# Robust Finite-TimeH∞ Control for Impulsive Switched Nonlinear Systems with State Delay

Authors: Jian Guo; Chao Liu; Zhengrong Xiang
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper investigates robust finite-timeH∞ control for a class of impulsive switched nonlinear systems with time-delay. Firstly, using piecewise Lyapunov function, sufficient conditions ensuring finite-time boundedness of the impulsive switched system are derived. Then, finite-time H∞  performance analysis for impulsive switched systems is developed, and a robust finite-time H∞ state feedback controller is proposed to guarantee that the resulting closed-loop system is finite-time bounded with H∞ disturbance attenuation. All the results are given in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

A switched system is a hybrid dynamical system consisting of a family of continuous-time or discrete-time subsystems and a switching law that orchestrates the switching between them [1]. In the last decades, in the stability analysis and stabilization for switched systems, lots of valuable results are established (see [2–5]). Most recently, on the basis of Lyapunov functions and other analysis tools, the stability problem of linear and nonlinear switched systems with time-delay has been further investigated (see [6–15]), and lots of valuable results are established for H∞ control problems (see [16–22]).It is well known that impulsive dynamical behaviors inevitably exist in some practical systems like physical, biological, engineering, and information science systems due to abrupt changes at certain instants during the dynamical process. Although hybrid system and switched system are important models for dealing with complex real systems, there is little work concerned with the above impulsive phenomena. Such a phenomenon can be modeled as an impulsive switched system, it is characteristic that their states change during the switching because of the occurrence of impulses [23].In recent years, the impulsive switched systems have drawn more and more attention and many useful conclusions have been obtained. Multiple Krasovskii-Lyapunov function approach is employed to study the problem of ISS stability of a class of impulsive switched systems with time-delay in [24]. By the Lyapunov-Razumikhin technique, a delay-independent criterion of the exponential stability is established on the minimum dwell time in [25]. The problem of robust H∞ stabilization of nonlinear impulsive switched system with time-delays is studied in [23].Usually, the stability of a system is defined over an infinite-time interval. But in many practical systems, we focus on the dynamical behavior of a system over a fixed finite-time interval. Based on this, finite-time stability is first proposed by Dorato in 1961 [26]. Compared with the classical Lyapunov stability, finite-time stability is proposed for the study of the transient performance of the system, which is a totally different concept. The so-called finite-time stability means the boundedness of the state of a system over a fixed finite-time interval. Finite-time stability problems can be found in [27–32]. The finite-time stability of linear impulsive systems is analyzed in [33], the finite-time stability and stabilization of impulsive dynamic systems are carried out in [34–36]. The finite-time stability and stabilization of switched systems are investigated in [37].Recently, robust finite-time control of switched systems is studied in [38, 39]. However, to the best of our knowledge, there are very few results on finite-time boundedness and robust H∞ control of the impulsive switched systems, which motivates the present study. The paper is organized as follows. In Section 2, problem formulation and some necessary lemmas are given. In Section 3, based on the dwell time approach, finite-time boundedness and finite-time H∞ performance for switched impulsive systems are addressed, and sufficient conditions for the existence of a robust finite-time H∞ state feedback controller are proposed in terms of a set of matrix inequalities. Numerical examples are provided to show the effectiveness of the proposed approach in Section 4. Concluding remarks are given in Section 5.Notations. The notations used in this paper are standard. The notation P>0 means that P is a real positive definite matrix; diag{⋯} stands for a block-diagonal matrix; λmax(P) and λmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively; ∥x(t)∥=xT(t)x(t) and ∥x(t)∥2=(∫0∞∥x(t)∥2dt)1/2.


## 2. Problem Formulation and Preliminaries

Consider the following impulsive switched system:(2.1a)x˙(t)=A^σ(t)x(t)+A^dσ(t)x(t-h)+B^1σ(t)u1(t)+fσ(t)(x(t))+B2σ(t)w(t),t≠tk(2.1b)Δx=Eσ(t)x(t)+u2(t),t=tk,k=1,2,3,…(2.1c)z(t)=Cσ(t)x(t)+Dσ(t)u1(t),(2.1d)x(t)=φ(t),t∈[t0-h,t0],where x(t)∈Rn is the state vector, z(t)∈Rr is the controlled output, w(t)∈Rp is the disturbance input which belongs to L2[0,∞), u1(t)∈Rm, t≠tk is the switched control input, u2(tk)∈Rn is the impulsive control input at tk, on the other hand, u2(t)=0, t≠tk, k=1,2,3,…. σ(t):[t0,+∞)→N-={1,2,…,N} is a switching signal. t∈(tk,tk+1], σ(t)=ik, ik∈N-, k=0,1,2,3,…. Δx(t)=x(t+)-x(t-),x(t+)=limh→0+x(t+h),x(t)=x(t-)=limh→0+x(t-h). tk, k=0,1,2,3,… are the impulsive jumping points or switching points. t0 is the initial time, t0<t1<⋯<tk<⋯, and limk→∞tk=+∞. h>0 is the time-delay which is a positive constant. fi(·):Rn→Rn,i∈N- is nonlinear vector-valued function. φ(t),t∈[t0-h,t0] is a continuous vector-valued initial function. A^i, A^di, B^1i, i∈N- are uncertain real-valued matrices with appropriate dimensions, B2i, Ei, Ci, Di, i∈N- are known real constant matrices with appropriate dimensions.Assumption 2.1.
For eachi∈N-, A^i, A^di, B^1i are uncertain real-valued matrices with appropriate dimensions. We assume that the uncertainties are of the form(2.2a)A^i=Ai+ΔAi,A^di=Adi+ΔAdi,B^1i=B1i+ΔB1i,(2.2b)[ΔAiΔAdiΔB1i]=HiFi(t)[EAiEAdiEBi],where Ai, Adi, B1i, Hi, EAi, EAdi, and EBi are known real-valued constant matrices with appropriate dimensions, Fi(t) is the uncertain matrix satisfying
(2.3)FiT(t)Fi(t)≤I.Assumption 2.2.
For eachi∈N-, nonlinear vector-valued function fi satisfies Lipschitz condition
(2.4)∥fi(x(t))∥≤∥Uix(t)∥,
where Ui is the Lipschitz constant matrix.Assumption 2.3.
For a given time constantTf>t0, the external disturbance w(t) satisfies
(2.5)∫0TfwT(t)w(t)dt≤d2.Assumption 2.4.
For system (2.1a)–(2.1d), the impulsive jump matrices Ei satisfy that (I+Ei) are invertible.Definition 2.5 (see [32]).
For a given time constantTf>t0, impulsive switched system (2.1a), (2.1b), (2.1c) and (2.1d) with u1(t)≡0, u2(t)≡0, and w(t)≡0, is said to be finite-time stable with respect to (c12,c22,Tf,R,σ(t)) if the following inequality holds:
(2.6)supt0-h≤τ≤t0xT(τ)Rx(τ)≤c12⇒xT(t)Rx(t)<c22,t∈(t0,Tf],
where c2>c1>0, R is a positive definite matrix, and σ(t) is a switching signal.Remark 2.6.
Equation (2.6) stands for the boundedness of the state of a system over a fixed finite-time interval (t0,Tf], when the initial state is bounded.Definition 2.7 (see [40]).
For a given time constantTf, impulsive switched system (2.1a)–(2.1d) with u1(t)≡0,  u2(t)≡0, and w(t) satisfying (2.5), is said to be finite-time bounded with respect to (c12,c22,Tf,d2,R,σ(t))  if the condition (2.6) holds, where c2>c1>0, R is a positive definite matrix and σ(t) is a switching signal.Definition 2.8.
For anyT2>T1>0, let  Nσ(t)(T1,T2) denote the switching number of σ(t) on an interval (T1,T2). If Nσ(t)(T1,T2)≤N0+(T2-T1)/τa holds for given  N0≥0, τa>0, then the constant τa is called the average dwell time. In this paper we let N0=0.Definition 2.9.
For a given time constantTf, impulsive switched system (2.1a)–(2.1d) with u1(t)≡0, u2(t)≡0 is said to have finite-time H∞ performance with respect to (0,c22,Tf,d2,γ,R,σ(t)) if the system is finite-time bounded and the following inequality holds:
(2.7)∥z(t)∥2≤γ∥w(t)∥2,∀w(t)∈L2[0,∞),
where c2>0,γ>0, R is a positive definite matrix and σ(t)is a switching signal.Definition 2.10.
For a given time constantTf, impulsive switched system (2.1a)–(2.1d) is said to be robust finite-time stabilization with H∞ disturbance attenuation level γ, if there exists a switched controller u1(t)=Kσ(t)x(t), t≠tk and an impulsive controller u2(tk)=K-σ(t)x(tk),t=tk, where t∈(t0,Tf] such that(i)
the corresponding closed-loop system is finite-time bounded with respect to(0,c22,Tf,d2,R,σ(t));(ii)
under zero initial condition, inequality (2.7) holds for any w(t) satisfying (2.5).Lemma 2.11.
LetU, V, W, and X be real matrices of appropriate dimensions with X satisfying X=XT, then for all VTV≤I,
(2.8)X+UVW+WTVTUT<0,
if and only if there exists a scalar ε>0 such that
(2.9)X+εUUT+ε-1WTW<0.


