Chao Liu, Zheng Yang, Xiaoyang Liu, and Xianying Huang

Abstract—This paper investigates the stability of switched systems with time-varying delay and all unstable subsystems. According to the stable convex combination, we design a state-dependent switching rule. By employing Wirtinger integral inequality and Leibniz-Newton formula, the stability results of nonlinear delayed switched systems whose nonlinear terms satisfy Lipschitz condition under the designed state-dependent switching rule are established for different assumptions on time delay. Moreover,some new stability results for linear delayed switched systems are also presented. The effectiveness of the proposed results is validated by three typical numerical examples.

I. Introduction

A switched system is composed of a set of continuous time or discrete-time subsystems and a law which governs the switching among them [1]. As a class of typical hybrid systems, switched systems have widespread practical backgrounds and engineering value. On the one hand, many natural or engineering systems which generate different modes owing to sudden changes of their environment can be modeled as switched systems. For example, aircraft control systems [2],traffic management systems [3], robotic walking control systems [4], computer disk drives [5] and some chemical processes [6] can fall into switched systems. On the other hand,switching can be viewed as an effective way of solving certain control problems. Multi-controller switching scheme [7],Bang-Bang control [8], intermittent control [9] and supervisory control [10] are derived from the idea of switching. Until now, switched systems have been a popular research focus in the control field and many significant results have been deduced in recent years.

System switching can generate high nonlinearities, which indicates that stability analysis of switched systems is more complicated than those of ordinary dynamical systems. Under some typical examples, Decarloet al. have pointed out that a switched system with all unstable subsystems (or with all asymptotically stable subsystems), may be asymptotically stable (or unstable) under some specified switching rules [11].In [12], the authors show that the delayed random switched systems derived from exponentially stable subsystems may lose stability with decreases in dwell times. This is because switching may generate destabilizing effects which destroy the stability of switched systems. Therefore, we must concentrate on both subsystems and switching rules in the study of stability. Liberzon has summed up the stability investigation of switched systems as three basic problems[13]: 1) stability under arbitrary switching signals, 2) stability under constrained switching signals, and 3) the construction of switching signal to stabilize switched systems. In order to cope with the stability of switched systems, many effective analysis tools such as the common Lyapunov function [14],multiple Lyapunov functions [15], average dwell time [16],and mode-dependent average dwell time [17] have been employed to derive novel stability results.

Designing appropriate switching rules to stabilize switched systems with entirely unstable subsystems is a valuable and challenging problem. This problem can be solved with two strategies: time-dependent switching and state-dependent switching. For the first, dwell time of the admissible switching rules is required to have both upper and lower bounds.According to discretized Lyapunov function, the timedependent switching rule to guarantee the asymptotic stability of linear switched systems was designed [18]. Based on the same method, the switching rule is constructed in [19] to stabilize switched neural networks with time-varying delays and unstable subsystems. The bound of dwell time for the asymptotic stability of linear delayed switched systems is presented in [20]. In these results, the stabilization property of switching behaviors is utilized to compensate the divergent effect of unstable subsystems. When a switched system is not stable under any time-dependent switching rule, the only alternative is to employ a state-dependent switching rule to stabilize the switched system. As early as 2003, Liberzon has proved that linear switched systems with two unstable subsystems is asymptotically stable under some statedependent switching rule, if there exists a Hurwitz convex combination of coefficient matrices [13]. In [1], this result is extended to linear switched systems with multi-subsystems.At present, the switching rule design can be usually summarized with the following three steps: 1) based on Hurwitz linear convex combination, divide state space into several switching regions, 2) construct the state-dependent switching rule, and 3) establish the stability conditions that the switched system is asymptotically stable under the constructed state-dependent switching rule. According to the common Lyapunov function and mixed mode, Kimet al. designed state-dependent switching rules for linear switched systems with a constant time delay. However, the proposed results are only applicable for cases where the time delay is sufficiently small [21]. In [22], the new state-dependent switching rule is designed for linear switched systems with time delays to weaken the restriction on time delay. Under time delay approximation and small gain theorem, Liet al. present new stability results under the state-dependent switching rule for linear switched systems with time delay and show that time delay restriction is weaker [23]. Owing to common Lyapunov function and multiple Lyapunov functions, the statedependent switching rule for delayed switched Hopfield neural networks is designed in [24]. The analogous switching rule for linear positive switched systems is designed in [25].Furthermore, some design methods for state-dependent switching rules are presented in [26]–[28] by guaranteeing the convergence of Lyapunov function. However, these results are only valid for linear switched systems.

