- Open Access
A constructive way to design a switching rule and switching regions to mean square exponential stability of switched stochastic systems with non-differentiable and interval time-varying delay
© Rajchakit et al.; licensee Springer. 2013
- Received: 25 March 2013
- Accepted: 25 September 2013
- Published: 8 November 2013
This paper addresses a mean square exponential stability problem for a class of switched stochastic systems with time-varying delay. The time delay is any continuous function belonging to a given interval, but not necessary differentiable. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for the mean square exponential stability of switched stochastic systems with time-varying delay are first established in terms of LMIs. Numerical example is given to show the effectiveness of the obtained result.
MSC:15A09, 52A10, 74M05, 93D05.
- switching design
- mean square exponential stability
- switched stochastic systems
- scalar Wiener process
- Brownian motion
- interval delay
- Lyapunov function
- linear matrix inequalities
In the past decades, the problem of stability for neutral differential systems, which have delays in both their state and the derivatives of their states, has been widely investigated by many researchers. Such systems are often encountered in engineering, biology, and economics. The existence of time delay is frequently a source of instability or poor performance in the systems. Recently, some stability criteria for a neutral system with time delay have been given [1–25]. Stability analysis of linear systems with time-varying delays is fundamental to many practical problems and has received considerable attention [1–7]. In [8–17], which are not based on the method of Lyapunov functional, one of them uses the diagonal equations for reducing systems of delay differential equations to ones of integral equations and estimates the norms or spectral radii of corresponding integral operators obtained on the basis of the results in the book. Most of the known results on this problem are derived assuming only that the time-varying delay is a continuously differentiable function, satisfying some boundedness condition on its derivative: . In delay-dependent stability criteria, the main concern is to enlarge the feasible region of stability criteria in a given time-delay interval. Interval time-varying delay means that a time delay varies in an interval in which the lower bound is not restricted to be zero. By constructing a suitable argument, Lyapunov functional and utilizing free weight matrices, some less conservative conditions for asymptotic stability are derived in [18–24] for systems with time delay varying in an interval. However, the shortcoming of the method used in these works is that the delay function is assumed to be differential and its derivative is still bounded: . To the best of our knowledge, a constructive way to design a switching rule, switching regions, and mean square exponential stability of switched stochastic systems with interval time-varying delay, non-differentiable time-varying delays, which are important in both theory and applications, have not been fully studied yet (see, e.g., [25–38] and the references therein). This motivates our research.
This paper gives the improved results for the mean square exponential stability of switched stochastic systems with interval time-varying delay. The time delay is assumed to be a time-varying continuous function belonging to a given interval, but not necessary differentiable. Specifically, our goal is to develop a constructive way to design a switching rule to exponential stability of switched stochastic systems with interval time-varying delay. By constructing a Lyapunov functional combined with the LMI technique, we propose new criteria for the mean square exponential stability of switched stochastic systems with interval time-varying delay. The delay-dependent mean square exponential stability conditions are formulated in terms of LMIs, being thus solvable by utilizing Matlab’s LMI control toolbox available in the literature to date.
The paper is organized as follows. Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results. Delay-dependent mean square exponential stability conditions of switched stochastic systems with interval time-varying delay are presented in Section 3. Numerical example is provided to illustrate the theoretical results in Section 4, and the conclusions are drawn in Section 5.
The following notations will be used in this paper. denotes the set of all real non-negative numbers; denotes the n-dimensional space with the scalar product and the vector norm ; denotes the space of all matrices of -dimensions; denotes the transpose of matrix A; A is symmetric if ; I denotes the identity matrix; denotes the set of all eigenvalues of A; ; , ; denotes the set of all -valued continuous functions on ; matrix A is called semi-positive definite () if for all ; A is positive definite () if for all ; means . ∗ denotes the symmetric term in a matrix.
where is the state; is the switching rule, which is a function depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. Moreover, implies that the system realization is chosen as the i th system, . It is seen that system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state hits predefined boundaries. , , are given constant matrices, and is the initial function with the norm .
where and , , are known constant scalars. For simplicity, we denote by , respectively.
The mean square stability problem for switched stochastic system (2.1) is to construct a switching rule that makes the system mean square exponentially stable.
Definition 2.2 The system of matrices , , is said to be strictly complete if for every , there is such that .
We end this section with the following technical well-known propositions, which will be used in the proof of the main results.
Proposition 2.1 
If , then the above condition is also necessary for the strict completeness.
Proposition 2.2 (Cauchy inequality)
Proposition 2.3 
Proposition 2.4 [, p.89-90]
Proposition 2.5 (Schur complement lemma )
The following is the main result of the paper, which gives sufficient conditions for mean square exponential stability problem for a class of switched stochastic systems (2.1) with time-varying delay.
, , ,
By Definition 2.1, system (2.1) is exponentially stable in the mean square. The proof is complete. □
To illustrate the obtained result, let us give the following numerical example.
In this paper, we have proposed new delay-dependent conditions for the mean square exponential stability of switched stochastic systems with time-varying delay. Based on the improved Lyapunov-Krasovskii functional and the linear matrix inequality technique, a switching rule for the mean square exponential stability of switched stochastic systems with time-varying delay has been established in terms of LMIs.
This work was supported by the Thailand Research Fund Grant, the Commission for Higher Education and Faculty of Science, Maejo University, Thailand. The second author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.
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