## 3. Main Results

### 3.1. Finite-Time Boundedness Analysis

In this subsection, we focus on the finite-time boundedness of the following impulsive switched system:(3.1a)x˙(t)=Aσ(t)x(t)+Adσ(t)x(t-h)+fσ(t)(x(t))+B2σ(t)w(t),t≠tk(3.1b)Δx=Eσ(t)x(t),t=tk,k=1,2,3,…(3.1c)x(t)=φ(t),t∈[t0-h,t0].Before proceeding to Lemma3.2, we first introduce a function v(t). For given positive definite matrices Qik, ik∈N-, by Assumption 2.4, there exists a real number ρik≥1, ρ*=max{ρik,ik∈N-} such that
(3.2)Qik-1≤ρik(I+Eik-1)TQik(I+Eik-1).
Furthermore, we define the following function
(3.3)vk(t)=ρik-(t-tk)2(tk+1-tk)2(ρik-1),t∈(tk,tk+1].
Finally, a piecewise continuous function v(t) is as follows:
(3.4)v(t)=vk(t),t∈(tk,tk+1].
Consider the function v(t), for each interval (tk,tk+1], v(tk+)=ρik, v(tk+1)=1, and v(t) is monotonically nonincreasing and bounded function, v(tk+1)≤v(t)≤v(tk+).Remark 3.1.
Note that the previous works require the conditionQik-1≤(I+Eik-1)TQik(I+Eik-1) (see [23, 41]), which can be obtained by setting ρik=1 in (3.2). Thus, the proposed approach may provide more relaxed conditions.Lemma 3.2.
Consider the following Lyapunov functional candidate:(3.5)V(t)=xT(t)Pσ(t)x(t)+∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds
for system (3.1a), (3.1b), and (3.1c), where Pi and Qi, i∈N- are symmetric positive definite matrices with appropriate dimensions.
The following inequality is derived:(3.6)V˙(t)≤2xT(t)Pikx˙(t)+α∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds+v(t)xT(t)Qikx(t)-v(t-h)xT(t-h)Qik-mx(t-h)eαht∈(tk,tk+1],t-h∈(tk-m,tk-m+1],m∈{0,1,2,3,…}.Proof.
(i) Whentk+h≥tk+1,
(3.7)V(t)=xT(t)Pσ(t)x(t)+∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds=xT(t)Pikx(t)+∫t-htk-m+1v(s)xT(s)eα(t-s)Qik-mx(s)ds+∫tk-m+1tk-m+2v(s)xT(s)eα(t-s)Qik-m+1x(s)ds⋯+∫tktv(s)xT(s)eα(t-s)Qikx(s)ds,V˙(t)=2xT(t)Pikx˙(t)+α∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds+v(tk-m+1)xT(tk-m+1)eα(t-tk-m+1)Qik-mx(tk-m+1)-v(t-h)xT(t-h)eαhQik-mx(t-h)+v(tk-m+2)xT(tk-m+2)eα(t-tk-m+2)Qik-m+1x(tk-m+2)-v(tk-m+1+)xT(tk-m+1+)eα(t-tk-m+1)Qik-m+1x(tk-m+1+)⋯+v(tk)xT(tk)eα(t-tk)Qik-1x(tk)-v(tk-1+)xT(tk-1+)eα(t-tk)Qik-1x(tk-1+)+v(t)xT(t)Qikx(t)-v(tk+)xT(tk+)eα(t-tk)Qikx(tk+),V˙(t)=2xT(t)Pikx˙(t)+α∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds+v(t)xT(t)Qikx(t)-v(t-h)xT(t-h)eαhQik-mx(t-h)+xT(tk-m+1)eα(t-tk-m+1)[Qik-m-ρik-m+1(I+Eik-m)TQik-m+1(I+Eik-m)]x(tk-m+1)…+xT(tk)eα(t-tk)[Qik-1-ρik(I+Eik-1)TQik(I+Eik-1)]x(tk).
From (3.2), we can obtain that
(3.8)Qik-m-ρik-m+1(I+Eik-m)TQik-m+1(I+Eik-m)≤0Qik-m-ρik-m+1(I+Eik-m)T⋮Qik-1-ρik(I+Eik-1)TQik(I+Eik-1)≤0.
Combining (3.7) and (3.8), (3.6) is obtained.(ii)
Whentk+h<tk+1,(1)
t∈(tk,tk+h], the proof is similar to the proof line in the situation (i).(2)
t∈(tk+h,tk+1],
(3.9)V(t)=xT(t)Pσ(t)x(t)+∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds=xT(t)Pikx(t)+∫t-htv(s)xT(s)eα(t-s)Qikx(s)ds.The proof for this situation is omitted.
The proof is completed.Lemma 3.3.
Consider the following Lyapunov function:(3.10)V(t)=xT(t)Pσ(t)x(t)+∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds
for system (3.1a), (3.1b), and (3.1c), where Pi and Qi,i∈{1,2,…,N} are symmetric positive definite matrices with appropriate dimensions. Under the condition
(3.11)[-eαhρ*PjI+EjTEiT*-Pi-10**-e-αh(ρ*)-1Qi-1]<0,∀i,j∈N-,
we have
(3.12)V(tk+)<eαhρ*V(tk),
where ρ*=max{ρik,ik∈N-}.Proof.
Without loss of generality, letσ(tk+)=i,  σ(tk)=j. Then, we have
(3.13)V(tk+)=xT(tk+)Pσ(tk+)x(tk+)+∫tk+-htk+v(s)xT(s)eα(tk+-s)Qσ(s)x(s)ds≤xT(tk+)Pσ(tk+)x(tk+)+eαhρ*∫tk+-htk+xT(s)Qσ(s)x(s)ds≤xT(tk)(I+Ej)TPi(I+Ej)x(tk)+eαhρ*xT(tk)EjTQiEjx(tk)+eαhρ*∫tk-htkv(s)xT(s)eα(tk-s)Qσ(s)x(s)ds,(3.14)V(tk)=xT(tk)Pjx(tk)+∫tk-htkv(s)xT(s)eα(tk-s)Qσ(s)x(s)ds.
Combining (3.13) with (3.14), we have
(3.15)V(tk+)-eαhρ*V(tk)≤xT(tk)(I+Ej)TPi(I+Ej)x(tk)+eαhρ*xT(tk)EjTQiEjx(tk)-eαhρ*xT(tk)Pjx(tk)=xT(tk)Σijx(tk),
where
(3.16)∑ij=(I+Ej)TPi(I+Ej)+eαhρ*EjTQiEj-eαhρ*Pj.
Using Schur complement, (3.11) is equivalent to
(3.17)Σij<0orV(tk+)-eαhρ*V(tk)<0.
The proof is completed.Theorem 3.4.
R is a positive definite matrix. Let P~i=R-1/2PiR-1/2, Q~i=R-1/2QiR-1/2, For  all i∈N-, if there exist positive scalars ρi≥1, i∈N-, ρ*=max{ρi,i∈N-}, α, λ1, λ2, λ3 and symmetric positive matrices Pi,Pj,Qi,Ti, i,j∈N- such that
(3.18)1ρ*(I+Ei)-1Q~j(I+Ei)-T-Q~i≤0,∀i,j∈N-(3.19)[P~iAiT+AiP~i-αP~i+IAdiQ~iB2iP~i*-eαhQ~j00**-Ti0***-(ρ*-1Q~i+UikTUik)]<0,∀i,j∈N-(3.20)[-eαhρ*P~jP~j(I+EjT)P~jEjT*-P~i0**-e-αh(ρ*)-1Q~i]<0,∀i,j∈N-(3.21)λ1R-1<P~i<R-1,λ2R-1<Q~i,Ti<λ3I,∀i∈N-(3.22)[-c22e-αTf+d2λ3c1c1*-λ10**-1ρ*he-αhλ2]<0
hold, under the average dwell time scheme
(3.23)τa>τa*=Tf(αh+lnρ*)ln(c22e-αTf)-ln[(1/λ1+hρ*eαh/λ2)c12+d2λ3],
system (3.1a)–(3.1c) is finite-time bounded with respect to (c12,c22,Tf,d2,R,σ(t)).Proof.
Assuming that whent∈(tk,tk+1], σ(t)=ik, ik∈N-, k=0,1,2,3,….
Choose the following Lyapunov functional candidate:(3.24)V(t)=xT(t)P~σ(t)-1x(t)+∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds.
Whent∈(tk,tk+1], according to (3.18) and Lemma 3.2, we have
(3.25)V˙(t)≤2xT(t)P~ik-1x˙(t)+α∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds+v(t)xT(t)Q~ik-1x(t)-v(t-h)xT(t-h)Q~ik-m-1x(t-h)eαh,V˙(x(t))-αV(x(t))-wT(t)Tikw(t)≤2xT(t)P~ik-1x˙(t)+α∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds+v(t)xT(t)Q~ik-1x(t)-v(t-h)xT(t-h)Q~ik-m-1x(t-h)eαh-αxT(t)P~ik-1x(t)-α∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds-wT(t)Tikw(t)≤2xT(t)P~ik-1x˙(t)+ρ*xT(t)Q~ik-1x(t)-xT(t-h)Q~ik-m-1x(t-h)eαh-αxT(t)P~ik-1x(t)-wT(t)Tikw(t).
According to (3.1a)–(3.1c), and (3.25), Assumption 2.2, and the fallowing inequality:
(3.26)2xT(t)P~ik-1f(x(t))≤fikT(x(t))fik(x(t))+xT(t)P~ik-1P~ik-1x(t)≤xT(t)UikTUikx(t)+xT(t)P~ik-1P~ik-1x(t),
we have
(3.27)V˙(x(t))-αV(x(t))-wT(t)Tikw(t)≤XT(t)ΞkX(t),
where XT(t)=(xT(t)xT(t-h)wT(t)),
(3.28)Ξk=[ΔkP~ik-1AdikP~ik-1B2ik*-eαhQ~ik-m-10**-Tik],Δk=AikTP~ik-1+P~ik-1Aik+ρ*Q~ik-1-αP~ik-1+UikTUik+P~ik-1P~ik-1.
Using Schur complement, we obtain from (3.19) that
(3.29)[OiP~i-1AdiP~i-1B2i*-eαhQ~j-10**-Ti]<0,
where
(3.30)Oi=AiTP~i-1+P~i-1Ai+ρ*Q~i-1-αP~i-1+UiTUi+P~i-1P~i-1.
Noticing that the above inequality holds for all i,j∈N-, then we have Ξk<0 for ik,ik-1∈N-. Thus,
(3.31)V˙(x(t))-αV(x(t))-wT(t)Tikw(t)<0.
When t∈(tk,tk+1], according to Lemma 3.3, we can obtain (3.12) from condition (3.20).
Combining (3.31) and (3.12), we can obtain that
(3.32)V(t)<eα(t-tk)V(tk+)+∫tkteα(t-s)wT(s)Tikw(s)ds<eα(t-tk)eαhρ*V(tk)+∫tkteα(t-s)wT(s)Tikw(s)ds<eα(t-tk)eαhρ*[eα(tk-tk-1)V(tk-1+)+∫tk-1tkeα(tk-s)wT(s)Tik-1w(s)ds]+∫tkteα(t-s)wT(s)Tikw(s)ds<⋯<eα(t-t0)(eαhρ*)Nσ(t0,t)V(t0)+(eαhρ*)Nσ(t0,t)∫t0t1eα(t-s)wT(s)Ti0w(s)ds+(eαhρ*)Nσ(t1,t)∫t1t2eα(t-s)wT(s)Ti1w(s)ds+⋯+eαhρ*∫tk-1tkeα(t-s)wT(s)Tik-1w(s)ds+∫tkteα(t-s)wT(s)Tikw(s)ds=eα(t-t0)(eαhρ*)Nσ(t0,t)V(t0)+∫t0teα(t-s)(eαhρ*)Nσ(s,t)wT(s)Tikw(s)ds<eαt(eαhρ*)Nσ(t0,t)V(t0)+(eαhρ*)Nσ(t0,t)eαt∫t0twT(s)Tikw(s)ds<eαTf(eαhρ*)Nσ(t0,t)[V(t0)+∫t0TfwT(s)Tikw(s)ds]<eαTf(eαhρ*)Nσ(t0,t)[V(t0)+λmax(Tik)d2].
Noticing that Nσ(t0,Tf)<Tf/τa and according to (3.21), we have
(3.33)V(t)<e(α+αh/τa)Tf(ρ*)Tf/τα[V(t0)+λ3d2],V(t)≥xT(t)P~ik-1x(t)=xT(t)R1/2Pik-1R1/2x(t)≥λmin(Pik-1)xT(t)Rx(t)=1λmax(Pik)xT(t)Rx(t).
Because λ1R-1<P~i<R-1, we have
(3.34)V(t)>xT(t)Rx(t).
According to the Lyapunov function that we have chosen, we have
(3.35)V(t0)=xT(t0)P~i-1x(t0)+∫t0-ht0v(s)xT(s)e-α(t0-s)Q~i-1x(s)ds≤maxi∈N-λmax(Pi-1)xT(t0)Rx(t0)+heαhρ*maxi∈N-λmax(Qi-1)supt0-h≤θ≤t0xT(θ)Rx(θ)≤(1mini∈N-λmin(Pi)+ρ*heαhmini∈N-λmin(Qi))supt0-h≤θ≤t0xT(θ)Rx(θ).
According to (3.21), the following inequality is derived:
(3.36)V(t0)<(1λ1+ρ*heαhλ2)c12.
Combining (3.33), (3.34), and (3.36), we can obtain that
(3.37)xT(t)Rx(t)<V(t)<e(α+αh/τa)Tf(ρ*)Tf/τα[(1λ1+ρ*heαhλ2)c12+λ3d2].
Using Schur complement, (3.22) is equivalent to
(3.38)(1λ1+ρ*heαhλ2)c12+λ3d2<c22e-αTf.
From (3.38), we can obtain that τa>0.
Substituting (3.23) into (3.37) leads to
(3.39)xT(t)Rx(t)<c22.
Thus, system (3.1a)–(3.1c) is finite-time bounded with respect to (c12,c22,Tf,d2,R,σ(t)).
The proof is completed.Corollary 3.5.
R is a positive definite matrix, let w(t)≡0, P~i=R-1/2PiR-1/2, Q~i=R-1/2QiR-1/2 for  all  i∈N-. If there exist positive scalars ρi≥1, i∈N-, ρ*=max{ρi,i∈N-}, α,λ1,λ2 and symmetric positive matrices Pi, Pj, Qi  for  alli,j∈N- with appropriate dimensions such that
(3.40)1ρ*(I+Ei)-1Q~j(I+Ei)-T-Q~i≤0,∀i,j∈N-[P~iAiT+AiP~i-αP~i+IAdiQ~iP~i*-eαhQ~j0**-(ρ*-1Q~i+UikTUik)]<0,∀i,j∈N-[-eαhρ*P~jP~j(I+EjT)P~jEjT*-P~i0**-e-αh(ρ*)-1Q~i]<0,∀i,j∈N-λ1R-1<P~i<R-1,λ2R-1<Q~i,∀i∈N-[-c22e-αTfc1c1*-λ10**-1ρ*he-αhλ2]<0
hold with average dwell time
(3.41)τa>τa*=Tf(αh+lnρ*)ln(c22e-αTf)-ln[(1/λ1+ρ*heαh/λ2)c12].
System (3.1a)–(3.1c) with w(t)≡0 is finite-time stable with respect to (c12,c22,Tf,R,σ(t)).


