# RobustH∞ Control for Switched Nonlinear System with Multiple Delays

Authors: Juan Liu; Qingdao Huang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper focuses on the problem of robustH∞ control for a class of switched nonlinear systems with multiple time delays which are allowed to appear not only in state but also in input. We will design a controller and a switching law under which the system is “robust stable” and has performance L2-gain γ. Besides, multiple Lyapunov-Razumikhin functions which are relatively less conservative are introduced to investigate the stability of the candidate switched system. In detail, firstly, for uncertain nonlinear switched system with time-varying delays, sufficient condition is given by inequalities which are equivalent to LMIs, so it is easy to be solved; then, as a special case, nonlinear switched system with constant time delay is also considered. Finally, by providing numerical examples, the feasibility of the proposed approach is demonstrated.


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## Body

## 1. Introduction

Switched systems have constituted a very active field of current scientific research; many real world processes and systems can be modeled as switched systems, such as chemical processes and computer controlled systems. Besides, switched systems are extensively applied in many domains, including mechanical systems, automation, aircraft and air traffic control, and many other fields. And stability analysis and control, as the most important topics for the study of switched systems, have been studied widely [1–11]. Up to now, some methods are given to solve the stability analysis for switched system. For example, by constructing a common Lyapunov function, stability under arbitrary switching can be obtained. However, most switched systems still may be stable under certain switching laws, though they do not own a common Lyapunov function. The key idea to select such switching laws is the dwell time [12, 13]. Lately, H∞ control theory, which is another effective tool to solve robust stabilization problems, has been well built [14].On the other hand, time-delay phenomenon is very common in many kinds of engineering systems, for instance, long-distance transportation systems, hydraulic pressure systems, network control systems, and so on. It is regularly a source of instability and often causes undesirable performance and even makes the system out of control. So time-delay systems have also drawn more and more attention [15–23]. Based on the above two aspects, the problem of stability analysis and controller synthesis for switched system with time delay (switched delay system) has aroused growing interest. And switched delay system, as a new brand, can be found in many applications and there are quite a lot of related results [24–34]. However, in the above papers, systems with only one delay are considered. And system with multiple delays, which is widely used in practice, has gained more and more attention [35–38]. For switched systems with multiple time delays, the problem of robust stability is considered in [39]. In [39], delays are constant and appear only in state. In fact, in engineering control design for the actual system, time-varying delays are universal, and due to various reasons delays can also appear in control input. On this occasion, the system will become more complicated and then the response of state may also be affected. In [40], the problem of robust H∞ control for nonswitched system with input delays is considered. Motivated by this, it is meaningful to investigate the following problem: if the system in [40] is switched, how is its stability and also how do we design a controller and a switching law to make the system stable? As yet, as far as we know, it has not been fully investigated despite its potential in practical applications, and the purpose of this paper is therefore to shorten such a gap by providing a rather general framework.The contribution of this paper lies in the following aspects. First, we address the problem ofH∞ control for a more general class of switched systems with multiple time-varying delays, in which delays not only appear in the state but also appear in the input. Second, by choosing a new set of multiple Lyapunov-Razumikhin functionals (which is relatively less conservative than the CLF approach), we derive sufficient conditions of the existence of H∞ state-feedback controller for a class of switching signals with average dwell time scheme; besides, the specific expression of controller is also given in the paper. Third, in this paper, the inequality we derived is easy to compute for it can be rewritten as LMIs.The remainder of this paper is organized as follows. Firstly, problem formulation and preliminaries are stated in Section2. In the following section, the main results are addressed. And a numerical example is given in Section 4. Section 5 draws the conclusion.Notations. Throughout this paper, the notation P>0(≥,≤,<) means that P is a positive definite (positive-semidefinite, negative-semidefinite, and negative definite) matrix P. λmax⁡(P) and λmin⁡(P) denote the maximum and minimum eigenvalues of P, respectively; the superscript “T” stands for the transpose; x(t) denotes the Euclidean norm; L20,∞ represents the space of square integrable vector functions over 0,∞; I is an identity matrix with appropriate dimension, and if not stated, matrices are assumed to have compatible dimensions. AB∗C denotes a symmetric matrix.


