# Nonfragile Guaranteed Cost Control and Optimization for Interconnected Systems of Neutral Type

Authors: Heli Hu; Dan Zhao; Qingling Zhang
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

The design and optimization problems of the nonfragile guaranteed cost control are investigated for a class of interconnected systems of neutral type. A novel scheme, viewing the interconnections with time-varying delays as effective information but not disturbances, is developed to decrease the conservatism. Many techniques on decomposing and magnifying the matrices are
utilized to obtain the guaranteed cost of the considered system. Also, an algorithm is proposed to solve the nonlinear problem of the interconnected matrices. Based on this algorithm, the minimization of
the guaranteed cost of the considered system is obtained by optimization. Further, the state feedback control is extended to the case in which the underlying system is dependent on uncertain parameters. Finally, two numerical examples are given to illustrate the proposed method, and some comparisons are made to show the advantages of the schemes of dealing with the interconnections.


---

## Body

## 1. Introduction

Time delays often arise in the processing state, input or related variables of dynamic systems. Particularly, when the state derivative also contains time delay, the considered systems are called neutral systems [1]. The outstanding characteristic of neutral systems is the fact that such systems contain the same highest order derivatives for the state vector x(t), at both time t and past time(s) ts≤t. Many engineering systems can be represented as neutral equation [2–10], such as heat exchangers, robots in contact with rigid environments [11], distributed networks containing lossless transmission lines [12], and population ecology [13]. Therefore, great interest has been devoted to analysis and synthesis of a class of neutral delay systems. The delay-dependent stability criteria for stochastic systems of neutral type are studied in [3, 6]. The difference between them is that the exponential stability problem is investigated in the former, and the robust stochastic stability, stabilization, and H∞ control problems are considered in the other. Furthermore, the improved stability criteria for neutral systems are established by the method of a memory state feedback control [2] and by the method of a robust H∞ reduced order filter in [4]. In the context of infinite-dimensional linear systems modeled by neutral functional differential equations, a periodic output feedback is studied in [14] and the stabilization of neutral systems with delayed control is the main work. As the further results, in [15–17], the stability and H∞ performance analysis, the finite-time H∞ control, and the reliable stabilization for uncertain switched systems of neutral type are investigated, respectively.On the other hand, interconnected systems appear in a variety of engineering applications including power systems, large structures and manufacturing systems, and their applications, such as [18–21]. In [18], Mukaidani investigates the stability of a class of nonlinear large-scale systems and proposes a suboptimal guaranteed cost control instead of solving the nonconvex optimization problem. But the time delays are invariant and not involved in the interconnections. Furthermore, the scheme of counteracting the interconnections to simplify the problem may add conservatism in some cases. In [19], Mahmoud and Xia propose a generalized approach to stabilization of systems which are composed of linear time delay subsystems coupled by linear time-varying interconnections. The decentralized structure of dissipative state-feedback controllers is designed to render the closed-loop interconnected system delay-dependent asymptotically stable with strict dissipativity. However, the optimization problem for the dissipativity βj is not taken into account. In [20], a decentralized control scheme for a class of linear large-scale interconnected systems with norm-bounded time-varying parameter uncertainties is designed under a class of control failures. It is worth noting that the considered systems do not include any time delay, and the optimization problem for the guaranteed cost J(x,u) is not investigated.To the best of the authors’ knowledge, the nonfragile guaranteed cost control and optimization for neutral interconnected systems have not yet been investigated, which motivates the present study. One contribution of this paper is that a novel scheme, viewing the interconnections with time-varying delays as effective information but not disturbances, is developed to decrease the conservatism. The other contribution lies in the fact that an algorithm is proposed to solve the nonlinear constraint problem caused by the interconnected matrices. In this paper, the designed control is the state feedback control with gain perturbations. Also, the guaranteed cost of the considered system can be obtained by solving the corresponding matrix inequality. Based on the proposed algorithm, the minimization of the guaranteed cost of the considered system can be obtained by optimization. particuraly, the matrixEi1/2 is introduced to denote the square root matrix of symmetric positive semidefinite matrix Ei≥0, that is, Ei1/2=ViHi1/2ViT with Vi the eigenvector matrix of Ei satisfying ViViT=I and Hi the diagonal eigenvalues matrix of Ei.The remainder of the paper is organized as follows. The nonfragile control problem formulation is described in Section2. In Section 3, the guaranteed cost control with gain perturbations and optimization are investigated for unperturbed and uncertain neutral interconnected systems. The numerical examples, the simulation results, and some comparisons are presented in Section 4. The conclusion is provided in Section 5.


## 2. Problem Formulation

Consider the following uncertain neutral interconnected systems composed ofN subsystems:
(1)x˙i(t)-∑j=1,j≠iN‍Aijxj(t-τij(t))=[Ai+ΔAi(t)]xi(t)+[Aiσi+ΔAiσi(t)]xi(t-σi(t))+[Aiηi+ΔAiηi(t)]x˙i(t-ηi(t))+[Bi+ΔBi(t)]ui(t)+[Biδi+ΔBiδi(t)]ui(t-δi(t)),xi(t)=ϕi(t),t∈[-l,0],i=1,2,…,N,
where xi(t)∈ℜni and ui(t)∈ℜmi are the state vector and the input vector of the ith subsystem, respectively. ∑j=1,j≠iNAijxj(t-τij(t)) is the interconnections between the ith subsystem and the other N-1 subsystems, in which Aij is known interconnected matrices of appropriate dimensions, and xj(t-τij(t)) implies the interconnections between the ith subsystem and the other N-1 subsystems have different time-varying delays τij(t), j=1,2,…,N, j≠i. Ai, Aiσi, Aiηi, Bi, and Biδi are known constant matrices of appropriate dimensions. ϕi(t) is the initial condition. Assume that there exist constants fi0, gi0, hi0, li0, fi, gi, hi, li, and l satisfying
(2)0≤σi(t)≤fi0,0≤ηi(t)≤gi0,0≤δi(t)≤hi0,0≤τij(t)≤li0,σ˙i(t)≤fi<1,η˙i(t)≤gi<1,δ˙i(t)≤hi<1,τ˙ij(t)≤li<1,l=max{fi0,gi0,hi0,li0},i,j=1,2…,N,j≠i.
Time-varying parametric uncertainties ΔAi(t), ΔAiσi(t), ΔAiηi(t), ΔBi(t), and ΔBiδi(t) are assumed to be of the following form:
(3)[ΔAi(t)ΔAiσi(t)ΔAiηi(t)ΔBi(t)ΔBiδi(t)]=CiFi(t)[Di1DiσiDiηiDi2Diδi],
where Ci, Di1, Diσi, Diηi, Di2, and Diδi are constant matrices of appropriate dimensions, and Fi(t) is the unknown matrix function satisfying FiT(t)Fi(t)≤Ini, for all t≥0.Construct the following state feedback control with gain perturbations:(4)ui(t)=-(Ki+ΔKi)xi(t),
where Ki∈ℜmi×ni is the control gain to be designed, and ΔKi is a perturbed matrix satisfying ΔKi=MiF¯i(t)Ni, where Mi and Ni are known matrices of appropriate dimensions, and F¯i(t) satisfies F¯iT(t)F¯i(t)≤Imi, for all t≥0; the resulting closed-loop uncertain neutral interconnected systems are obtained:
(5)x˙i(t)-∑j=1,j≠iN‍Aijxj(t-τij(t))=[Ai-Bi(Ki+ΔKi)+ΔAi(t)aaaaii-ΔBi(t)(Ki+ΔKi)]xi(t)+[Aiσi+ΔAiσi(t)]xi(t-σi(t))+[Aiηi+ΔAiηi(t)]x˙i(t-ηi(t))+[-Biδi(Ki+ΔKi)-ΔBiδi(t)(Ki+ΔKi)]×xi(t-δi(t)).Define the following quadratic cost function:(6)J=∑i=1N‍∫0∞‍[xiT(t)Si1xi(t)+uiT(t)Si2ui(t)]dt,
where Si1∈ℜni×ni and Si2∈ℜmi×mi are two given symmetric positive definite matrices.One objective of this paper is to design a control (4) and determine a scalar Ju satisfying the following two conditions:(a)
the closed-loop system (5) is asymptotically stable,(b)
J≤Ju.If the aforementioned control gainKi and constant Ju exist, control (4) is the decentralized nonfragile guaranteed cost control and Ju is the guaranteed cost for the considered system.The other is to find outJ*, the minimization of the guaranteed cost Ju.Lemma 1 (see [8]).
LetZ, X, S, and Y be matrices of appropriate dimensions. Assuming that Z is symmetric and STS≤I, then Z+XSY+YTSTXT<0 if and only if there exists a scalar ε>0 satisfying
(7)Z+εXXT+ε-1YTY=Z+ε-1(εX)(εX)T+ε-1YTY<0.Lemma 2 (see [8]).
For any constant matrixP>0 and differentiable vector function xi(t) with appropriate dimensions, one has
(8)[∫t-σi(t)t‍x˙i(s)ds]TP[∫t-σi(t)t‍x˙i(s)ds]≤fi0∫t-σi(t)t‍x˙iT(s)Px˙i(s)ds≤fi0∫t-fi0t‍x˙iT(s)Px˙i(s)ds.


