# Input-to-State Stability of Nonlinear Switched Systems via Lyapunov Method Involving Indefinite Derivative

Authors: Peng Li; Xiaodi Li; Jinde Cao
Journal: Complexity
 (2018)

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## Abstract

This paper studies the input-to-state stability (ISS) of nonlinear switched systems. By using Lyapunov method involving indefinite derivative and average dwell-time (ADT) method, some sufficient conditions for ISS are obtained. In our approach, the time-derivative of the Lyapunov function is not necessarily negative definite and that allows wider applications than existing results in the literature. Examples are provided to illustrate the applications and advantages of our general results and the proposed approach.


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## Body

## 1. Introduction

Switched systems are a special subclass of hybrid systems which consist of two components: a family of systems and a switching signal. The systems in the family are described by a collection of indexed differential or difference equations. The switching signal selects an active mode at every instant of time, that is, the system from the family that is currently being followed. As a special class of hybrid systems, switched systems arise in a variety of applications, such as biological systems [1], automobiles and locomotives with different gears [2], DC-DC converters [3], manufacturing processes [4], and shrimp harvesting mode [5]. Many interesting results for switched systems have been reported in the literature [6–9]. Qualitative behaviour of switched systems depends not only on the behaviour of individual subsystems in the family, but also on the switching signal. For instance, divergent trajectories can be generated by switching appropriately among stable subsystems, while a proper switching signal may ensure stability of a switched system even when all the subsystems are unstable. Due to such interesting features, stability of switched systems has attracted considerable research attention over the past few decades; see [10–15].When investigating stability of a system, it is important to characterize the effects of external inputs. The concepts of input-to-state stability (ISS) introduced by Sontag et al. in [16, 17] have been proved useful in this regard. Roughly speaking, the ISS property means that no matter what the size of the initial state is, the state will eventually approach a neighborhood of the origin whose size is proportional to the magnitude of the input. Many interesting results on ISS properties of various systems such as discrete systems, switched systems, and hybrid systems have been reported; see [18–29]. For example, [19] presented converse Lyapunov theorems for input-to-state stability and integral input-to-state stability (iISS) of switched nonlinear systems; [22, 23] studied the ISS of nonlinear systems subject to delayed impulses; [29] dealt with the ISS of discrete-time nonlinear systems. However, one may observe that most of them, such as those in [18–31], require the derivative of Lyapunov functions to be negative definite in order to derive the desired ISS property. Recently, [32] proposed a new approach for ISS property of nonlinear systems. It presents a new comparison principle for estimating an upper bound on the state of the system in which the derivative of the Lyapunov function may be indefinite, rather than negative definite, which improves the previous work on this topic greatly. The authors of [33] developed the idea to delayed systems and established a class of continuously differentiable Lyapunov-Krasovskii functionals involving indefinite derivative, which generalizes the classic Lyapunov-Krasovskii functional method. However, the approach used there only applies for systems without switched structures. Moreover, to the best of our knowledge, there are few results on ISS of switched systems based on Lyapunov method involving indefinite derivative.Motivated by the above discussions, in this paper, we shall study the ISS property for switched systems via Lyapunov method involving indefinite derivative. Some sufficient conditions based on ADT method are derived. It is worth mentioning that, although the method used in this paper is based on [32], the results in this paper are more general than [32], even for the case of systems without switched structures. The rest of this paper is organized as follows. In Section 2 the problem is formulated and some notations and definitions are given. In Section 3, we present some new characterizations of ISS based on Lyapunov method involving indefinite derivative. Examples are given in Section 4. Finally, the paper is concluded in Section 5.


