Journal of Applied Mathematics

Volume 2012, Article ID 956370, 17 pages

http://dx.doi.org/10.1155/2012/956370

## Robust Stability of Switched Delay Systems with Average Dwell Time under Asynchronous Switching

^{1}School of Automation Engineering, University of Electronic Science and Technology of China, Sichuan, Chengdu 611731, China^{2}School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan, Chengdu 611731, China^{3}Key Laboratory for Neuroinformation of Ministry of Education, University of Electronic Science and Technology of China, Sichuan, Chengdu 611731, China

Received 10 August 2012; Accepted 7 September 2012

Academic Editor: Chong Lin

Copyright © 2012 Jun Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problem of robust stability of switched delay systems with average dwell time under asynchronous switching is investigated. By taking advantage of the average dwell-time method and an integral inequality, two sufficient conditions are developed to guarantee the global exponential stability of the considered switched system. Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed techniques.

#### 1. Introduction

In recent years, there has been increasing interest in the analysis and switched systems because of its applications in a variety of areas such as signal processing, signal estimation, pattern recognition, communications, control application, and many practical control systems. Switched linear systems comprise a collection of linear subsystems described by differential or difference equations and a switching law to specify the switching among these subsystems. A switched system is a combination of discrete and continuous dynamical systems. All of these systems arise as models for phenomena which cannot be described by exclusively continuous or exclusively discrete processes. Most recently, on the basis of Lyapunov functions and other analysis tools, the stability or stabilization for linear or nonlinear switched systems has been further investigated, and many valuable results have been obtained, for a recent survey on this topic and related questions has attracted increasing attention [1–10]. The average dwell-time is an effective method for the switched systems which do not exist common Lyapunov function. Time delay commonly exists in engineering. Because of time delay, the system can become unstable or less capable, it is significant to study time delay. There are two kinds of stability for switched systems with time delay: time delay-independent stability and time delay-dependent stability. The time delay-independent stability is obviously conservative to the bounded time delay or small time delay, many results are obtained [11–13]. At present, there has been increasing interest in time-delay switched systems, and many valuable results have been obtained [14–18].

It is worth noting that the aforementioned results are all based on the basic assumption that the switching instants are simultaneous with those of the system. However, in actual operation, there inevitably exists asynchronous switching between the controllers and the practical subsystems, that is to say, the real switching instants of the controller exceed or lag behind those of the system, which will deteriorate performance of the systems. In fact, the necessity of taking into consideration the asynchronous switching is shown in efficient controller design in many mechanical and chemical systems. There are some results presented on control synthesis under asynchronous switching which have been proposed [19–28]. However, to the best of our knowledge, the issue of switched delay systems under asynchronous switching has not fully been investigated, which motivated this study for us.

In this paper, we deal with the problem of robust stability and -gain of switched delay systems under asynchronous switching. In terms of the average dwell-time method and an integral inequality, two sufficient conditions are developed to guarantee the global exponential stability of the considered switched system. Finally, a numerical example is provided to illustrate the effectiveness and feasibility.

*Notations.* Throughout this paper, denotes the -dimensional Euclidean space and refers to the set of all real matrices. For real symmetric matrices and , the notation (resp., ) mean that the matrix , are positive semidefinite, (resp., and positive definite). is the identity matrix with appropriate dimensions. represents the elements below the main diagonal of a symmetric matrix. The superscripts and stand for matrix transposition and matrix inverse, respectively; denotes the Euclidean norm. and denote the maximum and minimum eigenvalues of matrix , respectively. The shorthand denotes a block diagonal matrix with diagonal blocks being the matrices . In this paper, if not explicit, matrices are assumed to have compatible dimensions.

#### 2. Preliminaries

In this paper, we consider the following time-delay system described by where is the state of the system, is the noise signal. Switching signal is a piecewise constant function of time , and we take values in a finite set denotes the number of subsystems. means the th subsystem is active, and the corresponding subsystem matrices are denoted by known constant matrices , , and with appropriate dimensions and is the unknown time-varying delay satisfying where , and are constants; , and are known real constant matrices; is a compatible vector-valued initial function on .

The switching signal discussed in this paper is time dependent, that is, , where is the initial instant. In this paper, we denote represent the th switching instant. for convenience, is used to denote the practical switching signal which can be written as where , . Then there exist a matched period time interval and mismatched period time interval because of asynchronous switching. For simplicity, we assume that , .

