-Gain Analysis for Switched Systems with Time-Varying Delay

For the switched systems, switching behavior always affects the finite-time stability (FTS) property, which was neglected by most previous studies. This paper is mainly concerned with the problem of delay-dependent finite-time and L 2 -gain analysis for switched systems with time-varying delay. Several less conservative sufficient conditions related to finite-time stability and boundness of switched system with time-varying delays are proposed; the system trajectory stays within a special bound with the information of switching signal. At last, a numerical example is also given to illustrate the efficiency of the developed method.


Introduction
During the past decades, the systems with time delays have received much attention since they often fall in various practical systems, for instance, neural networks, networked control systems, engineering systems, biology, economics, and other fields.However, time delay is always frequently the major cause of oscillation and instability; the stability for systems with time delays has been devoted to a lot of effort.Nowadays, the stability of time delay is parted into two classes: delayindependent stability criteria and delay-dependent ones.It should be pointed out that delay-independent criteria turn out to be more conservative, especially for the small-size delays.The problem of delay-dependent stability analysis for time-delay systems has received considerable attention, and lots of important results have also been reported [1][2][3][4][5][6][7][8][9][10][11][12].
In recent years, there has been an increasing interest in analysis of hybrid and switched systems due to their importance both in theory and its applications.As an important class of hybrid systems, switched linear systems comprise a collection of linear subsystems described by differential equations as well as a switching law to specify the switching during these subsystems.A switched system is a type of hybrid system with a combination of the discrete and continuous dynamical systems.These systems are recognized as models for appearance which cannot be denoted by exclusively continuous or discrete processes.Recently, based on the Lyapunov functions and other analysis tools, the corresponding stability and stabilization for both linear and nonlinear and switched systems have been investigated and many good results have been obtained; for a recent survey on this issue and related matters, one can also refer to [13][14][15][16][17][18][19][20][21][22][23][24][25][26].
Most of the existing results have concentrated on Lyapunov-based asymptotic stability for the switched systems; the behavior of that is above the infinite time interval.However, in practical engineering, there exists the bad transient characteristics; the system is asymptotically stable without unusable.For the systems which work in a short time interval, for instance, control system, missile system, and communication network system, which mainly care about the behavior over a finite time interval.To deal with this problem, in 1961, Dorato presented the idea of finite-time stability in [27].Over the past few years, many study efforts have been dedicated to the finite-time stability (FTS) of switched systems due to its wide applications.Comparing with the classical Lyapunov stability, which currently is the focus on a large and growing interdisciplinary area of research.To study the transient behavior of systems, FTS concerns the stability of a system over a finite interval of time and plays an important role.It is important to emphasize the disconnection between classical Lyapunov stability and finite-time stability.The problem about finite-time stabilization has been widely learned in the literature [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44].It is worth pointing out that there is a difference between finite-time stability and Lyapunov asymptotic stability, and they are also independent of each other.
Recently, some papers are found to be related to finitetime stability for switched systems.For example, based on the technique of average dwell time, the problem of finite-time boundedness for the switched linear system with time delays was investigated in [36].The issue of finite-time stability and  ∞ stabilization for switched nonlinear discrete-time systems was developed in [37].The problem of finite-time quantized  ∞ control problem for a class of discrete-time switched time-delay systems with time-varying exogenous disturbances was also discussed in [38].But, to the best of the authors' knowledge, the finite-time  2 -gain problems for switched systems have not been completely studied.The previous research has some conservatism of stability criterion; it is natural to look for an alternative way to reduce the conservatism of corresponding stability criteria.This idea motivates to our study.
The main contribution of this paper is that we present a novel approach for finite-time stability of the given switched system.Moreover, several sufficient conditions ensuring the finite-time stability and boundness are proposed with different information we know about the switching signal.It shows the less conservative results when more information about the switching signal is available.By selecting the appropriate Lyapunov-Krasovskii functional, the sufficient conditions are obtained to guarantee finite-time stability of the systems and the closed-loop system trajectory stays in a special bound.The finite-time stability criteria can also be dealt with in the form of linear matrix inequalities and average dwell time.Finally, a numerical example is given to illustrate the effectiveness of the developed techniques.
Corresponding to the switching signal (), we get the follwing switching sequence: where  0 is the initial time when   ∈ [  ,  +1 ), ( 0 ) is the initial state and   th subsystem is active.Therefore, the trajectory () of the switched system (1) is called the trajectory of the   th subsystem.As assumed before, we get rid of Zeno behavior for all types related to switching signal.Throughout this paper, we assumed that the state of the switched system (1) does not jump at the switching instants; that is, the trajectory () is continuous everywhere.Firstly, we will give the definitions and lemmas about switched system (1), which plays an major role in the derivation of our results.Definition 1. Switch system (1) is said to be finite-time bounded with respect to ( 1 ,  2 , , , ), if condition (6) holds: max where  2 >  1 ≥ 0 and  > 0.
Definition 3 (see [21]).For any /  holds for  0 ≥ 0 and   > 0, then  0 and   are called chattering bound and average dwell time, respectively.Here we assume  0 = 0 for simplicity as supported by reference.
Remark 8.It should be mentioned that to reduce the conservatism, Remark 9.One can clearly see that criteria given in Theorem 7 are delay-dependent.It is well known that the delay-dependent criteria are less conservative than delayindependent criteria when the delay is small or belongs to a given interval.Interval time-varying delay is a time delay that varies in an interval in which the lower bound is not restricted to zero.Remark 10.In Theorem 7, the free weight matrix technology is employed, which reduces the conservativeness of the stability condition.
Remark 13.In many real applications, the minimum value of  2 min is of interest.In Theorem 11, with a fixed  and ,  min can be obtained through following procedure: min  2 s.t.LMIs (( 37)-( 39)). (60)

Illustrative Example
Example 1.Consider the system (1) with the following data: Through Theorem 7, when  2 = 50, we can see that the admissible maximum ℎ computed by [36] is 0.2, and the maximal value of  in this paper is 0.262.It is obvious that our method shows less conservatism result than that in [36].Moreover, when  = 0.2, the optimal bound with minimum value of  2 min relies on the parameter .One can obtain clearly that Theorem 7 in our paper can indeed provide much smaller admissible  2 min than the stability criteria in [36] which shows the less conservative result in this paper.

Conclusions
In this paper, we have examined the problems of delaydependent finite-time and  2 -gain analysis for switched systems with time-varying delay by allowing new Lyapunov-Krasovskii functional and average dwell time.A numerical example has also been given to demonstrate the effectiveness of the proposed approach.It is possible to extend the main results obtained in this paper to real systems which contain some practical constrains, for example, the failures of controllers.Details will be reported in our follow-up work in the future.

𝐴 1
values of  1 ,  2 , ,  and matrix  are given as follows: