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

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.

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) 𝑡_𝑠≤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 𝛽_𝑗 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 matrix 𝐸_𝑖^1/2 is introduced to denote the square root matrix of symmetric positive semidefinite matrix 𝐸_𝑖≥0, that is, 𝐸_𝑖^1/2=𝑉_𝑖𝐻_𝑖^1/2𝑉^𝑇_𝑖 with 𝑉_𝑖 the eigenvector matrix of 𝐸_𝑖 satisfying 𝑉_𝑖𝑉^𝑇_𝑖=𝐼 and 𝐻_𝑖 the diagonal eigenvalues matrix of 𝐸_𝑖.

The remainder of the paper is organized as follows. The nonfragile control problem formulation is described in Section 2. 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 of 𝑁 subsystems: 𝑥̇_𝑖(𝑡)−^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)=𝐴_𝑖+Δ𝐴_𝑖(𝑡)𝑥_𝑖(𝑡)+𝐴_𝑖𝜎𝑖+Δ𝐴_𝑖𝜎𝑖(𝑡)𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+𝐴_𝑖𝜂𝑖+Δ𝐴_𝑖𝜂𝑖(𝑡)𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)+𝐵_𝑖+Δ𝐵_𝑖(𝑡)𝑢_𝑖(𝑡)+𝐵_𝑖𝛿𝑖+Δ𝐵_𝑖𝛿𝑖(𝑡)𝑢_𝑖𝑡−𝛿_𝑖(𝑡),𝑥_𝑖(𝑡)=𝜙_𝑖(𝑡),𝑡∈[−𝑙,0],𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖(𝑡)∈ℜ^𝑛𝑖 and 𝑢_𝑖(𝑡)∈ℜ^𝑚𝑖 are the state vector and the input vector of the 𝑖th subsystem, respectively. ∑^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)) is the interconnections between the 𝑖th subsystem and the other 𝑁−1 subsystems, in which 𝐴_𝑖𝑗 is known interconnected matrices of appropriate dimensions, and 𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)) implies the interconnections between the 𝑖th subsystem and the other 𝑁−1 subsystems have different time-varying delays 𝜏_𝑖𝑗(𝑡), 𝑗=1,2,…,𝑁, 𝑗≠𝑖. 𝐴_𝑖, 𝐴_𝑖𝜎𝑖, 𝐴_𝑖𝜂𝑖, 𝐵_𝑖, and 𝐵_𝑖𝛿𝑖 are known constant matrices of appropriate dimensions. 𝜙_𝑖(𝑡) is the initial condition. Assume that there exist constants 𝑓_𝑖0, 𝑔_𝑖0, ℎ_𝑖0, 𝑙_𝑖0, 𝑓_𝑖, 𝑔_𝑖, ℎ_𝑖, 𝑙_𝑖, and 𝑙 satisfying 0≤𝜎_𝑖(𝑡)≤𝑓_𝑖0,0≤𝜂_𝑖(𝑡)≤𝑔_𝑖0,0≤𝛿_𝑖(𝑡)≤ℎ_𝑖0,0≤𝜏_𝑖𝑗(𝑡)≤𝑙_𝑖0,𝜎̇_𝑖(𝑡)≤𝑓_𝑖<1,𝜂̇_𝑖(𝑡)≤𝑔_𝑖𝛿̇<1,_𝑖(𝑡)≤ℎ_𝑖<1,𝜏̇_𝑖𝑗(𝑡)≤𝑙_𝑖𝑓<1,𝑙=max_𝑖0,𝑔_𝑖0,ℎ_𝑖0,𝑙_𝑖0,𝑖,𝑗=1,2…,𝑁,𝑗≠𝑖. (2) Time-varying parametric uncertainties Δ𝐴_𝑖(𝑡), Δ𝐴_𝑖𝜎𝑖(𝑡), Δ𝐴_𝑖𝜂𝑖(𝑡), Δ𝐵_𝑖(𝑡), and Δ𝐵_𝑖𝛿𝑖(𝑡) are assumed to be of the following form: Δ𝐴_𝑖(𝑡)Δ𝐴_𝑖𝜎𝑖(𝑡)Δ𝐴_𝑖𝜂𝑖(𝑡)Δ𝐵_𝑖(𝑡)Δ𝐵_𝑖𝛿𝑖(𝑡)=𝐶_𝑖𝐹_𝑖(𝑡)𝐷_𝑖1𝐷_𝑖𝜎𝑖𝐷_𝑖𝜂𝑖𝐷_𝑖2𝐷_𝑖𝛿𝑖, (3) where 𝐶_𝑖, 𝐷_𝑖1, 𝐷_𝑖𝜎𝑖, 𝐷_𝑖𝜂𝑖, 𝐷_𝑖2, and 𝐷_𝑖𝛿𝑖 are constant matrices of appropriate dimensions, and 𝐹_𝑖(𝑡) is the unknown matrix function satisfying 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼_𝑛𝑖, for all 𝑡≥0.

