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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 320703, 12 pages

http://dx.doi.org/10.1155/2013/320703

## Observer-Based Robust Control for Switched Stochastic Systems with Time-Varying Delay

^{1}School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China^{2}Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

Received 6 June 2013; Accepted 25 July 2013

Academic Editor: Lixian Zhang

Copyright © 2013 Guoxin Chen 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

This paper investigates the problem of observer-based robust control for a class of switched stochastic systems with time-varying delay. Based on the average dwell time method, an exponential stability criterion for switched stochastic delay systems is proposed. Then, performance analysis and observer-based robust controller design for the underlying systems are developed. Finally, a numerical example is presented to illustrate the effectiveness of the proposed approach.

#### 1. Introduction

Switched systems are a kind of hybrid dynamical systems composed of a set of continuous-time subsystems or discrete-time subsystems and a switching law that orchestrates the switching between them. Switched systems have attracted increasing attention during the past decades because of their wide applications in real-world systems, such as robot control systems [1], networked control systems [2, 3]. Many useful results on stability analysis and control synthesis for such systems have been reported in [4–8]. For example, control of switched linear discrete-time systems with polytopic uncertainties was investigated in [8].

It is well known that the time delay phenomenon is frequently encountered in engineering and social systems, and the existence of which may cause instability or undesirable system performance. Therefore, many research efforts have been devoted to the study of switched time delay systems [9–15]. On the other hand, stochastic systems have attracted considerable attention during the past several decades. Early results can be found in [16], and the control problem of stochastic systems with time delay was investigated in [17, 18]. The study on control of stochastic system was developed in [19]. Stability analysis on stochastic system with multiple delays was proposed in [20]. Moreover, some results on switched stochastic systems with and without time delay have been obtained (see [21–25] and the references cited therein).

In many real-world systems, state feedback control will fail to guarantee the stabilization because the states of the systems are not all measurable [26]. One of the key approaches to solve the problem is to reconstruct the states of the systems and realize the required feedback control. Hence, the observer-based control has been an interesting topic in control theory. Some results on observer-based control for stochastic delay systems or Markovian jump systems have been presented in [27–29]. However, to the best of our knowledge, the problem of observer-based robust control for switched stochastic systems with time delay has not been fully studied, which motivates the present study.

In this paper, we aim to design an observer-based robust controller for switched stochastic systems with time delay such that the closed-loop system is mean-square exponentially stable with performance. The major contributions of the work can be summarized as follows: a new Lyapunov-Krasovskii functional candidate is introduced to derive the exponential stability of switched stochastic systems with time delay, and the free-weighting matrix method is employed to reduce the conservatism; an observer-based robust controller design scheme for the underlying systems 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, the main results are presented. 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.* In this paper, the superscript “*T*” denotes the transpose, and the symmetric term in a matrix is denoted by *∗*. The notation () means that is positive definite (positive semidefinite, respectively). denotes the -dimensional Euclidean space. denotes the Euclidean norm. denotes the absolute value of . is the space of square integrable functions on , and is the initial time. and denote the maximum and minimum eigenvalues of , respectively. denotes the Moore-Penrose pseudoinverse of . is the identity matrix. denotes a diagonal matrix with the diagonal elements .

#### 2. Problem Formulation and Preliminaries

Consider the following switched stochastic system with time delay:where is the state vector, is the initial state function, is the control input, is the disturbance input which is assumed to belong to , is the measurable output, is the controlled output, and is a one-dimensional zero-mean Wiener process on a probability space and satisfies 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 time delay satisfying where and are known constants.

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 instant and denotes the th switching instant. Moreover, means that the th subsystem is activated. For all , , , , and are known real-value matrices with appropriate dimensions, , , and are uncertain real matrices with appropriate dimensions and can be written as where , , , , , , , and are known real-value matrices with appropriate dimensions, and is an unknown time-varying matrix that satisfies

The state feedback controller is designed as . In actual operation, however, the states of the systems are not all measurable. The following switched system is constructed to estimate the state of system (1a), (1b), (1c), and (1d):where is the estimation of , is the observer output, and is the initial observer state function. The real state feedback controller becomes . and are the observer gains and controller gains to be determined, respectively.

