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Available online at www.sciencedirect.com

Journal of the Franklin Institute ] (]]]]) ]]]–]]]

Stabilization of discrete-time switched singular

time-delay systems under asynchronous switching

Jinxing Lin

, Shumin Fei

, Zhifeng Gao

College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China

Key Laboratory of Measurement and Control of CSE (School of Automation, Southeast University), Ministry of

Education, Nanjing 210096, PR China

Received 18 May 2011; received in revised form 14 February 2012; accepted 15 February 2012

Abstract

This paper is concerned with the problem of state feedback stabilization of a class of discrete-time

switched singular systems with time-varying state delay under asynchronous switching. The

asynchronous switching considered here means that the switching instants of the candidate

controllers lag behind those of the subsystems. The concept of mismatched control rate is introduced.

By using the multiple Lyapunov function approach and the average dwell time technique, a sufﬁcient

condition for the existence of a class of stabilizing switching laws is ﬁrst derived to guarantee the

closed-loop system to be regular, causal and exponentially stable in the presence of asynchronous

switching. The stabilizing switching laws are characterized by a upper bound on the mismatched

control rate and a lower bound on the average dwell time. Then, the corresponding solvability

condition for a set of mode-dependent state feedback controllers is established by using the linear

matrix inequality (LMI) technique. Finally, two numerical examples are provided to illustrate the

effectiveness of the proposed method.

1. Introduction

Switched systems have drawn considerable attention since the 1990s, due to their great

ﬂexibility in modeling and control of practical systems, for example, event-driven systems,

www.elsevier.com/locate/jfranklin

doi:10.1016/j.jfranklin.2012.02.009

This work was supported by the National Natural Science Foundation of China under Grant 60904020 and

the Scientiﬁc Research Foundation of Nanjing University of Posts and Telecommunications (NY210080).

Corresponding author.

E-mail address: jxlin2004@126.com (J. Lin).

Please cite this article as: J. Lin, et al., Stabilization of discrete-time switched singular time-delay systems under

asynchronous switching, Journal of the Franklin Institute. (2012), doi:10.1016/j.jfranklin.2012.02.009

logic-based systems, parameter- or structure-varying systems, and highly complex

nonlinear syst ems, etc.; for details, see [1–3] and the references therein. A switched system

consists of a collection of continuous- or discrete-time subsystems and a switching rule

specifying the switching among them. According to the features of the switching rule,

switched systems can be class iﬁed into two classes: systems under uncontrolled switching

and systems under controlled switching. In the ﬁrst class the attention is focused on

stability analysis and synthesis of stabilizing controllers with given switching signals,

including arbit rary switching and stochastic switching governed by Markov chains. The

other, which is of interest in this paper, is on synthesizing a stabilizing switching signal or

even corresponding controllers for a given collection of subsystems. It has been proved

that the multiple Lyapunov function approach [4] and average dwell time technique [5] are

two power ful and effective tools to deal with switched systems under controlled switching.

Besides switching properties, many engineering systems always involve time-delay

phenomenon due to various reasons such as inherent phenomena like mass transport

ﬂow and recycling and/or by-products of computational delays. Therefore, the study of

switched systems with time delays has become a hot topic in control community during the

last decade, see, e.g. [6–12] and the refer ences therein.

As an important class of switched systems, switched singular (SS) systems have found

many practical application in industry, for example, electrical networks [13], DC motor

[13], networked control systems [14], etc., and even in economic systems [15]. Singular

systems, also known as descriptor, implicit or differential-algebraic systems, are much

superior to systems represented by state-space models due to their capacity to describe the

algebraic constraints between physical variables [16]. Compared with switched state-space

models, the study of SS systems is more arduous, since not only stability, but also

regularity and impulse elimination (for continuous-time SS systems) and causality (for

discrete-time SS systems) shou ld be considered simultaneously. The last decade has

witnessed a rapidly growing interest in SS systems, and many important results have been

reported in [17–23], and references therein. Speciﬁcally, the control problems for discrete-

time SS systems with or without time delays under arbitrary switching is investigated in

[18–20], and for discrete-time SS time-delay systems under stochastic switching is

addressed in [22].

It should be pointed out that, in all the afore-mentioned results on control of SS systems,

it is implicitly assumed that the con trollers are switched synchronously with the switching

of the subsystems. In actual operation of switched systems, however, this assumption may

be unfeasible. The reason is mainly twofold. Firstly, the temporary failure of component or

the transmission delay will inevitably impede detecting the change of the subsystem’s

switching signal instantly, but after a time period, which results in the switching signals

available to the controller be a delayed version of the subsystem’s switching signals [24,25].