Construct the following state feedback control with gain perturbations: 𝑢_𝑖(𝑡)=−𝐾_𝑖+Δ𝐾_𝑖𝑥_𝑖(𝑡), (4) where 𝐾_𝑖∈ℜ^{𝑚𝑖×𝑛𝑖} is the control gain to be designed, and Δ𝐾_𝑖 is a perturbed matrix satisfying Δ𝐾_𝑖=𝑀_𝑖𝐹_𝑖(𝑡)𝑁_𝑖, where 𝑀_𝑖 and 𝑁_𝑖 are known matrices of appropriate dimensions, and 𝐹_𝑖(𝑡) satisfies 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼_𝑚𝑖, for all 𝑡≥0; the resulting closed-loop uncertain neutral interconnected systems are obtained: 𝑥̇_𝑖(𝑡)−^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)=𝐴_𝑖−𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖+Δ𝐴_𝑖(𝑡)−Δ𝐵_𝑖(𝑡)𝐾_𝑖+Δ𝐾_𝑖𝑥_𝑖(𝑡)+𝐴_𝑖𝜎𝑖+Δ𝐴_𝑖𝜎𝑖(𝑡)𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+𝐴_𝑖𝜂𝑖+Δ𝐴_𝑖𝜂𝑖(𝑡)𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡+)−𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖−Δ𝐵_𝑖𝛿𝑖(𝑡)𝐾_𝑖+Δ𝐾_𝑖×𝑥_𝑖𝑡−𝛿_𝑖(𝑡). (5)

Define the following quadratic cost function: 𝐽=^𝑁_𝑖=1^∞₀𝑥^𝑇_𝑖(𝑡)𝑆_𝑖1𝑥_𝑖(𝑡)+𝑢^𝑇_𝑖(𝑡)𝑆_𝑖2𝑢_𝑖(𝑡)𝑑𝑡, (6) where 𝑆_𝑖1∈ℜ^{𝑛𝑖×𝑛𝑖} and 𝑆_𝑖2∈ℜ^{𝑚𝑖×𝑚𝑖} are two given symmetric positive definite matrices.

One objective of this paper is to design a control (4) and determine a scalar 𝐽_𝑢 satisfying the following two conditions:(a)the closed-loop system (5) is asymptotically stable,(b)𝐽≤𝐽_𝑢.

If the aforementioned control gain 𝐾_𝑖 and constant 𝐽_𝑢 exist, control (4) is the decentralized nonfragile guaranteed cost control and 𝐽_𝑢 is the guaranteed cost for the considered system.

The other is to find out 𝐽^∗, the minimization of the guaranteed cost 𝐽_𝑢.

Lemma 1 (see [8]). Let 𝑍, 𝑋, 𝑆, and 𝑌 be matrices of appropriate dimensions. Assuming that 𝑍 is symmetric and 𝑆^𝑇𝑆≤𝐼, then 𝑍+𝑋𝑆𝑌+𝑌^𝑇𝑆^𝑇𝑋^𝑇<0 if and only if there exists a scalar 𝜀>0 satisfying 𝑍+𝜀𝑋𝑋^𝑇+𝜀⁻¹𝑌^𝑇𝑌=𝑍+𝜀⁻¹(𝜀𝑋)(𝜀𝑋)^𝑇+𝜀⁻¹𝑌^𝑇𝑌<0. (7)

Lemma 2 (see [8]). For any constant matrix 𝑃>0 and differentiable vector function 𝑥_𝑖(𝑡) with appropriate dimensions, one has 𝑥̇^𝑡_{𝑡−𝜎𝑖(𝑡)𝑖}(𝑠)𝑑𝑠^𝑇𝑃^𝑡_{𝑡−𝜎𝑖(𝑡)}𝑥̇_𝑖(𝑠)𝑑𝑠≤𝑓_𝑖0^𝑡_{𝑡−𝜎𝑖(𝑡)}𝑥̇^𝑇_𝑖(𝑠)𝑃𝑥̇_𝑖(𝑠)𝑑𝑠≤𝑓_𝑖0^𝑡_{𝑡−𝑓𝑖0}𝑥̇^𝑇_𝑖(𝑠)𝑃𝑥̇_𝑖(𝑠)𝑑𝑠. (8)