## 2. Preliminaries

Notations. Let Z+ denote the set of positive integer numbers, R the set of real numbers, R+ the set of all nonnegative real numbers, and Rn and Rm×n the n-dimensional and m×n-dimensional real spaces equipped with the Euclidean norm |·|, respectively. a∧b and a∨b are the minimum and maximum of a and b, respectively. P={1,2,…,m},m∈Z+, is an index set, C(J,S)={φ:J→Siscontinuous}, F={φ:[t0,∞)→P,isapiecewiseconstantfunction}. The notations AT and A-1 denote the transpose and the inverse of A, respectively. I denotes the identity matrix with appropriate dimensions.Consider the following switched system:(1)x˙=fσtt,x,u,where x∈Rn is the system state, u∈Rm is a measurable locally bounded disturbance input, x˙ denotes the right-hand derivative of x, and σ∈F denotes the switching function, which is assumed to be a piecewise constant function continuous from the right. When σ(t)=i,1≤i≤m, we say that the mode x˙=fi is activated. A sequence of discrete times {tn},n∈Z+, called the switching times, determines when the switching occurs. Throughout this paper, we assume that it satisfies 0≤t0<t1<⋯<tk→+∞ as k→+∞ (t1 is the first switching time). In particular, we exclude the possibility of the {tk} having a finite accumulation point, often referred to as chattering. It indicates that a switching signal ({tn},σ) has at most finite switching times over a finite time interval. fσ(t)∈C(R+×Rn×Rm,Rn) with local Lipschitz, and fσ(t)(t,0,0)≡0,t∈R+. In order to study the ISS, in the following we assume that the solution of system (1) with an initial condition x(t0)=x0 exists on [t0,+∞) uniquely.By the ideas proposed by Hespanha and Morse [34] for switched systems, we say that a switching signal ({tn},σ) has average dwell-time (ADT) τ if there exist numbers N0≥0 and τ>0 such that (2)NσT,t≤N0+T-tτ,∀T≥t≥t0,where N0 is called the “chatter bound” and Nσ(T,t) is the number of switches occurring in the interval [t,T). We denote such kind of switching signals by set Fτ. Denote the switching times in the interval (t,T] by t1,t2,…,tNσ(t,t0) and the index of the system that is active in the interval [tn,tn+1) by pn.A functionα:[0,∞)→[0,∞) is of class K if α is continuous and strictly increasing and α(0)=0. If α is also unbounded, it is of class K∞. A function β:[0,∞)×[0,∞)→[0,∞) is of class KL if β(·,t) is of class K for each fixed t≥0, and β(r,t) decreases to 0 as t→∞ for each fixed r≥0.Definition 1 (see [16]).
Suppose that a switching signal({tn},σ) is given. The system (1) is said to be ISS if there exist functions γ∈K∞ and β∈KL such that for each t0≥0, x0∈Rn and for each input u, the solution satisfies (3)xt≤βx0,t-t0+γut0,t,for all t≥t0, where |·|J denotes the supremum norm on the interval J. This definition depends on the choice of the switching signal; however, it is often of interest to characterize ISS over classes of switching signals. We say that system (1) is uniformly input-to-state stable (UISS) over the class Fτ (of switching signal) if for any ({tn},σ)∈Fτ condition (3) is satisfied with the same γ and β for every ({tn},σ)∈Fτ.