*Remark 1. *It is noted that the observer-based control for stochastic systems or Markovian jump systems was considered in [27–29]. However, the results in the aforementioned papers cannot be directly applied to the switched stochastic system considered in the paper. This motivates our study. Also, the proposed observer in (6a), (6b), and (6c) is a switching observer, which is different from the existing ones given in [27–29].

From systems (1a), (1b), (1c), and (1d) and (6a), (6b), and (6c), we can obtain the following augmented closed-loop system:where , denotes the state estimated error.

For , the parameters of system (7a), (7b), and (7c) are given as follows:

*Assumption 2. * is full row rank, for all .

*Definition 3. *System (1a), (1b), (1c), and (1d) with is said to be mean-square exponentially stable under the switching signal if there exist scalars and , such that the solution of the system satisfies

*Definition 4 (see [24]). *For any , let denote the switching number of on an interval . If
holds for given constants and , then the constant is called the average dwell time. As commonly used in the literature, some chooses .

*Definition 5 (see [30]). *For any and , system (7a), (7b), and (7c) is said to be mean-square exponentially stable with a prescribed weighted performance level if the following conditions are satisfied:(1)when , system (7a), (7b), and (7c) is mean-square exponentially stable;(2)under the zero initial condition, the output satisfies

Lemma 6 (see [31]). *For any positive symmetric constant matrix and a scalar , if there exists a vector function such that integrations in the following are well defined, then the following inequality holds
*

Lemma 7 (see [32]). *Let , , , and be constant matrices of appropriate dimensions with satisfying , then for all , if and only if there exists a scalar such that .*

The objective of this paper is to design an observer-based robust controller for switched stochastic delay system (1a), (1b), (1c), and (1d) such that the augmented closed-loop system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted performance level .

#### 3. Main Results

##### 3.1. Stability Analysis

In this subsection, in order to obtain the main results, we first focus on the problem of stability analysis for the following switched stochastic systems with time delay

Theorem 8. *For a given scalar , if there exist symmetric positive definite matrices , , and and any matrices such that
**
then system (13a) and (13b) is mean-square exponentially stable under arbitrary switching signal with the average dwell time
**
where , , and satisfy
*

*Proof. *Let , then (13a) can be described as

Choose the following Lyapunov functional candidate for the th subsystem
where

For the sake of simplicity, is written as in this paper. According to Itô’s formula, along the trajectory of the th subsystem, we have
where

According to Lemma 6, we have

Integrating both sides of (18) from to , we have

Thus,
where .

Combining (21)–(25) leads to
where

By using the Schur complement, we obtain from (14) that

According to (26), one obtains that

Then, taking mathematical expectation, we have

From (16) and (19), we obtain that

Let . Then, using formula, we can obtain that for ,

Notice that

Thus, it can be obtained that
where

Noticing that , one gets

For any , from (16) and (36), it follows that

When (15) holds, noticing that , one has
where , .

The proof is completed.

*Remark 9. *In the derivation of Theorem 8, a new Lyapunov-Krasovskii functional candidate is constructed for the stability analysis of switched stochastic systems with time delay, and it is different from the ones given in [9–15]. Also, the free-weighting matrix method is utilized to reduce the conservatism.

*Remark 10. *If in (15), which leads to , , and , for all , then system (13a) and (13b) possesses a common Lyapunov function, and the switching signals can be arbitrary.

When , system (13a) and (13b) becomes the following switched system with time delay:

From Theorem 8, we can readily get the exponential stability criterion for switched system (39a) and (39b).

Corollary 11. *Consider system (39a) and (39b), for a given scalar , if there exist symmetric positive definite matrices , , and and any matrices such that
**
then system (39a) and (39b) is exponentially stable under arbitrary switching signal with the average dwell time scheme (15).*

##### 3.2. Performance Analysis

In the sequel, we will investigate the problem of performance analysis for switched stochastic systems with time delay. Consider the following system:

Theorem 12. *For a given scalar , if there exist symmetric positive definite matrices , , and and any matrices such that
**
then system (41a), (41b), and (41c) is mean-square exponentially stable with a weighted prescribed performance level under arbitrary switching signal with the average dwell time
**
where , , and satisfy
*

*Proof. *We can easily obtain that (14) is satisfied if (42) holds. Thus, system (41a), (41b), and (41c) with is mean-square exponentially stable.

When , let

Choosing the same Lyapunov functional candidate as (19) and following the proof line of Theorem 8, we have
where
and satisfies

Using the Schur complement, from (42), we get

It follows that

From (44), we obtain that

For any , using formula and taking the mathematical expectation, one has

Under the zero initial condition, we obtain that

According to (46), one has

Multiplying both sides of (55) by leads to

Noticing that and , one obtains that

When , the following inequality is derived:

The proof is completed.

##### 3.3. Observer-Based Robust Stabilization

Now, we are in a position to design an observer-based robust controller for system (1a), (1b), (1c), and (1d) such that the augmented closed-loop system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted performance level . Based on Theorem 12, a sufficient condition for the existence of such a controller is presented in the following theorem.

Theorem 13. *Consider system (1a), (1b), (1c), and (1d), for a given scalar , if there exist scalars , symmetric positive definite matrices , , and , and any matrices , , and such that, for all ,**then there exists an observer-based controller such that system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted performance level under arbitrary switching signal with the average dwell time**
where , , and satisfy
**
Moreover, if the above conditions have a feasible solution, the controller gain matrices and the observer gain matrices can be obtained by and .*

*Proof. *According to Theorem 12, we get that system (7a), (7b), and (7c) is mean-square exponentially stable with a weighted performance level if the following inequality is satisfied:
where , and , , and are symmetric positive definite matrices with appropriate dimensions.

Then, we have
where

From Lemma 7, we have
Choose , , and , and let and . By using the Schur complement, we can obtain that (67) is equivalent to (59).

Thus, according to Theorem 12, we obtain from (59)–(62) that system (7a), (7b), and (7c) is mean-square exponentially stable with a weighted performance level . Moreover, we can obtain the controller gain matrices and the observer gain matrices .

The proof is completed.

*Remark 14. *An observer-based controller design scheme is proposed in the paper. Compared with the existing results presented in [27–29], a remark advantage of the work is that the proposed observer is mode-dependent, which means that each subsystem has its individual observer. Moreover, the proposed observer not only ensures the convergence of the estimated error of each subsystem, but also guarantees that the estimated error of the whole system converges to zero exponentially.

In Theorem 13, when we choose that , it is not difficult to get that and , for all . Then, the following corollary is derived.

Corollary 15. *Consider system (1a), (1b), (1c), and (1d), for a given scalar , if there exist scalars , symmetric positive definite matrices , and matrices , , and such that, for all ,** then there exists an observer-based robust controller such that system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted performance level under arbitrary switching signal with the average dwell time
**
where , , and satisfy
**
Moreover, the controller gain matrices are , and the observer gain matrices are .*

#### 4. Numerical Example

Consider system (1a), (1b), (1c), and (1d) with the following parameters

The disturbance input , and ; by calculation, we can obtain that and . Choosing , and solving the LMIs in Corollary 15, we have

Then, we can obtain the following observer gain matrices: and the controller gain matrices

Moreover, we have , , , and . Taking , and letting , , , and , and simulation results are shown in Figures 1–5.

Figure 1 shows the switching signal of the switched system with the average dwell time . Figures 2 and 3 illustrate the state trajectories of the closed-loop system. The estimated errors are plotted in Figures 4 and 5, respectively. We can see from Figures 2–5 that the proposed observer can guarantee the convergence of the estimated error and the designed controller can guarantee the stability of the corresponding closed-loop system. This demonstrates the effectiveness of the proposed method.

In addition, some observer-based controller design approaches proposed in the existing literature [27–29] are only applicable to stochastic systems or Markovian jump systems and they cannot be used to stabilize the system considered in this section, which also shows the advantage of the proposed method.

#### 5. Conclusions

In this paper, the problem of observer-based robust stabilization for stochastic switched systems with time delay has been investigated. By using the average dwell time method, sufficient conditions which guarantee the mean-square exponential stability of switched stochastic systems with time delay are derived. Then, performance analysis and observer-based control for such systems are developed. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach.

#### Acknowledgment

This work was supported by the Nationa