## 3. Main Result

### 3.1. Nonfragile Guaranteed Cost Control and Optimization for Unperturbed Neutral Interconnected Systems

For convenience, firstly consider the following unperturbed neutral interconnected systems with time-varying delays:(9)x˙i(t)-∑j=1,j≠iN‍Aijxj(t-τij(t))=Aixi(t)+Aiσixi(t-σi(t))+Aiηix˙i(t-ηi(t))+Biui(t)+Biδiui(t-δi(t)),xi(t)=ϕi(t),t∈[-l,0],i=1,2,…,N.Now a sufficient condition for existence of the decentralized nonfragile guaranteed cost control (4) for unperturbed neutral interconnected systems (9) with cost function (6) is presented in the following results.Theorem 3.
Assume∥Aiηi∥<1. If there exist a positive number εi1, some symmetric positive definite matrices Qik(k=0,1,2),W¯ji,Wij, and matrix Xi such that the following inequality holds:
(10)Γi=[Γ11iΓ12i0Γ14iΓ15iΓ16i*Γ22i0Γ24i0Γ26i**Γ33i000***Γ44i0Γ46i****Γ55i0*****Γ66i]<0;
then control (4) with Ki=XiQi0-1 is the decentralized nonfragile guaranteed cost control of unperturbed neutral interconnected systems (9) with the following guaranteed cost:
(11)Ju=∑i=1N‍[∑j=1,j≠iN‍ϕiT(0)Qi0-1ϕi(0)aaaaaaaia+∫-σi(0)0‍ϕiT(s)Qi0-1Qi1Qi0-1ϕi(s)dsaaaaaaaia+11-gi∫-ηi(0)0‍ϕ˙iT(s)ϕ˙i(s)dsaaaaaaia+fi0∫-fi00‍(s+fi0)ϕ˙iT(s)ϕ˙i(s)dsaaaaaaaia+∫-δi(0)0‍ϕiT(s)Qi0-1Qi2Qi0-1ϕi(s)dsaaaaaaia+11-li∑j=1,j≠iN‍∫-τij(0)0‍ϕjT(s)Wijϕj(s)ds],
where(12)Γ11i=AiQi0+Qi0AiT-BiXi-XiTBiT+Qi1+Qi2+Ei,Γ12i=[AiσiQi0Aiηi0-BiδiXi],Γ16i=[εi1BiMiεi1BiδiMi-Qi0NiT0],Γ14i=[Qi0AiT-XiTBiTQi0-XiTQi0AiTEi1/2-XiTBiTEi1/2000],Γ22i=diag{-(1-fi)Qi1,-Ini,-Ini,-(1-hi)Qi2},Γ24i=[Qi0AiσiT0000Qi0AiσiTEi1/200AiηiT00000AiηiTEi1/2000000000-XiTBiδiT000000-XiTBiδiTEi1/2],Γ15i=[Qi0⋯Qi0︸N-1],Γ26i=[000000000000000-Qi0NiT],Ei=∑j=1,j≠iN‍AijAijT,Γ33i=[Ψ11i⋯Ψ1i-1iΨ1i+1i⋯Ψ1Ni⋮⋱⋮⋮⋮⋮**Ψi-1i-1iΨi-1i+1i⋯Ψi-1Ni***Ψi+1i+1i⋯Ψi+1Ni⋮⋮⋮⋮⋱⋮*****ΨNNi],Γ44i=diag{-ei1-1Ini,-Si1-1,-Si2-1,-ei1-1Ini,-ei1-1Ini,-ei1-1Ini,-ei1-1Ini,-ei1-1Ini},Γ46i=[εi1BiMiεi1BiδiMi000000εi1Mi0000000εi1Ei1/2BiMi000000000000εi1Ei1/2BiδiMi00],ei1=11-gi+fi02,ei2=1+51-gi+5fi02,Γ55i=diag{-(1-l1)W¯1i,…,-(1-li-1)W¯i-1i,-(1-li+1)W¯i+1i,…,-(1-lN)W¯Ni},Γ66i=diag{-εi1Ini,-εi1Ini,-εi1Ini,-εi1Ini},Ψ11i=ei2In1-Wi1+ei1Ai1TAi1,Ψ1i-1i=ei1Ai1TAii-1,Ψ1i+1i=ei1Ai1TAii+1,Ψ1Ni=ei1Ai1TAiN,Ψi-1i-1i=ei2Ini-1-Wii-1+ei1Aii-1TAii-1,Ψi-1i+1i=ei1Aii-1TAii+1,Ψi-1Ni=ei1Aii-1TAiN,Ψi+1i+1i=ei2Ini+1-Wii+1+ei1Aii+1TAii+1,Ψi+1Ni=ei1Aii+1TAiN,ΨNNi=ei2InN-WiN+ei1AiNTAiN,W¯ji=Wji-1,i,j=1,2…,N,j≠i.Proof.
ChoosePi0=Qi0-1, Pi1=Qi0-1Qi1Qi0-1, and Pi2=Qi0-1Qi2Qi0-1, and construct the following Lyapunov functional:
(13)V(x(t),t)=∑i=1N‍[∑j=1,j≠iN‍xiT(t)Pi0xi(t)aaaaaaaaa+∫t-σi(t)t‍xiT(s)Pi1xi(s)dsaaaaaaaaa+11-gi∫t-ηi(t)t‍x˙iT(s)x˙i(s)dsaaaaaaaaa+fi0∫t-fi0t‍(s-(t-fi0))x˙iT(s)x˙i(s)dsaaaaaaaaa+∫t-δi(t)t‍xiT(s)Pi2xi(s)dsaaaaaaaaa+11-li∑j=1,j≠iN‍∫t-τij(t)t‍xjT(s)Wijxj(s)ds].
Obviously,V(x(t),t)>0 for all xi(t)≠0. Differentiating V(x(t),t) along the trajectories of the unperturbed neutral interconnected systems (9) with control (4) and applying (2) and Lemma 2 yield
(14)V˙(x(t),t)≤∑i=1N‍{(t-τij(t))∑j=1,j≠iN‍xiT(t)(Pi0Ai+AiTPi0)xi(t)+2xiT(t)Pi0Aiσixi(t-σi(t))+2xiT(t)Pi0Aiηix˙i(t-ηi(t))-2xiT(t)Pi0Bi(Ki+ΔKi)xi(t)-2xiT(t)Pi0Biδi(Ki+ΔKi)×xi(t-δi(t))+2xiT(t)Pi0×∑j=1,j≠iN‍Aijxj(t-τij(t))+xiT(t)Pi1xi(t)-(1-fi)xiT×(t-σi(t))Pi1xi(t-σi(t))+(11-gi+fi02)×[∑j=1,j≠iN‍xiT(t)AiT+xiT(t-σi(t))AiσiT+x˙iT(t-ηi(t))AiηiT-xiT(t)(Ki+ΔKi)TBiT-xiT×(t-δi(t))(Ki+ΔKi)TBiδiT+∑j=1,j≠iN‍xjT(t-τij(t))AijT]×[∑j=1,j≠iN‍Aixi(t)+Aiσixi(t-σi(t))+Aiηix˙i(t-ηi(t))-Bi(Ki+ΔKi)xi(t)-Biδi(Ki+ΔKi)xi(t-δi(t))+∑j=1,j≠iN‍Aijxj(t-τij(t))]-x˙iT(t-ηi(t))x˙i(t-ηi(t))-[∫t-σi(t)t‍x˙i(s)ds]T[∫t-σi(t)t‍x˙i(s)ds]+xiT(t)Pi2xi(t)-(1-hi)xiT×(t-δi(t))Pi2xi(t-δi(t))+11-li∑j=1,j≠iN‍xjT(t)Wijxj(t)-∑j=1,j≠iN‍xjT(t-τij(t))Wijxj(t-τij(t))}.
According to Lemma1 and the following the fact:
(15)∑i=1N‍11-li∑j=1,j≠iN‍xjT(t)Wijxj(t)=∑i=1N‍xiT(t)∑j=1,j≠iN‍11-ljWjixi(t),
one can obtain
(16)∑i=1N‍{∑j=1,j≠iN‍2xiT(t)Pi0∑j=1,j≠iN‍Aijxj(t-τij(t))+2ei1aaaaaa×[xiT(t)AiT+xiT(t-σi(t))AiσiTaaaaaaaaa+x˙iT(t-ηi(t))AiηiT-xiT(t)aaaaaaaaa×(Ki+ΔKi)TBiT-xiT(t-δi(t))aaaaaaaaa×(Ki+ΔKi)TBiδiT]aaaaaa×∑j=1,j≠iN‍Aijxj(t-τij(t))+ei1aaaaaa×∑j=1,j≠iN‍xjT(t-τij(t))AijTaaaaaa×∑j=1,j≠iN‍Aijxj(t-τij(t))aaaaaa+11-li∑j=1,j≠iN‍xjT(t)Wijxj(t)aaaaaa-∑j=1,j≠iN‍xjT(t-τij(t))aaaaaa×Wijxj(t-τij(t))∑j=1,j≠iN‍}≤∑i=1N‍{xiT(t)[∑j=1,j≠iN‍Pi0EiPi0+ei1AiTEiAiaaaaaaaaaaaaaaiia+ei1(Ki+ΔKi)Taaaaaaaaaaaaiia×BiTEiBi(Ki+ΔKi)aaaaaaaaaaaaiia+∑j=1,j≠iN‍11-ljWji]aaaaaaaa×xi(t)+ei1xiT(t-σi(t))aaaaaaaa×AiσiTEiAiσixi(t-σi(t))aaaaaaaa+ei1xiT(t-δi(t))(Ki+ΔKi)Taaaaaaaa×BiδiTEiBiδi(Ki+ΔKi)xi(t-δi(t))aaaaaaaa+ei1x˙iT(t-ηi(t))AiηiTEiAiηix˙i(t-ηi(t))aaaaaaaa+∑j=1,j≠iN‍xjT(t-τij(t))aaaaaaaa×(ei2I-Wij)xj(t-τij(t))+ei1aaaaaaaa×∑j=1,j≠iN‍xjT(t-τij(t))AijTaaaaaaaa×∑j=1,j≠iN‍Aijxj(t-τij(t))}.
Therefore, it follows from (14) and (16) that
(17)V˙(x(t),t)+∑i=1N‍[xiT(t)Si1xi(t)+xiT(t)(Ki+ΔKi)T×Si2(Ki+ΔKi)xi(t)xiT]≤∑i=1N‍ξiTΥiξi,
where(18)ξiT=[xiT(t)xiT(t-σi(t))x˙iT(t-ηi(t))∫t-σi(t)tx˙iT(s)dsxiT(t-δi(t))x1T(t-τi1(t))⋯xi-1T(t-τii-1(t))xi+1T(t-τii+1(t))⋯xNT(t-τiN(t))∫t-σi(t)tx˙iT(s)ds],Υi=[Ωi0*Γ33i]+[Gi10](ei1Ini)[Gi1T0]+[Gi20]Si1[Gi2T0]+[Gi30]Si2[Gi3T0],Ωi=[Ω11iΩ12iΩ13i0Ω15i*Ω22i000**Ω33i00***Ω44i0****Ω55i],Ω12i=Pi0Aiσi,Ω13i=Pi0Aiηi,Ω44i=-Ini,Ω11i=Pi0Ai+AiTPi0-Pi0Bi(Ki+ΔKi)-(Ki+ΔKi)TBiTPi0+Pi1+Pi2+Pi0EiPi0+ei1AiTEiAi+ei1(Ki+ΔKi)TBiTEiBi(Ki+ΔKi)+∑j=1,j≠iN‍11-ljWji,Ω15i=-Pi0Biδi(Ki+ΔKi),Ω22i=-(1-fi)Pi1+ei1AiσiTEiAiσi,Ω33i=-Ini+ei1AiηiTEiAiηi,Ω55i=-(1-hi)Pi2+ei1(Ki+ΔKi)TBiδiTEiBiδi(Ki+ΔKi),Gi1=[Ai-Bi(Ki+ΔKi)AiσiAiηi0-Biδi(Ki+ΔKi)]T,Gi2=[Ini0000]T,Gi3=[-(Ki+ΔKi)0000]T.
Define(19)Υ¯i=[Ωi0Υ¯13i*Γ33i0**Υ¯33i],
where Υ¯13i=[Gi1Gi2Gi3], Υ¯33i=diag{-ei1-1Ini,-Si1-1,-Si2-1}.
Pre- and postmultiplying the matrixΥ¯i in (19) by UiT and Ui, where Ui=diag{Qi0,Qi0,Ini,Ini,Qi0,In1,…,Ini-1,Ini+1,…,InN,Ini,Ini,Imi}, the following matrix is obtained:
(20)Υ̿i=[Ω¯i0Υ̿13i*Γ33i0**Υ¯33i],
where
(21)Ω¯i=[Ω¯11iΩ¯12iΩ¯13i0Ω¯15i*Ω¯22i000**Ω33i00***Ω44i0****Ω¯55i],Ω¯12i=AiσiQi0,Ω¯13i=Aiηi,Ω¯11i=AiQi0+Qi0AiT-Bi(Ki+ΔKi)Qi0-Qi0(Ki+ΔKi)TBiT+Qi1+Qi2+Ei+ei1Qi0AiTEiAiQi0+ei1Qi0×(Ki+ΔKi)TBiTEiBi(Ki+ΔKi)Qi0+Qi0∑j=1,j≠iN‍11-ljWjiQi0,Ω¯15i=-Biδi(Ki+ΔKi)Qi0,Ω¯22i=-(1-fi)Qi1+ei1Qi0AiσiTEiAiσiQi0,Ω¯55i=-(1-hi)Qi2+ei1Qi0×(Ki+ΔKi)TBiδiTEiBiδi(Ki+ΔKi)Qi0,Υ̿13i=[Qi0AiT-Qi0(Ki+ΔKi)TBiTQi0-Qi0(Ki+ΔKi)TQi0AiσiT00AiηiT00000-Qi0(Ki+ΔKi)TBiδiT00].
Define(22)Θi=[Ω̿i0Υ̿13iΞiΘ15i*Γ33i000**Υ¯33i00***Θ44i0****Θ55i],
where
(23)Ω¯¯i=[Ω̿11iΩ¯12iΩ¯13i0Ω¯15i*Ω̿22i000**Ω̿33i00***Ω44i0****Ω̿55i],Ξi=[Ξ11iΞ12i00000Ξ23i00000Ξ34i0000000000Ξ55i],Θ15i=[Γ15i0000],Θ44i=diag{-ei1-1Ini,-ei1-1Ini,-ei1-1Ini,aaaaaaaaaa-ei1-1Ini,-ei1-1Ini},Θ55i=Γ55i,Ω̿11i=AiQi0+Qi0AiT-Bi(Ki+ΔKi)Qi0-Qi0(Ki+ΔKi)TBiT+Qi1+Qi2+Ei,Ω̿22i=-(1-fi)Qi1,Ω̿33i=-Ini,Ω̿55i=-(1-hi)Qi2,Ξ11i=Qi0AiTEi1/2,Ξ12i=-Qi0(Ki+ΔKi)TBiTEi1/2,Ξ23i=Qi0AiσiTEi1/2,Ξ34i=AiηiTEi1/2,Ξ55i=-Qi0(Ki+ΔKi)TBiδiTEi1/2.
The following equality is obvious:(24)Θi=[Γ11iΓ12i0Γ14iΓ15i*Γ22i0Γ24i0**Γ33i00***Γ44i0****Γ55i]+ΛiTRi1$i+$iTRi1TΛi,
where
(25)Λi=[MiTBiT00Λ14i0MiTBiδiT00Λ24i0],$i=[-NiQi000000$22i000],Ri1=diag{F¯i(t),F¯i(t)},Λ14i=[MiTBiT0MiT0MiTBiTEi1/2000],Λ24i=[MiTBiδiT000000MiTBiδiTEi1/2],$22i=[000-NiQi0].
By Lemma1 and Schur complement formula, the condition Γi<0 in (10) is equivalent to Θi<0 in (24). By Schur complement formula with Θi<0, one can obtain Υ̿i<0 in (20). The condition Υ̿ii<0 is equivalent to Υ¯i<0. Again, by Schur complement formula with Υ¯i<0, one can obtain Υi<0. From the condition Υi<0 in (17), there exists a constant ρi>0, such that
(26)V˙(x(t),t)≤∑i=1N‍-ρi∥xi(t)∥2.