## 3. ISS Theorems

In this section, we shall present some ADT results for ISS of switched system (1) based on Lyapunov method involving indefinite derivative.Theorem 2.
Assume that there exist functionsα1,α2∈K∞, ρ∈K, a continuous function ϕ:R+→R, continuous differentiable functions Vp:R+×Rn→R+, and constants η>0,μ≥1 such that, for all t∈R+, x∈Rn, and all p,q∈P, (4)α1x≤Vpt,x≤α2x;(5)∂Vp∂xfpt,x,u+∂Vp∂t≤ϕtVpt,xwheneverVp≥ρuexp∫t0tϕsds;(6)Vpt,x≤μVqt,x;(7)∫t0tϕs+ηds≤0.Then the switched system (1) is UISS over the class Fτ, where ADT constant τ>0 satisfies (8)τ>ln⁡μη.Proof.
Letx(t) be a solution of system (1). Define Vσ(t)(t)=Vσ(t)(t,x(t)). If (9)Vpt≥ρuexp∫t0tϕsds,during some interval [t′,t′′), in this case, suppose that there exists switching signal {tn} such that t′<tn<tn+1<⋯<tn+m<t′′. For t∈[t′,tn), it follows from (5) that (10)Vpn-1t≤Vpn-1t′exp∫t′tϕsds.For t∈[tn,tn+1), it follows from (5), (6), and (10) that (11)Vpnt≤Vpntnexp∫tntϕsds≤μVpn-1t′exp∫t′tϕsds.Then it can be deduced that (12)Vσtt≤μNσt′,texp∫t′tϕsdsVσt′t′,∀t∈t′,t′′.Since μ≥1, it follows from the ADT condition (2) that (13)Vσtt≤μN0+t-t′/τexp∫t′tϕsdsVσt′t′=μN0exp∫t′tln⁡μτ+ϕsdsVσt′t′.We denote the first time when (14)Vσtt≤ρuexp∫t0tϕsdsby ť; that is, (15)ť1=inft≥t0:Vσtt≤ρuexp∫t0tϕsds.If  ť1=∞, then it holds that (16)Vσtt≤μN0exp∫t0tln⁡μτ+ϕsdsVσt0t0,∀t≥t0.It follows from (8) that there exists ε>0 small enough such that (17)ln⁡μτ+ε≤η,which together with (7) yields that (18)Vσtt≤μN0exp-εt-t0Vσt0t0,∀t≥t0.It then follows from (4) that (19)xt≤α1-1μN0exp-εt-t0α2xt0.Thus x(t) is bounded by a KL-class function, which implies that system (1) is ISS. Hence we only need to consider the case that ť1<∞. It follows from (18) that (20)Vσtt≤μN0exp-εt-t0Vσt0t0,t∈t0,ť1.For t≥ť1, we denote the first time when (21)Vσtt>ρuexp∫t0tϕsdsby t^1; that is,(22)t^1=inft≥ť1:Vσtt>ρuexp∫t0tϕsds.If    t^1=∞, then it is obvious that system (1) is ISS. Assuming that t^1<∞, then (23)Vσtt≤ρuexp∫t0tϕsds≤μN0ρut0,texp∫t0tϕsds,t∈ť1,t^1.Then we further denote the second time when (24)Vσtt≤ρuexp∫t0tϕsdsby ť2; that is, (25)ť2=inft≥t^1:Vσtt≤ρuexp∫t0tϕsds.Due to the continuity of Vσ(t)(t) and the monotonicity of ρ(|u(t)|), when ť2<∞, it holds that (26)Vσt^1t^1=ρut^1exp∫t0t^1ϕsds,Vσtt≤μN0exp⁡∫t^1tln⁡μτ+ϕsdsVσt^1t^1≤μN0exp⁡∫t^1tln⁡μτ+ϕsdsρut^1exp⁡∫t0t^1ϕsds≤μN0exp⁡∫t^1tln⁡μτdsρut^1,texp⁡∫t0tϕsds≤μN0exp⁡∫t0tln⁡μτ+ϕsdsρut0,t≤μN0exp⁡-εt-t0ρut0,t≤μN0ρut0,t,∀t∈t^1,ť2.By this way, it can be deduced that, for every t≥ť1, it holds that (27)Vσtt≤μN0λρut0,t,where (28)λ=exp∫t0tϕsds∨1.It follows from (20) and (27) that (29)Vσtt≤μN0exp⁡-εt-t0Vσt0t0+μN0λρut0,t,for all t≥t0, which together with (4) yields that (30)xt≤α1-12μN0exp⁡-εt-t0α2xt0+α1-12μN0λρut0,t≔βxt0,t-t0+γut,t0,for all t≥t0. This indicates that system (1) is UISS over the class Fτ. The proof is completed.In particular, if system (1) is given in the form of (31)x˙=ft,x,u,which is a general case without switched structure, by Theorem 2, one may derive the following corollary.Corollary 3.
Assume that there exist functionsα1,α2∈K∞, ρ∈K, a continuous function ϕ:R+→R, a continuous differentiable function V:R+×Rn→R+, and a constant η>0 such that, for all t∈R+, x∈Rn, (7) and the following conditions hold: (32)α1x≤Vt,x≤α2x;∂V∂xft,x,u+∂V∂t≤ϕtVt,xwheneverV≥ρuexp∫t0tϕsds.Then system (31) is ISS.Remark 4.
