# Nonfragile Finite-Time Extended Dissipative Control for a Class of Uncertain Switched Neutral Systems

Authors: Hui Gao; Jianwei Xia; Guangming Zhuang; Zhen Wang; Qun Sun
Journal: Complexity
 (2017)

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## Abstract

This paper is concerned with finite-time extended dissipative analysis and nonfragile control for a class of uncertain switched neutral systems with time delay, and the controller is assumed to have either additive or multiplicative form. By employing the average dwell-time and linear matrix inequality technique, sufficient conditions for finite-time boundedness of the switched neutral system are provided. Then finite-time extended dissipative performance for the switched neutral system is addressed, where we can solveH∞, L2-L∞, Passivity, and (Q,S,R)-dissipativity performance in a unified framework based on the concept of extended dissipative. Furthermore, nonfragile state feedback controllers are proposed to guarantee that the closed-loop system is finite-time bounded with extended dissipative performance. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Switched system is an important class of hybrid systems, which consists of a family of subsystems and a logical rule that orchestrates switching between them. For its practical importance, switched systems have received considerable attention in the last decades [1–5]. Meanwhile, time delay exists widely in many practical systems and may cause undesirable system performance or even instability [6–9]. Switched system with time delay is a main issue in recent years. As a special time delay system, switched neutral systems have received much attention [10–14]. For example, the problem of stability analysis and H∞ control for several switched neutral systems were considered in [10] and [13], respectively.Up to now, most researches for switched neutral systems focus on Lyapunov asymptotic stability, which is defined over an infinite time interval. However, in practice, the transient performance of a system is also of great significance. In many practical applications such as missile systems and robot control systems, the main concern is the system behavior over a finite-time interval. Therefore, finite-time analysis of switched systems is worth researching. Recently, some related research results were published in the literatures [15–20]. More specifically, finite-time H∞ control of switched systems was addressed in [16], and finite-time stabilization and boundedness of switched linear system were investigated in [19].On the other hand, the controller coefficients are generally exact values when designing a desired controller. However, in practice, uncertainty cannot be avoided in controller design, and it may be caused by many reasons, such as numerical round-off errors and actuator degradation. The existence of uncertainty motivates the study of nonfragile control. Over decades, much attention has been devoted to the issue of controller fragility and related remedies [21–24]. To name a few, the problem of passivity-based nonfragile control for Markovian jump systems with aperiodic sampling is studied in [22], and nonfragile H∞ control for linear systems with multiplicative controller gain variations is investigated in [24], respectively. More recently, an effective tool named extended dissipative was firstly proposed by Zhang et al. in [25] to deal with the problem of robust control. By adjusting weighting matrices, the extended dissipative covers some well-known performance indices such as H∞ performance, L2-L∞ performance, Passivity performance, and (Q,S,R)-dissipativity performance. This concept has been successfully applied to the stability analysis for several neural networks [26–30]. Could this concept be applied to switched systems? To the best of our knowledge, the topic of nonfragile finite-time extended dissipative control for a class of uncertain switched neutral systems has not been investigated yet, which motivates our study.This paper is organized as follows. In Section2, preliminaries and problem statement are formulated and some necessary lemmas are given. In Section 3, by employing the average dwell-time and linear matrix inequality approach, some sufficient conditions of finite-time boundness and finite-time extended dissipative performance for switched neutral systems are established. Furthermore, existence and the design method of the nonfragile state feedback controllers are proposed. All of the results are in terms of a set of linear matrix inequalities which can be easily resolved using the LMIs toolbox. In Section 4, numerical examples are given to show the effectiveness of the proposed approach. The main contributions of this paper include the following. (1) We firstly apply the concept of extended dissipative to the nonfragile finite-time analysis and control to the uncertain switched neutral systems. (2) More general switched systems are considered in our paper, including the time-varying delay and distributed delay, neutral parameters, and additive and multiplicative form controller.Notation. The notations used in this paper are standard. Rn denotes the n-dimensional Euclidean space and MT represents the transpose of the matrix M. The notation X>0 (≥0) is used to denote a symmetric positive definite (positive-semidefinite) matrix. X represents the Euclidean norm of the matrix X; λmin(P) and λmax(P) denote the minimum and maximum eigenvalue of matrix P, respectively. I is the identity matrix with appropriate dimension. diag{·} stands for a block-diagonal matrix. The asterisk ∗ in a matrix is used to denote a term that is induced by symmetry.


## 2. Preliminaries and Problem Statement

Consider the following switched neutral system with time-varying delay:(1)x˙t-C^σtx˙t-τt=A^σtxt+B^σtxt-ht+Dσtwt+Eσtut+G^σt∫t-rttxsds,zt=Fσtxt,xθ=φθ,∀θ∈-τ,0,where x(t)∈Rn is the state vector, u(t) is the control input, w(t) is the exogenous disturbance which belongs to L2[0,∞), and z(t)∈Rn is the output. The switching signal σ(t):[0,∞)↦M={1,2,…,l} is a piecewise continuous function, where l is the number of subsystems and σ(t)=i means that the ith subsystem is activated. φ(θ) is the initial condition, and h(t), r(t), and τ(t) denote the time-varying delay and satisfy(2)0≤ht≤hm,h˙t≤h^<1,0≤τt≤τm,τ˙t≤τ^<1,0≤rt≤rm.Furthermore, -τ=max⁡{hm,τm}, and for each σ(t)=i, A^i, B^i, C^i, and G^i are uncertain real-valued matrices with appropriate dimensions. We assume that the uncertainties are norm-bounded and of the form(3)A^i,B^i,C^i,G^i=Ai,Bi,Ci,Gi+LiΞitM1i,M2i,M3i,M4i,where Ai, Bi, Ci, Gi, M1i, M2i, M3i, and M4i are known real-valued constant matrices with appropriate dimensions and Ξi(t) is unknown and possibly time-varying matrix with Lebesgue measurable elements satisfying ΞiT(t)Ξi(t)≤I.In this paper, we consider the following nonfragile state feedback controller:u(t)=Kσ(t)(t)x(t), where Kσ(t)(t)=Ki+ΔKi(t), Ki is the controller gain and ΔKi(t) is a perturbed matrix with the following forms.Case 1.
Δ
K
i
(
t
) has an additive uncertainty which is assumed to be(4)ΔKit=J1iUitJ2i,where J1i and J2i are known real constant matrices with appropriate dimensions and the time-varying matrix Ui(t) satisfies Ui(t)UiT(t)≤I.Case 2.
Δ
K
i
(
t
) has a multiplicative uncertainty(5)ΔKit=k11iσ1i+σ11i⋯k1niσ1i+σ1nik21iσ2i+σ21i⋯k2niσ2i+σ2ni⋮⋱⋮km1iσmi+σm1i⋯kmniσmi+σmni,where kpqi is the element of Ki and σpi and σpqi(p=1,…,m;q=1,…,n) are real uncertain parameters which satisfy(6)σpi≤σ¯1pi≤1,σpqi≤σ¯pqi≤σ¯2pi≤1.Assumption 1.
For a given time constantTf, the external disturbance satisfies (7)∫0TfwTtwtdt≤d,d≥0.Assumption 2.
For a given time constantTf, the state vector x(t) is time-varying and satisfies the constraint (8)∫0TfxTtxtdt≤k,where k is a fixed sufficient large constant number.Assumption 3 (see [25]).
Matricesψ1, ψ2, ψ3, and ψ4 satisfy the following conditions:(1)
ψ1=ψ1T≤0,ψ3=ψ3T>0,ψ4=ψ4T≥0;(2)
(ψ1+ψ2)ψ4=0.Assumption 4.
For∀α≥0,μ≥1,∀t∈[0,Tf], we have (9)eαtμNσ0,t≤b,where Nσ(0,t) denotes the switching number of σ(t) over (0,t) and b denotes a positive number.Definition 5 (see [25]).
For given matricesψ1, ψ2, ψ3, and ψ4 satisfying Assumption 3, system (1) is said to be extended dissipative if the following inequality holds for any Tf≥0 and all μ(t)∈L2[0,∞):(10)∫0TfJtdt-sup0≤t≤Tf⁡zTtψ4zt≥0,where(11)Jt=zTtψ1zt+2zTtψ2wt+wTtψ3wt.Remark 6.
The concept of extended dissipative introduced in Definition5 contains a few of well-known performance indices as special cases by setting the weighting matrices:(1)
L2-L∞ performance: ψ1=0, ψ2=0, ψ3=γ2I, and ψ4=I;(2)
H∞ performance: ψ1=-I, ψ2=0, ψ3=γ2I, and ψ4=0;(3)
Passivity performance:ψ1=0, ψ2=I, ψ3=γI, and ψ4=0;(4)
(Q,S,R)-dissipativity performance: ψ1=Q, ψ2=S, ψ3=R-βI, and ψ4=0.Definition 7 (see [17]).
Given three positive constantsc1, c2, and Tf with c1<c2, a positive definite matrix R and a switching signal σ(t), assume that μ(t)≡0, ∀t∈[0,Tf], and switched neutral system (1) is said to be finite-time bounded with respect to (c1,c2,R,Tf,σ), if, ∀t∈[0,Tf],(12)sup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ≤c1⟹xTtRxt≤c2. Furthermore, if the condition above holds with w(t)≡0, ∀t∈[0,Tf], the system is said to be finite-time stable.Definition 8 (see [17]).
For anyT2>T1≥0, let Nσ(T1,T2) denote the switching number of σ(t) over (T1,T2). If(13)NσT1,T2≤N0+T2-T1τaholds for τa>0 and an integer N0≥0, then τa is called an average dwell-time. Without loss of generality, in this paper we choose N0=0.Lemma 9 (see [29]).
Leta and b be real matrices of appropriate dimensions and satisfy 2aTb≤aTa+bTb.Lemma 10 (see [31]).
For any positive definite symmetric matrixN∈Rn×n, scalar τ>0, and a vector function x(•):[-τ,0]→Rn, the following integral inequality is satisfied: (14)-τ∫t-τtxTsNxsds≤-∫t-τtxTsdsN∫t-τtxsds.Lemma 11 (see [32]).
LetJ and L be real matrices of appropriate dimensions. Then, for any scalar ϵ>0, one has (15)JUtL+JUtLT≤ϵJJT+ϵ-1LTL,when U(t) satisfies UT(t)U(t)≤I.Lemma 12 (see [33]).
LetG be real positive definite symmetric matrices and let B and L be appropriate dimensional real matrices. Then, one has (16)BL+BLT≤BG-1BT+LTGL.


## 3. Main Results

### 3.1. Finite-Time Boundedness Analysis

Consider the following unforced switched neutral system without uncertainties:(17)x˙t-Cσtx˙t-τt=Aσtxt+Bσtxt-ht+Dσtwt+Gσt∫t-rttxsds,zt=Fσtxt,xt0+θ=φθ,∀θ∈-τ,0.In this section, the problem of finite-time boundedness analysis of the switched neutral system is proposed, by using the average dwell-time approach, sufficient conditions are derived by solving some linear matrix inequalities, and the results are shown as follows.Theorem 13.
For given positive scalarsα, h^, τ^, hm, and rm, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices Ni, X11i, X12i, X22i, H1i, and H2i with appropriate dimensions, then(18)Xi=X11iX12i∗X22i≥0,(19)Δi=X11iX12iH1i∗X22iH2i∗∗T~i≥0,(20)Θi=ϕ11ϕ12ϕ13ϕ14P~iGi+AiTZ~iGi+hmAiTT~iGi∗ϕ22ϕ23BiTZ~iDi+hmBiTT~iDiBiTZ~iGi+hmBiTT~iGi∗∗ϕ33CiTZ~iDi+hmCiTT~iDiCiTZ~iGi+hmCiTT~iGi∗∗∗ϕ44DiTZ~iGi+hmDiTT~iGi∗∗∗∗GiTZ~iGi+hmGiTT~iGi-M~irm<0,hold, where (21)ϕ11=-αP~i+P~iAi+AiTP~i+Q~i+AiTZ~iAi+hmAiTT~iAi+rmM~i+H1i+H1iT+hmX11i,ϕ12=P~iBi+AiTZ~iBi+hmAiTT~iBi-H1iT+H2i+hmX12i,ϕ13=P~iCi+AiTZ~iCi+hmAiTT~iCi,ϕ14=P~iDi+AiTZ~iDi+hmAiTT~iDi,ϕ22=-1-h^Q~i+BiTZ~iBi+hmBiTT~iBi-H2i-H2iT+hmX22i,ϕ23=BiTZ~iCi+hmBiTT~iCi,ϕ33=-1-τ^Z~i+CiTZ~iCi+hmCiTT~iCi,ϕ44=-Ni+DiTZ~iDi+hmDiTT~iDi,(22)λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d<c2λ1e-αTf;meanwhile, the average dwell-time satisfies(23)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d-αTf.We define (24)P~i=R1/2PiR1/2,Q~i=R1/2QiR1/2,Z~i=R1/2ZiR1/2,T~i=R1/2TiR1/2,M~i=R1/2MiR1/2,(25)λminPi=λ1,λmaxPi=λ2,λmaxQi=λ3,λmaxZi=λ4,λmaxTi=λ5,λmaxMi=λ6,λmaxNi=λ7,where μ>1 satisfies(26)P~i<μP~j,Q~i<μQ~j,Z~i<μZ~j,T~i<μT~j,M~i<μM~j,∀i,j∈M.Then, switched system (17) is finite-time bounded with respect to (c1,c2,d,R,Tf,σ).Proof.
Choose the piecewise Lyapunov-Krasovskii functional candidate as(27)Vt=Vσtt=Vit=V1it+V2it+V3it+V4it+V5it,where(28)V1it=xTtP~ixt,V2it=∫t-htteαt-sxTtQ~ixtds,V3it=∫t-τtteαt-sx˙TtZ~ix˙tds,V4it=∫-hm0∫t+ϵteαt-sx˙TtT~ix˙tdsdϵ,V5it=∫-rm0∫t+ϵteαt-sxTtM~ixtdsdϵ,in which α is a given scalar and P~i, Q~i, Z~i, T~i, and M~i are positive definite matrices to be determined. Taking the derivative of V(t) with respect to t along the trajectory of system (17) yields (29)V˙1it=2xTtP~ix˙t,V˙2it=αV2it+xTtQ~ixt-eαht1-h˙txTt-htQ~ixt-ht≤αV2it+xTtQ~ixt-1-h^xTt-htQ~ixt-ht,V˙3it=αV3it+x˙TtZ~ix˙t-eατt1-τ˙tx˙Tt-τtZ~ix˙t-τt≤αV3it+x˙TtZ~ix˙t-1-τ^x˙Tt-τtZ~ix˙t-τt,V˙4it=αV4it+hmx˙TtT~ix˙t-∫t-htteαt-sx˙TsT~ix˙sds≤αV4it+hmx˙TtT~ix˙t-∫t-httx˙TsT~ix˙sds,V˙5it=αV5it+rmxTtM~ixt-∫t-rtteαt-sxTsM~ixsds≤αV5it+rmxTtM~ixt-∫t-rttxTsM~ixsds.Using the Leibniz-Newton formula, we have(30)2xTtH1i+xTt-htH2ixt-∫t-httx˙sds-xt-ht=0. Let (31)χt=xTtxTt-htT, and it obviously holds that(32)hmχTtXiχt-∫t-httχTtXiχtds≥0. By Lemma 10, it is easy to obtain(33)-∫t-rttxTsM~ixsds≤-1rm∫t-rttxTsdsM~i∫t-rttxsds.Thus we have (34)V˙t-αVt-wTtNiwt≤XTtΘiXt-∫t-httϑTt,sΔiϑt,sds, where(35)Xt=xTtxTt-htx˙Tt-τtwTt∫t-rttxTsdsT,ϑt,s=xTtxTt-htx˙TsT.Considering (19) and (20), we can obtain that(36)V˙t-αVt-wTtNiwt<0.Integrating (36), it can be obtained from (26) and (36) that, ∀t∈[tk,tk+1),(37)Vt<eαt-tkVtk+∫tkteαt-swTsNiwsds<eαt-tkμVtk-+∫tkteαt-swTsNiwsds<eαt-tkμeαtk-tk-1Vtk-1+∫tk-1tkeαtk-swTsNiwsds+∫tkteαt-swTsNiwsds=eαt-tk-1μVtk-1+μ∫tk-1tkeαt-swTsNiwsds+∫tkteαt-swTsNiwsds<⋯<eαt-0μNσ0,tV0+μNσ0,t∫0t1eαt-swTsNiwsds+μNσt1,t∫t1t2eαt-swTsNiwsds+⋯+μ∫tk-1tkeαt-swTsNiwsds+∫tkteαt-swTsNiwsds=eαt-0μNσ0,tV0+∫0teαt-sμNσs,twTsNiwsds<eαtμNσ0,tV0+μNσ0,teαt∫0twTsNiwsds<eαTfμNσ0,TfV0+∫0TfwTsNiwsds<eαTfμNσ0,TfV0+λmaxNid.From Definition 8, we can deduce that Nσ(0,Tf)<Tf/τa, and then we can obtain(38)Vt<eα+ln⁡μ/τaTfV0+λ7d.On the other hand(39)Vt>xTtP~ixt=xTtR1/2PiR1/2xt≥λminPixTtRxt=λ1xTtRxt,V0≤λmaxPixT0Rx0+hmeαhmλmaxQisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ+τmeατmλmaxZisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ+hmeαhmλmaxTisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ+rmeαrmλmaxMisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ≤λmaxPi+hmeαhmλmaxQi+τmeατmλmaxZi+hmeαhmλmaxTi+rmeαrmλmaxMisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ≤λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1.From (38)-(39), we can obtain(40)xTtRxt≤Vtλ1<λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7dλ1eαTfμTf/τa.When μ=1, it is obvious that xT(t)Rx(t)<c2 by (22).
Whenμ>1, by virtue of (22), we have that (41)ln⁡λ1c2-ln⁡λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d-αTf>0. From (23), we can obtain(42)Tfτa<ln⁡λ1c2-ln⁡λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d-αTfln⁡μ=ln⁡λ1c2e-αTf/λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7dln⁡μ.Substituting (42) into (40) yields (43)xTtRxt<λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7dλ1eαTfλ1c2e-αTfλ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d=c2.The proof is completed.Based on Theorem13, when we set w(t)=0, the following corollary is proposed to solve the finite-time stable problem.Corollary 14.
Consider system (17) with w(t)=0. For given positive scalars α, h^, τ^, hm, and rm, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices X11i, X12i, X22i, H1i, and H2i with appropriate dimensions, then(44)X11iX12i∗X22i≥0,X11iX12iH1i∗X22iH2i∗∗T~i≥0,Δ11Δ12Δ13P~iGi+AiTZ~iGi+hmAiTT~iGi∗Δ22Δ23BiTZ~iGi+hmBiTT~iGi∗∗Δ33CiTZ~iGi+hmCiTT~iGi∗∗∗GiTZ~iGi+hmGiTT~iGi-M~irm<0hold, where (45)Δ11=-αP~i+P~iAi+AiTP~i+Q~i+AiTZ~iAi+hmAiTT~iAi+rmM~i+H1i+H1iT+hmX11i,Δ12=P~iBi+AiTZ~iBi+hmAiTT~iBi-H1iT+H2i+hmX12i,Δ13=P~iCi+AiTZ~iCi+hmAiTT~iCi,Δ22=-1-h^Q~i+BiTZ~iBi+hmBiTT~iBi-H2i-H2iT+hmX22i,Δ23=BiTZ~iCi+hmBiTT~iCi,Δ33=-1-τ^Z~i+CiTZ~iCi+hmCiTT~iCi,λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1<c2λ1e-αTf;meanwhile, the average dwell-time satisfies(46)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1-αTf,where μ>1 satisfying (26) and P~i, Q~i, Z~i, T~i, M~i, λ1, λ2, λ3, λ4, λ5, and λ6 are defined just the same as (25). Then the switched system (17) is finite-time stable with respect to (c1,c2,d,R,Tf,σ).Proof.
The proof is similar to that of Theorem13, and it is omitted here.


### 3.2. Finite-Time Extended Dissipative Analysis

In this section, the finite-time extended dissipative analysis is considered in the following theorem.Theorem 15.
For given positive scalarsα, h^, τ^, hm, rm, and b, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices X11i, X12i, X22i, H1i, and H2i with appropriate dimensions, then(47)1bP~i-FiTψ4Fi>0,(48)Xi=X11iX12i∗X22i≥0,(49)Δi=X11iX12iH1i∗X22iH2i∗∗T~i≥0,(50)Φi=Φ11Φ12Φ13Φ14P~iGi+AiTZ~iGi+hmAiTT~iGi∗Φ22Φ23BiTZ~iDi+hmBiTT~iDiBiTZ~iGi+hmBiTT~iGi∗∗Φ33CiTZ~iDi+hmCiTT~iDiCiTZ~iGi+hmCiTT~iGi∗∗∗Φ44DiTZ~iGi+hmDiTT~iGi∗∗∗∗GiTZ~iGi+hmGiTT~iGi-M~irm<0hold, where (51)Φ11=-αP~i+P~iAi+AiTP~i+Q~i+AiTZ~iAi+hmAiTT~iAi-FiTψ1Fi+rmM~i+H1i+H1iT+hmX11i,Φ12=P~iBi+AiTZ~iBi+hmAiTT~iBi-H1iT+H2i+hmX12i,Φ13=P~iCi+AiTZ~iCi+hmAiTT~iCi,Φ14=P~iDi+AiTZ~iDi+hmAiTT~iDi-FiTψ2,Φ22=-1-h^Q~i+BiTZ~iBi+hmBiTT~iBi-H2i-H2iT+hmX22i,Φ23=BiTZ~iCi+hmBiTT~iCi,Φ33=-1-τ^Z~i+CiTZ~iCi+hmCiTT~iCi,Φ44=-ψ3+DiTZ~iDi+hmDiTT~iDi; meanwhile, the average dwell-time satisfies(52)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ8k+λ9+λ10d-αTf,where(53)λminPi=λ1,λmaxFiTFi=λ8,λmaxψ2Tψ2=λ9,λmaxψ3=λ10,P~i=R1/2PiR1/2,Q~i=R1/2QiR1/2,Z~i=R1/2ZiR1/2,T~i=R1/2TiR1/2,M~i=R1/2MiR1/2.Then, system (17) is finite-time bounded with extended dissipative performance with respect to (0,c2,d,R,Tf,σ).Proof.
Choose the same Lyapunov-Krasovskii function as in Theorem13, similar to the proof of Theorem 13, and we obtain (54)V˙t-αVt-Jt≤XTtΦiXt-∫t-httϑTt,sΔiϑt,sds, where (55)Xt=xTtxTt-htx˙Tt-τtwTt∫t-rttxTsdsT,ϑt,s=xTtxTt-htx˙TsT;by virtue of (49)-(50) we can obtain that (56)V˙t-αVt-Jt<0 follows the proof line of (37); it is easy to obtain the following inequality: (57)Vt<eαtμNσ0,tV0+∫0teαt-sμNσs,tJsds; under zero initial condition V(0)=0, it can be calculated that (58)Vt<eαtμNσ0,t∫0tJsds, and it is equivalent to (59)VteαtμNσ0,t<∫0tJsds; by Assumption 4, we have (60)Vtb<∫0tJsds, so we obtain (61)∫0tJsds>Vtb>1bxTtP~ixt>0, considering inequality (62)∫0TfJtdt-sup0≤t≤Tf⁡zTtψ4zt≥0; when ψ4=0, one obtains (63)∫0TfJtdt≥0; when ψ4>0, by Assumption 3 we have ψ1=0, ψ2=0, and ψ3>0, and then we obtain (64)∫0tJsds=∫0twTsψ3wsds; thus, for ∀t∈[0,Tf], we have (65)∫0TfJsds>∫0tJsds≥1bxTtP~ixt>0; it follows from (47) that (66)∫0TfJsds≥1bxTtP~ixt≥xTtFiTψ4Fixt=zTtψ4zt, so we get (67)∫0TfJtdt-sup0≤t≤Tf⁡zTtψ4zt≥0. Thus the proof of extended dissipative is completed.
Next, we proof finite-time boundedness. Following the proof above, we can deduce that(68)Vt<eαtμNσ0,t∫0tJsds,Vt<eα+ln⁡μ/τaTf∫0TfJsds.When ψ1≤0, we can obtain (69)∫0TfJsds≤∫0Tf2zTsψ2ws+wTsψ3wsds,Vt<eα+ln⁡μ/τaTf∫0Tf2zTsψ2ws+wTsψ3wsds,so we get (70)xTtRxt≤Vtλ1<eα+ln⁡μ/τaTfλ1∫0Tf2zTsψ2ws+wTsψ3wsds, by (71)∫0Tf2zTsψ2ws+wTsψ3wsds=∫0Tf2xTsFiTψ2ws+wTsψ3wsds, and by Lemma 9, we have (72)2xTsFiTψ2ws≤xTsFiTFixs+wTsψ2Tψ2ws. According to (106), we can obtain (73)xTtRxt≤Vtλ1<eα+ln⁡μ/τaTfλ1∫0Tf2zTsψ2ws+wTsψ3wsds<eα+ln⁡μ/τaTfλ1∫0TfxTsFiTFixs+wTsψ2Tψ2ws+wTsψ3wsds<eα+ln⁡μ/τaTfλ1λ8k+λ9+λ10d.From (52), we can conclude that xT(t)Rx(t)≤c2. Thus the proof is completed.