### 3.2.H∞ Performance Analysis

In this subsection,H∞ performance of the following system is investigated:(3.42a)x˙(t)=Aσ(t)x(t)+Adσ(t)x(t-h)+fσ(t)(x(t))+B2σ(t)w(t),t≠tk(3.42b)Δx=Eσ(t)x(t),t=tk,k=1,2,3,…(3.42c)z(t)=Cσ(t)x(t),(3.42d)x(t)=φ(t),t∈[t0-h,t0]Theorem 3.6.
R is a positive definite matrix. Let P~i=R-1/2PiR-1/2, Q~i=R-1/2QiR-1/2 for alli∈N-. Suppose that there exist positive scalars ρi≥1,i∈N-, ρ*=max{ρi,i∈N-}, α,γ,ε and symmetric positive matrices Pi,Pj,Qi for alli,j∈N- such that
(3.43)1ρ*(I+Ei)-1Q~j(I+Ei)-T-Q~i≤0,∀i,j∈N-(3.44)[P~iAiT+AiP~i-αP~i+IAdiQ~iB2iP~iP~iCiT*-eαhQ~j000**-γ200***-(ρ*-1Q~i+UikTUik)0****-I]<0,∀i,j∈N-(3.45)[-eαhρ*P~jP~j(I+EjT)P~jEjT*-P~i0**-e-αh(ρ*)-1Q~i]<0,∀i,j∈N-(3.46)P~i<R-1,∀i∈N-(3.47)-c22+eαTfγ2d2<0
hold with average dwell time
(3.48)τa>τa*=max{Tf(αh+lnρ*)ln(c22)-ln(eαTfγ2d2),hε}.
Then, system (3.42a)–(3.42d) is finite-time bounded and has H∞ performance with respect to (0,c22,Tf,d2,γ-,R,σ(t)), where γ-2=e(1+ε)αTf(ρ*)εTf/hγ2.Proof.
Whent∈(tk,tk+1], σ(t)=ik, ik∈N-, k=0,1,2,3,…. Choose the following Lyapunov functional candidate for system (3.42a)–(3.42d)
(3.49)V(t)=xT(t)P~σ(t)-1x(t)+∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds.
When t∈(tk,tk+1],
(3.50)V˙(x(t))-αV(x(t))+zT(t)z(t)-γ2wT(t)w(t)≤XT(t)ΨkX(t),
where XT(t)=(xT(t)xT(t-h)wT(t)),
(3.51)Ψk=[ΔkP~ik-1AdikP~ik-1B2ik*-eαhQ~ik-m-10**-γ2I],Δk=AikTP~ik-1+P~ik-1Aik+ρ*Q~ik-1-αP~ik-1+UikTUik+P~ik-1P~ik-1+CikTCik.
Using Schur complement, we obtain from (3.44) that
(3.52)[EiP~i-1AdiP~i-1B2i*-eαhQ~j-10**-γ2I]<0,
where Ei=AiTP~i-1+P~i-1Ai+ρ*Q~i-1-αP~i-1+UiTUi+P~i-1P~i-1+CiTCi.
Noticing that the above inequality holds for  alli,j∈N-, then we have Ψk<0, for ik,ik-m∈N-.
Thus,(3.53)V˙(x(t))-αV(x(t))+zT(t)z(t)-γ2wT(t)w(t)<0,
Letγ2wT(s)w(s)-zT(s)z(s)=Δ(s), from (3.32), we have
(3.54)V(t)<eα(t-t0)(eαhρ*)Nσ(t0,t)V(t0)+∫t0teα(t-s)(eαhρ*)Nσ(s,t)Δ(s)ds.
Under zero initial condition, we have
(3.55)0<∫t0teα(t-s)(eαhρ*)Nσ(s,t)Δ(s)ds,
that is,
(3.56)∫t0teα(t-s)(eαhρ*)Nσ(s,t)zT(s)z(s)ds<∫t0teα(t-s)(eαhρ*)Nσ(s,t)γ2wT(s)w(s)ds.
Noticing that
(3.57)∫t0teα(t-s)(eαhρ*)Nσ(s,t)zT(s)z(s)ds>∫t0tzT(s)z(s)ds.
Then, we have
(3.58)∫t0teα(t-s)(eαhρ*)Nσ(s,t)γ2wT(s)w(s)ds<eαt(eαhρ*)Nσ(t0,t)∫t0tγ2wT(s)w(s)ds.
Let  t=Tf, because τa>h/ε, we have
(3.59)∫t0TfzT(s)z(s)ds<e(1+ε)αTf(ρ*)εTf/hγ2∫t0TfwT(s)w(s)ds,
then
(3.60)∫t0TfzT(s)z(s)ds<γ-2∫t0TfwT(s)w(s)ds.
Thus, system (3.42a)–(3.42d) is finite-time bounded and has H∞ performance with respect to (0,c22,Tf,d2,γ-,R,σ(t)), where  γ-2=e(1+ε)αTf(ρ*)εTf/hγ2.
The proof is completed.Remark 3.7.
Whenρ*=1, Theorem 3.6 degenerates to the result of [41], which cannot guarantee the finite-time boundedness of the addressed system if ρ*>1.


### 3.3. Robust Finite-TimeH∞ Control

Consider system (2.1a)–(2.1d), under the switching controller u1(t)=Kσ(t)x(t), t≠tk and impulsive controller u2(tk)=K-σ(t)x(tk),   t=tk, the corresponding closed-loop system is given by(3.61a)x˙(t)=(A^σ(t)+B^1σ(t)Kσ(t))x(t)+A^dσ(t)x(t-h)+fσ(t)(x(t))+B2σ(t)w(t),t≠tk(3.61b)Δx=(Eσ(t)+K-σ(t))x(t),t=tk,k=1,2,3,…(3.61c)z(t)=(Cσ(t)+Dσ(t)Kσ(t))x(t),(3.61d)x(t)=φ(t),t∈[t0-h,t0].Theorem 3.8.
Consider impulsive switched system (2.1a)–(2.1d), let P~i=R-1/2PiR-1/2, Q~i=R-1/2QiR-1/2  for  all   i∈N-. If  there exist positive scalars ρi≥1,   i∈N-,ρ*=max{ρi,i∈N-}α,γ,ε,δi and positive definite symmetric matrices Pi,Qi, and matrices Yi,   i∈N-,  with appropriate dimensions, such that the following inequalities hold
(3.62)1ρ*(I+Ei)-1Q~j(I+Ei)-T-Q~i≤0,∀i,j∈N-(3.63)[ΓiAdiQ~iB2iP~iP~iCiT+YiTDiTYiTEBiT+P~iEAiT*-eαhQ~j000Q~iEAdiT**-γ2000***-(ρ*-1Q~i+UikTUik)00****-I0*****-δi]<0,∀i,j∈N-,
where
(3.64)Γi=P~iAiT+YiTB1iT+AiP~i+B1iYi-αP~i+I+δiHiHiT,[-eαhρ*P~jP~j*-P~i]<0,∀i,j∈N-(3.65)P~i<R-1,i∈N-,(3.66)-c22+eαTfγ2d2<0.
Then, under the controller Ki=YiP~i-1, K-i=-Ei, and the following average dwell time scheme
(3.67)τa>τa*=max{Tf(αh+lnρ*)ln(c22)-ln(eαTfγ2d2),hε},
the corresponding closed-loop system is finite-time bounded with H∞ performance with respect to  (0,c22,Tf,d2,γ-,R,σ(t)) and γ-2=e(1+ε)αTf(ρ*)εTf/hγ2.Proof.
According to Assumption2.1, we have
(3.68)A^i+B^1iKi=(Ai+B1iKi)+HiFi(EAi+EBiKi),A^di=Adi+HiFiEAdi.
Now replacing Ai, Adi, Ci in the left side of (3.44) with A^i+B^1iKi,A^di, Ci+DiKi, we can obtain that
(3.69)Θij=[Ωi(Adi+HiFiEAdi)Q~iB2iP~iP~i(Ci+DiKi)T*-eαhQ~j000**-γ200***-(ρ*-1Q~i+UikTUik)0****-I],
where
(3.70)Ωi=[(Ai+B1iKi)+HiFi(EAi+EBiKi)]P~i+P~i[(Ai+B1iKi)+HiFi(EAi+EBiKi)]T-αP~i+I.
From (3.69), we know that
(3.71)Θij=Π1ij+Π2ij,
where
(3.72)Π1ij=[Υ1iAdiQ~iB2iP~iP~i(Ci+DiKi)T*-eαhQ~j000**-γ200***-(ρ*-1Q~i+UikTUik)0****-I],Π2ij=[Υ2iHiFiEAdiQ~i000*0000**000***00****0],
with
(3.73)Υ1i=P~i(Ai+B1iKi)T+(Ai+B1iK)P~i-αP~i+I,Υ2i=P~i(EAi+EBiKi)TFiTHiT+HiFi(EAi+EBiKi)P~i,
let Yi=KiP~i, then
(3.74)Υ1i=P~iAiT+YiTB1iT+AiP~i+B1iYi-αP~i+I,Υ2i=(YiTEBiT+P~iEAiT)FiTHiT+HiFi(EAiP~i+EBiYi).
From Lemma 2.11, we can obtain that
(3.75)Θij=Π1ij+[Hi0000]Fi[EAiP~i+EBiYiEAdiQ~i000]+[YiTEBiT+P~iEAiTQ~iEAdiT000]Fi[HiT0000]≤Π1ij+δi[Hi0000][HiT0000]+1δi[YiTEBiT+P~iEAiTQ~iEAdiT000][EAiP~i+EBiYiEAdiQ~i000].
Using Schur complement lemma, we get from (3.63) that
(3.76)Θij<0.
Now we choose K-i=-Ei, and replacing Ei in (3.45) with Ei+K-i, we know that
(3.77)[-eαhρ*P~jP~j0*-P~i0**-e-αh(ρ*)-1Q~i]<0,
by (3.64), we know that the condition(3.45) hold.
Then, system (2.1a)–(2.1d) is robust finite-time bounded with H∞ performance with respect to  (0,c22,Tf,d2,γ-,R,σ(t)), and γ-2=e(1+ε)αTf(ρ*)εTf/hγ2.
The proof is completed.Remark 3.9.
In order to eliminate the impulsive jump, we design an impulsive feedback controllerK-i=-Ei,  t=tk. Then the system becomes a switched system with continuous states.