In this paper, we denote and as follows:

Let

This way, system (2.1) can be rewritten as where

First of all, we will give some definitions and lemmas about system (2.6) which plays an important role in the derivation of our result.

*Definition 2.1 (see [29]). *The unforced system is said to be exponential stable if there exist constants and such that

*Definition 2.2 (see [30]). *For , the switched system (2.1) is said to have weighted -gain, if under zero initial condition , it holds that

*Definition 2.3 (see [30]). *For any , let denote the switching number of discontinuities of during on an intercal . If holds for and , then and are called chattering bound and average dwell time, respectively. Here we assume for simplicity as commonly used in the literature.

Lemma 2.4 (see [29]). *For any given symmetric positive definite matrix , and scalars , if there exists a vector function such that the following integration is well defined, then
*

Lemma 2.5 (see [28]). *Let and . If there exists a real-value continuous function , , such that the differential inequality
**
holds, then
**
where , and satisfies .*

Lemma 2.6 (see [9]). * If a real scalar function satisfies the following differential inequality:
**
where , then
*

#### 3. Main Results

The following theorem presents a sufficient stability condition for system (2.1). We first present conditions , the corresponding closed-loop system is given by where

Theorem 3.1. *For given scalars , , then the system (2.1) is exponentially stable, if there exist positive-definite matrices , and such that the following LMIs hold:
**
with
**In this case, for any switching signal with the average dwell-time satisfying
**
System (3.1) is exponentially stable with satisfying that
*

*Proof. *When , we consider the following Lyapunov-Krasovskii functional:
where

Taking the time derivative of for along the trajectory of the system (2.1) turns out to be

On the other hand, according to Lemma 2.4, we get that

Then we can get
where
with

In view of Schur complement, (3.3) implies that . Then we have for all .

Then during the matched period, by Lemma 2.5, satisfy

When , we consider the following Lyapunov-Krasovskii functional:
where
where .

Similarly we have that
where
with

Through Lemma 2.5, we also have

When , we have the relationship , and it follows that

From (3.7), we can obtain

Then through (3.23) and (3.24), we can easily get

By (3.9) and (3.25), then we have
which means
where

Similarly, when , we can also have
where

For convenience, let , , through (3.27) and (3.29), we have

Theorem 3.2. *For given scalars , then the system (2.1) is exponentially stable with -gain, if there exist positive-definite matrices , and , such that the following LMIs hold:
**
with
*

*Proof. *For all nonzero and a scalar , then we establish system (2.1) with -gain performance . For convenience, denoting .

When , by the system (2.1), we can obtain

From (3.32), we can easily get

Integrate this inequality during , it is known that

When , by the system (2.1), by the same way, we can obtain

Then we have

When , it follows that

Under the zero initial condition, Let , (3.40) implies

Integrate (3.41) during , then we can obtain

When , by the same mathematical operations, we have .

From which we can get . This proof is completed.

*Remark 3.3. *If , which implies that , by (3.3)-(3.4) and (3.32)-(3.33), we have , then it requires a common Lyapunov functional for all subsystems, and the switching signals can be arbitrary. If , we get from (3.3)-(3.4) and (3.32)-(3.33) that there is no switching, that is, switching signal will have a great dwell-time on the average.

#### 4. Illustrative Example

In this section, a numerical example is given to illustrate the effectiveness of the obtained results.

*Example 4.1. *Consider the system (2.1) with parameters as follows:

is fixed and assumed to be 0.2. The initial condition is assumed to be , . Then by solving the LMIs in Theorem 3.2, different and for different , , and can be obtained in Table 1. It can be seen that, for the given , the upper bounds of the time delay and the minimal average dwell-time are dependent on , , and . Then the simulation result of the system is shown in Figures 1, 2, and 3. The switching signal with average dwell-time is shown in Figure 1. Figures 2–3 indicate that the state response of the switched system without asynchronous switching and with asynchronous switching, respectively.

#### 5. Conclusions

In terms of the LMI approach, the problem of robust stabilization of switched delay systems with average dwell-time under asynchronous switching has been considered. Two sufficient conditions are developed to guarantee the global exponential stability of the considered switched system. At last, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed techniques.

#### Acknowledgment

This work was supported by the Fundamental Research Funds for the Central Universities (103.1.2E022050205).

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