a known nonlinear function. Ai, Adi, Bi, Dfi, Eji and Ci, i; j 2 N; i–j, are known real constant matrices with appro-priate dimensions.

Remark 1. It should be noted that Eq. (1b) has been introduced in the literature [25–26,33] to describe the impulsivedynamical behavior of some practical switched systems due to sudden changes in the state of the system at certain instantsof switching.

Remark 2. In the paper, it is assumed that the switching occurs at the instant kb. The switching sequence of the system canbe described as

R ¼ fðk0;rðk0ÞÞ; ðk1;rðk1ÞÞ; ðk2;rðk2ÞÞ; � � � ; ðkb;rðkbÞÞ; � � �g; ð2Þ

where kb denotes the b-th switching instant.We construct the following discrete-time impulsive switched systems to estimate the state of system (1):

xðkþ 1Þ ¼ ArðkÞxðkÞ þ AdrðkÞxðk� dðkÞÞ þ BrðkÞuðkÞ þ LrðkÞðyðkÞ � yðkÞÞ þ DfrðkÞ f rðkÞðxðkÞÞ; k–kb � 1; b 2 Zþ; ð3aÞ

xðkþ 1Þ ¼ Hrðkþ1ÞrðkÞxðkÞ; k ¼ kb � 1; b 2 Zþ; ð3bÞ

yðkÞ ¼ CrðkÞxðkÞ; ð3cÞ

xðk0 þ hÞ ¼ 0; h 2 ½�d2; 0�; ð3dÞ

where xðkÞ 2 Rn is the estimated state vector of x(k), yðkÞ 2 Rp is the observer output vector. For each i 2 N, f iðxðkÞÞ 2 Rn is aknown nonlinear function. Li 2 Rn�p and Hij 2 Rn�n, 8i; j 2 N; i–j, are the matrices to be determined.

Let ~xðkÞ ¼ xðkÞ � xðkÞ be the estimated error, then we can obtain the following error system:

~xðkþ 1Þ ¼ ðArðkÞ � LrðkÞCrðkÞÞ~xðkÞ þ AdrðkÞ~xðk� dðkÞÞ þ DfrðkÞðfrðkÞðxðkÞÞ � f rðkÞðxðkÞÞÞ; k–kb � 1; b 2 Zþ; ð4aÞ

~xðkþ 1Þ ¼ Erðkþ1ÞrðkÞxðkÞ � Hrðkþ1ÞrðkÞxðkÞ; k ¼ kb � 1; b 2 Zþ; ð4bÞ

~xðk0 þ hÞ ¼ /ðhÞ; h 2 ½�d2;0� ð4cÞ

Before ending this section, we introduce the following definitions and lemmas.

X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339 329

Definition 1 [4]. For any k > k0, let Nr(k0, k) denote the switching number of r(k), during the interval [k0, k). If there existN0 P 0 and sa P 0 such that, Nrðk0; kÞ 6 N0 þ k�k0

sa; then sa and N0 are called the average dwell time and the chatter bound,

respectively.

Definition 2. System (1) is said to be exponentially stable, if its solution satisfies

kxðkÞk 6 Kkxðk0Þkhqðk�k0Þ;8k P k0 ð5Þ

for any initial conditions u(h), where kxðk0Þkh ¼ sup�d26h60fk/ðhÞk; k/ðhþ 1Þ � /ðhÞkg;K > 0 is the decay coefficient, and0 < q < 1 is the decay rate.

Lemma 1 [34]. For a given matrix S ¼ S11 S12

ST12 S22

� �, where S11 and S22 are square matrices, the following conditions are

equivalent:

(i) S < 0;

(ii) S11 < 0; S22 � ST12S�1

11 S12 < 0;

(iii) S22 < 0; S11 � S12S�122 ST

12 < 0:

Lemma 2 [35]. For a matrix G of full rank and a positive definite symmetric matrix S, we have

ðS� GÞT S�1ðS� GÞP 0;

which is equivalent to

GTðSÞ�1G P GT þ G� S:

Without loss of generality, we make the following assumption.

