The new stability criteria of discrete-time switched systems with an improved mode dependent average dwell time approach

Abstract

This paper investigates the stability analysis of discrete-time switched systems with two subsystems (DSTSs). By the modified mode-dependent average dwell time (MDADT) approach, new stability criteria of DSTSs are obtained in both nonlinear and linear contexts, which are less conservative and more applicable in practice than those existing results. The design of stabilizing controller for linear DSTSs also is addressed by using LMIs. At last, three numerical examples with some comparisons to the existing results are involved to illustrate the superiority of the results in the paper.

Introduction

Loosely speaking, a switched system is one of hybrid systems in nature, and its compositions are some subsystems and a switching signal. It can model many dynamic systems, and frequently occurs in some fields, for example in biological systems, computer communication network, chemical processes, power electronics [1], neural network [2], etc. Stability analysis of switched systems has received a lot of attention over the past decades, and it has gotten some interesting results, see the literatures [3], [4], [5]. As we know, most of the existing results are based and focused on the continuous-time case, although switched systems with discrete-time subsystems appear frequently in the real world, for example, a piecewise affine system, networks with TCP, and flight systems [6]. In the last decade, discrete-time switched systems (DTSSs) have more drawn considerable interests and attentions in system and control communities [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].

So far, most stability criteria for DTSSs, just like continuous-time switched systems, fall into two categories, i.e., the stability with the switching signal being free and the stability with the signal being constrained. A common Lyapunov function is a basic method for the former, while the method called multiple Lyapunov functions (MLFs) may obtain some excellent results with the latter. Since slow switching, one of constrained switching, for DTSSs is frequently encountered in practice, the stability problem under slow switching becomes a hot spot. There are three main methods for slow switching, namely dwell time (DT) [8], [9], average dwell time (ADT) [10], [11], [12], [13] and MDADT [14], [15], [16], [17], [18], [19], [20]. These results are conservative to some extent. Most literatures consider the ADT scheme as a special case of the MDADT scheme, but the two are quite different. It should be noticed that the MDADT scheme does not present a more general result than the ADT. Different from ADT, MDADT is dependent on each subsystem, which has two sides: advantages and disadvantages [24]. A new stability scheme compatible with the two is worth further investigation. As far as the authors know, few literatures have been done on this issue.

Motivated by the above literatures, we investigate a more general stability scheme for DSTSs, which is compatible with ADT and MDADT. There are two reasons for us to consider DTSSs with two subsystems. One is that the structure is simple to study. The other is that the results obtained can be easily extended to DTSSs with multiple subsystems under certain conditions. The contribution of this paper lies in two aspects. On the one hand, a more general MDADT scheme is given, which is compatible with the existing schemes. On the other hand, the new stability conclusions with less conservatism have been obtained for DSTSs. In addition, comparing with those exciting results, the advantages and compatibility of the results in the paper are presented. It should be pointed out that, although the new scheme in this paper is based on the premise that each subsystem is stable and time-delay-free, it can easily be extended directly to DSTSs with unstable subsystems and time-delay by some simple modifications.

The brief paper is organized as follows. In Section 2, some preliminaries and problem formulation are given. Section 3 obtains some stability and stabilization criteria of DSTSs. Based on three examples, comparisons between the existing results and ours are presented in Section 4. Finally, we conclude the article in Section 5.

In this paper, $\mathbb{R}$ denotes the set of real numbers, $\mathbb{Z}$ denotes the set of integers, $\mathbb{N}$ denotes the set of positive integers. ${\mathbb{R}}^n$ represents the set of n dimensional real column vectors, and ${\mathbb{R}}^{n\times n}$ stands for the space of n × n matrices with real entries. For $x\in {\mathbb{R}}^n,$ ‖x‖ is Euclidean vector norm of x. In addition, ‖A‖ denotes the spectral norm of A, where $A\in {\mathbb{R}}^{n\times n}$. C¹ denotes the space of continuously differentiable functions. Class κ_∞ stands for a unbounded function $\alpha :[0,+\infty ]\to [0,+\infty ],$ if it is continuous, strictly increasing, and $\alpha \left(0\right)=0.$ The notation  ∈ ( ∉ ) means “in” (“not in”). ∀ represents “for all”. In the set X with finite elements, max {X} (min {X}) means the maximum (minimum) value of X. In addition, the notation ⇒(⇔) means “imply” (“be equivalent to”) and lim denotes the limit inferior. If A, B is two given sets, then A ∩ B stands for the intersection of A and B. For $P\in {\mathbb{R}}^{n\times n},$ P > 0 (P ≥ 0) denotes the matrix P being positive definite (positive semidefinite).