## 3.1. Finite-Time Boundedness Analysis

In this subsection, we focus on the finite-time boundedness of the following impulsive switched system:(3.1a)x˙(t)=Aσ(t)x(t)+Adσ(t)x(t-h)+fσ(t)(x(t))+B2σ(t)w(t),t≠tk(3.1b)Δx=Eσ(t)x(t),t=tk,k=1,2,3,…(3.1c)x(t)=φ(t),t∈[t0-h,t0].Before proceeding to Lemma3.2, we first introduce a function v(t). For given positive definite matrices Qik, ik∈N-, by Assumption 2.4, there exists a real number ρik≥1, ρ*=max{ρik,ik∈N-} such that
(3.2)Qik-1≤ρik(I+Eik-1)TQik(I+Eik-1).
Furthermore, we define the following function
(3.3)vk(t)=ρik-(t-tk)2(tk+1-tk)2(ρik-1),t∈(tk,tk+1].
Finally, a piecewise continuous function v(t) is as follows:
(3.4)v(t)=vk(t),t∈(tk,tk+1].
Consider the function v(t), for each interval (tk,tk+1], v(tk+)=ρik, v(tk+1)=1, and v(t) is monotonically nonincreasing and bounded function, v(tk+1)≤v(t)≤v(tk+).Remark 3.1.
Note that the previous works require the conditionQik-1≤(I+Eik-1)TQik(I+Eik-1) (see [23, 41]), which can be obtained by setting ρik=1 in (3.2). Thus, the proposed approach may provide more relaxed conditions.Lemma 3.2.
Consider the following Lyapunov functional candidate:(3.5)V(t)=xT(t)Pσ(t)x(t)+∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds
for system (3.1a), (3.1b), and (3.1c), where Pi and Qi, i∈N- are symmetric positive definite matrices with appropriate dimensions.
The following inequality is derived:(3.6)V˙(t)≤2xT(t)Pikx˙(t)+α∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds+v(t)xT(t)Qikx(t)-v(t-h)xT(t-h)Qik-mx(t-h)eαht∈(tk,tk+1],t-h∈(tk-m,tk-m+1],m∈{0,1,2,3,…}.Proof.
(i) Whentk+h≥tk+1,
(3.7)V(t)=xT(t)Pσ(t)x(t)+∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds=xT(t)Pikx(t)+∫t-htk-m+1v(s)xT(s)eα(t-s)Qik-mx(s)ds+∫tk-m+1tk-m+2v(s)xT(s)eα(t-s)Qik-m+1x(s)ds⋯+∫tktv(s)xT(s)eα(t-s)Qikx(s)ds,V˙(t)=2xT(t)Pikx˙(t)+α∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds+v(tk-m+1)xT(tk-m+1)eα(t-tk-m+1)Qik-mx(tk-m+1)-v(t-h)xT(t-h)eαhQik-mx(t-h)+v(tk-m+2)xT(tk-m+2)eα(t-tk-m+2)Qik-m+1x(tk-m+2)-v(tk-m+1+)xT(tk-m+1+)eα(t-tk-m+1)Qik-m+1x(tk-m+1+)⋯+v(tk)xT(tk)eα(t-tk)Qik-1x(tk)-v(tk-1+)xT(tk-1+)eα(t-tk)Qik-1x(tk-1+)+v(t)xT(t)Qikx(t)-v(tk+)xT(tk+)eα(t-tk)Qikx(tk+),V˙(t)=2xT(t)Pikx˙(t)+α∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds+v(t)xT(t)Qikx(t)-v(t-h)xT(t-h)eαhQik-mx(t-h)+xT(tk-m+1)eα(t-tk-m+1)[Qik-m-ρik-m+1(I+Eik-m)TQik-m+1(I+Eik-m)]x(tk-m+1)…+xT(tk)eα(t-tk)[Qik-1-ρik(I+Eik-1)TQik(I+Eik-1)]x(tk).
From (3.2), we can obtain that
(3.8)Qik-m-ρik-m+1(I+Eik-m)TQik-m+1(I+Eik-m)≤0Qik-m-ρik-m+1(I+Eik-m)T⋮Qik-1-ρik(I+Eik-1)TQik(I+Eik-1)≤0.
Combining (3.7) and (3.8), (3.6) is obtained.(ii)
Whentk+h<tk+1,(1)
t∈(tk,tk+h], the proof is similar to the proof line in the situation (i).(2)
t∈(tk+h,tk+1],
(3.9)V(t)=xT(t)Pσ(t)x(t)+∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds=xT(t)Pikx(t)+∫t-htv(s)xT(s)eα(t-s)Qikx(s)ds.The proof for this situation is omitted.
The proof is completed.Lemma 3.3.
Consider the following Lyapunov function:(3.10)V(t)=xT(t)Pσ(t)x(t)+∫t-htv(s)xT(s)eα(t-s)Qσ(s)x(s)ds
for system (3.1a), (3.1b), and (3.1c), where Pi and Qi,i∈{1,2,…,N} are symmetric positive definite matrices with appropriate dimensions. Under the condition
(3.11)[-eαhρ*PjI+EjTEiT*-Pi-10**-e-αh(ρ*)-1Qi-1]<0,∀i,j∈N-,
we have
(3.12)V(tk+)<eαhρ*V(tk),
where ρ*=max{ρik,ik∈N-}.Proof.
Without loss of generality, letσ(tk+)=i,  σ(tk)=j. Then, we have
(3.13)V(tk+)=xT(tk+)Pσ(tk+)x(tk+)+∫tk+-htk+v(s)xT(s)eα(tk+-s)Qσ(s)x(s)ds≤xT(tk+)Pσ(tk+)x(tk+)+eαhρ*∫tk+-htk+xT(s)Qσ(s)x(s)ds≤xT(tk)(I+Ej)TPi(I+Ej)x(tk)+eαhρ*xT(tk)EjTQiEjx(tk)+eαhρ*∫tk-htkv(s)xT(s)eα(tk-s)Qσ(s)x(s)ds,(3.14)V(tk)=xT(tk)Pjx(tk)+∫tk-htkv(s)xT(s)eα(tk-s)Qσ(s)x(s)ds.
Combining (3.13) with (3.14), we have
(3.15)V(tk+)-eαhρ*V(tk)≤xT(tk)(I+Ej)TPi(I+Ej)x(tk)+eαhρ*xT(tk)EjTQiEjx(tk)-eαhρ*xT(tk)Pjx(tk)=xT(tk)Σijx(tk),
where
(3.16)∑ij=(I+Ej)TPi(I+Ej)+eαhρ*EjTQiEj-eαhρ*Pj.
Using Schur complement, (3.11) is equivalent to
(3.17)Σij<0orV(tk+)-eαhρ*V(tk)<0.
The proof is completed.Theorem 3.4.
R is a positive definite matrix. Let P~i=R-1/2PiR-1/2, Q~i=R-1/2QiR-1/2, For  all i∈N-, if there exist positive scalars ρi≥1, i∈N-, ρ*=max{ρi,i∈N-}, α, λ1, λ2, λ3 and symmetric positive matrices Pi,Pj,Qi,Ti, i,j∈N- such that