Assumption 1. For each i 2 N, fi(x(k)) satisfies

f Ti ðxðkÞÞðfiðxðkÞÞ � aixðkÞÞ 6 0; ð6Þ

and fiðxðkÞÞ � f iðxðkÞÞ satisfies

ðfiðxðkÞÞ � f iðxðkÞÞÞTðfiðxðkÞÞ � f iðxðkÞÞ � aiðxðkÞ � xðkÞÞÞ 6 0; ð7Þ

where ai is a known real constant.The object of the paper is to determine the matrices Li and Hij(i, j 2 N, i – j) in (3) such that error system (4) is exponen-

tially stable.

3. Main results

3.1. Stability analysis

To obtain the main results, we first consider the exponential stability of the following system

xðkþ 1Þ ¼ ArðkÞxðkÞ þ AdrðkÞxðk� dðkÞÞ þ DfrðkÞfrðkÞðxðkÞÞ; k–kb � 1; b 2 Zþ; ð8aÞ

xðkþ 1Þ ¼ Erðkþ1ÞrðkÞxðkÞ; k ¼ kb � 1; b 2 Zþ; ð8bÞ

xðk0 þ hÞ ¼ /ðhÞ; h 2 ½�d2;0�: ð8cÞ

Before giving the stability result, we present the following lemma which is essential for our later development.

Lemma 3. Suppose that there exists a C1 function V : Rn ! R, and two positive scalars k1 and k2 such that the following inequalityholds

k1 xðkÞk k26 VðxðkÞÞ 6 k2 xðkÞk k2

h; 8k P k0 ð9Þ

where kxðkÞk2h ¼ sup�d26h60 kxðkþ hÞk2

; kxðkþ hþ 1Þ � xðkþ hÞk2n o

, along the solution of system (8), we have the followinginequalities

Vðxðkþ 1ÞÞ 6 bVðxðkÞÞ; 8k 2 ½kb�1; kb � 1Þ; b 2 Zþ; ð10Þ

330 X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339

Vðxðkþ 1ÞÞ 6 lVðxðkÞÞ; k ¼ kb � 1; b 2 Zþ; ð11Þ

where 0 < b < 1 and l P 1, then system (8) is exponentially stable for any switching signals rðkÞ with the average dwelltime satisfying

sa > s�a ¼ 1� lnlln b

: ð12Þ

Proof. Similar to the proof line of Lemma 2 in [33], it can be deduced from (10) and (11) that

VðxðkÞÞ 6 ðlb�1ÞN0þk�k0sa bk�k0 Vðxðk0ÞÞ ¼ ðlb�1ÞN0qk�k0 Vðxðk0ÞÞ; ð13Þ

where q ¼ l 1sa b1� 1

sa .It follows from (13) that

xðkÞk k 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik�1

1 k2ðlb�1ÞN0

qq

k�k02 xðk0Þk kh;8k P k0:

One can verify that 0 < q < 1. Hence, system (8) is exponentially stable under the average dwell time scheme (12). Thiscompletes the proof.

The following theorem provides a stability criterion for system (8). h

Theorem 1. For a given positive scalar 0 < b < 1, if there exist positive definite symmetric matrices Pi, Qi, Ri, Zi, and any matrices

Xi ¼Xi11 Xi12

� Xi22

� �> 0; Ni2 ¼

Ni21

Ni22

� �; Ni1 ¼

Ni11

Ni12

� �ði 2 NÞ with appropriate dimensions, such that

Ci11 Ci12 �bd2 Ni21ai2 I AT

i Pi ðAi � IÞT Zi

� Ci22 �bd2 Ni22 0 ATdiPi AT

diZi

� � �bd2 Ri 0 0 0� � � �I DT

fiPi DTfiZi

� � � � �Pi 0� � � � � �d�1

2 Zi

26666666664

37777777775< 0; ð14Þ

Xi Ni1

� Zi

� �P 0;

Xi Ni2

� Zi

� �P 0; ð15Þ

where

Ci11 ¼ �bPi þ ðd2 � d1 þ 1ÞQ i þ Ri þ bd2 NTi11 þ Ni11

� �þ d2b

d2 Xi11;

Ci12 ¼ bd2 �Ni11 þ NTi12 þ Ni21

� �þ d2b

d2 Xi12;