## 2. Problem Formulation and Preliminaries

Consider a class of uncertain switched delay systems of the form(1)x˙t=Aσtxt+A1σtxt-τ1t+Eσtf0σtxt,t+E1σtf1σtxt-τ2t,t+E2σtf2σtxt-τ3t,t+Bσtωt+B1σtuσtt+B2σtuσtt-τ4t,zt=Cσtxt,xt=0,t≤0,where x(t)∈Rn, u(t)∈Rq, and z(t)∈Rp are the state vector, input, and output, respectively. ω(t)∈Rl which is assumed to belong to L20,∞ is external disturbance input; σ(t):[0,∞)→M={1,2,…,m} is the switching signal; Ai, A1i, Bi, Ei, and E1i are given constant matrices with appropriate dimensions; τi(t) (i=1,2,3,4) are the time-varying delays which satisfy the following assumption.Assumption 1.
Consider0≤τi(t)≤di<∞, τ˙i(t)≤αi<1, i=1,2,3,4, for all t>0.Remark 2.
With regard to the switching signalσ(t), there exists a switching sequence {xt0;(i0,t0),…,(ik,tk),…,∣ik∈M,k=0,1,…}, which means that when t∈[tk,tk+1), the ikth subsystem is activated.Remark 3.
In general, the nonlinear uncertaintiesf0σ(t)(x(t),t), f1σt(x(t-τ2t)), and f2σt(x(t-τ3t)) which refer to the nonlinear internal parameter uncertainties with respect to the current state and delayed states are assumed to satisfy the following inequalities.Assumption 4.
Consider(2)f0σtxt,t≤Fσtxt,f1σtxt-τ2t≤F1σtxt-τ2t,f2σtxt-τ3t≤F2σtxt-τ3t,where Fσ(t), F1σ(t), and F2σ(t) are known constant matrices.Remark 5.
Compared with the switched system in [39] (where the control input is not contained), in this paper, delays are time varying and control input is also contained with delays, so system (1) is more general and more practical for we can choose a controller to make an unstable system stable.In this paper, we are interested in designing a state-feedback controller which is described byu(t)=Kx(t) and a switching law σ(t) such that system (1) is “robust stable.” Firstly, the relevant definitions are given below.Definition 6.
System (1) is said to be exponentially stable under controller u(t) and switching law σ(t), if, with ω=0, the solution x(t) of system (1) satisfies (3)xt≤kxt0e-λt-t0,∀t≥t0,where k≥1 and λ>0.Definition 7 (see [11]).
For anyT2>T1≥0, let Nσ(T1,T2) represent the switching number of σ(t) on (T1,T2). If Nσ(T1,T2)≤N0+(T2-T1)/τa holds for τa>0, N0≥0, then τa is called average dwell time.Remark 8.
The concept of average dwell time, which was an effective tool for the stability analysis of switched systems, was put forward for continuous switched systems firstly by Hespanha and Morse (see [11]). As commonly used in the literature, we choose N0=0 in this paper.Definition 9 (see [41]).
Forγ>0, α>0, consider system (1) and design a switching law i=σ(t) and controller u(t), if it holds that(1)
system (1) is exponentially stable when ω(t)≡0;(2)
under zero initial conditions system (1) satisfies (4)∫0∞e-αszTszsds≤γ2∫0∞ωTsωsds.Then system (1) is said to have exponential H∞ performance γ (weighted L2-gain).Now the following lemma which will be used to draw the main results in this paper is presented.Lemma 10 (see [40]).
For any function vectorF(x), symmetric matrix P, and constant matrix D, it holds that (5)2xTPDFx≤β2xTPDDTPx+β-2FTxFx,∀x∈Rn,∀β>0.