The discussed content indicates that the stability of switched systems with all unstable subsystems (under the designed state-dependent switching rule) has been studied extensively.However, some inadequacies still remain. First, until now,most stability results have been suitable for linear switched systems with or without time delays [1], [13], [20]–[23],[25]–[28]. Unfortunately, the researchers gave less attention on the stability for nonlinear switched systems with time delay under state-dependent switching rules. Although the stability results for delayed Hopfield neural networks are presented in[24], they may not be valid for more general nonlinear switched systems. Second, the rigorous restrictions on time delay in the existing results should be further weakened,especially for the case that the bound of the derivative of time delay is known.

In this paper, we also cope with the stability of delayed switched systems with all unstable subsystems. First, we focus on the nonlinear delayed switched systems whose nonlinear terms satisfy Lipschitz condition. Based on the Hurwitz linear convex combination, we design the state-dependent switching rule. Then, some stability results under different assumptions are derived by the Wirtinger integral inequality and Leibniz-Newton formula. Second, by viewing linear delayed switched systems as special nonlinear delayed switched systems, some new stability results for linear switched systems with timevarying delay are also proposed. Three numerical examples are employed to show the effectiveness of the proposed results. The main contributions of this paper are listed as follows.

1) Certain stability results for nonlinear switched systems with all unstable subsystems and time-varying delay under state-dependent switching rule are derived.

2) For linear switched systems with all unstable subsystems and time-varying delay, the proposed stability results weaken the restriction on time delay if the bound of the derivative of time delay is known.

II. Preliminaries

This paper concerns delayed switched systems with the form

wherex(t)∈Rnis the state vector, σ(t)∈M={1,2,...,m} is a piecewise continuous function called switching signal,Ap,Bp∈Rn×n, τ(t) is the time-varying delay, ?(s) is a piecewise continuous function,fp(z)=(fp1(z1),...,fpn(zn)) is a nonlinear function. If σ(t)=p, we say that thepth subsystemx˙(t)=Ap(t)x(t)+Bp fp(x(t?τ(t))) is activated. Timetis called as a switching instant if σt+σt?.

In this paper, we always assume that for anyp∈Mandi=1,2,...,n,fpi(0)=0. Moreover, we assume that the functionfpisatisfies Lipschitz condition. Namely, there exists positive constantlpisuch that

For convenience, we denoteThen,system (1) can be rewritten as

We always assume that there exists a Hurwitz linear convex combinationFofNamely

where αp∈[0,1] andFor a given symmetric positive definite matrix Ξ, there exists some symmetric positive definite matrixPsuch that

Define region Θpas

Under the proof of [21, Proposition 2], it is easy to see

The state space can be divided into the following switching areas

Then, the switching rule for system (1) can be designed as

The above switching rule indicates that, for any instantt, the index of the activated subsystem isp, ifx(t) is located in the area. Noting that there is no restriction ont, the switching rule (8) is thus only dependent on the state.

The main purpose of this paper is to deduce the sufficient conditions for the global asymptotic stability of system (1)under the state-dependent switching rule (8). The following assumptions are essential for our results.

Assumption 1:There exist positive constants τ ,and constantsuch that

Assumption 2:There exists positive constant τ such that

Now we give a lemma which is the core of this investigation.

Lemma 1 (Wirtinger-based integral inequality, [29]):For a given matrixR>0, positive constantst1andt2wheret1

III. Main Results

This section proposes some stability results for the delayed switched systems under state-dependent switching rule (8) via the Lyapunov functionals. In Section III-A, we give the stability results for system (1) with Assumptions 1 and 2.Some stability results for linear delayed switched systems are derived in Section III-B.