Ci22 ¼ �bd2 Q i � bd2 NTi12 þ Ni12 � Ni22 � NT

i22

� �þ d2b

d2 Xi22;

then system (8) is exponentially stable for any switching signals r(k) with the average dwell time satisfying

sa > s�a ¼ 1� lnlln b

; ð16Þ

where l P 1 satisfies

ETijPiEij � lPj þ ðd2 � d1 þ 1ÞQ i þ Ri þ d2Zi < 0; i; j 2; i–j; ð17Þ

bQi 6 lQ j;bRi 6 lRj; bZi 6 lZj; i; j 2; i–j: ð18Þ

Proof. Denote

gðkÞ ¼ xðkþ 1Þ � xðkÞ ¼ ArðkÞxðkÞ þ AdrðkÞxðk� dðkÞÞ þ DfrðkÞfrðkÞðxðkÞÞ � xðkÞ¼ ðArðkÞ � IÞxðkÞ þ AdrðkÞxðk� dðkÞÞ þ DfrðkÞfrðkÞðxðkÞÞ ð19Þ

For system (8), choose the following piece-wise Lyapunov functional candidate

VðxðkÞÞ ¼ VrðkÞðxðkÞÞ;

X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339 331

the form of each Vr(k)(x(k)) is given by

VrðkÞðxðkÞÞ ¼ VrðkÞ1ðxðkÞÞ þ VrðkÞ2ðxðkÞÞ þ VrðkÞ3ðxðkÞÞ þ VrðkÞ4ðxðkÞÞ þ VrðkÞ5ðxðkÞÞ; ð20Þ

where

Vi1ðxðkÞÞ ¼ xTðkÞPixðkÞ;

Vi2ðxðkÞÞ ¼Xk�1

r¼k�dðkÞbk�r�1xTðrÞQixðrÞ;

Vi3ðxðkÞÞ ¼X�d1

s¼�d2þ1

Xk�1

r¼kþs

bk�r�1xTðrÞQ ixðrÞ;

Vi4ðxðkÞÞ ¼Xk�1

r¼k�d2

bk�r�1xTðrÞRixðrÞ;

Vi5ðxðkÞÞ ¼X�1

s¼�d2

Xk�1

r¼kþs

bk�r�1gTðrÞZigðrÞ;

where g(r) = x(r + 1) � x(r), and Pi, Qi, Ri and Zi ði 2 NÞ are positive definite matrices to be determined.By (20), we can easily find two positive scalars k1 and k2 such that

k1 xðkÞk k26 VðxðkÞÞ 6 k2 xðkÞk k2

h; ð21Þ

where

k1 ¼mini2NðkminðPiÞÞ; k2 ¼max

i2NkmaxðPiÞ þ d2kmaxðQiÞð þðd2 � d1Þ2kmaxðQ iÞ þ d2kmaxðRiÞ þ d2

2kmaxðZiÞ�

When k 2 ½kb; kbþ1 � 1Þ, let r(k) = i (i 2 N), along the trajectory of system (8), we have

Vi1ðxðkþ 1ÞÞ � bVi1ðxðkÞÞ ¼ xTðkþ 1ÞPixðkþ 1Þ � bxTðkÞPixðkÞ

¼ AixðkÞ þ Adixðk� dðkÞÞ þ DfifiðxðkÞÞ� �T Pi½AixðkÞ þ Adixðk� dðkÞÞ þ DfifiðxðkÞÞ�� bxTðkÞPixðkÞ; ð22Þ

Vi2ðxðkþ 1ÞÞ � bVi2ðxðkÞÞ ¼Xk

r¼k�dðkþ1Þþ1

bk�rxTðrÞQ ixðrÞ �Xk�1

r¼k�dðkÞbk�rxTðrÞQixðrÞ;

6 xTðkÞQ ixðkÞ þXk�d1

r¼k�d2þ1

bk�rxTðrÞQ ixðrÞ � bd2 xTðk� dðkÞÞQ ixðk� dðkÞÞ; ð23Þ

Vi3ðxðkþ 1ÞÞ � bVi3ðxðkÞÞ ¼ ðd2 � d1ÞxTðkÞQixðkÞ �Xk�d1

r¼k�d2þ1

bk�rxTðrÞQixðrÞ; ð24Þ

Vi4ðxðkþ 1ÞÞ � bVi4ðxðkÞÞ ¼ xTðkÞRixðkÞ � bd2 xTðk� d2ÞRixðk� d2Þ; ð25Þ

Vi5ðxðkþ 1ÞÞ � bVi5ðxðkþ 1ÞÞ ¼X�1

s¼�d2

gTðkÞZigðkÞ � b�sgTðkþ sÞZigðkþ sÞ� �

¼ d2gTðkÞZigðkÞ �Xk�1

r¼k�d2

bk�rgTðrÞZigðrÞ: ð26Þ

From (6) in Assumption 1, we can obtain that

uTi ðkÞ

0 � ai2 I

� I

� �uiðkÞ 6 0; ð27Þ

where uTi ðkÞ ¼ xTðkÞ f T

i ðxðkÞÞ� �

:

DenotingnT

i ðkÞ ¼ xTðkÞ xTðk� dðkÞÞ xTðk� d2Þ f Ti ðxðkÞÞ

� �, we obtain from (22)–(27) that

Viðxðkþ 1ÞÞ � bViðxðkÞÞ 6 nTi ðkÞðwi þ wi1 þ wi2ÞniðkÞ �

Xk�1

r¼k�d2

bk�rgTðrÞZigðrÞ; ð28Þ

332 X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339

where

wi ¼

�bPi þ ðd2 � d1 þ 1ÞQ i þ Ri 0 0 ai2 I

� �bd2 Q i 0 0� � �bd2 Ri 0� � � �I

266664

377775;

wi1 ¼

ATi

ATdi

0DT

fi

266664

377775Pi Ai Adi 0 Dfi½ �;

wi2 ¼ d2

ðAi � IÞT

ATdi

0DT

fi

266664

377775Zi½Ai � I Adi 0 Dfi �

� � � �

In addition, the following equations hold for any matrices Ni1 ¼

Ni11

Ni12and Ni2 ¼

Ni21

Ni22with appropriate dimensions:2 3

0 ¼ 2bd2 xTðkÞNi11 þ xTðk� dðkÞÞNi12� �

� xðkÞ � xðk� dðkÞÞ �Xk�1

r¼k�dðkÞgðrÞ4 5; ð29Þ

0 ¼ 2bd2 xTðkÞNi21 þ xTðk� dðkÞÞNi22� �

� xðk� dðkÞÞ � xðk� d2Þ �Xk�dðkÞ�1

r¼k�d2

gðrÞ" #

ð30Þ� �

On the other hand, for any matrices Xi ¼Xi11 Xi12

� Xi22> 0, the following equation also holds:

0 ¼ d2bd2

xðkÞxðk� dðkÞÞ

� �T

XixðkÞ

xðk� dðkÞÞ

� �� bd2

Xk�1

r¼k�dðkÞ

xðkÞxðk� dðkÞÞ

� �T

XixðkÞ

xðk� dðkÞÞ

� �� bd2

Xk�dðkÞ�1

r¼k�d2

xðkÞxðk� dðkÞÞ

� �T

XixðkÞ

xðk� dðkÞÞ

� �: ð31Þ

From (28)–(31), we have

Viðxðkþ 1ÞÞ � bViðxðkÞÞ 6 nTi ðkÞðw

0i þ wi1 þ wi2ÞniðkÞ � bd2

Xk�1

r¼k�dðkÞ

xðkÞxðk� dðkÞÞ

gðrÞ

264

375

T

Xi Ni1

NTi1 Zi

� � xðkÞxðk� dðkÞÞ

gðrÞ

264

375

�bd2Xk�dðkÞ�1

r¼k�d2

xðkÞxðk� dðkÞÞ

gðrÞ

264

375

T

Xi Ni2

NTi2 Zi

� � xðkÞxðk� dðkÞÞ

gðrÞ

264

375; ð32Þ

where

w0i ¼

Ci11 Ci12 �bd2 Ni21ai2 I

� Ci22 �bd2 Ni22 0� � �bd2 Ri 0� � � �I

266664

377775:

By Lemma 1, we can obtain that w0i þ wi1 þ wi2 < 0 is equivalent to

Ci11 Ci12 �bd2 Ni21ai2 I AT

i ðAi � IÞT

� Ci22 �bd2 Ni22 0 ATdi AT

di

� � �bd2 Ri 0 0 0� � � �I DT

fi DTfi

� � � � �P�1i 0

� � � � � �d�12 Z�1

i

266666666664

377777777775< 0 ð33Þ

Using diagfI; I; I; Pi; I; Zig to pre- and post-multiply both sides of (33), it is easy to get that (14) is equivalent to (33). Thus itcan be obtained from (14) and (15) that

