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

The problems of mean-square exponential stability and robust H ∞ 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 (cid:2) LMIs (cid:3) . Then, H ∞ performance is studied and robust H ∞ controller is designed. Finally, a numerical example is given to illustrate the e ﬀ ectiveness of the proposed approach.


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 2 Journal of Applied Mathematics 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 . H ∞ 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 H ∞ 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 H ∞ 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 H ∞ 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, H ∞ performance is studied and H ∞ 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 H ∞ performance of the switched stochastic systems with time delay are investigated. Then, robust H ∞ 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 "T " denotes the transpose, and the symmetric terms in a matrices are denoted by * . The notation X > Y X ≥ Y means that matrix X − Y is positive definite positive semidefinite, resp. . R n denotes the n dimensional Euclidean space. x t denotes the Euclidean norm. L 2 t 0 , ∞ is the space of square integrable functions on t 0 , ∞ . λ max P and λ min P denote the maximum and minimum eigenvalues of matrix P , respectively. I is an identity matrix with appropriate dimension. diag{a i } denotes diagonal matrix with the diagonal elements a i , i 1, 2, . . . , n.

Problem Formulation and Preliminaries
Consider the following stochastic switched systems with time-delay: Journal of Applied Mathematics 3 where x t ∈ R n is the state vector, ϕ t ∈ R n is the initial state function, u t ∈ R l is the control input, v t ∈ R p is the disturbance input which is assumed belong to L 2 t 0 , ∞ , z t ∈ R q is the signal to be estimated, w t ∈ R is a zero-mean Wiener process on a probability space Ω F P satisfying where Ω is the sample space, F is σ-algebras of subsets of the sample space, P is the probability measure on F, and E{·} is the expectation operator. h t is the system state delay satisfying where h d is a known constant. The function σ t : t 0 , ∞ → N {1, 2, . . . , N} is a switching signal which is deterministic, piecewise constant, and right continuous. The switching sequence can be described as σ : where t 0 is the initial time and t k denotes the kth switching instant. Moreover σ t i means that the ith subsystem is activated.
For each for all i ∈ N, C i , G i , and M i are known real-value matrices with appropriate dimensions, and A i , B i , and D i are uncertain real matrix with appropriate dimensions, which can be written as where A i , B i , and D i are known real-value matrices with appropriate dimensions, and F i t 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 σ t , if there exist scalars κ > 0 and α > 0, such that the solution x t of Moreover, α is called the decay rate.
holds for given N 0 ≥ 0, T α > 0, then the constant T α is called the average dwell time. As commonly used in the literature, we choose N 0 0.
Then system 2.1 is said to be robustly exponentially stabilizable with a prescribed weighted H ∞ performance, where λ > 0.
The following lemmas play an important role in the later development.

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: Journal of Applied Mathematics 5 for all i ∈ N, then system 3.1 is mean-square exponentially stable under arbitrary switching signal with the average dwell time: Proof. Consider the following Lyapunov functional for the ith subsystem: For the sake of simplicity, V i t, x t is written as V i t in this paper. According to Itô formula, along the trajectory of system 3.1 , we have where

3.8
According to 2.3 , we can obtain that Journal of Applied Mathematics 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 ith subsystem is activated during t k , t k 1 and jth subsystem is activated during t k−1 , t k , respectively. Using Itô formula and according to 3.13 -3.15 , we have, for any t ∈ t k , t k 1 ,

3.16
Journal of Applied Mathematics 7 According to Lemma 2.4 and when 3.3 holds, we have Moreover, we can obtain where κ max i∈N λ max P i hλ max Q i /min i∈N λ min P i , and λ 1/2 α − ln μ/T α is the decay rate.
The proof is completed.

Remark 3.2.
The exponential stability criterion of stochastic switched systems with timevarying delay is given in Theorem 3.1. When w t 0, system 3.1 is degenerated to the switched system with time-varying delay, which can be described aṡ

3.19
Using the same method, we can obtain the following exponential stability criterion of switched system 3.19 .

H ∞ Performance Analysis
In this subsection, we will investigate the H ∞ performance of switched stochastic systems with time-varying delay. Consider the following switched stochastic system:

dx t A σ t x t B σ t x t − h t G σ t v t dt D σ t x t dw t ,
x t ϕ t , t ∈ t 0 − h, t 0 , z t M σ t x t .
hold for all i ∈ N, system 3.21 is said to have weighted H ∞ performance γ under arbitrary switching signal with the average dwell time: Proof. By Theorem 3.1, we can readily obtain that system 3.21 is mean-square exponential stable when v t 0. Assume that the ith subsystem is activated during t k , t k 1 . Choose the following Lyapunov functional candidate for the ith subsystem: Using Itô formula, along the trajectory of system 3.21 ; we have where 28 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

3.32
According to 3.25 -3.27 , we have Using Itô formula, we have, for any t ∈ t k , t k 1 , Multiplying both sides of 3.36 by e −N σ t 0 ,t ln μ leads to Noticing When t → ∞, it leads to The proof is completed.
Remark 3.5. When dw t 0, system 3.21 is reduced to a switched delay system, which can be described asẋ

3.40
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 α > 0, if there exist symmetric positive definite matrices P i , Q i > 0 such that hold for all i ∈ N, system 3.40 is said to have weighted H ∞ performance γ under arbitrary switching signal with the average dwell time scheme 3.23 .

Design of Robust H ∞ Controller
In this subsection, the following robust H ∞ controller u t K σ t x t 3.42 will be designed for system 2.1 . Then the corresponding closed-loop system can be described as

3.43
Theorem 3.7. Considering system 2.1 , for given scalars α, holds for all i ∈ N, with the average dwell time: where μ ≥ 1 satisfies

3.48
Proof. By Theorem 3.4, system 3.43 is mean-square exponentially stable with weighted H ∞ performance γ if the following inequalities are satisfied: to pre-and postmultiply Λ i , we have Furthermore,

3.54
According to Lemma 2.5, we have

3.55
Substituting 3.55 to 3.52 , and using Schur complement, we can obtain that 3.52 is equivalent to 3.44 . Denoting X i P −1 i , and Y i P −1 i Q i P −1 i , 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 H ∞ performance; when the robust H ∞ control

Conclusions
In this paper, the exponential stability analysis and robust H ∞ 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 H ∞ performance analysis are presented. Furthermore, robust H ∞ 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.