By conditions (13) and (26) and ∥Aiηi∥<1, one can conclude that system (9) with (2) and (4) is asymptotically stable. From (17) with Υi<0, one can obtain
(27)∫0∞‍V˙(x(t),t)dt=limt→∞V(x(t),t)-V(x(0),0)≤-∑i=1N‍∫0∞‍[xiT(t)Si1xi(t)+uiT(t)Si2ui(t)]dt.
Therefore, the following equalities hold:(28)limt→∞V(x(t),t)=0,∑i=1N‍∫0∞‍[xiT(t)Si1xi(t)+uiT(t)Si2ui(t)]dt≤V(x(0),0)=Ju.
This completes the proof.Remark 4.
It is obvious that for every subsystem, the correspondingΓi in (10) is an LMI with obtained matrices W¯ji(W¯ji=Wji-1) and Wij in the last inequality (i.e., the inequality Γi-1<0). Hence, the decentralized nonfragile control (4) and the guaranteed cost Ju in (11) can be obtained by finding feasible set to Γi<0 with feasp in [22] one by one.Remark 5.
Obviously, the guaranteed costJu in (11) cannot be directly optimized by using the toolbox of mincx in [22]. One reason is that inequalities (10) with variable matrices Wij and W¯ji(W¯ji=Wji-1) are not a group of LMIs but N coupled nonlinear inequalities. Another reason is that Ju is a nonconvex function with respect to the optimization variables.The following algorithm is given to solve the nonlinear problem of inequalities (10).Algorithm 6.
Choose constant matricesWij>0 and W¯ji>0 satisfying Ψjji<0 in Γi, where W¯ji=Wji-1, i,j=1,2…,N,j≠i.
It is needed to simultaneously selectN×(N-1) constant matrices Wij>0 and W¯ji>0(W¯ji=Wji-1) satisfying Ψjji<0. For simplicity, one can choose Wij and W¯ji to be positive definite diagonal matrices according to the eigenvalues of ei2Inj+ei1AijTAij due to Ψjji=ei2Inj-Wij+ei1AijTAij. The chosen entries need to be as small as possible, because
(29)11-li∑j=1,j≠iN‍∫-τij(0)0‍xjT(s)Wijxj(s)ds
is involved in Ju. However, if there is no solution to inequalities (10), the large scalars can be considered.
In the sequel, instead of solving the nonconvex optimization problem, a suboptimal method of minimizing the guaranteed costJu, based on Algorithm 6, is presented.Theorem 7.
Consider unperturbed system (9) with cost function (6), and assume ∥Aiηi∥<1. If the following optimization problem:
(30)min∑i=1N‍[αi+Tr(Ui1TΦi1Ui1+Ui2TΦi2Ui2)]
subject to LMI (10) with Algorithm 6, and
(31)[-αiϕiT(0)*-Qi0]<0,[-2Qi0+Qi1Ini*-Φi1]<0,[-2Qi0+Qi2Ini*-Φi2]<0
has a solution set (αi>0,  εi1>0,  Qik>0(k=0,1,2),  Φi1>0,  Φi2>0,  Xi), where ∫-σi(0)0‍ϕi(s)ϕiT(s)ds=Ui1Ui1T, ∫-δi(0)0‍ϕi(s)ϕiT(s)ds=Ui2Ui2T, (1/(1-gi))∫-ηi(0)0‍ϕ˙iT(s)ϕ˙i(s)ds=Li1, fi0∫-fi00‍(s+fi0)ϕ˙iT(s)ϕ˙i(s)ds=Li2, (1/(1-li))∑j=1,j≠iN∫-τij(0)0‍ϕjT(s)Wijϕj(s)ds=Li3, then control (4) with Ki=XiQi0-1 is the decentralized nonfragile guaranteed cost control of unperturbed system (9) with the minimization of the guaranteed cost Ju as follows:
(32)J*=min(∑i=1N‍[αi+Tr(Ui1TΦi1Ui1+Ui2TΦi2Ui2)])+∑i=1N‍(Li1+Li2+Li3).Proof.
Applying the Schur complement formula to LMIs (31) leads to ϕiT(0)Qi0-1ϕi(0)<αi,  -2Qi0+Qi1+Φi1-1<0,  -2Qi0+Qi2+Φi2-1<0, respectively.
Noting that [8]
(33)[Qi0-Φi1-1]Φi1[Qi0-Φi1-1]=Qi0Φi1Qi0-2Qi0+Φi1-1≥0,[Qi0-Φi2-1]Φi2[Qi0-Φi2-1]=Qi0Φi2Qi0-2Qi0+Φi2-1≥0,
the following inequalities are obtained
(34)Pi1=Qi0-1Qi1Qi0-1<Φi1,Pi2=Qi0-1Qi2Qi0-1<Φi2.
Further, one can obtain(35)∫-σi(0)0‍ϕiT(s)Pi1ϕi(s)ds=Tr(∫-σi(0)0‍ϕi(s)ϕiT(s)Pi1)ds=Tr(Ui1TPi1Ui1)≤Tr(Ui1TΦi1Ui1),∫-δi(0)0‍ϕiT(s)Pi2ϕi(s)ds=Tr(∫-δi(0)0‍ϕi(s)ϕiT(s)Pi2)ds=Tr(Ui2TPi2Ui2)≤Tr(Ui2TΦi2Ui2),11-gi∫-ηi(0)0‍ϕ˙iT(s)ϕ˙i(s)ds=Li1,fi0∫-fi00‍(s+fi0)ϕ˙iT(s)ϕ˙i(s)ds=Li2,11-li∑j=1,j≠iN‍∫-τij(0)0‍ϕjT(s)Wijϕj(s)ds=Li3.
Therefore, it follows from (11) that
(36)Ju≤∑i=1N‍[αi+Tr(Ui1TΦi1Ui1+Ui2TΦi2Ui2)]+∑i=1N‍(Li1+Li2+Li3).
The minimization of the right hand of inequality (36) implies the minimization of the guaranteed cost Ju for unperturbed system (9). This completes the proof.


