Journal of Applied Mathematics

Volume 2012, Article ID 202916, 17 pages

http://dx.doi.org/10.1155/2012/202916

## Stability Analysis and Robust Control of Switched Stochastic Systems with Time-Varying Delay

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China

Received 9 December 2011; Accepted 6 March 2012

Academic Editor: Vu Phat

Copyright © 2012 Zhengrong Xiang and Guoxin Chen. 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 problems of mean-square exponential stability and robust control of switched stochastic systems with time-varying delay are investigated in this paper. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criterion of such system is derived in terms of linear matrix inequalities (LMIs). Then, performance is studied and robust controller is designed. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

#### 1. Introduction

Switched systems, a special hybrid system, are composed of a set of continuous-time or discrete-time subsystems and a rule orchestrating the switching between the subsystems. In the last two decades, there has been increasing interest in the stability analysis and control design for such switched systems since many real-world systems such as chemical systems [1], robot control systems [2], traffic systems [3], and networked control systems [4, 5] can be modeled as such systems. The past decades have witnessed an enormous interest in the stability analysis and control synthesis of switched systems [6–11].

It is well known that time delay phenomenon exists in many engineering systems such as networked systems and long-distance transportation systems. Such phenomenon may cause the system unstable if it cannot be handled properly, which motivates many scientists to involve themselves in researching switched systems with time delay. Many results have been reported for stability analysis of switched systems with time delay [12, 13], where the asymptotical stability criteria are given by using common Lyapunov function approach in [12], and the exponential stability criteria under average dwell time switching signals are proposed in [13]. Moreover, the problem of delay-dependent global robust asymptotic stability of switched uncertain Hopfield neural networks with time delay in the leakage term is discussed in [14]. control of continuous-time switched systems with time delay and discrete-time switched systems with time delay are investigated in [15, 16], respectively.

On the other hand, stochastic systems have attracted considerable attention during the past decades because stochastic disturbance exists in many actual operations. Many useful results on the stability analysis of stochastic systems are reported in [17–21]. The problem of robust control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays is considered [22]. Based on the results of stochastic systems and switched systems, the stability analysis and stabilization of switched stochastic systems are investigated in [23, 24]. Furthermore, the problems of reliable control and reliable control for switched stochastic systems under asynchronous switching are studied in [25, 26], respectively. Recently, these results are extended to stochastic switched systems with time delay, and the exponential stability criteria are addressed [27, 28]. However, these results are very complex, which make it more difficult for us to solve many issues such as controller design under asynchronous switching and actuator failures. Therefore, there is a lot of work to do in such field. This motivates the present study.

In this paper, we focus on the mean-square exponential stability analysis and robust control of switched stochastic systems with time-varying delay. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criterion is derived. Moreover, performance is studied and state feedback controller is proposed. The remainder of the paper is organized as follows. In Section 2, problem statement and some useful lemmas are given. In Section 3, based on the average dwell time method and Gronwall-Bellman inequality, the mean-square exponential stability and performance of the switched stochastic systems with time delay are investigated. Then, robust controller is designed. In Section 4, a numerical example is given to illustrate the effectiveness of the proposed approach. Finally, concluding remarks are provided in Section 5.

*Notation. *Throughout this paper, the superscript ‘‘’’ denotes the transpose, and the symmetric terms in a matrices are denoted by *∗*. The notation means that matrix is positive definite (positive semidefinite, resp.). denotes the dimensional Euclidean space. denotes the Euclidean norm. is the space of square integrable functions on . and denote the maximum and minimum eigenvalues of matrix , respectively. is an identity matrix with appropriate dimension. denotes diagonal matrix with the diagonal elements .

#### 2. Problem Formulation and Preliminaries

Consider the following stochastic switched systems with time-delay: where is the state vector, is the initial state function, is the control input, is the disturbance input which is assumed belong to , is the signal to be estimated, is a zero-mean Wiener process on a probability space satisfying where is the sample space, is -algebras of subsets of the sample space, is the probability measure on , and is the expectation operator. is the system state delay satisfying where is a known constant. The function is a switching signal which is deterministic, piecewise constant, and right continuous. The switching sequence can be described as , , where is the initial time and denotes the th switching instant. Moreover means that the th subsystem is activated.

For each for all , , , and are known real-value matrices with appropriate dimensions, and , and are uncertain real matrix with appropriate dimensions, which can be written as where , , and are known real-value matrices with appropriate dimensions, and is unknown time-varying matrix that satisfies

*Definition 2.1. *System (2.1) is said to be mean-square exponentially stable with under switching signal , if there exist scalars and , such that the solution of system (2.1) satisfies , for all . Moreover, is called the decay rate.

*Definition 2.2. *For any , let denote the switching number of on an interval . If
holds for given , , then the constant is called the average dwell time. As commonly used in the literature, we choose .

*Definition 2.3. *Let be a positive constant, for system (2.1), if there exists a controller and a switching signal , such that(1)when , system (2.1) is mean-square exponentially stable;(2)under zero initial condition , , the output satisfies
Then system (2.1) is said to be robustly exponentially stabilizable with a prescribed weighted performance, where .

The following lemmas play an important role in the later development.

Lemma 2.4 (see [29] (Gronwall-Bellman Inequality)). * Let and be real-valued nonnegative continuous functions with domain ; is a nonnegative scalar; if the following inequality
**
holds, for , then .*

Lemma 2.5 (see [30]). *Let , , , and be constant matrices of appropriate dimensions with satisfying , then for all , , if and only if there exists a scalar such that .*

#### 3. Main Results

##### 3.1. Stability Analysis

In this subsection, we will focus on the exponential stability analysis of switched stochastic systems with time-varying delay.

Consider the following switched stochastic system:

Theorem 3.1. *Considering system (3.1), for a given scalar , if there exist symmetric positive definite matrices satisfying
**
for all , then system (3.1) is mean-square exponentially stable under arbitrary switching signal with the average dwell time:
**
where satisfies
*

*Proof. *Consider the following Lyapunov functional for the th subsystem:
where
For the sake of simplicity, is written as in this paper.

According to Itô formula, along the trajectory of system (3.1), we have
where
According to (2.3), we can obtain that
where
Using Schur complement, it is not difficult to get that if inequality (3.2) is satisfied, the following inequality can be obtained:
Combining (3.7) with (3.11) leads to
Noticing (2.2) and taking the expectation to (3.12), we have
According to (3.4)–(3.6), we have
Assume that the th subsystem is activated during and th subsystem is activated during , respectively. Using Itô formula and according to (3.13)–(3.15), we have, for any ,
According to Lemma 2.4 and when (3.3) holds, we have
Moreover, we can obtain
where , and is the decay rate.

The proof is completed.

*Remark 3.2. *The exponential stability criterion of stochastic switched systems with time-varying delay is given in Theorem 3.1. When , system (3.1) is degenerated to the switched system with time-varying delay, which can be described as
Using the same method, we can obtain the following exponential stability criterion of switched system (3.19).

Corollary 3.3. *Considering system (3.19), for a given scalar , if there exist symmetric positive definite matrices satisfying
**
for all , system (3.19) is exponentially stable under arbitrary switching signal with average dwell time satisfying (3.3).*

##### 3.2. Performance Analysis

In this subsection, we will investigate the performance of switched stochastic systems with time-varying delay.

Consider the following switched stochastic system:

Theorem 3.4. *Considering system (3.21), for a given scalar , if there exist symmetric positive definite matrices such that
**
hold for all , system (3.21) is said to have weighted performance under arbitrary switching signal with the average dwell time:
**
where satisfies
*

*Proof. *By Theorem 3.1, we can readily obtain that system (3.21) is mean-square exponential stable when .

Assume that the th subsystem is activated during . Choose the following Lyapunov functional candidate for the th subsystem:
where
Using Itô formula, along the trajectory of system (3.21); we have
where
Let ; we have
where , and
Combining (3.22) with (3.29)–(3.30), and using Schur complement, we have
Noticing (2.2) and taking the expectation to (3.27), we have
According to (3.25)–(3.27), we have
Using Itô formula, we have, for any ,
Under zero initial condition, we can obtain
Moreover, we have
Multiplying both sides of (3.36) by leads to
Noticing and , we have
When , it leads to
The proof is completed.

*Remark 3.5. *When , system (3.21) is reduced to a switched delay system, which can be described as
Using the method proposed in Theorem 3.4, we can obtain the following conclusion.

Corollary 3.6. *Considering system (3.40), for a given scalar , if there exist symmetric positive definite matrices such that
**
hold for all , system (3.40) is said to have weighted performance under arbitrary switching signal with the average dwell time scheme (3.23).*

##### 3.3. Design of Robust Controller

In this subsection, the following robust controller will be designed for system (2.1). Then the corresponding closed-loop system can be described as

Theorem 3.7. *Considering system (2.1), for given scalars , , , if there exist matrix , symmetric positive definite matrices , > 0 such that
**
holds for all , with the average dwell time:
**
where satisfies
**
then, there exists a robust controller:
**
which can render the corresponding closed-loop system (3.43) mean-square exponentially stable with weighted performance , where
*

*Proof. *By Theorem 3.4, system (3.43) is mean-square exponentially stable with weighted performance if the following inequalities are satisfied:
where .

Then using to pre- and postmultiply , we have

Furthermore,
where , , , and .

Combining (3.51) with (2.4), we have
where

Moreover,
According to Lemma 2.5, we have
Substituting (3.55) to (3.52), and using Schur complement, we can obtain that (3.52) is equivalent to (3.44). Denoting , and , it is easy to get that (3.46) is equivalent to (3.24).

The proof is completed.

*Remark 3.8. *Theorem 3.4 presents the sufficient conditions which could guarantee that the switched stochastic delay system is stable with performance; when the robust control problem is considered, we can solve the problem by substituting the closed-loop uncertain parameters , , to Theorem 3.4.

#### 4. Numerical Example

In this section, a numerical example is given to illustrate the effectiveness of the proposed approach. Consider system (2.1) with the following parameters: the disturbance input .

Let , , , ; then solving the LMIs in Theorem 3.7, we have Then we obtain that . Thus, under the average dwell time , the designed controller can guarantee that the corresponding closed-loop system is mean-square exponentially stable with weighted performance.

Simulation results are shown in Figures 1–3, where the initial state , , and . The switching signal with the average dwell time is shown in Figure 1, and Figures 2 and 3 show state and of the closed-loop system, respectively.

#### 5. Conclusions

In this paper, the exponential stability analysis and robust control for switched stochastic time delay systems have been investigated. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criteria and performance analysis are presented. Furthermore, robust controller is designed to guarantee that the corresponding closed-loop system is mean-square exponentially stable. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach. The proposed method provides a powerful tool to solve many other problems such as controller design under asynchronous switching and actuator failures. These problems are the topics of the future research.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 60974027 and NUST Research Funding (2011YBXM26).

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