### 3.3. Nonfragile Finite-Time Extended Dissipative Control

Consider system (1), under the controller μ(t)=Kσ(t)x(t), the corresponding closed-loop system is given by(74)x˙t-C^σtx˙t-τt=A^σt+EσtKσttxt+B^σtxt-ht+Dσtwt+G^σt∫t-rttxsds,zt=Fσtxt,xt0+θ=φθ,∀θ∈-τ,0.Firstly, for the additive gain variation model satisfying the form of Case1, that is, ΔKi(t)=J1iUi(t)J2i, we have the following theorem.Theorem 16.
For given positive scalarsα, h^, τ^, hm, rm, δ,ϵ, and b, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices Li, M1i, M2i, M3i, M4i, J1i, and J2i with appropriate dimensions, then(75)1bP~i-FiTψ4Fi>0,(76)X11iX12i∗X22i≥0,(77)X11iX12iH1i∗X22iH2i∗∗T~i≥0,(78)Θ11Θ12CiRiDi-RiFiTψ2GiΘ16Θ17RiM1iTRiJ2iTRiFiT∗Θ22000RiBiThmRiBiTRiM2iT00∗∗Θ3300RiCiThmRiCiTRiM3iT00∗∗∗-ψ30DiThmDiT000∗∗∗∗-M~irmGiThmGiTM4iT00∗∗∗∗∗Θ66Θ67000∗∗∗∗∗∗Θ77000∗∗∗∗∗∗∗-δI00∗∗∗∗∗∗∗∗-ϵI0∗∗∗∗∗∗∗∗∗ψ1-1<0hold, where (79)Θ11=-αRi+AiRi+RiAiT+EiYi+YiTEiT+Q^i+rmM^i+H^1i+H^1iT+hmX^11i+δLiLiT+ϵEiJ1iJ1iTEiT,Θ12=BiRi-H^1iT+H^2i+hmX^12i,Θ16=RiAiT+YiTEiT+δLiLiT+ϵEiJ1iJ1iTEiT,Θ17=hmRiAiT+hmYiTEiT+hmδLiLiT+ϵhmEiJ1iJ1iTEiT,Θ22=-1-h^Q^i-H^2i-H^2iT+hmX22i,Θ33=-1-τ^Z^i,Θ66=-Wi+δLiLiT+ϵEiJ1iJ1iTEiT,Θ67=hmδLiLiT+ϵhmEiJ1iJ1iTEiT,Θ77=-hmVi+hmδLiLiT+ϵhm2EiJ1iJ1iTEiT; the matrices are defined as follows: (80)KiP~i-1=Yi,P~i-1=Ri,Z~i-1=Wi,T~i-1=Vi,P~i-1Q~iP~i-1=Q^i,P~i-1Z~iP~i-1=Z^i,P~i-1M~iP~i-1=M^i,P~i-1H1iP~i-1=H^1i,P~i-1H2iP~i-1=H^2i,P~i-1X11iP~i-1=X^11i,P~i-1X12iP~i-1=X^12i,P~i-1X22iP~i-1=X^22i;meanwhile, the average dwell-time satisfies(81)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ8k+λ9+λ10d-αTf,where(82)λminPi=λ1,λmaxFiTFi=λ8,λmaxψ2Tψ2=λ9,λmaxψ3=λ10,P~i=R1/2PiR1/2,Q~i=R1/2QiR1/2,Z~i=R1/2ZiR1/2,T~i=R1/2TiR1/2,M~i=R1/2MiR1/2.Then the switched linear neutral system is finite-time bounded with extended dissipative performance. Furthermore, the nonfragile controller can be chosen by(83)ut=Kσttxt.Proof.
ReplacingAi, Bi, Ci, and Gi in (50) with A^i+EiKi+EiΔKi(t), B^i, C^i, and G^i and by Schur complement, we obtain(84)Λi=Λ11Λ12P~iC^iP~iDi-FiTψ2P~iG^iΛ16hmA^i+EiKi+EiΔKitTT~i∗Λ22000B^iTZ~ihmB^iTT~i∗∗Λ3300C^iTZ~ihmC^iTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmG^iTZ~ihmG^iTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, where (85)Λ11=-αP~i+P~iA^i+EiKi+EiΔKit+A^i+EiKi+EiΔKitTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Λ12=P~iB^i-H1iT+H2i+hmX12i,Λ16=A^i+EiKi+EiΔKitTZ~i,Λ22=-1-h^Q~i-H2i-H2iT+hmX22i,Λ33=-1-τ^Z~i;Λi can be rewritten as(86)Ωi+Γ1iUitΓ2i+Γ2iTUiTtΓ1iT, where(87)Ωi=Ω11Ω12P~iC^iP~iDi-FiTψ2P~iG^iΩ16hmA^i+EiKiTT~i∗Ω22000B^iTZ~ihmB^iTT~i∗∗Ω3300C^iTZ~ihmC^iTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmG^iTZ~ihmG^iTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, with (88)Ω11=-αP~i+P~iA^i+EiKi+A^i+EiKiTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Ω12=P~iB^i-H1iT+H2i+hmX12i,Ω16=A^i+EiKiTZ~i,Ω22=-1-h^Q~i-H2i-H2iT+hmX22i,Ω33=-1-τ^Z~i; by using Lemma 11, there exists a scalar ϵ>0, such that (89)Ωi+Γ1iUitΓ2i+Γ2iTUiTtΓ1iT<Ωi+ϵΓ1iΓ1iT+ϵ-1Γ2iTΓ2i, where (90)Γ1i=J1iTEiTP~i0000J1iTEiTZ~ihmJ1iTEiTT~iT,Γ2i=J2i000000.Here we consider the norm-bounded uncertainties, and we set Ωi=Ω1i+Ω2i, where(91)Ω1i=Ξ11Ξ12P~iCiP~iDi-FiTψ2P~iGiΞ16hmAi+EiKiTT~i∗Ξ22000BiTZ~ihmBiTT~i∗∗Ξ3300CiTZ~ihmCiTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmGiTZ~ihmGiTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, with (92)Ξ11=-αP~i+P~iAi+EiKi+Ai+EiKiTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Ξ12=P~iBi-H1iT+H2i+hmX12i,Ξ16=Ai+EiKiTZ~i,Ξ22=-1-h^Q~i-H2i-H2iT+hmX22i,Ξ33=-1-τ^Z~i,Ω2i=P~iLi0000Z~iLihmT~iLiΞitM1iM2iM3i0M4i00+M1iTM2iTM3iT0M4iT00ΞiTtLiTP~i0000LiTZ~ihmLiTT~i.By Lemma 11, there exists a scalar δ>0, such that(93)Ω2i≤δP~iLi0000Z~iLihmT~iLiLiTP~i0000LiTZ~ihmLiTT~i+δ-1M1iTM2iTM3iT0M4iT00M1iM2iM3i0M4i00.Then pre- and postmultiplying (78) by diag{P~i,P~i,P~i,I,I,Z~i,T~i,I,I,I}, we have(94)Π1i=Π11Π12P~iCiP~iDi-FiTψ2P~iGiΠ16Π17M1iTJ2iTFiT∗Π22000BiTZ~ihmBiTT~iM2iT00∗∗Π3300CiTZ~ihmCiTT~iM3iT00∗∗∗-ψ30DiTZ~ihmDiTT~i000∗∗∗∗-M~irmGiTZ~ihmGiTT~iM4iT00∗∗∗∗∗Π66Π67000∗∗∗∗∗∗Π77000∗∗∗∗∗∗∗-δI00∗∗∗∗∗∗∗∗-ϵI0∗∗∗∗∗∗∗∗∗ψ1-1<0,where (95)Π11=-αP~i+P~iAi+EiKi+Ai+EiKiTP~i+Q~i+rmM~i+H1i+H1iT+hmX11i+δP~iLiLiTP~i+ϵP~iEiJ1iJ1iTEiTP~i,Π12=P~iBi-H1iT+H2i+hmX12i,Π16=Ai+EiKiTZ~i+δP~iLiLiTZ~i+ϵP~iEiJ1iJ1iTEiTZ~i,Π17=hmAi+EiKiTT~i+hmδP~iLiLiTT~i+ϵhmP~iEiJ1iJ1iTEiTT~i,Π22=-1-h^Q~i-H2i-H2iT+hmX22i,Π33=-1-τ^Z~i,Π66=-Z~i+δZ~iTLiLiTZ~i+ϵZ~iTEiJ1iJ1iTEiTZ~i,Π67=hmδZ~iTLiLiTT~i+ϵhmZ~iTEiJ1iJ1iTEiTT~i,Π77=-hmT~i+hmδT~iTLiLiTT~i+ϵhm2T~iTEiJ1iJ1iTEiTT~i.Based on above discussion, from Π1i<0, by Schur complement, we can conclude that Λi<0. Similar to the proof of Theorem 15, we can obtain(96)V˙t-αVt-Jt≤XTtΛiXt-∫t-httϑTt,sΔiϑt,sds, where (97)Xt=xTtxTt-htx˙Tt-τtwTt∫t-rttxTsdsT,ϑt,s=xTtxTt-htx˙TsT;Δi is given in (77). The following proof is similar to that of Theorem 15; it is omitted here.Furthermore, for the multiplicative gain variation modelΔKi(t) with the form in (5) of Case 2, we have the following theorem.Theorem 17.
For given positive scalarsα, h^, τ^, hm, rm, δ, ϵ, b, and 0≤c≤1, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices Li, M1i, M2i, M3i, and M4i, then (98)1bP~i-FiTψ4Fi>0,(99)-cIWqiWqi-Qqi<0,(100)X11iX12i∗X22i≥0,(101)X11iX12iH1i∗X22iH2i∗∗T~i≥0,(102)Γ11Γ12CiRiΓ14GiΓ16Γ17YiTRiFiTRiM1iTΨ1iKi∗Γ22000RiBiThmRiBiT00RiM2iT0∗∗Γ3300RiCiThmRiCiT00RiM3iT0∗∗∗-ψ30DiThmDiT0000∗∗∗∗-M~irmGiThmGiT00M4iT0∗∗∗∗∗Γ66Γ67000Ψ2iKi∗∗∗∗∗∗Γ77000Ψ3iKi∗∗∗∗∗∗∗-ϵI000∗∗∗∗∗∗∗∗ψ1-100∗∗∗∗∗∗∗∗∗-δI0∗∗∗∗∗∗∗∗∗∗-∑q=1nσ¯2qiI<0,where (103)Γ11=-αRi+AiRi+RiAiT+EiYi+YiTEiT+Q^i+rmM^i+H^1i+H^1iT+hmX^11i+δLiLiT+ϵEi∅i2EiT+m∑q=1nσ¯2qiQ^qi,Γ12=BiRi-H^1iT+H^2i+hmX^12i,Γ14=Di-RiFiTψ2,Γ16=RiAiT+YiTEiT+δLiLiT+ϵEi∅i2EiT,Γ17=hmRiAiT+hmYiTEiT+hmδLiLiT+ϵhmEi∅i2EiT,Γ22=-1-h^Q^i-H^2i-H^2iT+hmX^22i,Γ33=-1-τ^Z^i,Γ66=-Wi+δLiLiT+ϵEi∅i2EiT,Γ67=ϵhmEi∅i2EiT+hmδLiLiT,Γ77=-hmVi+hmδLiLiT+ϵhm2Ei∅i2EiT,∅i=diagσ¯11i,σ¯12i,…,σ¯1mi,Ψ1i=EiV1iEiV2i⋯EiVmi,Ψ2i=EiV1iEiV2i⋯EiVmi,Ψ3i=hmEiV1ihmEiV2i⋯hmEiVmi,Vpi=diag0,0,…,0︸p-1,1,0,…,0;the matrices are defined as follows: (104)KiP~i-1=Yi,P~i-1=Ri,Z~i-1=Wi,T~i-1=Vi,P~i-1QqiP~i-1=Q^qi,P~i-1Q~iP~i-1=Q^i,P~i-1Z~iP~i-1=Z^i,P~i-1M~iP~i-1=M^i,P~i-1H1iP~i-1=H^1i,P~i-1H2iP~i-1=H^2i,P~i-1X11iP~i-1=X^11i,P~i-1X12iP~i-1=X^12i,P~i-1X22iP~i-1=X^22i;the average dwell-time satisfies(105)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ8k+λ9+λ10d-αTf,(106)λminPi=λ1,λmaxFiTFi=λ8,λmaxψ2Tψ2=λ9,λmaxψ3=λ10,(107)P~i=R1/2PiR1/2,Q~i=R1/2QiR1/2,Z~i=R1/2ZiR1/2,T~i=R1/2TiR1/2,M~i=R1/2MiR1/2.The controller gains can be given by Ki=YiP~i. Then the switched linear neutral system is finite-time bounded with extended dissipative performance under the nonfragile controller u(t)=Kσ(t)(t)x(t).Proof.
ReplacingAi, Bi, Ci, and Gi in (50) with A^i+EiKi+EiΔKi(t), B^i, C^i, and G^i and by Schur complement, we obtain(108)Γi=Λ11Λ12P~iC^iP~iDi-FiTψ2P~iG^iΛ16hmA^i+EiKi+EiΔKitTT~i∗Λ22000B^iTZ~ihmB^iTT~i∗∗Λ3300C^iTZ~ihmC^iTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmG^iTZ~ihmG^iTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i,where (109)Λ11=-αP~i+P~iA^i+EiKi+EiΔKit+A^i+EiKi+EiΔKitTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Λ12=P~iB^i-H1iT+H2i+hmX12i,Λ16=A^i+EiKi+EiΔKitTZ~i,Λ22=-1-h^Q~i-H2i-H2iT+hmX22i,Λ33=-1-τ^Z~i;(108) can be rewritten as(110)Ωi+P~iEiΔKit+EiΔKitTP~i0000EiΔKitTZ~ihmEiΔKitTT~i∗000000∗∗00000∗∗∗0000∗∗∗∗000∗∗∗∗∗00∗∗∗∗∗∗0,where(111)Ωi=Ω11Ω12P~iC^iP~iDi-FiTψ2P~iG^iΩ16hmA^i+EiKiTT~i∗Ω22000B^iTZ~ihmB^iTT~i∗∗Ω3300C^iTZ~ihmC^iTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmG^iTZ~ihmG^iTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, with (112)Ω11=-αP~i+P~iA^i+EiKi+A^i+EiKiTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Ω12=P~iB^i-H1iT+H2i+hmX12i,Ω16=A^i+EiKiTZ~i,Ω22=-1-h^Q~i-H2i-H2iT+hmX22i,Ω33=-1-τ^Z~i; considering (5), (110) can be rewritten as(113)Ωi+B~iH~iL~i+B~iH~iL~iT+∑p=1m∑q=1nσpqiK~pqiWqiI~i+K~pqiWqiI~iT, where (114)H~i=diagσ1i,σ2i,…,σmi,L~i=Ki000000,I~i=I000000,Vpi=diag0,0,…,0︸p-1,1,0,…,0,Wqi=diag0,0,…,0︸q-1,1,0,…,0,B~i=EiTP~i0000EiTZ~ihmEiTT~iT,K~pqi=KiTVpiTEiTP~i0000KiTVpiTEiTZ~ihmKiTVpiTEiTT~iT;based on (6) and Lemma 12, for some ϵ>0, Qqi>0 (q=1,…,n) and ∅i=diag{σ¯11i,σ¯12i,…,σ¯1mi}, it can be verified that (115)B~iH~iL~i+B~iH~iL~iT≤ϵB~i∅i2B~iT+ϵ-1L~iTL~i,K~pqiWqiI~i+K~pqiWqiI~iT≤K~pqiWqiQqi-1WqiTK~pqiT+I~iTQqiI~i,∑p=1m∑q=1nσpqiI~iTQqiI~i≤m∑q=1nσ¯2qiI~iTQqiI~i,∑p=1m∑q=1nσpqiK~pqiWqiQqi-1WqiTK~pqiT≤∑p=1m∑q=1nσ¯2qiK~pqiWqiQqi-1WqiTK~pqiT.On the other hand, by Schur complement, (116)-cIWqiWqi-Qqi<0is equal to WqiQqi-1Wqi<cI. Then WqiQqi-1Wqi<I holds. It can be proven that K~pqiWqiQqi-1WqiTK~pqiT≤K~pqiK~pqiT. Hence, we have (117)Ωi+B~iH~iL~i+B~iH~iL~iT+∑p=1m∑q=1nσpqiK~pqiWqiI~i+K~pqiWqiI~iT<Ωi+ϵB~i∅i2B~iT+ϵ-1L~iTL~i+m∑q=1nσ¯2qiI~iTQqiI~i+∑p=1m∑q=1nσ¯2qiK~pqiK~pqiT.Here we consider the norm-bounded uncertainties, and we set (118)Ωi=Ω1i+Ω2i, where(119)Ω1i=Ξ11Ξ12P~iCiP~iDi-FiTψ2P~iGiΞ16hmAi+EiKiTT~i∗Ξ22000BiTZ~ihmBiTT~i∗∗Ξ3300CiTZ~ihmCiTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmGiTZ~ihmGiTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, with (120)Ξ11=-αP~i+P~iAi+EiKi+Ai+EiKiTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Ξ12=P~iBi-H1iT+H2i+hmX12i,Ξ16=Ai+EiKiTZ~i,Ξ22=-1-h^Q~i-H2i-H2iT+hmX22i,Ξ33=-1-τ^Z~i,Ω2i=P~iLi0000Z~iTLihmT~iTLiΞitM1iM2iM3i0M4i00+M1iTM2iTM3iT0M4iT00ΞiTtLiTP~iT0000LiTZ~ihmLiTT~i.By Lemma 11, there exists a scalar δ>0, such that (121)Ω2i≤δP~iLi0000Z~iLihmT~iLiLiTP~i0000LiTZ~ihmLiTT~i+δ-1M1iTM2iTM3iT0M4iT00M1iM2iM3i0M4i00.Then pre- and postmultiplying (102) by diag{P~i,P~i,P~i,I,I,Z~i,T~i,I,I,I,I,…,I︸m}, we have (122)Π2i=Ψ11Ψ12P~iCiΨ14P~iGiΨ16Ψ17KiTFiTM1iTΔ1iKi∗Ψ22000BiTZ~ihmBiTT~i00M2iT0∗∗Ψ3300CiTZ~ihmCiTT~i00M3iT0∗∗∗-ψ30DiTZ~ihmDiTT~i0000∗∗∗∗-M~irmGiTZ~ihmGiTT~i00M4iT0∗∗∗∗∗Ψ66Ψ67000Δ2iKi∗∗∗∗∗∗Ψ77000Δ3iKi∗∗∗∗∗∗∗-ϵI000∗∗∗∗∗∗∗∗ψ1-100∗∗∗∗∗∗∗∗∗-δI0∗∗∗∗∗∗∗∗∗∗-∑q=1nσ¯2qiI<0,where (123)Ψ11=-αP~i+P~iAi+EiKi+Ai+EiKiTP~i+Q~i+rmM~i+H1i+H1iT+hmX11i+ϵP~iEi∅i2EiTP~i+m∑q=1nσ¯2qiQqi+δP~iLiLiTP~i,Ψ12=P~iBi-H1iT+H2i+hmX12i,Ψ14=P~iDi-FiTψ2,Ψ16=Ai+EiKiTZ~i+ϵP~iEi∅i2EiTZ~i+δP~iLiLiTZ~i,Ψ17=hmAi+EiKiTT~i+ϵhmP~iEi∅i2EiTT~i+hmδP~iLiLiTT~i,Ψ22=-1-h^Q~i-H2i-H2iT+hmX22i,Ψ33=-1-τ^Z~i,Ψ66=-Z~i+ϵZ~iTEi∅i2EiTZ~i+δZ~iTLiLiTZ~i,Ψ67=ϵhmZ~iTEi∅i2EiTT~i+hmδZ~iTLiLiTT~i,Ψ77=-hmT~i+ϵhm2T~iTEi∅i2EiTT~i+hmδT~iTLiLiTT~i,Δ1i=P~iEiV1iP~iEiV2i⋯P~iEiVmi,Δ2i=Z~iEiV1iZ~iEiV2i⋯Z~iEiVmi,Δ3i=hmT~iEiV1ihmT~iEiV2i⋯hmT~iEiVmi.Based on above discussion, from Π2i<0, by Schur complement, we can conclude that Γi<0. Similar to the proof of Theorem 15, we can obtain (124)V˙t-αVt-Jt≤XTtΓiXt-∫t-httϑTt,sΔiϑt,sds, where (125)Xt=xTtxTt-htx˙Tt-τtwTt∫t-rttxTsdsT,ϑt,s=xTtxTt-htx˙TsT;Δi is given in (101). The following proof is similar to that of Theorem 15; it is omitted here.Remark 18.
The concept of extended dissipative could be employed to lots of other systems, for example, the T-S fuzzy systems [34–37], which shows the effectiveness of the powerful tool.


