Abstract

The problem of exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays is studied. The constraint on the derivative of the time-varying delay is not required which allows the time delay to be a fast time-varying function. We study the stability properties of switched nonlinear systems consisting of both stable and unstable subsystems. Average dwell-time approached and improved piecewise Lyapunov functional combined with Leibniz-Newton are formulated. New delay-dependent sufficient conditions for the exponential stabilization of the switched systems are first established in terms of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.

1. Introduction

The switched systems are an important class of hybrid systems. They are described by a family of continuous or discrete-time subsystems and a rule that orchestrates the switching between the subsystems. Recently, switched systems have attracted much attention due to the widespread application in control, chemical engineering processing [1], communication networks, traffic control [2, 3], and control of manufacturing systems [4–6]. A switched nonlinear system with time delay is called switched nonlinear delay system, where delay may be contained in the system state, control input, or switching signals. In [7–9], some stability properties of switched linear delay systems composed of both stable and unstable subsystems have been studied by using an average dwell-time approach and piecewise Lyapunov functions. It is shown that when the average dwell time is sufficiently large and the total activation time of the unstable subsystems is relatively small compared with that of the Hurwitz stable subsystems, global exponential stability is guaranteed. The concept of dwell time was extended to average dwell time by Hespanha and Morse [10] with switching among stable subsystems. Furthermore, [11] generalized the results to the case where stable and unstable subsystems coexist.

The stability analysis of nonlinear time-delay systems has received increasing attention. Time-delay systems are frequently encountered in various areas such as chemical engineering systems, biological modeling, and economics. The stability analysis for nonlinear time-delay systems has been investigated extensively. Various approaches to such problems have been proposed, see [12–14] and the references therein. It is well known that the existences of time delay in a system may cause instability and oscillations system. Thus, the stability analysis of nonlinear time-delay systems has received considerable attention for the last few decades, see [15, 16].

Recently, the exponential stability problems for time-delay systems have been studied by many researchers, see for examples [17–21]. The problems have been dealt with for various control areas such as exponential stabilization for linear time-delay systems [22], exponential stabilization for uncertain time-delay systems [21, 23]. Hien and Phat [23] presented exponential stability and stabilization conditions for a class of uncertain linear system with time-varying delay, based on an improved Lyapunov-Krasovskii functional combined with Leibniz-Newton formula. The robust stability conditions are derived in terms of LMIs, but the time-varying delays are required to be differentiable and the lower bound is restricted to zero. However, in most cases, these conditions are difficult to satisfy. Therefore, in this paper, we will employ some new techniques so that the above conditions can be removed.

Stability analysis of nonlinear systems with distributed delays is of both practical and theoretical importance. For some systems, delay phenomena may not be simply considered as delays in the velocity terms and/or discrete delays in the states. Therefore, it is desirable to extend the system model to include distributed delays. Practical applications, modeled by systems with distributed delays, can be found in [24–26]. In [27], Gao et al. studied the problem of linear switching systems with discrete and continuous time delay. By constructing Lyapunov functional under a condition on the time delay, we show that it stabilizes the system for sufficiently small delays.

In this paper, the problem of exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays is studied. The discrete time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available. However, the delay function is not necessary to be differentiable. We study the stability properties of switched nonlinear systems consisting of both stable and unstable subsystems. Average dwell-time approached and improved piecewise Lyapunov functional combined with Leibniz-Newton's formula, and new delay-dependent sufficient conditions for the exponential stabilization of the switched systems are first established in terms of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.

The rest of this paper is organized as follows. In Section 2, we give notations, definition, propositions, and lemma which will be used in the proof of the main results. Delay-dependent sufficient conditions for the exponential stability of switched nonlinear time-varying delay systems are presented in Section 3. A Numerical example illustrated the obtained results is given in Section 4. The paper ends with conclusions in Section 5 and cited references.

2. Preliminaries

The following notation will be used in this paper. denotes the set of all real nonnegative numbers. denotes the dimensional space and the vector norm . denotes the space of all matrices.

We also let denote the transpose of matrix is symmetric if . denotes the identity matrix. denotes the set of all eigenvalues of . : denotes the set of all valued continuous functions on . denotes the set of all the valued square integrable functions on matrix is called semipositive definite if , for all is positive definite if for all means the symmetric term in a matrix is denoted by .

Consider a nonlinear system with interval time-varying and distributed delays of the form where is the state. are given matrices of appropriate dimensions. is piecewise constant and left continuous and is called a switching signal and is the initial function with the norm The time delays and denoted the discrete time-varying delay and distributed time-varying delay, respectively. They are assumed to satisfy the following conditions: The nonlinear perturbation satisfies the following growth condition:

We introduce the following technical well-known propositions, which will be used in the proof of our results.

Definition 2.1 (see [7]). For any switching signal and any , let denote the number of switching of over the interval . For given , if the inequality holds, then is call average dwell time and is called chattering bound. As commonly used in the literature, for convenience, we choose in this paper.

Proposition 2.2 (Cauchy inequality). For any symmetric positive definite matrix and one has

Proposition 2.3 (see [28]). For any symmetric positive definite matrix , scalar and vector function such that the integrations concerned are well defined, the following inequality holds:

Proposition 2.4 (see [28] (Schur complement lemma)). Given constant symmetric matrices with appropriate dimensions satisfying . Then if and only if

3. Main Results

3.1. Stability of Switched Nonlinear Time-Varying Delay Systems

First, we present a delay-dependent exponential stabilizability analysis conditions for the given nonlinear time-varying delay systems (2.1). We consider the case when the stable and unstable subsystems coexist. We assume that the subsystem is stable, where the positive integer satisfies and the subsystem is unstable. Let us set .

For the system (2.1), given , the following lemma provides a change estimation of Lyapunov-Krasovskii functional candidate: where

Lemma 3.1. For a given constant , suppose (2.3) holds. If there exist symmetric positive definite matrices such that the following LMI holds: where Then, along the trajectory of system (2.1), one has

Proof. Let . We consider the following Lyapunov-Krasovskii functional (3.1). By taking the derivative of Lyapunov-Krasovskii functional candidate (3.1) along the trajectory of the system (2.1) we have Applying Proposition 2.3 and the Leibniz-Newton formula, we have Note that Using Proposition 2.3 gives Let . Then Therefore, from (3.12) and (3.13), we obtain By using the following identity relation we have Applying Propositions 2.2 and 2.3 gives By using Proposition 2.2 and condition (2.4), we have Hence, according to (3.10)–(3.18), we have where and are defined in (3.3), (3.4), respectively, and Since , is a convex combination of and , therefore, is equivalent to and . Applying Shur complement lemma, the inequalities is equivalent to . Thus, it follows from (3.3)–(3.5) and (3.19), we obtain Thus, by the above differential inequality, we have