## 3. Main Results

In this paper, we focus on the switched delay system with time-varying delays and constant delays, respectively. And we will tackle the robustH∞ control problem for switched delay system (1). When t∈[tk,tk+1), the “ikth” subsystem of system (1) is activated. At this moment it can be seen as a “nonswitched” system corresponding to system (1). Based on this, we will first consider the nonswitched system:(6)x˙t=Axt+A1xt-τ1t+Ef0xt,t+E1f1xt-τ2t,t+E2f2xt-τ3t,t+Bωt+B1ut+B2ut-τ4t,zt=Cxt,xt=0,t≤0.Lemma 11.
Consider the nonswitched system (6). For a given positive definite matrix R>0 and constants α>0, γ>0, μ>0, and εi>0, i=1,2,3,4,5, define (7)Vx,t=xTtPxt+∑i=14∫t-τitteαs-txTsQixsdswith(8)Q1=eαd1ε121-α1I,Q2=eαd2ε321-α2F1TF1,Q3=eαd3ε421-α3F2TF2,Q4=eαd4ε521-α4KTK;the linear feedback controller is designed by u(t)=Kx(t), K=-1/2μR-1B1TP, such that the inequality (9)ATP+PA+αP-1μPB1R-1B1TP+CTC+Q~+PNP<0has a positive definite solution P, where (10)Q~=∑i=14Qi+ε2-2FTF,N=ε12A1A1T+ε22EET+ε32E1E1T+ε42E2E2T+ε52B2B2T+γ-2BBT.
Then, along the trajectory of the system, whenω(t)=0,(11)Vt≤e-αt-t0Vt0.For general ω(t), (12)Vt≤e-αt-t0Vt0-∫t0te-αt-sΓsds,where Γ(t)=z(t)2-γ2ω(t)2.Proof.
The following proof is motivated by the method in [40].
For the functionV(x,t) defined by (7),(13)V˙x,t+αVx,t+z2-γ2ω2=xTt·A+B1KTP+PA+B1K+αP+CTCxt+2xTt·PA1xt-τ1t+2xTtPB2Kxt-τ4t+2xTtPBωt+2xTtPEf0xt,t+2xTtPE1f1xt-τ2t+2xTt·PE2f2xt-τ3t+∑i=14xTtQixt-∑i=141-τ˙ite-ατitxTt-τitQixt-τit-γ2ω2.Using Assumptions 1 and 4 and Lemma 10, we get the following inequalities: (14)-1-τ˙it≤αi-1<0,2xTtPA1xt-τ1t≤ε12xTtPA1A1TPxt+ε1-2xTt-τ1txt-τ1t,2xTtPEf0xt,t≤xTtε22PEETP+ε2-2FTFxt,2xTtPE1f1xt-τ2t≤ε32xTtPE1E1TPxt+ε3-2xTt-τ2tF1TF1xt-τ2t,2xTtPE2f2xt-τ3t≤ε42xTtPE2E2TPxt+ε4-2xTt-τ3tF2TF2xt-τ3t,2xTtPBωt≤γ2ωTtωt+γ-2xTtPBBTPxt,2xTtPB2Kxt-τ4t≤ε52xTtPB2B2TPxt+ε5-2xTt-τ4tKTKxt-τ4t.Substituting inequalities (14) into (13), it holds that (15)V˙x,t+αVx,t+z2-γ2ω2≤xTtA+B1KTP+PA+B1K+αP+CTC+∑i=14Qi+ε2-2FTF+Pε12A1A1T+ε22EET+ε32E1E1T+ε42E2E2T+ε52B2B2T+γ-2BBTPxt+ε1-2xTt-τ1txt-τ1t+ε3-2xTt-τ2tF1TF1xt-τ2t+ε4-2xTt-τ3tF2TF2xt-τ3t+ε5-2xTt-τ4tKTKxt-τ4t-∑i=141-αi·e-αdixTt-τitQixt-τit.