## 3.1. Finite-Time Boundedness Analysis

Consider the following unforced switched neutral system without uncertainties:(17)x˙t-Cσtx˙t-τt=Aσtxt+Bσtxt-ht+Dσtwt+Gσt∫t-rttxsds,zt=Fσtxt,xt0+θ=φθ,∀θ∈-τ,0.In this section, the problem of finite-time boundedness analysis of the switched neutral system is proposed, by using the average dwell-time approach, sufficient conditions are derived by solving some linear matrix inequalities, and the results are shown as follows.Theorem 13.
For given positive scalarsα, h^, τ^, hm, and rm, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices Ni, X11i, X12i, X22i, H1i, and H2i with appropriate dimensions, then(18)Xi=X11iX12i∗X22i≥0,(19)Δi=X11iX12iH1i∗X22iH2i∗∗T~i≥0,(20)Θi=ϕ11ϕ12ϕ13ϕ14P~iGi+AiTZ~iGi+hmAiTT~iGi∗ϕ22ϕ23BiTZ~iDi+hmBiTT~iDiBiTZ~iGi+hmBiTT~iGi∗∗ϕ33CiTZ~iDi+hmCiTT~iDiCiTZ~iGi+hmCiTT~iGi∗∗∗ϕ44DiTZ~iGi+hmDiTT~iGi∗∗∗∗GiTZ~iGi+hmGiTT~iGi-M~irm<0,hold, where (21)ϕ11=-αP~i+P~iAi+AiTP~i+Q~i+AiTZ~iAi+hmAiTT~iAi+rmM~i+H1i+H1iT+hmX11i,ϕ12=P~iBi+AiTZ~iBi+hmAiTT~iBi-H1iT+H2i+hmX12i,ϕ13=P~iCi+AiTZ~iCi+hmAiTT~iCi,ϕ14=P~iDi+AiTZ~iDi+hmAiTT~iDi,ϕ22=-1-h^Q~i+BiTZ~iBi+hmBiTT~iBi-H2i-H2iT+hmX22i,ϕ23=BiTZ~iCi+hmBiTT~iCi,ϕ33=-1-τ^Z~i+CiTZ~iCi+hmCiTT~iCi,ϕ44=-Ni+DiTZ~iDi+hmDiTT~iDi,(22)λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d<c2λ1e-αTf;meanwhile, the average dwell-time satisfies(23)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d-αTf.We define (24)P~i=R1/2PiR1/2,Q~i=R1/2QiR1/2,Z~i=R1/2ZiR1/2,T~i=R1/2TiR1/2,M~i=R1/2MiR1/2,(25)λminPi=λ1,λmaxPi=λ2,λmaxQi=λ3,λmaxZi=λ4,λmaxTi=λ5,λmaxMi=λ6,λmaxNi=λ7,where μ>1 satisfies(26)P~i<μP~j,Q~i<μQ~j,Z~i<μZ~j,T~i<μT~j,M~i<μM~j,∀i,j∈M.Then, switched system (17) is finite-time bounded with respect to (c1,c2,d,R,Tf,σ).Proof.
Choose the piecewise Lyapunov-Krasovskii functional candidate as(27)Vt=Vσtt=Vit=V1it+V2it+V3it+V4it+V5it,where(28)V1it=xTtP~ixt,V2it=∫t-htteαt-sxTtQ~ixtds,V3it=∫t-τtteαt-sx˙TtZ~ix˙tds,V4it=∫-hm0∫t+ϵteαt-sx˙TtT~ix˙tdsdϵ,V5it=∫-rm0∫t+ϵteαt-sxTtM~ixtdsdϵ,in which α is a given scalar and P~i, Q~i, Z~i, T~i, and M~i are positive definite matrices to be determined. Taking the derivative of V(t) with respect to t along the trajectory of system (17) yields (29)V˙1it=2xTtP~ix˙t,V˙2it=αV2it+xTtQ~ixt-eαht1-h˙txTt-htQ~ixt-ht≤αV2it+xTtQ~ixt-1-h^xTt-htQ~ixt-ht,V˙3it=αV3it+x˙TtZ~ix˙t-eατt1-τ˙tx˙Tt-τtZ~ix˙t-τt≤αV3it+x˙TtZ~ix˙t-1-τ^x˙Tt-τtZ~ix˙t-τt,V˙4it=αV4it+hmx˙TtT~ix˙t-∫t-htteαt-sx˙TsT~ix˙sds≤αV4it+hmx˙TtT~ix˙t-∫t-httx˙TsT~ix˙sds,V˙5it=αV5it+rmxTtM~ixt-∫t-rtteαt-sxTsM~ixsds≤αV5it+rmxTtM~ixt-∫t-rttxTsM~ixsds.Using the Leibniz-Newton formula, we have(30)2xTtH1i+xTt-htH2ixt-∫t-httx˙sds-xt-ht=0. Let (31)χt=xTtxTt-htT, and it obviously holds that(32)hmχTtXiχt-∫t-httχTtXiχtds≥0. By Lemma 10, it is easy to obtain(33)-∫t-rttxTsM~ixsds≤-1rm∫t-rttxTsdsM~i∫t-rttxsds.Thus we have (34)V˙t-αVt-wTtNiwt≤XTtΘiXt-∫t-httϑTt,sΔiϑt,sds, where(35)Xt=xTtxTt-htx˙Tt-τtwTt∫t-rttxTsdsT,ϑt,s=xTtxTt-htx˙TsT.Considering (19) and (20), we can obtain that(36)V˙t-αVt-wTtNiwt<0.Integrating (36), it can be obtained from (26) and (36) that, ∀t∈[tk,tk+1),(37)Vt<eαt-tkVtk+∫tkteαt-swTsNiwsds<eαt-tkμVtk-+∫tkteαt-swTsNiwsds<eαt-tkμeαtk-tk-1Vtk-1+∫tk-1tkeαtk-swTsNiwsds+∫tkteαt-swTsNiwsds=eαt-tk-1μVtk-1+μ∫tk-1tkeαt-swTsNiwsds+∫tkteαt-swTsNiwsds<⋯<eαt-0μNσ0,tV0+μNσ0,t∫0t1eαt-swTsNiwsds+μNσt1,t∫t1t2eαt-swTsNiwsds+⋯+μ∫tk-1tkeαt-swTsNiwsds+∫tkteαt-swTsNiwsds=eαt-0μNσ0,tV0+∫0teαt-sμNσs,twTsNiwsds<eαtμNσ0,tV0+μNσ0,teαt∫0twTsNiwsds<eαTfμNσ0,TfV0+∫0TfwTsNiwsds<eαTfμNσ0,TfV0+λmaxNid.From Definition 8, we can deduce that Nσ(0,Tf)<Tf/τa, and then we can obtain(38)Vt<eα+ln⁡μ/τaTfV0+λ7d.On the other hand(39)Vt>xTtP~ixt=xTtR1/2PiR1/2xt≥λminPixTtRxt=λ1xTtRxt,V0≤λmaxPixT0Rx0+hmeαhmλmaxQisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ+τmeατmλmaxZisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ+hmeαhmλmaxTisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ+rmeαrmλmaxMisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ≤λmaxPi+hmeαhmλmaxQi+τmeατmλmaxZi+hmeαhmλmaxTi+rmeαrmλmaxMisup-τ≤θ≤0⁡xTθRxθ,x˙TθRx˙θ≤λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1.From (38)-(39), we can obtain(40)xTtRxt≤Vtλ1<λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7dλ1eαTfμTf/τa.When μ=1, it is obvious that xT(t)Rx(t)<c2 by (22).
Whenμ>1, by virtue of (22), we have that (41)ln⁡λ1c2-ln⁡λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d-αTf>0. From (23), we can obtain(42)Tfτa<ln⁡λ1c2-ln⁡λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d-αTfln⁡μ=ln⁡λ1c2e-αTf/λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7dln⁡μ.Substituting (42) into (40) yields (43)xTtRxt<λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7dλ1eαTfλ1c2e-αTfλ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1+λ7d=c2.The proof is completed.Based on Theorem13, when we set w(t)=0, the following corollary is proposed to solve the finite-time stable problem.Corollary 14.
Consider system (17) with w(t)=0. For given positive scalars α, h^, τ^, hm, and rm, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices X11i, X12i, X22i, H1i, and H2i with appropriate dimensions, then(44)X11iX12i∗X22i≥0,X11iX12iH1i∗X22iH2i∗∗T~i≥0,Δ11Δ12Δ13P~iGi+AiTZ~iGi+hmAiTT~iGi∗Δ22Δ23BiTZ~iGi+hmBiTT~iGi∗∗Δ33CiTZ~iGi+hmCiTT~iGi∗∗∗GiTZ~iGi+hmGiTT~iGi-M~irm<0hold, where (45)Δ11=-αP~i+P~iAi+AiTP~i+Q~i+AiTZ~iAi+hmAiTT~iAi+rmM~i+H1i+H1iT+hmX11i,Δ12=P~iBi+AiTZ~iBi+hmAiTT~iBi-H1iT+H2i+hmX12i,Δ13=P~iCi+AiTZ~iCi+hmAiTT~iCi,Δ22=-1-h^Q~i+BiTZ~iBi+hmBiTT~iBi-H2i-H2iT+hmX22i,Δ23=BiTZ~iCi+hmBiTT~iCi,Δ33=-1-τ^Z~i+CiTZ~iCi+hmCiTT~iCi,λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1<c2λ1e-αTf;meanwhile, the average dwell-time satisfies(46)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ2+hmeαhmλ3+τmeατmλ4+hmeαhmλ5+rmeαrmλ6c1-αTf,where μ>1 satisfying (26) and P~i, Q~i, Z~i, T~i, M~i, λ1, λ2, λ3, λ4, λ5, and λ6 are defined just the same as (25). Then the switched system (17) is finite-time stable with respect to (c1,c2,d,R,Tf,σ).Proof.
The proof is similar to that of Theorem13, and it is omitted here.