Viðxðkþ 1ÞÞ 6 bViðxðkÞÞ ð34Þ

X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339 333

When k ¼ kb � 1, denote rðkb � 1Þ ¼ j, (j 2 N), then we have

Vi1ðxðkbÞÞ � lVj1ðxðkb � 1ÞÞ ¼ ðEijxðkb � 1ÞÞT PiEijxðkb � 1Þ � lxTðkb � 1ÞPjxðkb � 1Þ; ð35Þ

Vi2ðxðkbÞÞ � lVj2ðxðkb � 1ÞÞ ¼Xkb�1

r¼kb�dðkbÞbkb�r�1xTðrÞQ ixðrÞ � l

Xkb�2

r¼kb�1�dðkb�1Þbkb�2�rxTðrÞQ jxðrÞ

6 xTðkb � 1ÞQ ixðkb � 1Þ þXkb�2

r¼kb�d1�1

bkb�r�2xTðrÞðbQi � lQ jÞxðrÞ

þXkb�d1�1

r¼kb�d2

bkb�r�1xTðrÞQ ixðrÞ � bd1 xTðkb � d1 � 1ÞQ ixðkb � d1 � 1Þ; ð36Þ

Vi3ðxðkbÞÞ � lVj3ðxðkb � 1ÞÞ 6 ðd2 � d1ÞxTðkb � 1ÞQixðkb � 1Þ þX�d1

s¼�d2þ1

Xkb�2

r¼kbþs

bkb�r�2xTðrÞðbQ i � lQ jÞxðrÞ

� lXkb�d1�1

r¼kb�d2

bkb�r�2xTðrÞQ jxðrÞ; ð37Þ

Vi4ðxðkbÞÞ � lVj4ðxðkb � 1ÞÞ ¼ xTðkb � 1ÞRixðkb � 1Þ þXkb�2

r¼kb�d2

bkb�r�2xTðrÞðbRi � lRjÞxðrÞ � lbd2�1xTðkb � 1� d2ÞRjxðkb � 1� d2Þ; ð38Þ

Vi5ðxðkbÞÞ � lVj5ðxðkb � 1ÞÞ 6 d2xTðkb � 1ÞZixðkb � 1Þ þX�1

s¼�d2

Xkb�2

r¼kbþs

bkb�r�2xTðrÞðbZi � lZjÞxðrÞ � lXkb�2

r¼kb�d2�1

bkb�r�2xTðrÞZjxðrÞ ð39Þ

Combining (35)–(39), we obtain from (17)–(18) that

VrðkbÞðxðkbÞÞ 6 lVrðkb�1Þðxðkb � 1ÞÞ: ð40Þ

Thus according to Lemma 3, system (8) is exponentially stable. The proof is completed. h

Remark 3. Compared with the existing result in the literature [33], the proposed stability condition is delay-dependent.Also, the system considered in this section contains nonlinearity and time-varying delay, which are universal in the appli-cation. On the other hand, our result is different from those presented in [25–26], where the delay is constant and the non-linearity is not considered.

Remark 4. When Dfi ¼ 0, the condition (14) will degenerate tod2 T T2 3

Ci11 Ci12 �b Ni21 Ai Pi ðAi � IÞ Zi

� Ci22 �bd2 Ni22 ATdiPi AT

diZi

� � �bd2 Ri 0 0� � � �Pi 0� � � � �d�1

2 Zi

666664777775 < 0

3.2. Observer design

In this subsection, we will focus on the design of observer (3) for system (1). The following theorem presents sufficientconditions for the existence of the observer.