(3.18)1ρ*(I+Ei)-1Q~j(I+Ei)-T-Q~i≤0,∀i,j∈N-(3.19)[P~iAiT+AiP~i-αP~i+IAdiQ~iB2iP~i*-eαhQ~j00**-Ti0***-(ρ*-1Q~i+UikTUik)]<0,∀i,j∈N-(3.20)[-eαhρ*P~jP~j(I+EjT)P~jEjT*-P~i0**-e-αh(ρ*)-1Q~i]<0,∀i,j∈N-(3.21)λ1R-1<P~i<R-1,λ2R-1<Q~i,Ti<λ3I,∀i∈N-(3.22)[-c22e-αTf+d2λ3c1c1*-λ10**-1ρ*he-αhλ2]<0
hold, under the average dwell time scheme
(3.23)τa>τa*=Tf(αh+lnρ*)ln(c22e-αTf)-ln[(1/λ1+hρ*eαh/λ2)c12+d2λ3],
system (3.1a)–(3.1c) is finite-time bounded with respect to (c12,c22,Tf,d2,R,σ(t)).Proof.
Assuming that whent∈(tk,tk+1], σ(t)=ik, ik∈N-, k=0,1,2,3,….
Choose the following Lyapunov functional candidate:(3.24)V(t)=xT(t)P~σ(t)-1x(t)+∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds.
Whent∈(tk,tk+1], according to (3.18) and Lemma 3.2, we have
(3.25)V˙(t)≤2xT(t)P~ik-1x˙(t)+α∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds+v(t)xT(t)Q~ik-1x(t)-v(t-h)xT(t-h)Q~ik-m-1x(t-h)eαh,V˙(x(t))-αV(x(t))-wT(t)Tikw(t)≤2xT(t)P~ik-1x˙(t)+α∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds+v(t)xT(t)Q~ik-1x(t)-v(t-h)xT(t-h)Q~ik-m-1x(t-h)eαh-αxT(t)P~ik-1x(t)-α∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds-wT(t)Tikw(t)≤2xT(t)P~ik-1x˙(t)+ρ*xT(t)Q~ik-1x(t)-xT(t-h)Q~ik-m-1x(t-h)eαh-αxT(t)P~ik-1x(t)-wT(t)Tikw(t).
According to (3.1a)–(3.1c), and (3.25), Assumption 2.2, and the fallowing inequality:
(3.26)2xT(t)P~ik-1f(x(t))≤fikT(x(t))fik(x(t))+xT(t)P~ik-1P~ik-1x(t)≤xT(t)UikTUikx(t)+xT(t)P~ik-1P~ik-1x(t),
we have
(3.27)V˙(x(t))-αV(x(t))-wT(t)Tikw(t)≤XT(t)ΞkX(t),
where XT(t)=(xT(t)xT(t-h)wT(t)),
(3.28)Ξk=[ΔkP~ik-1AdikP~ik-1B2ik*-eαhQ~ik-m-10**-Tik],Δk=AikTP~ik-1+P~ik-1Aik+ρ*Q~ik-1-αP~ik-1+UikTUik+P~ik-1P~ik-1.
Using Schur complement, we obtain from (3.19) that
(3.29)[OiP~i-1AdiP~i-1B2i*-eαhQ~j-10**-Ti]<0,
where
(3.30)Oi=AiTP~i-1+P~i-1Ai+ρ*Q~i-1-αP~i-1+UiTUi+P~i-1P~i-1.
Noticing that the above inequality holds for all i,j∈N-, then we have Ξk<0 for ik,ik-1∈N-. Thus,
(3.31)V˙(x(t))-αV(x(t))-wT(t)Tikw(t)<0.
When t∈(tk,tk+1], according to Lemma 3.3, we can obtain (3.12) from condition (3.20).
Combining (3.31) and (3.12), we can obtain that
(3.32)V(t)<eα(t-tk)V(tk+)+∫tkteα(t-s)wT(s)Tikw(s)ds<eα(t-tk)eαhρ*V(tk)+∫tkteα(t-s)wT(s)Tikw(s)ds<eα(t-tk)eαhρ*[eα(tk-tk-1)V(tk-1+)+∫tk-1tkeα(tk-s)wT(s)Tik-1w(s)ds]+∫tkteα(t-s)wT(s)Tikw(s)ds<⋯<eα(t-t0)(eαhρ*)Nσ(t0,t)V(t0)+(eαhρ*)Nσ(t0,t)∫t0t1eα(t-s)wT(s)Ti0w(s)ds+(eαhρ*)Nσ(t1,t)∫t1t2eα(t-s)wT(s)Ti1w(s)ds+⋯+eαhρ*∫tk-1tkeα(t-s)wT(s)Tik-1w(s)ds+∫tkteα(t-s)wT(s)Tikw(s)ds=eα(t-t0)(eαhρ*)Nσ(t0,t)V(t0)+∫t0teα(t-s)(eαhρ*)Nσ(s,t)wT(s)Tikw(s)ds<eαt(eαhρ*)Nσ(t0,t)V(t0)+(eαhρ*)Nσ(t0,t)eαt∫t0twT(s)Tikw(s)ds<eαTf(eαhρ*)Nσ(t0,t)[V(t0)+∫t0TfwT(s)Tikw(s)ds]<eαTf(eαhρ*)Nσ(t0,t)[V(t0)+λmax(Tik)d2].
Noticing that Nσ(t0,Tf)<Tf/τa and according to (3.21), we have
(3.33)V(t)<e(α+αh/τa)Tf(ρ*)Tf/τα[V(t0)+λ3d2],V(t)≥xT(t)P~ik-1x(t)=xT(t)R1/2Pik-1R1/2x(t)≥λmin(Pik-1)xT(t)Rx(t)=1λmax(Pik)xT(t)Rx(t).
Because λ1R-1<P~i<R-1, we have
(3.34)V(t)>xT(t)Rx(t).
According to the Lyapunov function that we have chosen, we have
(3.35)V(t0)=xT(t0)P~i-1x(t0)+∫t0-ht0v(s)xT(s)e-α(t0-s)Q~i-1x(s)ds≤maxi∈N-λmax(Pi-1)xT(t0)Rx(t0)+heαhρ*maxi∈N-λmax(Qi-1)supt0-h≤θ≤t0xT(θ)Rx(θ)≤(1mini∈N-λmin(Pi)+ρ*heαhmini∈N-λmin(Qi))supt0-h≤θ≤t0xT(θ)Rx(θ).
According to (3.21), the following inequality is derived:
(3.36)V(t0)<(1λ1+ρ*heαhλ2)c12.
Combining (3.33), (3.34), and (3.36), we can obtain that
(3.37)xT(t)Rx(t)<V(t)<e(α+αh/τa)Tf(ρ*)Tf/τα[(1λ1+ρ*heαhλ2)c12+λ3d2].
Using Schur complement, (3.22) is equivalent to
(3.38)(1λ1+ρ*heαhλ2)c12+λ3d2<c22e-αTf.
From (3.38), we can obtain that τa>0.
Substituting (3.23) into (3.37) leads to
(3.39)xT(t)Rx(t)<c22.
Thus, system (3.1a)–(3.1c) is finite-time bounded with respect to (c12,c22,Tf,d2,R,σ(t)).
The proof is completed.Corollary 3.5.
R is a positive definite matrix, let w(t)≡0, P~i=R-1/2PiR-1/2, Q~i=R-1/2QiR-1/2 for  all  i∈N-. If there exist positive scalars ρi≥1, i∈N-, ρ*=max{ρi,i∈N-}, α,λ1,λ2 and symmetric positive matrices Pi, Pj, Qi  for  alli,j∈N- with appropriate dimensions such that
(3.40)1ρ*(I+Ei)-1Q~j(I+Ei)-T-Q~i≤0,∀i,j∈N-[P~iAiT+AiP~i-αP~i+IAdiQ~iP~i*-eαhQ~j0**-(ρ*-1Q~i+UikTUik)]<0,∀i,j∈N-[-eαhρ*P~jP~j(I+EjT)P~jEjT*-P~i0**-e-αh(ρ*)-1Q~i]<0,∀i,j∈N-λ1R-1<P~i<R-1,λ2R-1<Q~i,∀i∈N-[-c22e-αTfc1c1*-λ10**-1ρ*he-αhλ2]<0
hold with average dwell time
(3.41)τa>τa*=Tf(αh+lnρ*)ln(c22e-αTf)-ln[(1/λ1+ρ*heαh/λ2)c12].
System (3.1a)–(3.1c) with w(t)≡0 is finite-time stable with respect to (c12,c22,Tf,R,σ(t)).