## 3.2. Finite-Time Extended Dissipative Analysis

In this section, the finite-time extended dissipative analysis is considered in the following theorem.Theorem 15.
For given positive scalarsα, h^, τ^, hm, rm, and b, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices X11i, X12i, X22i, H1i, and H2i with appropriate dimensions, then(47)1bP~i-FiTψ4Fi>0,(48)Xi=X11iX12i∗X22i≥0,(49)Δi=X11iX12iH1i∗X22iH2i∗∗T~i≥0,(50)Φi=Φ11Φ12Φ13Φ14P~iGi+AiTZ~iGi+hmAiTT~iGi∗Φ22Φ23BiTZ~iDi+hmBiTT~iDiBiTZ~iGi+hmBiTT~iGi∗∗Φ33CiTZ~iDi+hmCiTT~iDiCiTZ~iGi+hmCiTT~iGi∗∗∗Φ44DiTZ~iGi+hmDiTT~iGi∗∗∗∗GiTZ~iGi+hmGiTT~iGi-M~irm<0hold, where (51)Φ11=-αP~i+P~iAi+AiTP~i+Q~i+AiTZ~iAi+hmAiTT~iAi-FiTψ1Fi+rmM~i+H1i+H1iT+hmX11i,Φ12=P~iBi+AiTZ~iBi+hmAiTT~iBi-H1iT+H2i+hmX12i,Φ13=P~iCi+AiTZ~iCi+hmAiTT~iCi,Φ14=P~iDi+AiTZ~iDi+hmAiTT~iDi-FiTψ2,Φ22=-1-h^Q~i+BiTZ~iBi+hmBiTT~iBi-H2i-H2iT+hmX22i,Φ23=BiTZ~iCi+hmBiTT~iCi,Φ33=-1-τ^Z~i+CiTZ~iCi+hmCiTT~iCi,Φ44=-ψ3+DiTZ~iDi+hmDiTT~iDi; meanwhile, the average dwell-time satisfies(52)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ8k+λ9+λ10d-αTf,where(53)λminPi=λ1,λmaxFiTFi=λ8,λmaxψ2Tψ2=λ9,λmaxψ3=λ10,P~i=R1/2PiR1/2,Q~i=R1/2QiR1/2,Z~i=R1/2ZiR1/2,T~i=R1/2TiR1/2,M~i=R1/2MiR1/2.Then, system (17) is finite-time bounded with extended dissipative performance with respect to (0,c2,d,R,Tf,σ).Proof.
Choose the same Lyapunov-Krasovskii function as in Theorem13, similar to the proof of Theorem 13, and we obtain (54)V˙t-αVt-Jt≤XTtΦiXt-∫t-httϑTt,sΔiϑt,sds, where (55)Xt=xTtxTt-htx˙Tt-τtwTt∫t-rttxTsdsT,ϑt,s=xTtxTt-htx˙TsT;by virtue of (49)-(50) we can obtain that (56)V˙t-αVt-Jt<0 follows the proof line of (37); it is easy to obtain the following inequality: (57)Vt<eαtμNσ0,tV0+∫0teαt-sμNσs,tJsds; under zero initial condition V(0)=0, it can be calculated that (58)Vt<eαtμNσ0,t∫0tJsds, and it is equivalent to (59)VteαtμNσ0,t<∫0tJsds; by Assumption 4, we have (60)Vtb<∫0tJsds, so we obtain (61)∫0tJsds>Vtb>1bxTtP~ixt>0, considering inequality (62)∫0TfJtdt-sup0≤t≤Tf⁡zTtψ4zt≥0; when ψ4=0, one obtains (63)∫0TfJtdt≥0; when ψ4>0, by Assumption 3 we have ψ1=0, ψ2=0, and ψ3>0, and then we obtain (64)∫0tJsds=∫0twTsψ3wsds; thus, for ∀t∈[0,Tf], we have (65)∫0TfJsds>∫0tJsds≥1bxTtP~ixt>0; it follows from (47) that (66)∫0TfJsds≥1bxTtP~ixt≥xTtFiTψ4Fixt=zTtψ4zt, so we get (67)∫0TfJtdt-sup0≤t≤Tf⁡zTtψ4zt≥0. Thus the proof of extended dissipative is completed.
Next, we proof finite-time boundedness. Following the proof above, we can deduce that(68)Vt<eαtμNσ0,t∫0tJsds,Vt<eα+ln⁡μ/τaTf∫0TfJsds.When ψ1≤0, we can obtain (69)∫0TfJsds≤∫0Tf2zTsψ2ws+wTsψ3wsds,Vt<eα+ln⁡μ/τaTf∫0Tf2zTsψ2ws+wTsψ3wsds,so we get (70)xTtRxt≤Vtλ1<eα+ln⁡μ/τaTfλ1∫0Tf2zTsψ2ws+wTsψ3wsds, by (71)∫0Tf2zTsψ2ws+wTsψ3wsds=∫0Tf2xTsFiTψ2ws+wTsψ3wsds, and by Lemma 9, we have (72)2xTsFiTψ2ws≤xTsFiTFixs+wTsψ2Tψ2ws. According to (106), we can obtain (73)xTtRxt≤Vtλ1<eα+ln⁡μ/τaTfλ1∫0Tf2zTsψ2ws+wTsψ3wsds<eα+ln⁡μ/τaTfλ1∫0TfxTsFiTFixs+wTsψ2Tψ2ws+wTsψ3wsds<eα+ln⁡μ/τaTfλ1λ8k+λ9+λ10d.From (52), we can conclude that xT(t)Rx(t)≤c2. Thus the proof is completed.