Theorem 2. Consider system (1), for a given positive scalar 0 < b < 1, if there exist positive definite symmetric matrices

�Pi;�Qi; �Ri; �Zi; any matrices Wi of full rank, and any matrices Yi; �Ni1 ¼

�Ni11�Ni12

� �; �Ni2 ¼

�Ni21�Ni22

� �; �Xi ¼

�Xi11�Xi12

� �Xi22

� �> 0 ði 2 NÞ

with appropriate dimensions, such that

�Ci11�Ci12 �bd2 �Ni21

ai2 WT

i ðAiWi � YiÞT ðAiWi � Yi �WiÞT

� �Ci22 �bd2 �Ni22 0 WTi AT

di WTi AT

di

� � �bd2 �Ri 0 0 0� � � �I DT

fi DTfi

� � � � �WTi �Wi þ �Pi 0

� � � � � �d�12

�Zi

2666666664

3777777775< 0; ð41Þ

334 X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339

�Xi�Ni1

� WTi þWi � �Zi

" #P 0;

�Xi�Ni2

� WTi þWi � �Zi

" #P 0; ð42Þ

where

�Ci11 ¼ �b�Pi þ ðd2 � d1 þ 1Þ�Q i þ �Ri þ bd2 ð�NTi11 þ �Ni11Þ þ d2b

d2 �Xi11;

�Ci12 ¼ bd2 ð��Ni11 þ �NTi12 þ �Ni21Þ þ d2b

d2 �Xi12;

�Ci22 ¼ �bd2 �Q i � bd2 ð�NTi12 þ �Ni12 � �Ni22 � �NT

i22Þ þ d2bd2 �Xi22;

then there exists an observer in the form of (3) such that error system (4) is exponentially stable for any switching signalswith the average dwell time scheme (16), where l P 1 satisfies

�Ni11�Ni12 0

� �Ni22 HTijPi

� � �Pi

264

375 < 0; i; j 2 N; i–j; ð43Þ

bQi 6 lQ j; bRi 6 lRj; bZi 6 lZj; i; j 2 N; i–j; ð44Þ

where

�Ni11 ¼ ETijPiEij � lPj þ ðd2 � d1 þ 1ÞQi þ Ri þ d2Zi;

�Ni12 ¼ �ETijPiHij þ lPj � ðd2 � d1 þ 1ÞQ i � Ri � d2Zi;

�Ni22 ¼ �lPj þ ðd2 � d1 þ 1ÞQ i þ Ri þ d2Zi;

Pi ¼ ðWTi Þ�1�PiðWiÞ�1

; Ri ¼ ðWTi Þ�1�RiðWiÞ�1

; Q i ¼ ðWTi Þ�1 �Q iðWiÞ�1

; Zi ¼ �Z�1i

Moreover, if the above conditions (41), (42) have a feasible solution, the observer gain matrices Li can be obtained by

Li ¼ YiðCiWiÞþ; ð45Þ

where ðCiWiÞþ denotes the pseudo-inverse matrix of CiWi.

Proof. Applying Theorem 1 to system (4) and replacing Ai in Theorem 1 with Ai � LiCi, one can obtain

Ci11 Ci12 �bd2 Ni21ai2 I ðAi � LiCiÞT Pi ðAi � LiCi � IÞT Zi

� Ci22 �bd2 Ni22 0 ATdiPi AT

diZi

� � �bd2 Ri 0 0 0� � � �I DT

fiPi DTfiZi

� � � � �Pi 0� � � � � �d�1

2 Zi

26666666664

37777777775< 0 ð46Þ

Using diagfI; I; I; I;GTi ; Ig and diagfI; I; I; I;Gi; Ig to pre- and post-multiply both sides of (46), respectively, and applying

Lemma 2, we obtain that (46) holds if the following inequality is satisfied:

Ci11 Ci12 �bd2 Ni21ai2 I ðAi � LiCiÞT Gi ðAi � LiCi � IÞT

� Ci22 �bd2 Ni22 0 ATdiGi AT

di

� � �bd2 Ri 0 0 0� � � �I DT

fiGi DTfi

� � � � �GTi � Gi þ Pi 0

� � � � � �d�12 Z�1

i

266666666664

377777777775< 0 ð47Þ

Denote Wi ¼ ðGiÞ�1 and assign the following new matrices:

WTi Ni11Wi ¼ �Ni11; WT

i Ni12Wi ¼ �Ni12; WTi Ni21Wi ¼ �Ni21; WT

i Ni22Wi ¼ �Ni22;

WTi Xi11Wi ¼ �Xi11; WT

i Xi12Wi ¼ �Xi12; WTi Xi21Wi ¼ �Xi21; WT

i Xi22Wi ¼ �Xi22;

WTi PiWi ¼ �Pi; WT

i RiWi ¼ �Ri; WTi Q iWi ¼ �Q i; Yi ¼ LiCiWi; Zi ¼ �Z�1

i :