Considering (8), inequality (15) can be expressed as(16)V˙x,t+αVx,t+z2-γ2ω2≤xTtA+B1KTP+PA+B1K+αP+CTC+ε2-2FTF+∑i=14Qi+Pε12A1A1T+ε22EET+ε32E1E1T+ε42E2E2T+ε52B2B2T+γ-2BBTPxt.Let K=-1/2μR-1B1TP in (16); we can get (17)V˙x,t+αVx,t+z2-γ2ω2≤xTtATP+PA+αP-1μPB1R-1B1TP+CTC+ε2-2FTF+∑i=14Qi+Pε12A1A1T+ε22EET+ε32E1E1T+ε42E2E2T+ε52B2B2T+γ-2BBTPxt.
Hence(18)V˙x,t+αVx,t+z2-γ2ω2<0.When ω(t)=0, it yields that (19)V˙x,t+αVx,t<z2<0.Then, integrating inequalities (18) and (19), respectively, we can get inequalities (11) and (12) in Lemma 11. This completes the proof.When the delays in the nonswitched system (6) are constant, we assume that the following conditions hold: (20)f0xt,t≤β0xt,f1xt-τ2≤β1xt-τ2,f2xt-τ3≤β2xt-τ3,where β0, β1, and β2 are positive real constants.Lemma 12.
For given constantsα>0, γ>0, β0>0, β1>0, β2>0, p>0, and u(t)=Kx(t), such that (21)β0E+β1E1+β2E2<p2P,define (22)Vx,t=xTtPxt,where P is a positive symmetric definite matrix of the following inequality: (23)A+B1KTP+PA+B1K+αP+PA1A1T+B2KKTB2P+γ-2PBBTP+CTC<-pI.Then, along the trajectory of system (6), when ω(t)=0, (24)Vt≤e-αt-t0Vt0and, for general ω(t), (25)Vt≤e-αt-t0Vt0-∫t0te-αt-sΓsds,where Γ(t)=z(t)2-γ2ω(t)2.Proof.
The proof is similar to that of Lemma11.
ForV(x,t)=xT(t)Px(t) and u(t)=Kx(t),(26)V˙x,t+αVx,t+z2-γ2ω2=xTt·A+B1KTP+PA+B1K+αP+CTC·xt+2xTtPA1xt-τ1+2xTt·PEf0xt,t+2xTtPE1f1xt-τ2+2xTtPE2f2xt-τ3+2xTt·PB2Kxt-τ4+2xTtPBωt-γ2ω2.Using a similar step of the proof in Lemma 11, we can obtain the following equality: (27)V˙x,t+αVx,t+z2-γ2ω2≤xTt·A+B1KTP+PA+B1K+αP+CTC+PA1A1T+B2KKTB2TP+γ-2PBBTPxt+2β0PExt2+2β1PE1xt·xt-τ2+2β2PE2xtxt-τ3+xt-τ42+xt-τ12.If we assume (28)xt-τi≤qixt,qi>1,i=1,2,3,4,inequality (27) becomes(29)V˙x,t+αVx,t+z2-γ2ω2≤xTt·A+B1KTP+PA+B1K+αP+CTC+PA1A1T+B2KKTB2TP+γ-2BBTPxt+2β0PE+2β1PE1q2+2β2PE2q3+q12+q42xt2.Considering that qi>1, it holds that(30)-2β0PE+β1PE1∣q2+β2PE2q3-q12-q42<-2β0PE+β1PE1+β2PE2-q42-q12<-2β0PE+β1PE1+β2PE2=-2Pβ0E+β1E1+β2E2<-p.Hence from inequalities (29) and (30), we have (31)V˙x,t+αVx,t+z2-γ2ω2<0.Then, when ω(t)=0, (32)V˙x,t+αVx,t<0.Similar to Lemma 11, we can get (24) and (25).Remark 13.