## 3.3. Nonfragile Finite-Time Extended Dissipative Control

Consider system (1), under the controller μ(t)=Kσ(t)x(t), the corresponding closed-loop system is given by(74)x˙t-C^σtx˙t-τt=A^σt+EσtKσttxt+B^σtxt-ht+Dσtwt+G^σt∫t-rttxsds,zt=Fσtxt,xt0+θ=φθ,∀θ∈-τ,0.Firstly, for the additive gain variation model satisfying the form of Case1, that is, ΔKi(t)=J1iUi(t)J2i, we have the following theorem.Theorem 16.
For given positive scalarsα, h^, τ^, hm, rm, δ,ϵ, and b, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices Li, M1i, M2i, M3i, M4i, J1i, and J2i with appropriate dimensions, then(75)1bP~i-FiTψ4Fi>0,(76)X11iX12i∗X22i≥0,(77)X11iX12iH1i∗X22iH2i∗∗T~i≥0,(78)Θ11Θ12CiRiDi-RiFiTψ2GiΘ16Θ17RiM1iTRiJ2iTRiFiT∗Θ22000RiBiThmRiBiTRiM2iT00∗∗Θ3300RiCiThmRiCiTRiM3iT00∗∗∗-ψ30DiThmDiT000∗∗∗∗-M~irmGiThmGiTM4iT00∗∗∗∗∗Θ66Θ67000∗∗∗∗∗∗Θ77000∗∗∗∗∗∗∗-δI00∗∗∗∗∗∗∗∗-ϵI0∗∗∗∗∗∗∗∗∗ψ1-1<0hold, where (79)Θ11=-αRi+AiRi+RiAiT+EiYi+YiTEiT+Q^i+rmM^i+H^1i+H^1iT+hmX^11i+δLiLiT+ϵEiJ1iJ1iTEiT,Θ12=BiRi-H^1iT+H^2i+hmX^12i,Θ16=RiAiT+YiTEiT+δLiLiT+ϵEiJ1iJ1iTEiT,Θ17=hmRiAiT+hmYiTEiT+hmδLiLiT+ϵhmEiJ1iJ1iTEiT,Θ22=-1-h^Q^i-H^2i-H^2iT+hmX22i,Θ33=-1-τ^Z^i,Θ66=-Wi+δLiLiT+ϵEiJ1iJ1iTEiT,Θ67=hmδLiLiT+ϵhmEiJ1iJ1iTEiT,Θ77=-hmVi+hmδLiLiT+ϵhm2EiJ1iJ1iTEiT; the matrices are defined as follows: (80)KiP~i-1=Yi,P~i-1=Ri,Z~i-1=Wi,T~i-1=Vi,P~i-1Q~iP~i-1=Q^i,P~i-1Z~iP~i-1=Z^i,P~i-1M~iP~i-1=M^i,P~i-1H1iP~i-1=H^1i,P~i-1H2iP~i-1=H^2i,P~i-1X11iP~i-1=X^11i,P~i-1X12iP~i-1=X^12i,P~i-1X22iP~i-1=X^22i;meanwhile, the average dwell-time satisfies(81)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ8k+λ9+λ10d-αTf,where(82)λminPi=λ1,λmaxFiTFi=λ8,λmaxψ2Tψ2=λ9,λmaxψ3=λ10,P~i=R1/2PiR1/2,Q~i=R1/2QiR1/2,Z~i=R1/2ZiR1/2,T~i=R1/2TiR1/2,M~i=R1/2MiR1/2.Then the switched linear neutral system is finite-time bounded with extended dissipative performance. Furthermore, the nonfragile controller can be chosen by(83)ut=Kσttxt.Proof.
ReplacingAi, Bi, Ci, and Gi in (50) with A^i+EiKi+EiΔKi(t), B^i, C^i, and G^i and by Schur complement, we obtain(84)Λi=Λ11Λ12P~iC^iP~iDi-FiTψ2P~iG^iΛ16hmA^i+EiKi+EiΔKitTT~i∗Λ22000B^iTZ~ihmB^iTT~i∗∗Λ3300C^iTZ~ihmC^iTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmG^iTZ~ihmG^iTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, where (85)Λ11=-αP~i+P~iA^i+EiKi+EiΔKit+A^i+EiKi+EiΔKitTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Λ12=P~iB^i-H1iT+H2i+hmX12i,Λ16=A^i+EiKi+EiΔKitTZ~i,Λ22=-1-h^Q~i-H2i-H2iT+hmX22i,Λ33=-1-τ^Z~i;Λi can be rewritten as(86)Ωi+Γ1iUitΓ2i+Γ2iTUiTtΓ1iT, where(87)Ωi=Ω11Ω12P~iC^iP~iDi-FiTψ2P~iG^iΩ16hmA^i+EiKiTT~i∗Ω22000B^iTZ~ihmB^iTT~i∗∗Ω3300C^iTZ~ihmC^iTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmG^iTZ~ihmG^iTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, with (88)Ω11=-αP~i+P~iA^i+EiKi+A^i+EiKiTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Ω12=P~iB^i-H1iT+H2i+hmX12i,Ω16=A^i+EiKiTZ~i,Ω22=-1-h^Q~i-H2i-H2iT+hmX22i,Ω33=-1-τ^Z~i; by using Lemma 11, there exists a scalar ϵ>0, such that (89)Ωi+Γ1iUitΓ2i+Γ2iTUiTtΓ1iT<Ωi+ϵΓ1iΓ1iT+ϵ-1Γ2iTΓ2i, where (90)Γ1i=J1iTEiTP~i0000J1iTEiTZ~ihmJ1iTEiTT~iT,Γ2i=J2i000000.Here we consider the norm-bounded uncertainties, and we set Ωi=Ω1i+Ω2i, where(91)Ω1i=Ξ11Ξ12P~iCiP~iDi-FiTψ2P~iGiΞ16hmAi+EiKiTT~i∗Ξ22000BiTZ~ihmBiTT~i∗∗Ξ3300CiTZ~ihmCiTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmGiTZ~ihmGiTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, with (92)Ξ11=-αP~i+P~iAi+EiKi+Ai+EiKiTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Ξ12=P~iBi-H1iT+H2i+hmX12i,Ξ16=Ai+EiKiTZ~i,Ξ22=-1-h^Q~i-H2i-H2iT+hmX22i,Ξ33=-1-τ^Z~i,Ω2i=P~iLi0000Z~iLihmT~iLiΞitM1iM2iM3i0M4i00+M1iTM2iTM3iT0M4iT00ΞiTtLiTP~i0000LiTZ~ihmLiTT~i.By Lemma 11, there exists a scalar δ>0, such that(93)Ω2i≤δP~iLi0000Z~iLihmT~iLiLiTP~i0000LiTZ~ihmLiTT~i+δ-1M1iTM2iTM3iT0M4iT00M1iM2iM3i0M4i00.Then pre- and postmultiplying (78) by diag{P~i,P~i,P~i,I,I,Z~i,T~i,I,I,I}, we have(94)Π1i=Π11Π12P~iCiP~iDi-FiTψ2P~iGiΠ16Π17M1iTJ2iTFiT∗Π22000BiTZ~ihmBiTT~iM2iT00∗∗Π3300CiTZ~ihmCiTT~iM3iT00∗∗∗-ψ30DiTZ~ihmDiTT~i000∗∗∗∗-M~irmGiTZ~ihmGiTT~iM4iT00∗∗∗∗∗Π66Π67000∗∗∗∗∗∗Π77000∗∗∗∗∗∗∗-δI00∗∗∗∗∗∗∗∗-ϵI0∗∗∗∗∗∗∗∗∗ψ1-1<0,where (95)Π11=-αP~i+P~iAi+EiKi+Ai+EiKiTP~i+Q~i+rmM~i+H1i+H1iT+hmX11i+δP~iLiLiTP~i+ϵP~iEiJ1iJ1iTEiTP~i,Π12=P~iBi-H1iT+H2i+hmX12i,Π16=Ai+EiKiTZ~i+δP~iLiLiTZ~i+ϵP~iEiJ1iJ1iTEiTZ~i,Π17=hmAi+EiKiTT~i+hmδP~iLiLiTT~i+ϵhmP~iEiJ1iJ1iTEiTT~i,Π22=-1-h^Q~i-H2i-H2iT+hmX22i,Π33=-1-τ^Z~i,Π66=-Z~i+δZ~iTLiLiTZ~i+ϵZ~iTEiJ1iJ1iTEiTZ~i,Π67=hmδZ~iTLiLiTT~i+ϵhmZ~iTEiJ1iJ1iTEiTT~i,Π77=-hmT~i+hmδT~iTLiLiTT~i+ϵhm2T~iTEiJ1iJ1iTEiTT~i.Based on above discussion, from Π1i<0, by Schur complement, we can conclude that Λi<0. Similar to the proof of Theorem 15, we can obtain(96)V˙t-αVt-Jt≤XTtΛiXt-∫t-httϑTt,sΔiϑt,sds, where (97)Xt=xTtxTt-htx˙Tt-τtwTt∫t-rttxTsdsT,ϑt,s=xTtxTt-htx˙TsT;Δi is given in (77). The following proof is similar to that of Theorem 15; it is omitted here.Furthermore, for the multiplicative gain variation modelΔKi(t) with the form in (5) of Case 2, we have the following theorem.Theorem 17.
For given positive scalarsα, h^, τ^, hm, rm, δ, ϵ, b, and 0≤c≤1, if there exist positive definite symmetric matrices P~i, Q~i, Z~i, T~i, and M~i and matrices Li, M1i, M2i, M3i, and M4i, then (98)1bP~i-FiTψ4Fi>0,(99)-cIWqiWqi-Qqi<0,(100)X11iX12i∗X22i≥0,(101)X11iX12iH1i∗X22iH2i∗∗T~i≥0,(102)Γ11Γ12CiRiΓ14GiΓ16Γ17YiTRiFiTRiM1iTΨ1iKi∗Γ22000RiBiThmRiBiT00RiM2iT0∗∗Γ3300RiCiThmRiCiT00RiM3iT0∗∗∗-ψ30DiThmDiT0000∗∗∗∗-M~irmGiThmGiT00M4iT0∗∗∗∗∗Γ66Γ67000Ψ2iKi∗∗∗∗∗∗Γ77000Ψ3iKi∗∗∗∗∗∗∗-ϵI000∗∗∗∗∗∗∗∗ψ1-100∗∗∗∗∗∗∗∗∗-δI0∗∗∗∗∗∗∗∗∗∗-∑q=1nσ¯2qiI<0,where (103)Γ11=-αRi+AiRi+RiAiT+EiYi+YiTEiT+Q^i+rmM^i+H^1i+H^1iT+hmX^11i+δLiLiT+ϵEi∅i2EiT+m∑q=1nσ¯2qiQ^qi,Γ12=BiRi-H^1iT+H^2i+hmX^12i,Γ14=Di-RiFiTψ2,Γ16=RiAiT+YiTEiT+δLiLiT+ϵEi∅i2EiT,Γ17=hmRiAiT+hmYiTEiT+hmδLiLiT+ϵhmEi∅i2EiT,Γ22=-1-h^Q^i-H^2i-H^2iT+hmX^22i,Γ33=-1-τ^Z^i,Γ66=-Wi+δLiLiT+ϵEi∅i2EiT,Γ67=ϵhmEi∅i2EiT+hmδLiLiT,Γ77=-hmVi+hmδLiLiT+ϵhm2Ei∅i2EiT,∅i=diagσ¯11i,σ¯12i,…,σ¯1mi,Ψ1i=EiV1iEiV2i⋯EiVmi,Ψ2i=EiV1iEiV2i⋯EiVmi,Ψ3i=hmEiV1ihmEiV2i⋯hmEiVmi,Vpi=diag0,0,…,0︸p-1,1,0,…,0;the matrices are defined as follows: (104)KiP~i-1=Yi,P~i-1=Ri,Z~i-1=Wi,T~i-1=Vi,P~i-1QqiP~i-1=Q^qi,P~i-1Q~iP~i-1=Q^i,P~i-1Z~iP~i-1=Z^i,P~i-1M~iP~i-1=M^i,P~i-1H1iP~i-1=H^1i,P~i-1H2iP~i-1=H^2i,P~i-1X11iP~i-1=X^11i,P~i-1X12iP~i-1=X^12i,P~i-1X22iP~i-1=X^22i;the average dwell-time satisfies(105)τa>τa∗=Tfln⁡μln⁡λ1c2-ln⁡λ8k+λ9+λ10d-αTf,(106)λminPi=λ1,λmaxFiTFi=λ8,λmaxψ2Tψ2=λ9,λmaxψ3=λ10,(107)P~i=R1/2PiR1/2,Q~i=R1/2QiR1/2,Z~i=R1/2ZiR1/2,T~i=R1/2TiR1/2,M~i=R1/2MiR1/2.The controller gains can be given by Ki=YiP~i. Then the switched linear neutral system is finite-time bounded with extended dissipative performance under the nonfragile controller u(t)=Kσ(t)(t)x(t).Proof.
ReplacingAi, Bi, Ci, and Gi in (50) with A^i+EiKi+EiΔKi(t), B^i, C^i, and G^i and by Schur complement, we obtain(108)Γi=Λ11Λ12P~iC^iP~iDi-FiTψ2P~iG^iΛ16hmA^i+EiKi+EiΔKitTT~i∗Λ22000B^iTZ~ihmB^iTT~i∗∗Λ3300C^iTZ~ihmC^iTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmG^iTZ~ihmG^iTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i,where (109)Λ11=-αP~i+P~iA^i+EiKi+EiΔKit+A^i+EiKi+EiΔKitTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Λ12=P~iB^i-H1iT+H2i+hmX12i,Λ16=A^i+EiKi+EiΔKitTZ~i,Λ22=-1-h^Q~i-H2i-H2iT+hmX22i,Λ33=-1-τ^Z~i;(108) can be rewritten as(110)Ωi+P~iEiΔKit+EiΔKitTP~i0000EiΔKitTZ~ihmEiΔKitTT~i∗000000∗∗00000∗∗∗0000∗∗∗∗000∗∗∗∗∗00∗∗∗∗∗∗0,where(111)Ωi=Ω11Ω12P~iC^iP~iDi-FiTψ2P~iG^iΩ16hmA^i+EiKiTT~i∗Ω22000B^iTZ~ihmB^iTT~i∗∗Ω3300C^iTZ~ihmC^iTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmG^iTZ~ihmG^iTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, with (112)Ω11=-αP~i+P~iA^i+EiKi+A^i+EiKiTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Ω12=P~iB^i-H1iT+H2i+hmX12i,Ω16=A^i+EiKiTZ~i,Ω22=-1-h^Q~i-H2i-H2iT+hmX22i,Ω33=-1-τ^Z~i; considering (5), (110) can be rewritten as(113)Ωi+B~iH~iL~i+B~iH~iL~iT+∑p=1m∑q=1nσpqiK~pqiWqiI~i+K~pqiWqiI~iT, where (114)H~i=diagσ1i,σ2i,…,σmi,L~i=Ki000000,I~i=I000000,Vpi=diag0,0,…,0︸p-1,1,0,…,0,Wqi=diag0,0,…,0︸q-1,1,0,…,0,B~i=EiTP~i0000EiTZ~ihmEiTT~iT,K~pqi=KiTVpiTEiTP~i0000KiTVpiTEiTZ~ihmKiTVpiTEiTT~iT;based on (6) and Lemma 12, for some ϵ>0, Qqi>0 (q=1,…,n) and ∅i=diag{σ¯11i,σ¯12i,…,σ¯1mi}, it can be verified that (115)B~iH~iL~i+B~iH~iL~iT≤ϵB~i∅i2B~iT+ϵ-1L~iTL~i,K~pqiWqiI~i+K~pqiWqiI~iT≤K~pqiWqiQqi-1WqiTK~pqiT+I~iTQqiI~i,∑p=1m∑q=1nσpqiI~iTQqiI~i≤m∑q=1nσ¯2qiI~iTQqiI~i,∑p=1m∑q=1nσpqiK~pqiWqiQqi-1WqiTK~pqiT≤∑p=1m∑q=1nσ¯2qiK~pqiWqiQqi-1WqiTK~pqiT.On the other hand, by Schur complement, (116)-cIWqiWqi-Qqi<0is equal to WqiQqi-1Wqi<cI. Then WqiQqi-1Wqi<I holds. It can be proven that K~pqiWqiQqi-1WqiTK~pqiT≤K~pqiK~pqiT. Hence, we have (117)Ωi+B~iH~iL~i+B~iH~iL~iT+∑p=1m∑q=1nσpqiK~pqiWqiI~i+K~pqiWqiI~iT<Ωi+ϵB~i∅i2B~iT+ϵ-1L~iTL~i+m∑q=1nσ¯2qiI~iTQqiI~i+∑p=1m∑q=1nσ¯2qiK~pqiK~pqiT.Here we consider the norm-bounded uncertainties, and we set (118)Ωi=Ω1i+Ω2i, where(119)Ω1i=Ξ11Ξ12P~iCiP~iDi-FiTψ2P~iGiΞ16hmAi+EiKiTT~i∗Ξ22000BiTZ~ihmBiTT~i∗∗Ξ3300CiTZ~ihmCiTT~i∗∗∗-ψ30DiTZ~ihmDiTT~i∗∗∗∗-M~irmGiTZ~ihmGiTT~i∗∗∗∗∗-Z~i0∗∗∗∗∗∗-hmT~i, with (120)Ξ11=-αP~i+P~iAi+EiKi+Ai+EiKiTP~i+Q~i+rmM~i-FiTψ1Fi+H1i+H1iT+hmX11i,Ξ12=P~iBi-H1iT+H2i+hmX12i,Ξ16=Ai+EiKiTZ~i,Ξ22=-1-h^Q~i-H2i-H2iT+hmX22i,Ξ33=-1-τ^Z~i,Ω2i=P~iLi0000Z~iTLihmT~iTLiΞitM1iM2iM3i0M4i00+M1iTM2iTM3iT0M4iT00ΞiTtLiTP~iT0000LiTZ~ihmLiTT~i.By Lemma 11, there exists a scalar δ>0, such that (121)Ω2i≤δP~iLi0000Z~iLihmT~iLiLiTP~i0000LiTZ~ihmLiTT~i+δ-1M1iTM2iTM3iT0M4iT00M1iM2iM3i0M4i00.Then pre- and postmultiplying (102) by diag{P~i,P~i,P~i,I,I,Z~i,T~i,I,I,I,I,…,I︸m}, we have (122)Π2i=Ψ11Ψ12P~iCiΨ14P~iGiΨ16Ψ17KiTFiTM1iTΔ1iKi∗Ψ22000BiTZ~ihmBiTT~i00M2iT0∗∗Ψ3300CiTZ~ihmCiTT~i00M3iT0∗∗∗-ψ30DiTZ~ihmDiTT~i0000∗∗∗∗-M~irmGiTZ~ihmGiTT~i00M4iT0∗∗∗∗∗Ψ66Ψ67000Δ2iKi∗∗∗∗∗∗Ψ77000Δ3iKi∗∗∗∗∗∗∗-ϵI000∗∗∗∗∗∗∗∗ψ1-100∗∗∗∗∗∗∗∗∗-δI0∗∗∗∗∗∗∗∗∗∗-∑q=1nσ¯2qiI<0,where (123)Ψ11=-αP~i+P~iAi+EiKi+Ai+EiKiTP~i+Q~i+rmM~i+H1i+H1iT+hmX11i+ϵP~iEi∅i2EiTP~i+m∑q=1nσ¯2qiQqi+δP~iLiLiTP~i,Ψ12=P~iBi-H1iT+H2i+hmX12i,Ψ14=P~iDi-FiTψ2,Ψ16=Ai+EiKiTZ~i+ϵP~iEi∅i2EiTZ~i+δP~iLiLiTZ~i,Ψ17=hmAi+EiKiTT~i+ϵhmP~iEi∅i2EiTT~i+hmδP~iLiLiTT~i,Ψ22=-1-h^Q~i-H2i-H2iT+hmX22i,Ψ33=-1-τ^Z~i,Ψ66=-Z~i+ϵZ~iTEi∅i2EiTZ~i+δZ~iTLiLiTZ~i,Ψ67=ϵhmZ~iTEi∅i2EiTT~i+hmδZ~iTLiLiTT~i,Ψ77=-hmT~i+ϵhm2T~iTEi∅i2EiTT~i+hmδT~iTLiLiTT~i,Δ1i=P~iEiV1iP~iEiV2i⋯P~iEiVmi,Δ2i=Z~iEiV1iZ~iEiV2i⋯Z~iEiVmi,Δ3i=hmT~iEiV1ihmT~iEiV2i⋯hmT~iEiVmi.Based on above discussion, from Π2i<0, by Schur complement, we can conclude that Γi<0. Similar to the proof of Theorem 15, we can obtain (124)V˙t-αVt-Jt≤XTtΓiXt-∫t-httϑTt,sΔiϑt,sds, where (125)Xt=xTtxTt-htx˙Tt-τtwTt∫t-rttxTsdsT,ϑt,s=xTtxTt-htx˙TsT;Δi is given in (101). The following proof is similar to that of Theorem 15; it is omitted here.Remark 18.
The concept of extended dissipative could be employed to lots of other systems, for example, the T-S fuzzy systems [34–37], which shows the effectiveness of the powerful tool.