Using diagfWTi ;W

Ti ;W

Ti ; I;W

Ti ; Ig and diagfWi;Wi;Wi; I;Wi; Ig to pre- and post-multiply both sides of (47), respectively,

we get (41).In addition, using diagfWT

i ;WTi ;W

Ti g and diagfWi;Wi;Wig to pre- and post-multiply both sides of the two inequalities in

(15), respectively, and applying Lemma 2, we obtain (42).From (4b), one has that

X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339 335

Vi1ð~xðkbÞÞ � lVj1ð~xðkb � 1ÞÞ ¼ ðEijxðkb � 1Þ � Hijxðkb � 1ÞÞT PiðEijxðkb � 1Þ � Hijxðkb � 1ÞÞ

� lðxðkb � 1Þ � xðkb � 1ÞÞT Pjðxðkb � 1Þ � xðkb � 1ÞÞ ð48Þ

Then following the proof line of Theorem 1, one obtains that

VrðkbÞð~xðkbÞÞ 6 lVrðkb�1Þð~xðkb � 1ÞÞ

if the following inequalities hold

Ni11 Ni12

� Ni22

� �< 0; i; j 2 N; i–j; ð49Þ

where

Ni11 ¼ ETijPiEij � lPj þ ðd2 � d1 þ 1ÞQ i þ Ri þ d2Zi;

Ni12 ¼ �ETijPiHij þ lPj � ðd2 � d1 þ 1ÞQ i � Ri � d2Zi;

Ni22 ¼ HTijPiHij � lPj þ ðd2 � d1 þ 1ÞQ i þ Ri þ d2Zi

By Lemma 1, it is easy to get that (43) is equivalent to (49).This completes the proof. h

We are now in a position to give a procedure for determining the desired matrices Li and Hijðj; i 2 NÞ.Algorithm 1

Step 1: Given a constant 0 < b < 1, we can obtain the matrices �Pi; �Q i; �Ri; Zi;Wi;Yi; �Ni1; �Ni2 and �Xiði 2 NÞ by solving (41), (42).Step 2: With the matrices obtained in Step 1, we can get Li by (45).Step 3: Solving (43), (44), the matrices Hij (j, i 2 N) and l can be obtained. Then compute the dwell time in (16).

When the proposed observer (3) has the same jumps at switching instants as those in system (1), i.e., Hij = Eij (j,i 2 N), the condition (43) can be written as (17). In the following numerical simulations, Hij = Eij will be chosenand (17) will be used.

4. Numerical examples

In this section, we present two examples to illustrate the effectiveness of the proposed approach.

Example 1. Consider system (8) with the following parameters:

A1 ¼0:56 0

0 0:2

� �; Ad1 ¼

0:1 00:1 0

� �; Df 1 ¼

0:08 00 0:12

� �; E12 ¼

1:3 00 1:1

� �;

A2 ¼0:65 0:1�0:1 0:3

� �; Ad2 ¼

0:11 00 �0:1

� �; Df 2 ¼

�0:13 00:1 0:13

� �; E21 ¼

1:25 00 1:3

� �

Let b ¼ 0:8; d1 ¼ 1; d2 ¼ 3; a1 ¼ a2 ¼ 0:6. Solving the matrix inequalities in Theorem 1, we get l ¼ 3:7873. From (16), itcan be obtained that s�a ¼ 6:9677.

Take

f1ðxðkÞÞ ¼x1ða1 � a1 sinð0:5npkÞj jÞx2ða1 � a1 sinð0:5pkÞj jÞ

� �; f 2ðxðkÞÞ ¼

x1ða2 � a2 cosð0:5pkÞj jÞx2ða2 � a2 cosð0:5pkÞj jÞ

� �:

Choosing sa ¼ 7, simulation results are shown in Figs. 1 and 2, where the initial conditions arexð0Þ ¼ ½5 8 �; xðkÞ ¼ /ðkÞ ¼ 0; k 2 ½�3;0Þ. It can be seen that the system is exponentially stable under the switching signalwith the average dwell time sa ¼ 7.

In order to illustrate the delay effect, we give the admissible average dwell time s�a for different d2 in Table 1.It can be found that if the value of d2 gets larger, the value of s�a tends to get larger. When d2 P 6, the admissible average

dwell time s�a cannot be found by Theorem 1.

