Stabilization for switched stochastic neutral systems under asynchronous switching

Abstract

In this paper, the mean-square exponential stability of switched stochastic neutral systems with time-varying delay under asynchronous switching is investigated. A new Lyapunov function dependent on the controllers’ switching signal is constructed, which can counteract the difficulty of controller design to stabilize the system under asynchronous switching. Moreover, the value of the Lyapunov function is allowed to increase during mismatched periods. Based on the average dwell time approach, a sufficient condition expressed by a set of linear matrix inequalities (LMIs) is derived to guarantee the mean-square exponential stability for the closed-loop system, and the controllers with asynchronous switching are designed. Finally, a numerical example illustrates the effectiveness of the results.

Introduction

Switched systems are an important class of hybrid systems which are composed of several dynamical subsystems and a switching rule that orchestrates the switching among them to ensure stability and satisfied performance. Many physical and engineering systems can be modeled as switched systems, such as chemical processes, power electronic, automatic highway systems [3], [15], [20]. In the past few decades, switched systems have attracted considerable attention. Many remarkable achievements have been made on the issues such as controllability, observability, stability, stabilization, observer design, filtering, and fault detection [1], [4], [7], [14], [16], [17], [18]. Among them, stability analysis and stabilization control problems are the main concerns, and large numbers of excellent results have been published; see, e.g., [4], [12], [13], [17], [21] and the references therein.

As is well known, time delay as a source of instability and poor performance often appears in many dynamical systems, such as neural networks, nuclear reactors, manual control and microwave oscillator systems. Actually, there are many physical processes which can be described by differential equations of neutral type, i.e. neutral systems, where time delay appears both in state and state derivative. In the literature, various analysis techniques have been utilized to study the neutral systems, see for example [2], [11], [27]. However, due to the complicated behavior of switched neutral systems, only a few results on such systems have appeared [8], [26], [28]. On the other hand, stochastic phenomena exist widely in engineering applications and social systems. Stochastic systems have received much attention and many results have been reported [2], [6], [30]. It should be noted that so far the synthesis issue for switched stochastic neutral systems has not been fully investigated.

In the ideal case, the switching of the controllers coincides exactly with that of corresponding subsystems. In engineering application, however, since it inevitably takes some time to identify the active subsystem and apply for the matched controller, the switching time of controllers may lag behind that of practical subsystems, which results in asynchronous switching between the controllers and system modes. The necessities of considering asynchronous switching for efficient controller design have been shown in a class of chemical systems [10]. Some primary studies on the asynchronous switching problems for switched systems have been proposed [5], [23], [24], [25]. Recently, Lyapunov stability theory is used to investigate the problem [9], [22], [29]. In [9], asynchronous H_∞ filtering is considered for discrete-time switched systems. [22] analyzes the robust reliable control for a class of uncertain switched neutral systems under asynchronous switching. Sufficient conditions for the stability of such systems are given by two inequalities. In [29], stability, L₂-gain and asynchronous H_∞ control are considered for discrete-time switched systems.

Based on the above discussion, switched stochastic neutral time-delay systems under asynchronous switching are worth studying. To the best of our knowledge, few results on the stabilization for such systems have been reported, which motivates this study for us. Applying Itô’s differential formula and the Lyapunov stability theory, a sufficient condition with respect to mean-square exponential stability of the given system is obtained and the controllers with asynchronous switching are designed. The presented condition is applicable to the case where the subsystems are unstable during mismatched periods resulted from asynchronous switching between subsystems and the corresponding controllers. Furthermore, the switching signal of the Lyapunov function constructed in the paper is dependent on the controllers’ switching signal, which is convenient for the analysis of the proposed issue.

The rest of this paper is organized as follows. In Section 2, the problem to be studied is formulated and some definitions and a lemma are introduced. In Section 3, based on the average dwell time approach, the controllers are developed to ensure the mean-square exponential stability of the switched stochastic neutral system under asynchronous switching. A numerical example is provided to illustrate the method in Section 4 and we conclude this paper in Section 5.

Notations: Throughout this paper, the notations are standard. The superscript “T” denotes matrix transposition, “∗” the symmetric term in a symmetric matrix, diag{⋯} a block-diagonal matrix and I the identity matrix. ${\mathbb{R}}^n$ stands for the n-dimensional Euclidean space, $\mathcal{E}\{·\}$ the expectation operator and ∥ · ∥ the Euclidean norm. P > 0 means that P is real symmetric and positive definite, and λ_min(P) (λ_max(P)) is the minimum (maximum) eigenvalue of P.