## 4. Numerical Example

In this section, we present an example to illustrate the effectiveness of the controller design method.Example 1.
Consider system (1) with two subsystems with parameters as follows: (126)A1=2113,B1=1011,C1=300-1,D1=2122,E1=3012,G1=3315,F1=1203,L1=1001,M11=M21=M31=M41=0.2000.2,A2=3202,B2=1202,C2=-2102,D2=1103,E2=2402,G2=2024,F2=4022,L2=1001,M12=M22=M32=M42=0.4000.4,h^=0.01,τ^=0.01,α=0.01,hm=0.5,rm=0.5,δ=0.5,ϵ=0.5.Case 1.
WhenΔKi(t) satisfies additive form (4), we set (127)J11=0.1000.1,J21=0.1000.1,J12=0.2000.2,J22=0.2000.2.Case 2.
WhenΔKi(t) satisfies multiplicative form (5), we choose (128)σ¯111=0.2,σ¯121=0.2,σ¯211=0.4,σ¯221=0.2m=2,n=2,σ¯112=0.2,σ¯122=0.4,σ¯212=0.2,σ¯222=0.2m=2,n=2.Furthermore, just as the discussion in Remark 6, we choose the values for the extended dissipative parameters in Table 1.Table 1
Matrices for each case.
Analysis
Ψ
1
Ψ
2
Ψ
3
Ψ
4
L
2
-
L
∞ performance
0
0
γ
2
I
I
H
∞ performance
-
I
0
γ
2
I
0
Passivity
0
I
γ
0
Dissipativity
-
I
I
1
0
0.3
1
-
β
∗
I
0Then, solve the LMIs from (47) to (50) in Theorem 15, and we can get the results of optimized variables of four performances in Table 2.Table 2
Optimized variable for each case.
L
2
-
L
∞ performance
H
∞ performance
Passivity
Dissipativity
γ
1
m
i
n
2
=
1
∗
1
0
-
7
γ
1
m
i
n
2
=
1
∗
1
0
-
7
γ
1
m
i
n
=
1
∗
1
0
-
7
β
1
m
a
x
=
1.9999999Furthermore, solve the LMIs presented in Theorems16 and 17, and we can obtain the controller gain for the additive form controller uncertainty and the multiplicative form controller uncertainty in Tables 3 and 4, respectively.Table 3
Controller gain of the additive form controller uncertainty for each case.
Subsystem
1
2
L
2
-
L
∞ performance
K
1
=
1
0
3
∗
-
8.5600
-
0.2148
-
0.1787
-
9.6241
K
2
=
1
0
4
∗
-
1.4313
-
0.1736
-
0.1853
-
0.9176
H
∞ performance
K
1
=
1
0
4
∗
-
1.3762
-
0.8518
-
0.5214
-
5.3445
K
2
=
1
0
4
∗
-
4.8652
-
0.9760
-
1.4654
-
1.9109
Passivity
K
1
=
1
0
8
∗
-
0.3767
-
0.1899
-
0.1414
-
1.5588
K
2
=
1
0
8
∗
-
1.4961
-
0.4558
-
0.7175
-
0.5430
Dissipativity
K
1
=
1
0
8
∗
-
0.2581
-
0.1677
-
0.1381
-
1.2602
K
2
=
1
0
8
∗
-
1.0597
-
0.4132
-
0.5249
-
0.4398Table 4
Controller gain of the multiplicative form controller uncertainty for each case.
Subsystem
1
2
L
2
-
L
∞ performance
K
1
=
1
0
-
4
∗
-
0.8602
-
0.2082
-
0.2037
-
0.5895
K
2
=
1
0
-
3
∗
-
0.1000
-
0.1023
-
0.1017
-
0.1090
H
∞ performance
K
1
=
1
0
-
3
∗
-
0.1922
-
0.2047
-
0.0870
-
0.4234
K
2
=
1
0
-
3
∗
-
0.4842
-
0.2812
-
0.5055
-
0.3274
Passivity
K
1
=
-
0.2875
-
0.2582
-
0.1017
-
0.7210
K
2
=
-
1.2292
-
0.5381
-
1.2374
-
0.5245
Dissipativity
K
1
=
-
0.2711
-
0.4428
-
0.1380
-
1.1896
K
2
=
-
2.5753
-
0.8837
-
2.4237
-
0.9115


## 5. Conclusion

In this paper, we have investigated the problem of finite-time extended dissipative analysis and nonfragile control of switched neutral system with unknown time-varying disturbance. The average dwell-time approach is utilized for finite-time boundedness and extended dissipative performance analysis; controllers are designed to guarantee that the system is finite-time bounded and satisfies the extended dissipative performance. Based on extended dissipative performance, we can solveH∞, L2-L∞, Passivity, and (Q,S,R)-dissipativity performance in a unified framework. All the results are given in terms of linear matrix inequalities (LMIs), and numerical examples are provided to show the effectiveness of the proposed method. In our future research, the nonfragile control and extended dissipative performance will be extended to more complex systems, such as Markovian jump delayed systems, sliding control systems, and T-S fuzzy systems, which deserve further study.



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*Source: 6581308-2017-11-14.xml*