A. Stability for Nonlinear Delayed Switched Systems

Theorem 1:Under Assumption 1, if there exist matricesP>0,Z>0 ,Rl>0(l=1,2) , diagonal matricesNph=n×nmatricesYp j,Tpj,S pj,Upj,Hpj(j=1,2), and 5n×5nsymmetric matricessuch that

where

then, system (1) is globally asymptotically stable under the state-dependent switching rule (8).

Proof:We choose the Lyapunov functional as follows

where

Whenaccording to the switching rule (8), we know that thepth subsystem is activated. It follows from (5)and (6) that

DifferentiatingV1,V2andV3, we have

Under (3), we conclude that

According to Lemma 1, we gain

Based on (2), we have

which yields that

For any matrixYp j,Tpj,Up j,S pjandHp j,j=1,2, owing to the Leibniz-Newton formula, it is known that

and

For any matrix) satisfying (12), we have

and

where

Then, it follows from (17)–(22) and (24)–(28) that

On the grounds of (13) and (14), we know thatfor anyand. Accordingly, due to (16), system (1)is globally asymptotically stable under the state-dependent switching rule (8).

If we restrictR1=R2=R, the stability results under Assumption 2 can be derived.P>0,Z>0,R>0 , diagonal matrices0(p∈M,h=1,2),n×nmatricesYpj,Tpj,S pj,Upj,Hp j(j=1,2) , and 5n×5nsymmetric matricessuch

Theorem 2:Under Assumption 2, if there exists matrices that (12), (14), (15) and

Remark 1:In [24], the researchers have investigated the stability of delayed switched neural networks

under state-dependent switching, where?apn} withapi>0 ,i=1,...,n. The switching region Πpis constructed by

Remark 2:In many stability results for switched systems with time-varying delay, the derivative of the time delay must satisfy τ˙(t)<1 [22], [24], which implies thatt?τ(t) is monotonically increasing. Generally speaking, this restriction is rigorous. In order to remove this restriction, in Assumption 1 we required that the derivative of the delay has both upper and lower bounds, which is consistent with that proposed in[30].

B. Stability for Linear Delayed Switched Systems

By restrictingfp=Ifor anyp∈M, system (1) can be written as the following linear switched systems with timevarying delay

Similarly, the above system can be rewritten as

Analogous to Theorems 1 and 2, we can obtain the stability results for system (32) under Assumptions 1 and 2.

Theorem 3:Under Assumption 1, if there existn×nmatricesP>0,Z>0,Ri>0(i=1,2),n×nmatricesYp j,Tpj,and 4n×4nsymmetric matricessuch that

where

then, system (32) must be globally asymptotically stable under the state-dependent switching rule (8).

Theorem 4:Under Assumption 2, if there existn×nmatricesP>0,Z>0 ,R>0,n×nmatricesYp j,Tpj,S pj,Upj(p∈M,j=1,2) , and 4n×4nsymmetric matricessuch that (34), (36), (37) and

As we know, the stability of system (32) under statedependent switching rules have been discussed in [22]. In order to directly show the comparison between our results and the proposed ones, we give some stability results under the assumption presented in [22].

Assumption 3:There exist positive constants τ andsuch that

Corollary 1:Under Assumption 3, if there existn×nmatricesP>0,Z>0,Ri>0(i=1,2),n×nmatricesYp j,Tpj,S pj,Upj(p∈M,j=1,2) , and 4n×4nsymmetric matricessuch that (34), (36), (37) and

Remark 3:In [22], researchers coped with the stability of system (32) with state-dependent switching under Assumption 3. According to the proof of [22, Theorem 1], one can derive that the following two inequalities are employed

and the equations

IV. Examples

Example 1:Consider the switched system (1) withm=2,

It is easy to obtain thatL1=L2=IandAp+BpLp,p∈M, is unstable. Letting α1=α2=0.5, we obtain the Hurwitz linear convex combination

TABLE I The Upper Bound τ of Time Delay and Novs for Different in Example 1

TABLE I The Upper Bound τ of Time Delay and Novs for Different in Example 1

μ=τˉ=?τ? 0.1 0.2 0.5 1Novs 0.2449 0.2332 0.2180 0.1264 N1

Fig. 1. The stable response curves and the corresponding switching rules for system in Example 1 with