Assumption (28), that is, x(t-τi)≤qix(t), qi>1, i=1,2,3,4, in the above proof is commonly used in papers (see [42]).Now, we will design a controlleru(t) and a switching law σ(t), under which switched system (1) has exponential H∞ performance γ. Firstly, for system (1) with time-varying delays, the following theorem is given.Theorem 14.
For a given matrixRi>0 and constants α>0, γ>0, and εij>0, j=1,2,3,4,5, ν>0, if there exist matrices Pi>0, i∈M, and a linear state-feedback controller ui(t)=Kix(t), Ki=-1/2νRi-1B1iTPi, such that (33)AiTPi+PiAi+αPi-1νPiB1iRi-1B1iTPi+CiTCi+Q~i+PiNiPi<0has a positive definite solution Pi, where (34)Q~i=∑j=14Qji+εi2-2FiTFi,Ni=εi12A1iA1iT+εi22EiEiT+εi32E1iE1iT+εi42E2iE2iT+εi52B2iB2iT+γ-2BiBiT,then system (1) is exponentially stable and has weighted L2-gain γ for any switching signal with average dwell time satisfying τa>τa∗=(ln⁡⁡μ)/α, where μ≥1 satisfies Pi≤μPj, Q1i≤μQ1j, Q2i≤μQ2j, Q3i≤μQ3j, Q4i≤μQ4j, ∀i,j∈M, where(35)Q1i=eαd1εi121-α1I,Q2i=eαd2εi321-α2F1iTF1i,Q3i=eαd3εi421-α3F2iTF2i,Q4i=eαd4εi521-α4KiTKi.Proof.
Define a set of Lyapunov-Razumikhin function candidates as follows:(36)Vxt=Vσtxt=xTtPσtxt+∑i=14∫t-τitteαs-txTsQiσtxTsds.
Then whent∈[tk,tk+1) and ω(t)=0, it follows from Lemma 11 that(37)Vσtkxt≤e-αt-tkVσtkxtk,and from (36) the following inequality holds at switching instant ti:(38)Vσtixti≤μVσti-xti-.
Lettingk=Nσ(t0,t)≤(t-t0)/τa, it follows from inequalities (37) and (38) that(39)Vσtkxtxt≤e-αt-tkμVσtk-xtk-≤⋯e-αt-t0μkVσt0xt0≤e-α-ln⁡⁡μ/τat-t0Vσt0xt0.Let(40)a=min⁡∀i∈M⁡λmin⁡Pi,b=max⁡∀i∈M⁡λmax⁡Pi+d1εi121-α1I+d2εi321-α2F1iTF1i+d3εi421-α3F2iTF2i+d4εi521-α4KiTKi.Then(41)axt2≤Vxt,Vσt0xt0≤bxt02.Combining inequalities (39) and (41), we obtain(42)xt2≤Vxta≤abe-α-ln⁡⁡μ/τat-t0xt02which leads to the exponential stability of the switched system (1) with ω(t)=0.
