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CIRCUITS SYSTEMS SIGNAL PROCESSING

 Birkh

auser Boston (2007)

OL. 26, NO. 3, 2007, PP. 361–377 DOI: 10.1007/s00034-006-0414-x

STABILIZATION OF SWITCHED

LINEAR SYSTEMS WITH

TIME-VARYING DELAY IN

SWITCHING OCCURRENCE

DETECTION*

Zhijian Ji,

Xiaoxia Guo,

Shixu Xu,

and Long Wang

Abstract. In this paper, we study stabilization of switched linear systems with time-varying

delay in switching occurrence detection. The stabilization is realized through online and

ofﬂine feedback mechanisms, respectively, and a state feedback controller design approach

is proposed for each mechanism. To develop the main results, a more precise calculation

method is proposed to estimate overshoots of transition matrices, which can be viewed as

a reﬁnement of existing results.

Key words: Switched linear systems, stabilization, feedback, switching rules.

1. Introduction

In recent years, the study of switched systems has attracted considerable attention

because of its signiﬁcance in both academic research and practical application

[18], [4]. In spite of the fruitful results reported on the stability analysis and design

of switched systems, the theoretical framework established is far from complete.

There are several excellent survey papers such as [18], [13], [5] to which the

reader is referred for surveys of recent developments on switched systems.

The stabilization of switched systems is important and challenging and has

been extensively investigated in the literature during the last decade [17], [4].

Among various methodologies, stabilization under certain classes of switching

∗

Received April 14, 2006; revised September 17, 2006; published online June 25, 2007. This

work was supported by the National Natural Science Foundation of China (Nos. 60604032,

10601050, 60674050, 60528007) and National 973 Program (2002CB312200).

School of Automation Engineering, Qingdao University, Qingdao, 266071, China. E-mail for Ji:

[email protected] (corresponding author); E-mail for Xu: [email protected]

Department of Mathematics, Ocean University of China, 266071, China. E-mail for Guo:

[email protected]

Intelligent Control Laboratory, Center for Systems and Control, Department of Mechan-

ics and Engineering Science, Peking University, Beijing, 100871, China. E-mail for Wang:

[email protected]

362 JI,GUO,XU, AND WANG

sequences has been studied by many researchers, and it is also one of the three

basic problems outlined in [13]. The difﬁculty inherent in stability analysis and

stabilization of switched systems comes mainly from its hybrid nature [26]. To

manage the difﬁculty, one approach adopted is to constrain switching rules to

satisfy the dwell or average dwell time condition. This approach was ﬁrst pro-

posed by Morse [15], and then developed by Hespanha [6], [7], Zhai [23], and

Xie and Wang [20]. The key point of the approach is to keep a running subsystem

unswitched for a period long enough to allow the overshoot of the closed-loop

system in the transient phases to fade. Recently, concerning the stabilization of

switched linear systems, Cheng, Guo, Lin, and Wang put forward a concept of

switching frequency, which can be deemed a further development of the average

dwell time method [4]. The way in which the switching frequency is deﬁned al-

lows the acquired results to apply to the case when switching rules have some fast

switchings on some intervals, provided that the switching frequency is bounded

on average in the long run. The switching rules with dwell or average dwell

time belong to a class of time-dependent rules. Note that there is also a certain

class of state-dependent switching rules that are commonly used in the study of

stabilization of switched linear systems (see, e.g., [12], [22], [14]). The adoption

of this class of switching rules is usually associated to a constructive partition of

state space, and the computational burden involved is heavy, especially for the

high-dimension case.

Besides stabilization of switched systems under a certain class of constrained

switching rules, ﬁnding an appropriate switching rule to make switched systems

stable is another important and challenging stabilization problem. Wicks, Peleties,

and DeCarlo ﬁrst developed an elegant construction of a stabilizing switching rule

by examining the existence of a stable convex combination of subsystems [19].

This constructive design method for stabilizing switching rules has been utilized

for switched controller design and robustness [1], [17], [24], [25], [8]–[11]. Its

application is commonly related to a constructive partition of state space, and

multiple Lyapunov-like functions have been introduced to cope with the intrinsic

discontinuous nature of switched systems [8], [21], [2].

In this paper, we consider the stabilization of switched linear systems with

time-varying delay in detection of switching occurrence. In the context of prac-

tical application of switched systems, one cannot detect the changing value of

switching rules instantly, but only after a time period. However, most of the

above-mentioned results do not involve time delay in the associated switching

rules, and hence become ineffective in such a case. Xie and Wang studied the

stabilization of switched linear systems with a constant time delay in detecting the

switching signal [20]. Instead of the constant time-delay assumption, we deal with

a time-varying delay in this paper. The main results are to be established by ﬁrst

developing a much more precise estimation on overshoots of transition matrices

than that reported, which can be considered an enhancement of the Squashing

Lemma in [6]. This is different from the approach adopted in [20] to cope with a

SWITCHED LINEAR SYSTEMS WITH TIME-VARYING DELAY 363

constant time delay. Our approach can be used to deal with a time-varying delay,

and the results can be proved more concisely.

The paper is organized as follows. In Section 2, we present preliminaries in-

cluding the system description, deﬁnitions and a supporting lemma. Main results

are presented in Section 3. An illustrative example is given in Section 4. Section 5

brieﬂy concludes the work.

2. Problem formulation and preliminaries

We consider a stabilization problem for the following switched linear system

given by



˙x(t) = A

σ(t)

x(t) + B

σ(t)

u(t ),

γ(t) = σ(t − τ(t )),

(1)

where A

∈ R

n×n

, B

∈ R

n×m

. The right continuous function σ(t) :[t

, ∞) →

{1, 2,...,N } is the switching rule created by some unknown or nondeterminis-

tic function (e.g., unexpected fault, change of working points, etc.). Moreover,

σ(t) = i implies that the subsystem ( A

, B

) is activated, i = 1, 2,...,N . γ(t)

is the detection function of σ(t), the time-varying delay τ(t)>0 means that one

cannot detect which subsystem (A

, B

) is being activated at time t instantly, but

after a time period τ(t ). We assume that τ(t) is a piecewise constant signal that

commutes synchronously with the switched system dynamics, so that the resulting

signal γ(t) would present a single commutation between any two consecutive

switching occurrences.

Associated to the switching rule σ(t), the switching instants 0 = t

< t

< ··· of σ(t) are deﬁned recursively by t

m+1

= inf {t > t

|σ(t) = σ(t

)}.

The switching duration δ

is denoted by δ

= t

m+1

− t

, m = 0, 1, 2,..., and

the value of τ(t) at the switching instant t

is denoted by τ

= τ(t

). We make

the following assumptions.

Assumption 1. Each subsystem (A

, B

) is controllable , i = 1,...,N .

Assumption 2. If (A

, B

) is activated, it will hold at least for a period larger than

Assumption 3. The delay in the switching detection is available, and τ(t ) ≤

τ, ∀t > 0.

Remark 1. Assumption 1 is reasonable because controllability is generic for real

systems. Assumption 2 is reasonable because we require the switching rule to be

well deﬁned, i.e., the number of switchings is ﬁnite in any ﬁnite time interval

[17]. Assumption 2 means that if (A

, B

) is activated on instant t

, and held

on time interval [t

, t

i+1

), then the duration time δ

that (A

, B

) holds satisﬁes

> h

. Assumption 3 is also reasonable because the time-varying delay cannot

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