Recently, [32] has presented some sufficient conditions for ISS property of system (31) based on Lyapunov method involving indefinite derivative under the assumption that (33)∫t0∞ϕ+sds<∞,∫t0tϕ-sds≥ϵt-t0,where ϕ+(s)=ϕ(s)∨0,ϕ-(s)=[-ϕ(s)]∨0, and ϵ>0 is a constant, while our ISS result in Corollary 3 only requires that (7) holds, which has wider applications. For example, ϕ=sin⁡t-0.9 and t0=0, and it is easy to see that ϕ is a sign reversal function. In this case, one may choose η=0.1 such that (34)∫t0tϕs+ηds=∫0tsin⁡s-0.8ds<0,∀t>0,which implies that (7) holds. However, it is easy to see that (35)∫0∞ϕ+sds=∞.
Next we consider the time-varying linear switched system in the form of(36)x˙=Aσttxt+Bσttut,t≥0,where x(t)∈Rn is the system state, u∈Rm is locally bounded input, and Aσ(t)(t)∈Rn×n and Bσ(t)(t)∈Rm×n are time-varying functions. To ensure the ISS property of (36), we present the following result.Theorem 5.
Assume that there exist constantsη>0,μ≥1,ωp>0, and continuous functions ϕ:R+→R and ϕ-:R+→R such that, for all t∈R+ and all p,q∈P, ωp≤μωq, (7), and the following hold: (37)Apt+ApTt+BptBpTtexp⁡-∫0tϕ-sds+ωpexp⁡∫0tϕ-s-ϕsds·In×n≤ϕt·In×n.Then system (36) is UISS over the class Fτ, where ADT constant τ>0 satisfies (8).Proof.
Letx(t) be a solution of system (36) and define Vp(t)=Vp(t,x(t)). Then the proof of Theorem 5 is similar to Theorem 2. We only need to notice that the following are chosen: Vp(t)=ωpx(t)Tx(t) and ρ(|u|)=uT(t)u(t). It then follows from Theorem 2 that when Vp(t)≥ρ(|u|)exp⁡(∫0tϕ(s)ds), it holds that ωpxT(t)x(t)≥uT(t)u(t)exp(∫0tϕ(s)ds), which, together with (37), leads to the following:(38)V˙pt=2ωpxTtx˙t=2ωpxTtAptxt+Bptut=2ωpxTtAptxt+2ωpxTtBptut≤2ωpxTtAptxt+exp⁡-∫0tϕ-sdsωpxTt×BptBpTtxt+ωpuTtutexp⁡∫0tϕ-sds≤ωpxTtApt+ApTt+BptBpTt×exp⁡-∫0tϕ-sds+ωpexp⁡∫0tϕ-s-ϕsds·In×nxt≤ϕtVpt,which implies that condition (5) holds. Then it is easy to check that all conditions in Theorem 2 hold and thus Theorem 5 can be derived. The proof is completed.In particular, if we chooseϕ-(t)=ϕ(t), then the following corollary can be derived directly.Corollary 6.
Assume that there exist constantsη>0,μ≥1,ωp>0, and continuous function ϕ:R+→R such that, for all t∈R+ and all p,q∈P, ωp≤μωq, (7), and the following hold:(39)Apt+ApTt+BptBpTtexp-∫0tϕsds≤ϕt-ωp·In×n.Then the system (36) is UISS over the class Fτ, where ADT constant τ>0 satisfies (8).In addition, note that the ISS property guarantees the uniform asymptotic stability (UAS) of a system with a zero input. Consider the nonlinear switched system(40)x˙=fσtt,x,where σ∈F is the switching function, fσ∈C(R+×Rn,Rn) is local Lipschitz and fσ(t,0)=0,x(t,t0,x(t0)) is the solution for system (40) with the initial value x(t0)∈Rn and an initial time t0≥0. Then we have the following result for system (40).Corollary 7.
Assume that there exist functionsα1,α2∈K∞, a continuous function ϕ:R+→R, continuous differentiable functions Vp:R+×Rn→R+, and constants η∈R+,μ≥1 such that, for all t∈R+, x∈Rn, and all p,q∈P, (4), (6), (7), and the following condition hold: (41)∂Vp∂xfpt,x+∂Vp∂t≤ϕtVpt,x.Then system (40) is UAS in Lyapunov sense, where ADT constant τ satisfies (8).