Example 2. Consider system (1) with the following parameters:

A1 ¼0:56 0

0 0:2

� �; Ad1 ¼

0:1 00:1 0

� �; B1 ¼

0:20:1

� �; C1 ¼

0:5 00:6 0:8

� �; Df 1 ¼

0:08 00 0:12

� �; A2 ¼

0:65 0:1�0:1 0:3

� �;

Ad2 ¼0:11 0

0 �0:1

� �; B2 ¼

0:10:2

� �; C2 ¼

0:5 0:70 �0:6

� �;

Df 2 ¼�0:13 0

0:1 0:13

� �; E12 ¼

1:3 00 1:1

� �; E21 ¼

1:25 00 1:3

� �:

0 5 10 15 20 250

1

2

3

k

syst

em m

ode

Fig. 1. Switching signal with the average dwell time sa ¼ 7.

0 5 10 15 20 25-1

0

1

2

3

4

5

6

7

8

k

x

x1x2

Fig. 2. State x(k) of the system.

Table 1The admissible average dwell time s�a for different d2.

d2 3 4 5 d2 P 6

s�a 6.9677 9.4204 14.4190 Infeasible

336 X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339

Let b ¼ 0:8; d1 ¼ 1; d2 ¼ 3; a1 ¼ a2 ¼ 0:6. Solving (41), (42) in Theorem 2, we get

W1 ¼1:8233 0:11050:1182 2:2640

� �; W2 ¼

2:0135 �0:0169�0:0024 2:0000

� �; Y1 ¼

0:2131 0:02830:0939 �0:5301

� �; Y2 ¼

0:3509 0:2016�0:1397 �0:4630

� �

Then by (45), L1 and L2 can be obtained:

L1 ¼0:2227 �0:00850:4899 �0:2968

� �; L2 ¼

0:3488 0:2365�0:1393 0:2243

� �

From (16), (43), and (44), we get l ¼ 3:3261 and s�a ¼ 6:3858.

0 5 10 15 20 250

1

2

3

k

syst

em m

ode

Fig. 3. Switching signal with the average dwell time sa ¼ 6:5.

0 5 10 15 20 25-2

-1

0

1

2

3

4

5

6

7

8

k

stat

e

x1x2Estimation of x1Estimation of x2

Fig. 4. State x(k) of the system and its estimation xðkÞ.

0 5 10 15 20 25-1

0

1

2

3

4

5

6

7

8

k

erro

r sta

te

Estimated error of x1Estimated error of x2

Fig. 5. State of the error dynamic system.

X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339 337

338 X. Li, Z. Xiang / Applied Mathematics and Computation 229 (2014) 327–339

Take

f1ðxðkÞÞ ¼x1ða1 � a1 sinð0:5pkÞj jÞx2ða1 � a1 sinð0:5pkÞj jÞ

� �; f 2ðxðkÞÞ ¼

x1ða2 � a2 cosð0:5pkÞj jÞx2ða2 � a2 cosð0:5pkÞj jÞ

� �; f 1ðxðkÞÞ ¼

x1ða1 � a1 sinð0:5pkÞj jÞx2ða1 � a1 sinð0:5pkÞj jÞ

� �;

f 2ðxðkÞÞ ¼x1ða2 � a2 cosð0:5pkÞj jÞx2ða2 � a2 cosð0:5pkÞj jÞ

� �:

Choosing sa ¼ 6:5, simulation results are shown in Figs. 3–5, where xð0Þ ¼ ½5 8 �T ; xðhÞ ¼ ½0 0 �T ;h 2 ½�d2;0Þ; xðhÞ ¼ ½0 0 �; h 2 ½�d2;0�; and uðkÞ ¼ 3 sinð2kÞ.

We can see that the proposed observer can estimate the state of the system. This demonstrates the effectiveness of theproposed method.

5. Conclusion

In this paper, we have studied the problem of observer design for discrete-time nonlinear impulsive switched systemswith time-varying delays. A delay-dependent exponential stability criterion for the considered system is established byusing the average dwell time approach. Then, an observer is proposed. The designed observer can estimate the state ofthe system. Finally, two numerical examples are given to show the effectiveness of the proposed approach.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant No. 61273120.

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