## 3.2.H∞ Performance Analysis

In this subsection,H∞ performance of the following system is investigated:(3.42a)x˙(t)=Aσ(t)x(t)+Adσ(t)x(t-h)+fσ(t)(x(t))+B2σ(t)w(t),t≠tk(3.42b)Δx=Eσ(t)x(t),t=tk,k=1,2,3,…(3.42c)z(t)=Cσ(t)x(t),(3.42d)x(t)=φ(t),t∈[t0-h,t0]Theorem 3.6.
R is a positive definite matrix. Let P~i=R-1/2PiR-1/2, Q~i=R-1/2QiR-1/2 for alli∈N-. Suppose that there exist positive scalars ρi≥1,i∈N-, ρ*=max{ρi,i∈N-}, α,γ,ε and symmetric positive matrices Pi,Pj,Qi for alli,j∈N- such that
(3.43)1ρ*(I+Ei)-1Q~j(I+Ei)-T-Q~i≤0,∀i,j∈N-(3.44)[P~iAiT+AiP~i-αP~i+IAdiQ~iB2iP~iP~iCiT*-eαhQ~j000**-γ200***-(ρ*-1Q~i+UikTUik)0****-I]<0,∀i,j∈N-(3.45)[-eαhρ*P~jP~j(I+EjT)P~jEjT*-P~i0**-e-αh(ρ*)-1Q~i]<0,∀i,j∈N-(3.46)P~i<R-1,∀i∈N-(3.47)-c22+eαTfγ2d2<0
hold with average dwell time
(3.48)τa>τa*=max{Tf(αh+lnρ*)ln(c22)-ln(eαTfγ2d2),hε}.
Then, system (3.42a)–(3.42d) is finite-time bounded and has H∞ performance with respect to (0,c22,Tf,d2,γ-,R,σ(t)), where γ-2=e(1+ε)αTf(ρ*)εTf/hγ2.Proof.
Whent∈(tk,tk+1], σ(t)=ik, ik∈N-, k=0,1,2,3,…. Choose the following Lyapunov functional candidate for system (3.42a)–(3.42d)
(3.49)V(t)=xT(t)P~σ(t)-1x(t)+∫t-htv(s)xT(s)eα(t-s)Q~σ(s)-1x(s)ds.
When t∈(tk,tk+1],
(3.50)V˙(x(t))-αV(x(t))+zT(t)z(t)-γ2wT(t)w(t)≤XT(t)ΨkX(t),
where XT(t)=(xT(t)xT(t-h)wT(t)),
(3.51)Ψk=[ΔkP~ik-1AdikP~ik-1B2ik*-eαhQ~ik-m-10**-γ2I],Δk=AikTP~ik-1+P~ik-1Aik+ρ*Q~ik-1-αP~ik-1+UikTUik+P~ik-1P~ik-1+CikTCik.
Using Schur complement, we obtain from (3.44) that
(3.52)[EiP~i-1AdiP~i-1B2i*-eαhQ~j-10**-γ2I]<0,
where Ei=AiTP~i-1+P~i-1Ai+ρ*Q~i-1-αP~i-1+UiTUi+P~i-1P~i-1+CiTCi.
Noticing that the above inequality holds for  alli,j∈N-, then we have Ψk<0, for ik,ik-m∈N-.
Thus,(3.53)V˙(x(t))-αV(x(t))+zT(t)z(t)-γ2wT(t)w(t)<0,
Letγ2wT(s)w(s)-zT(s)z(s)=Δ(s), from (3.32), we have
(3.54)V(t)<eα(t-t0)(eαhρ*)Nσ(t0,t)V(t0)+∫t0teα(t-s)(eαhρ*)Nσ(s,t)Δ(s)ds.
Under zero initial condition, we have
(3.55)0<∫t0teα(t-s)(eαhρ*)Nσ(s,t)Δ(s)ds,
that is,
(3.56)∫t0teα(t-s)(eαhρ*)Nσ(s,t)zT(s)z(s)ds<∫t0teα(t-s)(eαhρ*)Nσ(s,t)γ2wT(s)w(s)ds.
Noticing that
(3.57)∫t0teα(t-s)(eαhρ*)Nσ(s,t)zT(s)z(s)ds>∫t0tzT(s)z(s)ds.
Then, we have
(3.58)∫t0teα(t-s)(eαhρ*)Nσ(s,t)γ2wT(s)w(s)ds<eαt(eαhρ*)Nσ(t0,t)∫t0tγ2wT(s)w(s)ds.
Let  t=Tf, because τa>h/ε, we have
(3.59)∫t0TfzT(s)z(s)ds<e(1+ε)αTf(ρ*)εTf/hγ2∫t0TfwT(s)w(s)ds,
then
(3.60)∫t0TfzT(s)z(s)ds<γ-2∫t0TfwT(s)w(s)ds.
Thus, system (3.42a)–(3.42d) is finite-time bounded and has H∞ performance with respect to (0,c22,Tf,d2,γ-,R,σ(t)), where  γ-2=e(1+ε)αTf(ρ*)εTf/hγ2.
The proof is completed.Remark 3.7.
Whenρ*=1, Theorem 3.6 degenerates to the result of [41], which cannot guarantee the finite-time boundedness of the addressed system if ρ*>1.


## 3.3. Robust Finite-TimeH∞ Control

Consider system (2.1a)–(2.1d), under the switching controller u1(t)=Kσ(t)x(t), t≠tk and impulsive controller u2(tk)=K-σ(t)x(tk),   t=tk, the corresponding closed-loop system is given by(3.61a)x˙(t)=(A^σ(t)+B^1σ(t)Kσ(t))x(t)+A^dσ(t)x(t-h)+fσ(t)(x(t))+B2σ(t)w(t),t≠tk(3.61b)Δx=(Eσ(t)+K-σ(t))x(t),t=tk,k=1,2,3,…(3.61c)z(t)=(Cσ(t)+Dσ(t)Kσ(t))x(t),(3.61d)x(t)=φ(t),t∈[t0-h,t0].Theorem 3.8.
Consider impulsive switched system (2.1a)–(2.1d), let P~i=R-1/2PiR-1/2, Q~i=R-1/2QiR-1/2  for  all   i∈N-. If  there exist positive scalars ρi≥1,   i∈N-,ρ*=max{ρi,i∈N-}α,γ,ε,δi and positive definite symmetric matrices Pi,Qi, and matrices Yi,   i∈N-,  with appropriate dimensions, such that the following inequalities hold
(3.62)1ρ*(I+Ei)-1Q~j(I+Ei)-T-Q~i≤0,∀i,j∈N-(3.63)[ΓiAdiQ~iB2iP~iP~iCiT+YiTDiTYiTEBiT+P~iEAiT*-eαhQ~j000Q~iEAdiT**-γ2000***-(ρ*-1Q~i+UikTUik)00****-I0*****-δi]<0,∀i,j∈N-,
where
(3.64)Γi=P~iAiT+YiTB1iT+AiP~i+B1iYi-αP~i+I+δiHiHiT,[-eαhρ*P~jP~j*-P~i]<0,∀i,j∈N-(3.65)P~i<R-1,i∈N-,(3.66)-c22+eαTfγ2d2<0.
Then, under the controller Ki=YiP~i-1, K-i=-Ei, and the following average dwell time scheme
(3.67)τa>τa*=max{Tf(αh+lnρ*)ln(c22)-ln(eαTfγ2d2),hε},
the corresponding closed-loop system is finite-time bounded with H∞ performance with respect to  (0,c22,Tf,d2,γ-,R,σ(t)) and γ-2=e(1+ε)αTf(ρ*)εTf/hγ2.Proof.
According to Assumption2.1, we have
(3.68)A^i+B^1iKi=(Ai+B1iKi)+HiFi(EAi+EBiKi),A^di=Adi+HiFiEAdi.
Now replacing Ai, Adi, Ci in the left side of (3.44) with A^i+B^1iKi,A^di, Ci+DiKi, we can obtain that
(3.69)Θij=[Ωi(Adi+HiFiEAdi)Q~iB2iP~iP~i(Ci+DiKi)T*-eαhQ~j000**-γ200***-(ρ*-1Q~i+UikTUik)0****-I],
where
(3.70)Ωi=[(Ai+B1iKi)+HiFi(EAi+EBiKi)]P~i+P~i[(Ai+B1iKi)+HiFi(EAi+EBiKi)]T-αP~i+I.
From (3.69), we know that
(3.71)Θij=Π1ij+Π2ij,
where
(3.72)Π1ij=[Υ1iAdiQ~iB2iP~iP~i(Ci+DiKi)T*-eαhQ~j000**-γ200***-(ρ*-1Q~i+UikTUik)0****-I],Π2ij=[Υ2iHiFiEAdiQ~i000*0000**000***00****0],
with
(3.73)Υ1i=P~i(Ai+B1iKi)T+(Ai+B1iK)P~i-αP~i+I,Υ2i=P~i(EAi+EBiKi)TFiTHiT+HiFi(EAi+EBiKi)P~i,
let Yi=KiP~i, then
(3.74)Υ1i=P~iAiT+YiTB1iT+AiP~i+B1iYi-αP~i+I,Υ2i=(YiTEBiT+P~iEAiT)FiTHiT+HiFi(EAiP~i+EBiYi).
From Lemma 2.11, we can obtain that
(3.75)Θij=Π1ij+[Hi0000]Fi[EAiP~i+EBiYiEAdiQ~i000]+[YiTEBiT+P~iEAiTQ~iEAdiT000]Fi[HiT0000]≤Π1ij+δi[Hi0000][HiT0000]+1δi[YiTEBiT+P~iEAiTQ~iEAdiT000][EAiP~i+EBiYiEAdiQ~i000].
Using Schur complement lemma, we get from (3.63) that
(3.76)Θij<0.
Now we choose K-i=-Ei, and replacing Ei in (3.45) with Ei+K-i, we know that
(3.77)[-eαhρ*P~jP~j0*-P~i0**-e-αh(ρ*)-1Q~i]<0,
by (3.64), we know that the condition(3.45) hold.
Then, system (2.1a)–(2.1d) is robust finite-time bounded with H∞ performance with respect to  (0,c22,Tf,d2,γ-,R,σ(t)), and γ-2=e(1+ε)αTf(ρ*)εTf/hγ2.
The proof is completed.Remark 3.9.
In order to eliminate the impulsive jump, we design an impulsive feedback controllerK-i=-Ei,  t=tk. Then the system becomes a switched system with continuous states.