## 4. Applications

In this section, we present two examples to illustrate our main results.Example 8.
Consider the switched system (1) with P={1,2}, t0=0, and (42)f1t,x,u=atx+btu,f2t,x,u=ctx+dtu,where a(t)=-cos⁡t-5/6,b(t)=1/6exp⁡(-sin⁡t-2/3t),c(t)=-cos⁡t-1/2exp⁡(cos⁡t-2/3), and d(t)=exp⁡(cos⁡t-sin⁡t-2/3t).
Note thata(t) and c(t) are sign reversal functions. Most of existing results, such as those in [18–23, 26–29], are inapplicable to switched system (1). Choose V1(t,x,u)=|x| and V2(t,x,u)=1/2|x| as ISS-Lyapunov functions. It is easy to see that condition (6) holds with μ=2. Let ρ(t)=t and ϕ(t)=-cos⁡t-2/3, and then when (43)V1t,x,u≥ρuexp⁡∫0tϕsds,that is, (44)x≥uexp⁡-sin⁡t-23t,it leads to (45)V˙1t,x,u≤-cos⁡t-56x+16x≤-cos⁡t-23V1t,x,u=ϕtV1t,x,u.Similarly, it can be deduced that V˙2(t,x,u)≤(-cos⁡t-2/3)V2(t,x,u) when V2(t,x,u)≥ρ(|u|)exp⁡(∫0tϕ(s)ds). Thus condition (5) is satisfied. Choose η=0.45 such that (46)∫t0tϕv+ηdv=∫0t-cos⁡v-0.2167dv<-0.02<0,∀t>0,which implies that (7) holds. Note that lnμ/η≈1.5403. Hence, the switched system (1) is UISS over the class Fτ with τ>1.5403. In particular, if we choose the switching sequence t2n-1=6n-4,t2n=6n,n∈Z+ and let x(0)=5, τ=3,u=sat(x), then Figures 1(a) and 1(b) illustrate the switching signal and the state trajectory of system (1), respectively.Figure 1
Simulation results for Example8.
(a)
(b)Remark 9.
Note that, if we chooseV2=|x| in the above example, then it can be deduced that V˙2≤(-cos⁡t+1/2exp⁡(cos⁡t)-2/3)V2, which goes against condition (5). It indicates that sometimes it is necessary and important to consider multiple Lyapunov functions for switched systems. In addition, it is easy to check that the ISS Theorems in [32] are invalid for the above example due to the stronger restriction on ϕ+.Example 10.
Consider the time-varying switched system (36) with P={1,2} and (47)x˙t=A1+ΔA1txt+B1+ΔB1tu1t,x˙t=A2+ΔA2txt+B2+ΔB2tu2t,where (48)A1=-0.10500-0.11,ΔA1t=a100a2,A2=-0.115-0.005-0.005-0.115,ΔA2t=b100b2,B1=B2=00,ΔB1t=0c,ΔB2t=cc,a1=a2=b1=-12sint,b2=12-sin⁡t-exp⁡-2t,c=110exp12cos⁡t-15t-1.
In this case, chooseω1=0.01,ω2=0.02,η=0.1,μ=2, and ϕ(t)=-sin⁡t-0.2. Then it is easy to see that (49)∫0tϕs+ηds=∫0t-sin⁡s-0.1ds<0,∀t>0,which implies that (7) holds. Moreover, note that (50)A1t+A1Tt+B1tB1Ttexp-∫0tϕsds=-sin⁡t-0.2100-sin⁡t-0.21,A2t+A2Tt+B2tB2Ttexp⁡-∫0tϕsds=-sin⁡t-0.2200-sin⁡t-exp⁡-2t-0.22.Note that lnμ/η≈6.931. Thus it follows from Corollary 6 that the switched system (36) is UISS over the class Fτ with τ>6.931. In particular, if we choose the switching sequence t2n-1=14n-1,t2n=14n,n∈Z+ and let xT(0)=(3,3), τ=7,u1=sinx,u2=sat(x), then Figures 2(a) and 2(b) illustrate the switching signal and the 2-norm of the state trajectory of system (36), respectively.Figure 2
Simulation results for Example10.
(a)
(b)


## 5. Conclusion

In this paper, we presented some new ADT-based sufficient conditions for ISS of switched systems via Lyapunov method involving indefinite derivative. The ISS property of the switched system can be guaranteed under the designed ADT scheme. Our results improved some recent work in the literature. Two examples were given to show the effectiveness and advantage of the obtained results. It should be pointed out that the main results of this paper are based on multiple Lyapunov functions, which are more general than existing results in some cases. Since complex factors such as nonlinearities, impulsive perturbations, and delays exist widely in various engineering systems [35], future work can be done to develop the Lyapunov method involving indefinite derivative to switched systems subject to complex factors.



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