### 3.2. Nonfragile Guaranteed Cost Control for Uncertain Neutral Interconnected Systems

Theorem 8.
Consider uncertain neutral interconnected systems (1) with (2), (3), and (4). If there exist positive numbers εi1>0, εi2>0, and εi3>0, some symmetric positive definite matrices Qik(k=0,1,2), W¯ji, Wij, and matrix Xi such that the following inequalities hold:
(37)Γ̿i=[ΓiΓ¯i*Γ~i]<0,(38)[-Ini+εi3DiηiTDiηiAiηiT0*-IniCi**-εi3Ini]<0,
then control (4) with Ki=XiQi0-1 is the decentralized nonfragile guaranteed cost control of uncertain neutral interconnected systems (1) with the guaranteed cost in (11), where Γ¯i=[εi2Πi1Πi2],
(39)Πi1=[Γ¯11i00Γ¯41i00],Πi2=[Γ¯12iΓ¯22i000Γ¯62i],Γ¯22i=[00Qi0DiσiT000000DiηiT000000000000-XiTDiδiT0],Γ¯11i=[CiCiCiCiCiCi],Γ¯12i=[Qi0Di1T-XiTDi2T0000],Γ¯41i=[CiCiCiCiCiCi000000000000Ei1/2Ci000000Ei1/2Ci000Ei1/2Ci00Ei1/2Ci000000Ei1/2Ci000000Ei1/2Ci0],Γ¯62i=[00000εi1MiTDi2T0000εi1MiTDiδiT0000000000000],Γ~i=diag{-εi2Ini,…,-εi2Ini︸12}.Proof.
From condition (10) with unperturbed neutral interconnected systems (9), one can obtain the corresponding condition to stabilize uncertain neutral interconnected systems (1) as follows:
(40)Σi=Γi+Πi1Ri2Πi2T+Πi2Ri2TΠi1T<0,
where Ri2=diag{Fi(t),Fi(t),Fi(t),Fi(t),Fi(t),Fi(t)}.
By Lemma1 and Schur complement formula, the condition Γ̿i<0 in (37) is equivalent to Σi<0 in (40). For the same reason, (38) is equivalent to
(41)[Aiηi+ΔAiηi(t)]T[Aiηi+ΔAiηi(t)]<Ini.
This implies that uncertain neutral interconnected systems (1) are Lipschitz in the term x˙i(t-ηi(t)) with Lipschitz constant less than 1 [8]. By the same derivation of Theorem 3, one can complete this proof.The decentralized nonfragile guaranteed cost control (4) and the minimization of the guaranteed cost Ju for uncertain neutral interconnected systems (1) are determined by the following theorem.Theorem 9.
Consider uncertain neutral interconnected systems (1) with (2), (3), (4), and cost function (6). If the following optimization problem:
(42)min∑i=1N‍[αi+Tr(Ui1TΦi1Ui1+Ui2TΦi2Ui2)]
is subject to LMI (37) with Algorithm 6, (38), and (31) has a solution set (αi>0, εi1>0, εi2>0, εi3>0, Qik>0(k=0,1,2), Φi1>0, Φi2>0, Xi), then control (4) with Ki=XiQi0-1 is the decentralized nonfragile guaranteed cost control for uncertain neutral interconnected systems (1) with the minimization J* of the guaranteed cost Ju in (32).Remark 10.
Reference [18] develops a scheme of counteracting the interconnections to simplify the problem, which may add conservatism in some cases. Compared with the approach of treating the interconnections in [18], we utilize an approach of magnifying the terms associated interconnections; for details, one can see the derivation of inequality (16). To some extent, it may reduce the conservatism of the results derived in the paper.