Now we will show the weightedL2-gain of system (1). Firstly for any t∈tk,tk+1, using Lemma 10, we have (43)Vxt≤e-αt-tkVtk-∫tkte-αt-sΓsds.Then considering both inequalities (38) and (43), it holds that (44)Vxt≤μVxtke-αt-tkVtk-∫tkte-αt-sΓsds≤μkVxt0e-αt-μk∫0t1e-αt-sΓsds-μk-1∫t1t2e-αt-sΓsds-⋯-∫tkte-αt-sΓsds=e-αt+Nσ0,tln⁡⁡μVx0-∫0te-αt-s+Nσs,tln⁡⁡μΓsds.This gives (45)-∫0te-αt-s+Nσs,tln⁡⁡μΓsds≥0.Multiplying both sides in (45) by e-Nσ(0,t)ln⁡⁡μ, it yields (46)∫0te-αt-s-Nσ0,sln⁡⁡μzTszsds≤∫0te-αt-s-Nσ0,sln⁡⁡μγ2ωTsωsds.Observing that Nσ(0,s)≤s/τa and τa>τa∗=(ln⁡⁡μ)/α, we have Nσ(0,s)ln⁡⁡μ≤αs; consequently (47)∫0te-αt-s-αszTszsds≤∫0te-αt-sγ2ωTsωsds.Integrating both sides of inequality (47) from t=0 to ∞, we obtain (48)∫0∞e-αszTszsds≤∫0∞γ2ωTsωsds.This ends the proof.Remark 15.
Inequality (33) in Theorem 14 is equivalent to LMIs as follows:(49)AiTPi+PiAi+αPi+CiTCi+Q~i-αd4εi521-α4KiTKiPN~iPB1i∗-N~i0∗0-1νRi<0,where N~i=Ni+(αd4/4ν2εi521-α4)B1iT(Ri-1)TRi-1B1i. So it can be calculated easily.When the delays in system (1) are constant, we can get the following corollary.Corollary 16.
For given constantsα>0, γ>0, β0>0, β1>0, β2>0, and p>0, i∈M, if there exists a linear state feedback controller ui(t)=Kix(t) such that (50)β0Ei+β1E1i+β2E2i<p2Pi,where Pi is a positive symmetric definite matrix of the following inequality: (51)Ai+B1iKiTPi+PiAi+B1iKi+αPi+PiA1iA1iT+B2iKiKiTB2iPi+γ-2PiBiBiTPi+CiTCi<-pI,then switched system (1) with constant delays is exponentially stable and has weighted L2-gain γ for any switching signal with average dwell time satisfying τa>τa∗=(ln⁡⁡μ)/α, where μ≥1 satisfies Pi≤μPj, ∀i,j∈M.Proof.
Define the Lyapunov-Razumikhin function candidate in Theorem14 as follows:(52)Vxt=Vσtxt=xTtPσtxt;the proof is similar to that of Theorem 14 by Lemma 12.Remark 17.
Inequality (51) is easy to be solved for it can be rewritten as a LMI: (53)GPiA1iPiB2iKiγ-2PiBi∗-I00∗∗-I0∗∗∗-I<0,where G=(Ai+B1iKi)TPi+Pi(Ai+B1iKi)+αPi+CiTCi+pI.Remark 18.
In [40], conditions are given to guarantee that the nonswitched system with input delays is asymptotically stable. Compared with [40], we get the conditions when the switched system is exponentially stable.


