Mathematical Problems in Engineering

Volume 2015, Article ID 191458, 9 pages

http://dx.doi.org/10.1155/2015/191458

## Stochastic Stability of Discrete-Time Switched Systems with a Random Switching Signal

^{1}School of Mathematics and Statistics, Zhengzhou University, China^{2}College of Science, Henan Institute of Engineering, Zhengzhou, China

Received 13 December 2014; Revised 15 April 2015; Accepted 4 May 2015

Academic Editor: Alexander Klimenko

Copyright © 2015 Kai Liu 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

Necessary and sufficient condition for stochastic stability of discrete-time linear switched system with a random switching signal is considered in this paper, assuming that the switching signal allows fixed dwell time before a Markov switch occurs. It is shown that the stochastic stability of the system is equivalent to that of an auxiliary system with state transformations at switching time, whose switching signal is a Markov chain. The stochastic stability is studied using a stochastic Lyapunov approach. The effectiveness of the proposed approach is demonstrated by a numerical example.

#### 1. Introduction

The study of hybrid system is motivated by several real world technological processes involving the interconnection of logical and discrete dynamics. The evolution of logical variables may be modeled either within a deterministic or within stochastic framework. Among stochastic hybrid systems, a widely investigated class is given by random switched linear system, which consists of a set of linear systems and a random switching signal. If the switching signal is a Markov process or Markov chain, Markov jump linear systems are considered [1, 2]. In [1], some stable conditions for mean square stability for discrete-time jump linear system with finite state Markov chain are presented and the stochastic stability is also considered. The analysis and synthesis problems of stochastic stability in Markov jump linear system have been extensively addressed, such as Markov jump Lur’e system in [3], state and mode detection delay system in [4], antilinear system in [5], and singular system in [6], in Borel space [7]. Due to the probabilistic description of communications, Markov jump systems are well suited to model changes induced by nature, for example, unexpected events and random faults.

Among different assumptions on the switching signal, the arbitrary switching framework is described by considering the switching signal to be an exogenous perturbation [8]. The properties of stability and performance must hold for any possible switching rules [9–11]. If the switching signal could be designed or governed by a supervisor, the deterministic models are more adequate. For example, in many hybrid systems, the switching signal may be designed in order to improve some properties of the systems [8].

To the best of our knowledge, there exists a vast literature on both stochastic and deterministic hybrid systems, but fewer contributions have investigated the stability of discrete-time hybrid systems subject to both stochastic jumps and deterministic switching. For continuous-time system, [12] considers the stability analysis of linear switched systems with a random switching signal, which could be partitioned into the deterministic part and random part. In [13], the stability of a class of Markov jump linear systems characterized by piecewise-constant transition rates and system dynamics is investigated. The switching signal proposed in this paper has a wide-ranging application; for example, in general models of queuing theory, the interarrival time is not exponentially distributed and may contain a deterministic component [14].

In this paper, the stochastic stability of a discrete-time switched linear system with a random switching signal is considered. The dwell time of each mode could consist of two parts: fixed dwell time and random dwell time, which means that the system almost surely stays in each mode for a few time instants before the Markov switch occurs. Through an auxiliary Markov jump linear system with state transitions at switching time, whose stochastic stability is equivalent to that of original system, a necessary and sufficient condition is proposed by using the stochastic Lyapunov approach. When the parameters of the random switching signal are known, the system stability can be checked by solving a set of coupled linear matrix inequalities. Noting that the fixed dwell time could be designed, one can change the dwell time to affect the stochastic stability of the system, which will be introduced in the numerical example.

Compared to the previous work [1–7], a new class of random switching signal is proposed and more general views of switching signals are given. Compared to [12], the stochastic stability is considered in discrete-time system. When the system matrices contain zero eigenvalues, the proof of equivalence between the stochastic stability of system and its auxiliary system is technically difficult, which is solved by using the Jordan decompositions of the system matrices. Moreover, the results in this paper also lay a foundation for novel hybrid controller design.

The remainder of this paper is organized as follows. The mathematical model of the concerned system is formulated and some preliminaries are given in Section 2. In Section 3, a Markov jump switched linear system with state transitions at switching time is proposed, whose stochastic stability is proved to be equivalent to that of the original system. A necessary and sufficient condition is given in Section 4. A numerical example is provided in Section 5.

*Notation.* The notation used throughout this paper is fairly standard. The superscript stands for matrix transposition. and denote the set of positive integers and the set of nonnegative integers, respectively. , , and denote the -dimensional Euclidean space, the set of real matrices, and the set of real symmetric positive definite matrices, respectively. The notation means that is real symmetric and positive definite, and means . For a matrix , denote and as the 2-norm and the set of eigenvalues of , respectively. Let the space be a complete probability space. and denote the mathematical expectation and the generated -algebra.

#### 2. Problem Information and Preliminaries

Consider the following discrete-time linear switched system defined on the complete probability space :where and is the state vector. Switching signal , governing the switching among different system modes, takes values in the finite set . Suppose that the system switches its operation mode to at time , which means that and , and the switching signal could be descried as follows. For the time , where , no switching is allowed almost surely; that is,For , mode switching occurs according to the mode transition probabilities given bywhere and . The mode transition probability matrix is denoted by . If the next switching occurs at time , we can define . The dwell time of the system in mode is defined as , which indicates the total time length of the system has been in mode .

*Remark 1. *The parameter , which is a fixed number for every mode , plays the roles of “dwell time” in deterministic switched systems and is called the* fixed dwell time* of system in (1). According to (3), is a random variable and is called the* random dwell time* of the system. The discrete-time switching signal is motivated by the continuous-time case in [12]. There are some differences between the discrete-time and the continuous-time case. In continuous-time case, at the time , the next switching might occur after a short time interval, but in discrete-time case, when the system switches its operation mode at time , the mode may not change in the time interval , which means , even if the parameter .

*Remark 2. *System (1) with the switching signal is no longer a traditional Markov jump linear system because of the fixed dwell time . This type of system has been studied in [1], in which the switching rule is a Markov chain. Here, a modified model setting technique, in which the fixed dwell time is needed for every mode , is proposed in order to relax the restrictions on switching signal. Obviously, if for every mode , , which means that there does not exist the fixed dwell time for every , then the switching signal reduces into a Markov chain.

The following Lemma is useful to study the property of random dwell time .

Lemma 3. *, we have , .*

*Proof. *
ConsiderThe proof is completed.

According to Lemma 3, we can get that is a random variable of geometric distribution with parameter .

An example is given to illustrate the property of the switching signal .

*Example 4. *Suppose that the mode set and the fixed dwell time , , and . A sample path of the switching signal is given in Figure 1, in which the symbols “+” and “” denote the modes at the fixed dwell time and random dwell time, respectively. When the system switches into mode 2 at time , from Figure 1, we can get that the mode switching will not happen almost surely at the time , which contains time intervals. After the time , the system is allowed to switch modes and obey the switching rules (3). Although the mode may not change in the time , which seems to be a part of the fixed dwell time interval, the time is set into the random dwell time interval.