## 3.1. Nonfragile Guaranteed Cost Control and Optimization for Unperturbed Neutral Interconnected Systems

For convenience, firstly consider the following unperturbed neutral interconnected systems with time-varying delays:(9)x˙i(t)-∑j=1,j≠iN‍Aijxj(t-τij(t))=Aixi(t)+Aiσixi(t-σi(t))+Aiηix˙i(t-ηi(t))+Biui(t)+Biδiui(t-δi(t)),xi(t)=ϕi(t),t∈[-l,0],i=1,2,…,N.Now a sufficient condition for existence of the decentralized nonfragile guaranteed cost control (4) for unperturbed neutral interconnected systems (9) with cost function (6) is presented in the following results.Theorem 3.
Assume∥Aiηi∥<1. If there exist a positive number εi1, some symmetric positive definite matrices Qik(k=0,1,2),W¯ji,Wij, and matrix Xi such that the following inequality holds:
(10)Γi=[Γ11iΓ12i0Γ14iΓ15iΓ16i*Γ22i0Γ24i0Γ26i**Γ33i000***Γ44i0Γ46i****Γ55i0*****Γ66i]<0;
then control (4) with Ki=XiQi0-1 is the decentralized nonfragile guaranteed cost control of unperturbed neutral interconnected systems (9) with the following guaranteed cost:
(11)Ju=∑i=1N‍[∑j=1,j≠iN‍ϕiT(0)Qi0-1ϕi(0)aaaaaaaia+∫-σi(0)0‍ϕiT(s)Qi0-1Qi1Qi0-1ϕi(s)dsaaaaaaaia+11-gi∫-ηi(0)0‍ϕ˙iT(s)ϕ˙i(s)dsaaaaaaia+fi0∫-fi00‍(s+fi0)ϕ˙iT(s)ϕ˙i(s)dsaaaaaaaia+∫-δi(0)0‍ϕiT(s)Qi0-1Qi2Qi0-1ϕi(s)dsaaaaaaia+11-li∑j=1,j≠iN‍∫-τij(0)0‍ϕjT(s)Wijϕj(s)ds],
where(12)Γ11i=AiQi0+Qi0AiT-BiXi-XiTBiT+Qi1+Qi2+Ei,Γ12i=[AiσiQi0Aiηi0-BiδiXi],Γ16i=[εi1BiMiεi1BiδiMi-Qi0NiT0],Γ14i=[Qi0AiT-XiTBiTQi0-XiTQi0AiTEi1/2-XiTBiTEi1/2000],Γ22i=diag{-(1-fi)Qi1,-Ini,-Ini,-(1-hi)Qi2},Γ24i=[Qi0AiσiT0000Qi0AiσiTEi1/200AiηiT00000AiηiTEi1/2000000000-XiTBiδiT000000-XiTBiδiTEi1/2],Γ15i=[Qi0⋯Qi0︸N-1],Γ26i=[000000000000000-Qi0NiT],Ei=∑j=1,j≠iN‍AijAijT,Γ33i=[Ψ11i⋯Ψ1i-1iΨ1i+1i⋯Ψ1Ni⋮⋱⋮⋮⋮⋮**Ψi-1i-1iΨi-1i+1i⋯Ψi-1Ni***Ψi+1i+1i⋯Ψi+1Ni⋮⋮⋮⋮⋱⋮*****ΨNNi],Γ44i=diag{-ei1-1Ini,-Si1-1,-Si2-1,-ei1-1Ini,-ei1-1Ini,-ei1-1Ini,-ei1-1Ini,-ei1-1Ini},Γ46i=[εi1BiMiεi1BiδiMi000000εi1Mi0000000εi1Ei1/2BiMi000000000000εi1Ei1/2BiδiMi00],ei1=11-gi+fi02,ei2=1+51-gi+5fi02,Γ55i=diag{-(1-l1)W¯1i,…,-(1-li-1)W¯i-1i,-(1-li+1)W¯i+1i,…,-(1-lN)W¯Ni},Γ66i=diag{-εi1Ini,-εi1Ini,-εi1Ini,-εi1Ini},Ψ11i=ei2In1-Wi1+ei1Ai1TAi1,Ψ1i-1i=ei1Ai1TAii-1,Ψ1i+1i=ei1Ai1TAii+1,Ψ1Ni=ei1Ai1TAiN,Ψi-1i-1i=ei2Ini-1-Wii-1+ei1Aii-1TAii-1,Ψi-1i+1i=ei1Aii-1TAii+1,Ψi-1Ni=ei1Aii-1TAiN,Ψi+1i+1i=ei2Ini+1-Wii+1+ei1Aii+1TAii+1,Ψi+1Ni=ei1Aii+1TAiN,ΨNNi=ei2InN-WiN+ei1AiNTAiN,W¯ji=Wji-1,i,j=1,2…,N,j≠i.Proof.
ChoosePi0=Qi0-1, Pi1=Qi0-1Qi1Qi0-1, and Pi2=Qi0-1Qi2Qi0-1, and construct the following Lyapunov functional:
(13)V(x(t),t)=∑i=1N‍[∑j=1,j≠iN‍xiT(t)Pi0xi(t)aaaaaaaaa+∫t-σi(t)t‍xiT(s)Pi1xi(s)dsaaaaaaaaa+11-gi∫t-ηi(t)t‍x˙iT(s)x˙i(s)dsaaaaaaaaa+fi0∫t-fi0t‍(s-(t-fi0))x˙iT(s)x˙i(s)dsaaaaaaaaa+∫t-δi(t)t‍xiT(s)Pi2xi(s)dsaaaaaaaaa+11-li∑j=1,j≠iN‍∫t-τij(t)t‍xjT(s)Wijxj(s)ds].
Obviously,V(x(t),t)>0 for all xi(t)≠0. Differentiating V(x(t),t) along the trajectories of the unperturbed neutral interconnected systems (9) with control (4) and applying (2) and Lemma 2 yield
(14)V˙(x(t),t)≤∑i=1N‍{(t-τij(t))∑j=1,j≠iN‍xiT(t)(Pi0Ai+AiTPi0)xi(t)+2xiT(t)Pi0Aiσixi(t-σi(t))+2xiT(t)Pi0Aiηix˙i(t-ηi(t))-2xiT(t)Pi0Bi(Ki+ΔKi)xi(t)-2xiT(t)Pi0Biδi(Ki+ΔKi)×xi(t-δi(t))+2xiT(t)Pi0×∑j=1,j≠iN‍Aijxj(t-τij(t))+xiT(t)Pi1xi(t)-(1-fi)xiT×(t-σi(t))Pi1xi(t-σi(t))+(11-gi+fi02)×[∑j=1,j≠iN‍xiT(t)AiT+xiT(t-σi(t))AiσiT+x˙iT(t-ηi(t))AiηiT-xiT(t)(Ki+ΔKi)TBiT-xiT×(t-δi(t))(Ki+ΔKi)TBiδiT+∑j=1,j≠iN‍xjT(t-τij(t))AijT]×[∑j=1,j≠iN‍Aixi(t)+Aiσixi(t-σi(t))+Aiηix˙i(t-ηi(t))-Bi(Ki+ΔKi)xi(t)-Biδi(Ki+ΔKi)xi(t-δi(t))+∑j=1,j≠iN‍Aijxj(t-τij(t))]-x˙iT(t-ηi(t))x˙i(t-ηi(t))-[∫t-σi(t)t‍x˙i(s)ds]T[∫t-σi(t)t‍x˙i(s)ds]+xiT(t)Pi2xi(t)-(1-hi)xiT×(t-δi(t))Pi2xi(t-δi(t))+11-li∑j=1,j≠iN‍xjT(t)Wijxj(t)-∑j=1,j≠iN‍xjT(t-τij(t))Wijxj(t-τij(t))}.
According to Lemma1 and the following the fact:
(15)∑i=1N‍11-li∑j=1,j≠iN‍xjT(t)Wijxj(t)=∑i=1N‍xiT(t)∑j=1,j≠iN‍11-ljWjixi(t),
one can obtain
(16)∑i=1N‍{∑j=1,j≠iN‍2xiT(t)Pi0∑j=1,j≠iN‍Aijxj(t-τij(t))+2ei1aaaaaa×[xiT(t)AiT+xiT(t-σi(t))AiσiTaaaaaaaaa+x˙iT(t-ηi(t))AiηiT-xiT(t)aaaaaaaaa×(Ki+ΔKi)TBiT-xiT(t-δi(t))aaaaaaaaa×(Ki+ΔKi)TBiδiT]aaaaaa×∑j=1,j≠iN‍Aijxj(t-τij(t))+ei1aaaaaa×∑j=1,j≠iN‍xjT(t-τij(t))AijTaaaaaa×∑j=1,j≠iN‍Aijxj(t-τij(t))aaaaaa+11-li∑j=1,j≠iN‍xjT(t)Wijxj(t)aaaaaa-∑j=1,j≠iN‍xjT(t-τij(t))aaaaaa×Wijxj(t-τij(t))∑j=1,j≠iN‍}≤∑i=1N‍{xiT(t)[∑j=1,j≠iN‍Pi0EiPi0+ei1AiTEiAiaaaaaaaaaaaaaaiia+ei1(Ki+ΔKi)Taaaaaaaaaaaaiia×BiTEiBi(Ki+ΔKi)aaaaaaaaaaaaiia+∑j=1,j≠iN‍11-ljWji]aaaaaaaa×xi(t)+ei1xiT(t-σi(t))aaaaaaaa×AiσiTEiAiσixi(t-σi(t))aaaaaaaa+ei1xiT(t-δi(t))(Ki+ΔKi)Taaaaaaaa×BiδiTEiBiδi(Ki+ΔKi)xi(t-δi(t))aaaaaaaa+ei1x˙iT(t-ηi(t))AiηiTEiAiηix˙i(t-ηi(t))aaaaaaaa+∑j=1,j≠iN‍xjT(t-τij(t))aaaaaaaa×(ei2I-Wij)xj(t-τij(t))+ei1aaaaaaaa×∑j=1,j≠iN‍xjT(t-τij(t))AijTaaaaaaaa×∑j=1,j≠iN‍Aijxj(t-τij(t))}.
Therefore, it follows from (14) and (16) that
(17)V˙(x(t),t)+∑i=1N‍[xiT(t)Si1xi(t)+xiT(t)(Ki+ΔKi)T×Si2(Ki+ΔKi)xi(t)xiT]≤∑i=1N‍ξiTΥiξi,
where(18)ξiT=[xiT(t)xiT(t-σi(t))x˙iT(t-ηi(t))∫t-σi(t)tx˙iT(s)dsxiT(t-δi(t))x1T(t-τi1(t))⋯xi-1T(t-τii-1(t))xi+1T(t-τii+1(t))⋯xNT(t-τiN(t))∫t-σi(t)tx˙iT(s)ds],Υi=[Ωi0*Γ33i]+[Gi10](ei1Ini)[Gi1T0]+[Gi20]Si1[Gi2T0]+[Gi30]Si2[Gi3T0],Ωi=[Ω11iΩ12iΩ13i0Ω15i*Ω22i000**Ω33i00***Ω44i0****Ω55i],Ω12i=Pi0Aiσi,Ω13i=Pi0Aiηi,Ω44i=-Ini,Ω11i=Pi0Ai+AiTPi0-Pi0Bi(Ki+ΔKi)-(Ki+ΔKi)TBiTPi0+Pi1+Pi2+Pi0EiPi0+ei1AiTEiAi+ei1(Ki+ΔKi)TBiTEiBi(Ki+ΔKi)+∑j=1,j≠iN‍11-ljWji,Ω15i=-Pi0Biδi(Ki+ΔKi),Ω22i=-(1-fi)Pi1+ei1AiσiTEiAiσi,Ω33i=-Ini+ei1AiηiTEiAiηi,Ω55i=-(1-hi)Pi2+ei1(Ki+ΔKi)TBiδiTEiBiδi(Ki+ΔKi),Gi1=[Ai-Bi(Ki+ΔKi)AiσiAiηi0-Biδi(Ki+ΔKi)]T,Gi2=[Ini0000]T,Gi3=[-(Ki+ΔKi)0000]T.
Define(19)Υ¯i=[Ωi0Υ¯13i*Γ33i0**Υ¯33i],
where Υ¯13i=[Gi1Gi2Gi3], Υ¯33i=diag{-ei1-1Ini,-Si1-1,-Si2-1}.
Pre- and postmultiplying the matrixΥ¯i in (19) by UiT and Ui, where Ui=diag{Qi0,Qi0,Ini,Ini,Qi0,In1,…,Ini-1,Ini+1,…,InN,Ini,Ini,Imi}, the following matrix is obtained:
(20)Υ̿i=[Ω¯i0Υ̿13i*Γ33i0**Υ¯33i],
where
(21)Ω¯i=[Ω¯11iΩ¯12iΩ¯13i0Ω¯15i*Ω¯22i000**Ω33i00***Ω44i0****Ω¯55i],Ω¯12i=AiσiQi0,Ω¯13i=Aiηi,Ω¯11i=AiQi0+Qi0AiT-Bi(Ki+ΔKi)Qi0-Qi0(Ki+ΔKi)TBiT+Qi1+Qi2+Ei+ei1Qi0AiTEiAiQi0+ei1Qi0×(Ki+ΔKi)TBiTEiBi(Ki+ΔKi)Qi0+Qi0∑j=1,j≠iN‍11-ljWjiQi0,Ω¯15i=-Biδi(Ki+ΔKi)Qi0,Ω¯22i=-(1-fi)Qi1+ei1Qi0AiσiTEiAiσiQi0,Ω¯55i=-(1-hi)Qi2+ei1Qi0×(Ki+ΔKi)TBiδiTEiBiδi(Ki+ΔKi)Qi0,Υ̿13i=[Qi0AiT-Qi0(Ki+ΔKi)TBiTQi0-Qi0(Ki+ΔKi)TQi0AiσiT00AiηiT00000-Qi0(Ki+ΔKi)TBiδiT00].
Define(22)Θi=[Ω̿i0Υ̿13iΞiΘ15i*Γ33i000**Υ¯33i00***Θ44i0****Θ55i],
where
(23)Ω¯¯i=[Ω̿11iΩ¯12iΩ¯13i0Ω¯15i*Ω̿22i000**Ω̿33i00***Ω44i0****Ω̿55i],Ξi=[Ξ11iΞ12i00000Ξ23i00000Ξ34i0000000000Ξ55i],Θ15i=[Γ15i0000],Θ44i=diag{-ei1-1Ini,-ei1-1Ini,-ei1-1Ini,aaaaaaaaaa-ei1-1Ini,-ei1-1Ini},Θ55i=Γ55i,Ω̿11i=AiQi0+Qi0AiT-Bi(Ki+ΔKi)Qi0-Qi0(Ki+ΔKi)TBiT+Qi1+Qi2+Ei,Ω̿22i=-(1-fi)Qi1,Ω̿33i=-Ini,Ω̿55i=-(1-hi)Qi2,Ξ11i=Qi0AiTEi1/2,Ξ12i=-Qi0(Ki+ΔKi)TBiTEi1/2,Ξ23i=Qi0AiσiTEi1/2,Ξ34i=AiηiTEi1/2,Ξ55i=-Qi0(Ki+ΔKi)TBiδiTEi1/2.
The following equality is obvious:(24)Θi=[Γ11iΓ12i0Γ14iΓ15i*Γ22i0Γ24i0**Γ33i00***Γ44i0****Γ55i]+ΛiTRi1$i+$iTRi1TΛi,
where
(25)Λi=[MiTBiT00Λ14i0MiTBiδiT00Λ24i0],$i=[-NiQi000000$22i000],Ri1=diag{F¯i(t),F¯i(t)},Λ14i=[MiTBiT0MiT0MiTBiTEi1/2000],Λ24i=[MiTBiδiT000000MiTBiδiTEi1/2],$22i=[000-NiQi0].
By Lemma1 and Schur complement formula, the condition Γi<0 in (10) is equivalent to Θi<0 in (24). By Schur complement formula with Θi<0, one can obtain Υ̿i<0 in (20). The condition Υ̿ii<0 is equivalent to Υ¯i<0. Again, by Schur complement formula with Υ¯i<0, one can obtain Υi<0. From the condition Υi<0 in (17), there exists a constant ρi>0, such that
(26)V˙(x(t),t)≤∑i=1N‍-ρi∥xi(t)∥2.
By conditions (13) and (26) and ∥Aiηi∥<1, one can conclude that system (9) with (2) and (4) is asymptotically stable. From (17) with Υi<0, one can obtain
(27)∫0∞‍V˙(x(t),t)dt=limt→∞V(x(t),t)-V(x(0),0)≤-∑i=1N‍∫0∞‍[xiT(t)Si1xi(t)+uiT(t)Si2ui(t)]dt.
Therefore, the following equalities hold:(28)limt→∞V(x(t),t)=0,∑i=1N‍∫0∞‍[xiT(t)Si1xi(t)+uiT(t)Si2ui(t)]dt≤V(x(0),0)=Ju.
This completes the proof.Remark 4.
It is obvious that for every subsystem, the correspondingΓi in (10) is an LMI with obtained matrices W¯ji(W¯ji=Wji-1) and Wij in the last inequality (i.e., the inequality Γi-1<0). Hence, the decentralized nonfragile control (4) and the guaranteed cost Ju in (11) can be obtained by finding feasible set to Γi<0 with feasp in [22] one by one.Remark 5.
Obviously, the guaranteed costJu in (11) cannot be directly optimized by using the toolbox of mincx in [22]. One reason is that inequalities (10) with variable matrices Wij and W¯ji(W¯ji=Wji-1) are not a group of LMIs but N coupled nonlinear inequalities. Another reason is that Ju is a nonconvex function with respect to the optimization variables.The following algorithm is given to solve the nonlinear problem of inequalities (10).Algorithm 6.
Choose constant matricesWij>0 and W¯ji>0 satisfying Ψjji<0 in Γi, where W¯ji=Wji-1, i,j=1,2…,N,j≠i.
It is needed to simultaneously selectN×(N-1) constant matrices Wij>0 and W¯ji>0(W¯ji=Wji-1) satisfying Ψjji<0. For simplicity, one can choose Wij and W¯ji to be positive definite diagonal matrices according to the eigenvalues of ei2Inj+ei1AijTAij due to Ψjji=ei2Inj-Wij+ei1AijTAij. The chosen entries need to be as small as possible, because
(29)11-li∑j=1,j≠iN‍∫-τij(0)0‍xjT(s)Wijxj(s)ds
is involved in Ju. However, if there is no solution to inequalities (10), the large scalars can be considered.
In the sequel, instead of solving the nonconvex optimization problem, a suboptimal method of minimizing the guaranteed costJu, based on Algorithm 6, is presented.Theorem 7.
Consider unperturbed system (9) with cost function (6), and assume ∥Aiηi∥<1. If the following optimization problem:
(30)min∑i=1N‍[αi+Tr(Ui1TΦi1Ui1+Ui2TΦi2Ui2)]
subject to LMI (10) with Algorithm 6, and
(31)[-αiϕiT(0)*-Qi0]<0,[-2Qi0+Qi1Ini*-Φi1]<0,[-2Qi0+Qi2Ini*-Φi2]<0
has a solution set (αi>0,  εi1>0,  Qik>0(k=0,1,2),  Φi1>0,  Φi2>0,  Xi), where ∫-σi(0)0‍ϕi(s)ϕiT(s)ds=Ui1Ui1T, ∫-δi(0)0‍ϕi(s)ϕiT(s)ds=Ui2Ui2T, (1/(1-gi))∫-ηi(0)0‍ϕ˙iT(s)ϕ˙i(s)ds=Li1, fi0∫-fi00‍(s+fi0)ϕ˙iT(s)ϕ˙i(s)ds=Li2, (1/(1-li))∑j=1,j≠iN∫-τij(0)0‍ϕjT(s)Wijϕj(s)ds=Li3, then control (4) with Ki=XiQi0-1 is the decentralized nonfragile guaranteed cost control of unperturbed system (9) with the minimization of the guaranteed cost Ju as follows:
(32)J*=min(∑i=1N‍[αi+Tr(Ui1TΦi1Ui1+Ui2TΦi2Ui2)])+∑i=1N‍(Li1+Li2+Li3).Proof.
Applying the Schur complement formula to LMIs (31) leads to ϕiT(0)Qi0-1ϕi(0)<αi,  -2Qi0+Qi1+Φi1-1<0,  -2Qi0+Qi2+Φi2-1<0, respectively.
Noting that [8]
(33)[Qi0-Φi1-1]Φi1[Qi0-Φi1-1]=Qi0Φi1Qi0-2Qi0+Φi1-1≥0,[Qi0-Φi2-1]Φi2[Qi0-Φi2-1]=Qi0Φi2Qi0-2Qi0+Φi2-1≥0,
the following inequalities are obtained
(34)Pi1=Qi0-1Qi1Qi0-1<Φi1,Pi2=Qi0-1Qi2Qi0-1<Φi2.
Further, one can obtain(35)∫-σi(0)0‍ϕiT(s)Pi1ϕi(s)ds=Tr(∫-σi(0)0‍ϕi(s)ϕiT(s)Pi1)ds=Tr(Ui1TPi1Ui1)≤Tr(Ui1TΦi1Ui1),∫-δi(0)0‍ϕiT(s)Pi2ϕi(s)ds=Tr(∫-δi(0)0‍ϕi(s)ϕiT(s)Pi2)ds=Tr(Ui2TPi2Ui2)≤Tr(Ui2TΦi2Ui2),11-gi∫-ηi(0)0‍ϕ˙iT(s)ϕ˙i(s)ds=Li1,fi0∫-fi00‍(s+fi0)ϕ˙iT(s)ϕ˙i(s)ds=Li2,11-li∑j=1,j≠iN‍∫-τij(0)0‍ϕjT(s)Wijϕj(s)ds=Li3.
Therefore, it follows from (11) that
(36)Ju≤∑i=1N‍[αi+Tr(Ui1TΦi1Ui1+Ui2TΦi2Ui2)]+∑i=1N‍(Li1+Li2+Li3).
The minimization of the right hand of inequality (36) implies the minimization of the guaranteed cost Ju for unperturbed system (9). This completes the proof.


## 3.2. Nonfragile Guaranteed Cost Control for Uncertain Neutral Interconnected Systems

Theorem 8.
Consider uncertain neutral interconnected systems (1) with (2), (3), and (4). If there exist positive numbers εi1>0, εi2>0, and εi3>0, some symmetric positive definite matrices Qik(k=0,1,2), W¯ji, Wij, and matrix Xi such that the following inequalities hold:
(37)Γ̿i=[ΓiΓ¯i*Γ~i]<0,(38)[-Ini+εi3DiηiTDiηiAiηiT0*-IniCi**-εi3Ini]<0,
then control (4) with Ki=XiQi0-1 is the decentralized nonfragile guaranteed cost control of uncertain neutral interconnected systems (1) with the guaranteed cost in (11), where Γ¯i=[εi2Πi1Πi2],
(39)Πi1=[Γ¯11i00Γ¯41i00],Πi2=[Γ¯12iΓ¯22i000Γ¯62i],Γ¯22i=[00Qi0DiσiT000000DiηiT000000000000-XiTDiδiT0],Γ¯11i=[CiCiCiCiCiCi],Γ¯12i=[Qi0Di1T-XiTDi2T0000],Γ¯41i=[CiCiCiCiCiCi000000000000Ei1/2Ci000000Ei1/2Ci000Ei1/2Ci00Ei1/2Ci000000Ei1/2Ci000000Ei1/2Ci0],Γ¯62i=[00000εi1MiTDi2T0000εi1MiTDiδiT0000000000000],Γ~i=diag{-εi2Ini,…,-εi2Ini︸12}.Proof.
From condition (10) with unperturbed neutral interconnected systems (9), one can obtain the corresponding condition to stabilize uncertain neutral interconnected systems (1) as follows:
(40)Σi=Γi+Πi1Ri2Πi2T+Πi2Ri2TΠi1T<0,
where Ri2=diag{Fi(t),Fi(t),Fi(t),Fi(t),Fi(t),Fi(t)}.
By Lemma1 and Schur complement formula, the condition Γ̿i<0 in (37) is equivalent to Σi<0 in (40). For the same reason, (38) is equivalent to
(41)[Aiηi+ΔAiηi(t)]T[Aiηi+ΔAiηi(t)]<Ini.
This implies that uncertain neutral interconnected systems (1) are Lipschitz in the term x˙i(t-ηi(t)) with Lipschitz constant less than 1 [8]. By the same derivation of Theorem 3, one can complete this proof.The decentralized nonfragile guaranteed cost control (4) and the minimization of the guaranteed cost Ju for uncertain neutral interconnected systems (1) are determined by the following theorem.Theorem 9.
Consider uncertain neutral interconnected systems (1) with (2), (3), (4), and cost function (6). If the following optimization problem:
(42)min∑i=1N‍[αi+Tr(Ui1TΦi1Ui1+Ui2TΦi2Ui2)]
is subject to LMI (37) with Algorithm 6, (38), and (31) has a solution set (αi>0, εi1>0, εi2>0, εi3>0, Qik>0(k=0,1,2), Φi1>0, Φi2>0, Xi), then control (4) with Ki=XiQi0-1 is the decentralized nonfragile guaranteed cost control for uncertain neutral interconnected systems (1) with the minimization J* of the guaranteed cost Ju in (32).Remark 10.
Reference [18] develops a scheme of counteracting the interconnections to simplify the problem, which may add conservatism in some cases. Compared with the approach of treating the interconnections in [18], we utilize an approach of magnifying the terms associated interconnections; for details, one can see the derivation of inequality (16). To some extent, it may reduce the conservatism of the results derived in the paper.


## 4. Illustrative Examples

In this section, some examples are presented to show the validity of the control approach and the advantages of the schemes of dealing with the interconnections.Example 1.
To illustrate the design method of the decentralized nonfragile guaranteed cost control and the optimization approach of the guaranteed cost for uncertain neutral interconnected system, consider uncertain neutral interconnected systems (1) composed of two third-order subsystems:
(43)A1=[1.122170.1414-5.12474.1437-1.12033.12432.4589-0.5671-2.2548],A1σ1=[-0.03210.0012-0.01230.1325-0.0321-0.02460.03480.00230.0236],A1η1=[0.2236-0.2011-0.03210.21340.0271-0.12820.01230.5621-0.0124],B1=[-2.1231-4.0126-1.12453.47250.1243-9.3417],B1δ1=[0.1012-0.02190.1427-0.0537-0.05310.05324],A12=[-0.08980.0161-0.0682-0.03590.0205-0.0542-0.02050.01760.0814],C1=[0.0680-0.06550.0283-0.0086-0.03810.08890.04220.00880.0366],D11=[-0.00510.04290.04640.0792-0.0749-0.0321-0.05790.09000.0946],D1σ1=[-0.02120.0481-0.0933-0.09910.08960.0941-0.0347-0.0204-0.0837],D12=[0.0928-0.0609-0.03900.0897-0.0874-0.0064],D1η1=[-0.05010.0860-0.0565-0.0311-0.09130.01850.00680.0621-0.0214],M1=[0.010.010.010.010.010.01],D1δ1=[-0.01750.00860.01720.06210.0142-0.0739],N1=diag{1,1,1},σ1(t)=0.1*(2+sin(t)),η1(t)=0.2*(1+cos(t)),δ1(t)=0.3*(1+sin(t)),τ12(t)=0.1*(1+cos(t)),A2=[8.19060.45712.5678-0.4724-4.45401.45270.4561-2.4561-5.9568],A2σ2=[0.06140.0973-0.0627-0.0819-0.0535-0.0848-0.0844-0.08950.0602],A2η2=[0.1147-0.02180.0157-0.12540.0282-0.05150.0919-0.0763-0.2169],B2=[9.7954-1.3341-7.5894-1.0482-0.0893-0.3494],B2δ2=[-0.0349-0.0189-0.0854-0.02310.03120.0993],A21=[-0.28980.3161-0.0682-0.03590.0205-0.0542-0.02050.31760.3814],C2=[0.03870.07380.0668-0.04410.0863-0.0823-0.09090.07500.0727],D21=[0.04170.03330.04830.0805-0.08880.09140.0205-0.06770.0988],D2σ2=[-0.0149-0.0275-0.0303-0.04880.0497-0.0892-0.06140.0711-0.0632],D22=[-0.0860-0.00730.03800.06730.04620.0578],D2η2=[-0.0103-0.0529-0.03890.01360.0132-0.00280.0579-0.0099-0.0987],D2δ2=[0.05660.0541-0.0304-0.03350.08940.0378],M2=M1,N2=N1,σ2(t)=0.12*(1+cos(t)),η2(t)=0.1*(2+sin(t)),δ2(t)=0.2*(1+cos(t)),τ21(t)=0.2*(2+sin(t)).
Let(44)S11=diag{0.3,0.3,0.3},S12=diag{0.5,0.5},S21=diag{0.1,0.1,0.1},S22=diag{1,1},
and give the following initial condition:
(45)ϕ1(t)=[-0.1e2t0.150.2t+0.05]T,ϕ2(t)=[0.2et-0.06+0.2t0.06]T.
According to Algorithm6, W12 and W21 are chosen as follows:
(46)W12=diag{7.7378,7.7123,7.7253},W21=diag{6.9514,9.0184,7.2122}.
Solving the optimization problem (42) subject to condition LMI (37) with Algorithm 6, (38), and (31), one can obtain
(47)K1=[-3.9594-1.2404-1.2278-0.6353-0.0710-0.6896],K2=[1.03130.05510.1222-4.53970.3862-1.8594],J*=1.3053.
The simulation results are shown in Figures 1–6 based on the above parameters. From Figures 1 and 2, one can see that the uncertain neutral systems (1) without controller are divergent. From Figures 3 and 4, one can see that the nonlinear neutral systems (1) with control law (4) are indeed well stabilized. The control signals u1(t) and u2(t) are rather smooth in Figures 5 and 6.Figure 1
State response of the first open-loop subsystem.Figure 2
State response of the second open-loop subsystem.Figure 3
State response of the first closed-loop subsystem.Figure 4
State response of the second closed-loop subsystem.Figure 5
Control signal of the first subsystem.Figure 6
Control signal of the second subsystem.Example 2.
To the best of the authors’ knowledge, the nonfragile control and optimization for neutral interconnected systems have not been studied. But in order to show the advantages of the schemes of dealing with the interconnections, the authors have to simplify the model of neutral interconnected systems (1) to compare with the existing results.
In contrast to the model of system(13a) in [18], let
(48)A1η1=0,D1η1=0,M1=0,N1=0,σ1(t)=0.3,η1(t)=0,δ1(t)=0.6,τ12(t)=0,A2η2=0,D2η2=0,M2=0,N2=0,σ2(t)=0.24,η2(t)=0,δ2(t)=0.3,τ21(t)=0,
and other parameters be the same as Example 1.
Solving the optimization problem (42) subject to condition LMI (37) with Algorithm 6 and (31), the minimization of the guaranteed cost Ju is given by
(49)J*=0.9429.
Since system (13a) in [18] is a nonlinear large-scale system, the authors choose the nonlinear vector function gij=Wijxj satisfying the assumptions in [18] and Gij=diag{1,1,1}, Dij=Eij=0. By Theorem  4.2 in [18], one can obtain the minimization J* as follows:
(50)J*=2.2627.Remark 11.
It is clear from Example2 that the minimization of the guaranteed cost provided by Theorem 9 in this paper is less than that of [18]. Viewing from this point, the results derived in this paper have the less conservatism.


## 5. Conclusion

The nonfragile guaranteed cost control and optimization are complex and challenging for uncertain interconnected systems of neutral type. In this paper, the sufficient conditions for the existence of the decentralized nonfragile guaranteed cost control for unperturbed and uncertain neutral interconnected systems are derived, which are presented in terms of coupled nonlinear inequalities. A novel algorithm is proposed to solve the nonlinear problems of coupled inequalities (10). Also, a good optimization scheme is introduced to solve the nonconvex problem of the guaranteed cost. Two numerical examples with the corresponding simulation results and the comparison results have elucidated the validity of the present control approach and the advantages of the schemes of dealing with the interconnections over the existing results in the literature.



---

*Source: 586091-2013-07-21.xml*