## 4. Examples

To illustrate the main results, we consider the following examples.Example 1.
Consider the time-varying delay switched system(54)x˙t=Aσtxt+A1σtxt-τ1t+Eσtf0σtxt,t+E1σtf1σtxt-τ2t,t+E2σtf2σtxt-τ3t,t+Bσtωt+B1σtuσtt+B2σtuσtt-τ4t,zt=Cσtxt,xt=0,t≤0,where i=1,2 and τ1(t)=0.2sin⁡t+0.3, τ2(t)=0.3cos⁡⁡t+0.4, τ3(t)=0.4sin⁡t+0.5, τ4(t)=0.5cos⁡⁡t+0.7; thus 0≤τ1(t)≤0.5, 0≤τ2(t)≤0.7, 0≤τ3(t)≤0.9, 0≤τ4(t)≤1.2, and, τ˙1(t)≤0.2<1, τ˙2(t)≤0.3<1, τ˙3(t)≤0.4<1, τ˙4(t)≤0.5<1. So we have α1=0.2, α2=0.3, α3=0.4, α4=0.5, d1=0.5, d2=0.7, d3=0.9, d4=1.2. And A1=-412-4, A11=10-13, E1=102-3, E11=E12=1001, E21=2-102, B1=1034, B11=1-103, B21=2-1-23, C1=24, R1=2002, A2=-311-2, A12=20-11, E22=2-102, B2=1043, B12=2-1-13, B22=3-2-13, C2=23, R2=3003, f0i=atsin⁡x1(t)0, f1i=0btsin⁡x2(t-τ2), f2i=ctcos⁡⁡x1(t-τ2)0, with at≤1, bt≤1, ct≤1. Then Fi=F1i=F2i=I2. Let εi1=1, εi2=εi3=εi4=εi5=0.01 and α=2, γ=1; solving (33) using Matlab software, we can get P1=(1.0e+0.003)5.65710.06190.06191.6519 and P2=(1.0e+0.003)4.61051.21501.21504.1704; besides Q11=Q21=2.1746002.1746, Q12=Q22=(1.0e+0.004)2.8386002.8386, Q13=Q23=(1.0e+0.004)3.6298003.6298, Q14=(1.0e+0.004)0.15-0.15-0.151.5, Q24=(1.0e+0.003)3.3333-3.3333-3.33336.6667. And we can get μ=20 satisfying the condition of Theorem 14.
Therefore according to Theorem14, system (54) is exponentially stable and has L2-gain γ for any switching signal with average dwell time satisfying τa>τa∗=(ln⁡⁡μ)/α=(ln⁡⁡20)/2. The simulation result is depicted in Figure 1.Figure 1
The state response of the closed-loop system (54) in Example 1.Example 2.
Consider the constant time-delay switched system withA1=-400-4, A11=00.50.50, B11=1000, B21=1010, C1=23, B1=23-11, A2=-410-4, A12=-2132, B12=1000, B22=-2010, C2=23, B2=1-3-1-2, E1=E2=E11=E12=E21=E22=I, τ1=1, τ2=1.5, τ3=2, τ4=0.5, f0i=atsin⁡x1(t)0, f1i=0btsin⁡x2(t-τ2), f2i=ctsin⁡x1(t-τ3)0, with at≤0.01, bt≤0.1, ct≤0.08. Then let β0=0.01, β1=0.1, and β2=0.08, and let p=1, α=2, γ=1, K1=-1-100, K2=-1200; solving (51) by the Matlab software, we can get P1=0.68960.80380.80381.9159 and P2=0.21150.34340.34341.9159 and P1=2.3137, P2=1.1453.
It is obvious thatβ0Ei+β1E1i+β2E2i<p/2Pi, i=1,2, and we can get μ=4 satisfying P2<μP1, P1<μP2. Therefore according to Corollary 16, this system is exponentially stable and has L2-gain γ for any switching signal with average dwell time satisfying τa>τa∗=(ln⁡⁡μ)/α. The simulation result is depicted in Figure 2.Figure 2
The state response of the closed-loop system (54) in Example 2.Remark 19.
In the figures, horizontal axis stands for time and vertical axis stands for the statesx1(t) and x2(t). The red and blue lines denote the state response of x1(t) and x2(t), respectively. We can see from the figures that the two systems are both exponentially stable under given controllers and switching law and this shows that the method proposed in this paper is feasible.Remark 20.
It is easy to verify that both subsystems are unstable. And using the method given in [40], controllers are only given to make the systems asymptotically stable. However, compared with [40], in this paper, we can see that, under given controllers, the switched systems in the examples are exponentially stable from the simulation results.


## 5. Summary

In this paper, we considered the problem of robustH∞ control for a class of uncertain nonlinear switched systems with state and input delays. The controller and switching law are designed and sufficient conditions have been presented for uncertain nonlinear switched delay systems under which the system is robust stable and has H∞ performance γ. Firstly, we considered the system with time-varying delays; then, as a special case, system with constant delays was also considered. The feasibility of the developed results has been proved using numerical examples.



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*Source: 852303-2015-07-26.xml*