Now we validate that the results presented in [24] are not valid for this example. Let α ∈[0,1]. The linear convex combination ofA1andA2can be expressed as

In order to ensure thatis Hurwitz, the following inequality must be satisfied

Namely,

Lety(α)=6α2?5α+169/144. According to the extreme value theorem, it is the case thaty(α)≥19/144 for any α ∈[0,1], which yields that (48) is not satisfied. This shows that all the linear convex combinations ofA1andA2are not Hurwitz. Therefore, one cannot design the state-dependent switching rule to stabilize this example under the results proposed in [24].

Example 2:Consider the switched system (32) withm=2 and

Similar to [22], [23], by choosing α1=0.6, α2=0.4, we have

TABLE II The Upper Bound τ of Time Delay and Novs for differentin Example 2

TABLE II The Upper Bound τ of Time Delay and Novs for differentin Example 2

μ=τˉ=?τ? 0.1 0.2 0.5 1Novs 0.0457 0.0430 0.0321 0.0183 N2

The stable time response curves for initial conditions ?(s)=(?3,2)Tand ?(s)=(2,?1)Tare shown in Fig. 2 under the corresponding state-dependent switching rules (8), which are shown in the subfigure of Fig. 2. This indicates the effectiveness of the proposed control strategy. It is worth noting thatis not satisfied; therefore, the results presented in [22] are not available for this case.

Fig. 2. The stable response curves and the corresponding switching rules for system in Example 2 with τ (t)=0.009?0.009sin

In order to show the superiority of our results, we can draw some comparisons with existing results. Table III gives the upper bound τ of time delay for differentand the Novs under the identical assumptions. It can be seen that the upper bound τ derived by our results is more precise than those obtained by the method proposed in [22] at the cost of greater computation. For example, for τ(t)=0.022?0.022sin[22, Theorem 1] fails because τ=0.044. However, owing to=0.1 and τ=0.044, by solving the LMI presented in Corollary 1 we obtain

which yields that this switched system is globally asymptotically stable under the state-dependent switching rule (8). The stable time response curves and the corresponding switching rules are plotted in Fig. 3 for different initial conditions.

Example 3:In order to show the application of the proposed results, we consider the water quality system, which is expressed as [31], [32]

where ρ(t) and ?(t) are the concentrations per unit volume of biological oxygen demand and dissolved oxygen, respectively,at timet. Whenm=2, τ (t)=0.05+0.05sin2t,

the two subsystems of (49) are unstable, which is shown inFig. 4. However, according to Theorem 3, one can obtain the feasible solution

TABLE III The Upper Bound τ of Time Delay and Novs for Different

TABLE III The Upper Bound τ of Time Delay and Novs for Different

CriterionˉτNovs 0 0.1 0.5 ˉτ is unknown Sun [22] 0.0202 0.0179 0.0176 0.0176 4mn2+mn+1.5n2+1.5n Corollary 1 0.0478 0.0457 0.0321 0.0236 24mn2+4mn+2n2+2n

Fig. 3. The stable response curves for the system in Example 2 with τ(t)=0.022?0.022sin

Fig. 4. The unstable response curves for the subsystems of (49).

This indicates that the water quality system is asymptotically stable under the state-dependent switching rule(8). Fig. 5 and its subfigure show the stable response curves of the water quality system (49) and the corresponding statedependent switching rules (8), respectively. This example indicates that the state-dependent switching strategy can effectively control the water quality.

Fig. 5. The stable response curves of the water quality system (49).

V. Conclusions

This paper has investigated the stability of switched systems with time-varying delay and all unstable subsystems under the state-dependent switching rule. Due to the Hurwitz linear convex combination, a state-dependent switching rule is designed. Based on Wirtinger integral inequality and Leibniz-Newton formula, the stability results for nonlinear delayed switched systems whose nonlinear terms satisfy Lipschitz condition and linear delayed switched systems have been derived, respectively. Through the numerical simulation, it has shown that the proposed results are more flexible than those presented in [24], and the restriction on time delay of the proposed results is weaker than those of the stability results in[22].