## 4. Numerical Examples

In this section, we present two examples to illustrate the effectiveness of the proposed approach.Example 4.1.
Consider system (2.1a)–(2.1d) with the following parameters.Subsystem 1
(4.1)
A
1
=
[
-
8
1
2
-
7
]
,
A
d
1
=
[
-
1.3
0.1
0.2
-
1
]
,
H
1
=
[
-
0.1
0.3
0.2
0.4
]
,
E
A
d
1
=
[
0.2
0.1
0.3
-
0.2
]
,
E
1
=
[
0.43
0
0
0.15
]
,
U
1
=
[
0.1
0
0
0.1
]
,
B
11
=
[
3
-
3
0
4
]
,
E
B
1
=
[
0.2
0.1
0
-
0.3
]
,
B
21
=
[
1
-
1
2
1
]
,
C
1
=
[
1
1
0
2
]
,
D
1
=
[
0.8
0
0
0.2
]
,
E
A
1
=
[
-
0.3
-
0.1
0.2
-
0.1
]
,
f
1
(
x
(
t
)
)
=
0.1
sin
x
(
t
), where ∥f1(x(t))∥<∥U1x(t)∥.Subsystem 2
(4.2)
A
2
=
[
-
7
2
1
-
6
]
,
A
d
2
=
[
-
1.2
0.1
0.3
-
1.1
]
,
H
2
=
[
-
0.1
0.2
-
0.2
-
0.1
]
,
E
A
d
2
=
[
-
0.3
0.1
0.2
-
0.3
]
,
E
2
=
[
0.15
0
0
0.4
]
,
U
2
=
[
0.2
0
0
0.18
]
,
B
12
=
[
4
-
1
1
6
]
,
E
B
2
=
[
-
0.3
0.1
0
0.2
]
,
B
22
=
[
-
1
0
2
0.8
]
,
C
2
=
[
2
1
0
3
]
,
D
2
=
[
0.8
0
1
-
1
]
,
E
A
2
=
[
0.1
0.3
0.2
-
0.2
]
,
f
2
(
x
(
t
)
)
=
0.18
cos
x
(
t
), where ∥f2(x(t))∥<∥U2x(t)∥.
ChoosingTf=12,h=0.2,d2=10,R=I,  α=0.1,C22=2,ε=0.1,  γ2=0.5441,  ρ*=1, solving the LMIs in (3.62)–(3.66) leads to
(4.3)Q~1=[1.3506-0.1265-0.12650.7891],Q~2=[0.50420.05250.05250.3221],Y1=[0.0234-0.35770.16310.2680],Y2=[-0.0001-0.52210.11090.0371],P~1=[0.98870.00110.00110.9921],P~2=[0.9995-0.0001-0.00011.0006],K1=[0.0241-0.36050.16470.2699],K2=[-0.0001-0.52180.11090.0371],Q~1-1-(I+E1)TQ~1-1(I+E1)≤0,Q~2-1-(I+E2)TQ~1-1(I+E2)≤0,Q~2-1-(I+E2)TQ~2-1(I+E2)≤0,Q~1-1-(I+E1)TQ~2-1(I+E1)≤0,τa>τa*=1.2049, we choose τa=2, γ-2=e(1+ε)αTf(ρ*)εαTfγ2=2.0368, then the system is finite-time bounded according to [41, Theorem 3].Example 4.2.
Consider system (2.1a)–(2.1d) with the following parameters.Subsystem 1
(4.4)
A
1
=
[
-
8
1
2
-
7
]
,
A
d
1
=
[
-
1.3
0.1
0.2
-
1
]
,
H
1
=
[
-
0.1
0.3
0.2
0.4
]
,
E
A
d
1
=
[
0.2
0.1
0.3
-
0.2
]
,
E
1
=
[
-
0.1
0
0
-
0.1
]
,
U
1
=
[
0.01
0
0
0.01
]
,
B
11
=
[
3
-
3
0
4
]
,
E
B
1
=
[
0.2
0.1
0
-
0.3
]
,
B
21
=
[
1
-
1
2
1
]
,
C
1
=
[
-
3
1
0
-
2
]
,
D
1
=
[
8
0
0
2
]
,
E
A
1
=
[
-
3
-
0.1
0.2
-
1
]
,
f
1
(
x
(
t
)
)
=
0.01
sin
x
(
t
).Subsystem 2
(4.5)
A
2
=
[
-
7
2
1
-
6
]
,
A
d
2
=
[
-
1.2
0.1
0.3
-
1.1
]
,
H
2
=
[
-
0.1
0.2
-
0.2
-
0.1
]
,
E
A
d
2
=
[
-
0.3
0.1
0.2
-
0.3
]
,
E
2
=
[
-
0.1
0
0
-
0.1
]
,
U
2
=
[
0.02
0
0
0.08
]
,
B
12
=
[
4
-
1
1
6
]
,
E
B
2
=
[
-
0.3
0.1
0
0.2
]
,
B
22
=
[
-
1
0
2
0.8
]
,
C
2
=
[
-
2
1
0
-
3
]
,
D
2
=
[
8
0
1
8
]
,
E
A
2
=
[
-
1
0.3
0.2
-
2
]
,
f
2
(
x
(
t
)
)
=
0.02
cos
x
(
t
).(1)
Leth=0.2,Tf=12,d2=10, R=I, α=0.001,  C22=21,  ρ*=1.3,γ2=0.9344. By solving the LMIs in (3.62)–(3.66), we can get
(4.6)Q~1=[0.42520.03870.03871.2272],Q~2=[0.43520.04700.04701.2369],Y1=[0.0866-0.4834-0.08630.5554],Y2=[0.1064-0.2575-0.12600.2934],P~1=[0.46060.04180.04180.9965],P~2=[0.5364-0.0611-0.06110.9884],K1=[0.2329-0.4949-0.23890.5673],K2=[0.1699-0.2500-0.20240.2844],
andτa>τa*=3.8340. We choose τa=4, ε=0.05, γ-2=0.9464, the initial condition x(t)=0, t∈[-h,0], the switching signal is shown in Figure 1, and state trajectories of the closed-loop system are shown in Figure 2.Figure 1
Switching signal.Figure 2
State trajectories of the closed-loop system.We can see from Figure2 that the states of the system are continuous due to the feedback K-i in impulsive instants.(2)
Leth=0.2, Tf=12, d2=10, R=I,  and   α=0.001. By solving the LMIs of [41, Theorem 3], we can get
(4.7)Q~1=[0.40150.03590.03591.0563],Q~2=[0.52240.11040.11041.0717],Y1=[0.1245-0.6523-0.09980.5952],Y2=[0.1279-0.2380-0.10060.2699],P~1=[0.55770.00990.00990.9992],P~2=[0.55770.00990.00990.9992],K1=[0.2349-0.6552-0.18960.5976],K2=[0.2336-0.2405-0.18520.2720],Q~1-1-(I+E1)TQ~1-1(I+E1)>0,Q~2-1-(I+E2)TQ~1-1(I+E2)>0,Q~2-1-(I+E2)TQ~2-1(I+E2)>0,Q~1-1-(I+E1)TQ~2-1(I+E1)>0.Obviously, the above inequalities do not satisfy the conditions of [41, Theorem 3]. Thus, we cannot draw the conclusion that the closed-loop system is finite-time bounded from Theorem 3 in [41].


## 5. Conclusions

This paper has investigated robust finite-timeH∞ control for a class of impulsive switched nonlinear systems with time-delay. Based on piecewise Lyapunov function, sufficient conditions which guarantee finite-time boundedness of the impulsive switched system are derived. Then, a feedback control scheme consisting of an impulsive feedback controller and a switching controller is proposed, and the proposed control strategy can guarantee that the closed-loop system is finite-time bounded with H∞ disturbance attenuation level. Finally, the results are illustrated by means of two numerical examples.



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*Source: 830154-2012-11-11.xml*