A typical example can be found in networked switched control systems, where the switched

plant and the switched con trollers are separated by a communication channel [26].Dueto

transmission delay, there inevitably exists asynchronous switching phenomena in the

closed-loop system. Secondly, in some situation, it may be necessary to design a robust

switching signal that can adapt the uncertain environment. One good example is the

switched control of nonlinear chemical systems whose mode of operation changes

according to a given time-depending switching rule [27]. So, from reliability as well as

performance point of view, it is quite necessary to design a switched control system that

could tolerate asynchronous switchi ng between the controllers and subsystems while still

J. Lin et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]2

Please cite this article as: J. Lin, et al., Stabilization of discrete-time switched singular time-delay systems under

asynchronous switching, Journal of the Franklin Institute. (2012), doi:10.1016/j.jfranklin.2012.02.009

retaining certain properties. In the past few years, some interesting results on stabilization

of switched state-space systems under asynchronous switching have been presented in the

literature; see [24,25,28–33] for non-delay cases and [34 ,35] for time-delay cases. However,

to the authors’ knowledge, the problem of stabilization of discrete-time SS time-delay

systems under asynchronous switching has not yet been investigated. Moreover, the

procedures given in [34,35] cannot be applied to the discrete-time case. This motivates the

present study.

In this paper, we investigate the state feedback stabilization problem for a class of

discrete-time SS time-delay systems under asynchronous switching. The concept of

mismatched control rate is introduced. By using the multiple Lya punov function approach

and the average dwell time technique, a sufﬁcient con dition for the existence of a

stabilizing switching law is derived at a ﬁrst attempt to guarantee the closed-loop system to

be regula r, causal and exponentially stabile in the presence of asynchronous switching. The

stabilizing switching law is characterized by a upper bound on the mismatched control rate

and a lower bound on the average dwell time. Then, the corresponding solvability

condition for a set of mode-dependent state feedback controllers is established by using the

linear matrix inequality (LMI) technique. Finally, two numerical examples are provided to

illustrate the e ffectiveness of the proposed method.

Notation: For real symmetric matrices P, P40 ðPZ0Þ means that matrix P is positive

deﬁnite (semi-positive deﬁnite). l

max

ðPÞ (l

min

ðPÞ) denotes the largest (smallest) eigenvalue

of the positive deﬁnite matrix P. R

is the n-dimensional real Euclidean space. R

mn

is the

set of all real m n matrices. Z

represents the sets of all non-nega tive integers. The

superscript ‘T’ represents matrix transposition, and ‘

n’ in a matrix is used to represent the

term which is induced by symm etry. J  J refers to the Euclidean vector norm. diagfg

stands for a block-diagonal matrix. SymfAg is the shorthand notation for AþA

2. Description of problem and preliminaries

2.1. Problem formulation

Consider a class of discrete-time SS time-delay systems of the form

kþ1

¼A

sðkÞ

þ A

dsðkÞ

kdðkÞ

þ B

sðkÞ

¼f

, k ¼d , ...,1; 0

(

ð1Þ

where x

2 R

is the system state, u

2 R

is the control input, and f

, k ¼d, ...,1; 0is

the initial condition sequence. d(k) is a time-varying delay and satisﬁes

d rdðkÞrd , where

d and d are constant positive scalars representing the minimum and maximum delays,

respectively. sðkÞ : Z

-I ¼f1; 2, ...,Ng (N is the number of subsystems or modes) is the

switching signal, which is a piecewise continuous (from the right) function of time. The

matrix E is singular and rank E ¼ron. A

, A

and B

, 8i 2 I , are co nstant matrices. As

often assumed in the switched system literature, we exclude Zeno behavior for the

switching signal here. Corresponding to the switching signal sðkÞ, we denote the switching

sequence by S ¼fði

Þ,ði

Þ, ...,ði

Þ, ...ji

2 I ,p 2 Z

g with k

¼0, which means

that the i

th subsystem is activated when k

rkok

jþ1

, or equivalently, sðkÞ¼i

when

rkok

jþ1

. The quadruple-matrix ðE,A

Þ, i

2 I , represents the i

th subsystem or

th mode of system (1).

J. Lin et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 3

Please cite this article as: J. Lin, et al., Stabilization of discrete-time switched singular time-delay systems under

asynchronous switching, Journal of the Franklin Institute. (2012), doi:10.1016/j.jfranklin.2012.02.009

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