A Novel Stability Analysis of Uncertain Switched Systems with Time-Varying Delays

This paper deals with the stability of switched systems with time-varying delay. The time-varying system parameters are assumed to be norm-bounded. Based on a novel switched time-varying Lyapunov functional method, some new LMI-based sufficient conditions have been obtained to ensure the exponential stability for the uncertain switched delays systems. Finally, the proposed method is applied to a numerical example and the simulative results are also given.


Introduction
Switched systems are essentially a kind of hybrid systems, which are made up of several subsystems and a switching rule.There are many reality systems that can be viewed as switched systems, for instance, robotic manufacture systems [1], power systems [2], transportation systems [3], and network control [4].In addition, switched control has shown its advantages in stabilization of many nonlinear systems compared to using static feedback control strategy.Thus the problem of stability and control design of switched systems is of theoretical and practical importance.
As the foremost issue in the research of switched systems, stability analysis has gained a lot of attention and many important results have also been obtained during the several decades; see [5][6][7][8][9] and the references therein.Lyapunov function method plays a key role in stability analysis for switched systems.In general, there are two main methods: one is called common Lyapunov function (CLF) approach.The aim of this method is to construct a common Lyapunov function, which can ensure that the systems are stable under all possible switching signals [10][11][12][13].Another more general technique is the so-called multiple Lyapunov function (MLF) method [14].Different from the CLF approach, MLF method is to find one or more Lyapunov functions for each subsystem and combine them together to constitute a novel Lyapunov function.The research results have shown that MLF method can obtain less conservative results than by using CLF method [15][16][17][18].
As is well known, many reality systems' instability and poor behaviors are due in large to time delays.Switched delay systems are referred to as a class of switched systems with time delays in each subsystem, which has extensive application in engineering, such as networked control systems [19,20] and power systems [21].Because of their importance and widespread occurrence, stability analysis for switched systems with time delays has been a very active field in the last two decades [22][23][24][25].It is worth noticing that there are few stability results reporting the uncertain switched system with time-varying delays.In [26], delay-dependent stability conditions and  2 -gain are obtained for switched systems by using average dwell time method combining LMI techniques.In [27], by using Lyapunov-Razumikhin functions method, stability conditions involving minimum dwell time are obtained for a class of switched time delays systems.Different from [26,27], some delay-independent LMIbased sufficient conditions, which guarantee the stability of uncertain switched systems with state delays, are derived by constructing two novel Lyapunov functions/functionals in [28].Exponential stability of a class of switched delays systems with state jumps is investigated in [29]; two stability results are obtained based on dwell time method.Recently, by constructing a new Lyapunov-Krasovskii functional and combining with average dwell time approach, [30] investigates the stability of a class of switched nonlinear systems with disturbance and delays; some conditions are proposed to ensure the input-to-state stability for the system.
In addition, note that stability of switched systems relates to the switching sequence besides subsystem [5].This leads us to believe that an appropriate Lyapunov function/functional, which relates to the switching sequence, will be able to obtain less conservative results.However, CLF method or time-invariant MLF method is mainly used in the aforesaid results, and both methods often cannot make use of the information from the switching time.Thus, the convergence conditions based on CLF or time-invariant MLF approach may be conservative [31].
Inspired by the previous discussion, we will address the problem of stability for uncertain switched delays systems in this paper.We first develop a new type of piecewise time-varying Lyapunov functional and then use the novel Lyapunov functional combining Lyapunov-Krasovskii techniques to derive the exponential stability conditions for the switched systems.It will be shown that the new approach can get better results than the exiting results, and these stability conditions can be expressed in the form of linear matrix inequalities.Finally, an example will be given to show the effectiveness of the proposed approach.

Problem Formulation
In what follows, for matrix , we use the notation  > 0 (≥, <, ≤) to denote a symmetric positive-definite (positivesemidefinite, negative, and negative-semidefinite) matrix, and if not otherwise specified, all matrices are assumed to have suitable dimensions.The symbol ‖ ⋅ ‖ denotes the Euclidean norm for a vector.The symbol  denotes an identity matrix, and N is a set of positive integers.
Consider the following switched system with uncertain and time delays: where () ∈ R  is the system state vector, () : [0, ∞) → F ≜ {1, 2, . . ., } is the switched signal function, and we define () =   ∈ F for  ∈ [  ,  +1 ), where {  } is the th switching time instant.The time-varying matrices   (),   () express parameter uncertainties and satisfy where   ,   ,   ,   , and   are known constant matrices which characterize the structure of the th subsystem and unknown time-varying matrices   () satisfy (),  ∈ F, denote the continuous time-varying delays of the th subsystem.In this paper, the time-varying delays   () are assumed to satisfy the following condition: As mentioned before, stability performance of the switched system often depends on the switched signal.In this paper, we will consider the following two types of switching sequence: where constants  1 and  2 satisfy 0 <  1 ≤  2 .Definition 1. If, for arbitrary switching signal () ∈ ( 1 ,  2 ) (or ( 1 , ∞)) and any uncertainties satisfying (2), there exist constants ,  > 0 such that then system ( 1) is said to be robustly uniformly exponentially stable (RUES) over ( 1 ,  2 ) (or ( 1 , ∞)).
(iii) If  −   > 0, then The main purpose of this paper is to develop new Lyapunov functional methods to analyze the stability of switched system (1) and to derive some conditions which can ensure switched system (1) is RUES over ( 1 ,  2 ) (or ( 1 , ∞)).
Remark 4. The stability analysis of system ( 1) is performed by applying the Lyapunov functional () given in (12).In (), the time-varying matrices  () () and () are time-varying convex combinations of two constant positive matrices, in which the combination are related to the switched time sequences.However, the traditional Lyapunov functional is usually chosen the case of () when  (),1 =  (),2 (i.e.,  1 =  2 ,  ∈ F) and  1 =  2 .Therefore, compared with the traditional Lyapunov functional, the time-varying Lyapunov functional can capture the switched characteristic of the considered system and bring less conservative results.
In the case of {  } ∈ ( 1 , ∞), based on Theorem 3, we have the following corollary.
Remark 6.In the earlier discussion, we have assumed that there are no impulsive effects at the switching points.However, similar approach can also be used in the stability analysis for impulsive switched systems.
Remark 7. We can also make a further study in the problem of stabilization for system (1) by using the method proposed in this paper.In the stabilization setting, different subsystem has different controller gain and Lyapunov matrix, so a key question is how to convert the stabilization issue to solve a set of LMIs; at this point, one can refer to the recent paper [31], where the problem of observers design for switched linear systems has been tackled.

Numerical Examples
In this section, we will give an example to show the effectiveness of our new approach.
Example 1.Consider switched system (1) with two subsystems, and the parameters are given as follows: In order to compare our results with the existing ones, we calculate the lower bound  1 on the minimum dwell time from three cases as followings.
In the results of [27,28], it was found that the corresponding lower bound of the minimum dwell time is 3.42 and 34.8579, respectively.
Based on the analysis above, our result is much less conservative than the results of [26][27][28].For simulation studies, we consider Case 1. Choose dwell time  1 = 3.14.With the initial functions (0) = [8 − 5] T , the time response curves of the system are plotted in Figure 1 and the switching signal is shown in Figure 2.

Conclusion
By introducing a new type of Lyapunov functional method, some delay-dependent sufficient conditions of stability for a class of switched systems with uncertain and state delays have been presented, and these conditions can be expressed in the form of LMI.The numerical analysis results have shown that the proposed approach can obtain better results than the earlier results.

Figure 1 :
Figure 1: State responses of the system.