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: 𝑥̇_𝑖(𝑡)−^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)=𝐴_𝑖𝑥_𝑖(𝑡)+𝐴_𝑖𝜎𝑖𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+𝐴_𝑖𝜂𝑖𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)+𝐵_𝑖𝑢_𝑖(𝑡)+𝐵_𝑖𝛿𝑖𝑢_𝑖𝑡−𝛿_𝑖(𝑡,𝑥_𝑖)(𝑡)=𝜙_𝑖(𝑡),𝑡∈[−𝑙,0],𝑖=1,2,…,𝑁. (9)

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 ‖𝐴_𝑖𝜂𝑖‖<1. If there exist a positive number 𝜀_𝑖1, some symmetric positive definite matrices 𝑄_𝑖𝑘(𝑘=0,1,2),𝑊_𝑗𝑖,𝑊_𝑖𝑗, and matrix 𝑋_𝑖 such that the following inequality holds: Γ_𝑖=Γ^𝑖₁₁Γ^𝑖₁₂0Γ^𝑖₁₄Γ^𝑖₁₅Γ^𝑖₁₆∗Γ^𝑖₂₂0Γ^𝑖₂₄0Γ^𝑖₂₆∗∗Γ^𝑖₃₃000∗∗∗Γ^𝑖₄₄0Γ^𝑖₄₆∗∗∗∗Γ^𝑖₅₅0∗∗∗∗∗Γ^𝑖₆₆<0; (10) then control (4) with 𝐾_𝑖=𝑋_𝑖𝑄⁻¹_𝑖0 is the decentralized nonfragile guaranteed cost control of unperturbed neutral interconnected systems (9) with the following guaranteed cost: 𝐽_𝑢=^𝑁_𝑖=1𝜙^𝑇_𝑖(0)𝑄⁻¹_𝑖0𝜙_𝑖(0)+⁰_{−𝜎𝑖(0)}𝜙^𝑇_𝑖(𝑠)𝑄⁻¹_𝑖0𝑄_𝑖1𝑄⁻¹_𝑖0𝜙_𝑖(𝑠)+1𝑑𝑠1−𝑔_𝑖⁰_{−𝜂𝑖(0)}𝜙̇^𝑇_𝑖(𝑠)𝜙̇_𝑖(𝑠)𝑑𝑠+𝑓_𝑖0⁰_−𝑓𝑖0𝑠+𝑓_𝑖0𝜙̇^𝑇_𝑖(𝑠)𝜙̇_𝑖(𝑠)+𝑑𝑠_{0−𝛿𝑖(0)}𝜙^𝑇_𝑖(𝑠)𝑄⁻¹_𝑖0𝑄_𝑖2𝑄⁻¹_𝑖0𝜙_𝑖(𝑠)+1𝑑𝑠1−𝑙_𝑖^𝑁_{𝑗=1,𝑗≠𝑖}⁰_{−𝜏𝑖𝑗(0)}𝜙^𝑇_𝑗(𝑠)𝑊_𝑖𝑗𝜙_𝑗(𝑠)𝑑𝑠, (11) whereΓ^𝑖₁₁=𝐴_𝑖𝑄_𝑖0+𝑄_𝑖0𝐴^𝑇_𝑖−𝐵_𝑖𝑋_𝑖−𝑋^𝑇_𝑖𝐵^𝑇_𝑖+𝑄_𝑖1+𝑄_𝑖2+𝐸_𝑖,Γ^𝑖₁₂=𝐴_𝑖𝜎𝑖𝑄_𝑖0𝐴_𝑖𝜂𝑖0−𝐵_𝑖𝛿𝑖𝑋_𝑖,Γ^𝑖₁₆=𝜀_𝑖1𝐵_𝑖𝑀_𝑖𝜀_𝑖1𝐵_𝑖𝛿𝑖𝑀_𝑖−𝑄_𝑖0𝑁^𝑇_𝑖0,Γ^𝑖₁₄=𝑄_𝑖0𝐴^𝑇_𝑖−𝑋^𝑇_𝑖𝐵^𝑇_𝑖𝑄_𝑖0−𝑋^𝑇_𝑖𝑄_𝑖0𝐴^𝑇_𝑖𝐸_𝑖^1/2−𝑋^𝑇_𝑖𝐵^𝑇_𝑖𝐸_𝑖^1/2,Γ000_𝑖22=diag−1−𝑓_𝑖𝑄_𝑖1,−𝐼_𝑛𝑖,−𝐼_𝑛𝑖,−1−ℎ_𝑖𝑄_𝑖2,Γ^𝑖₂₄=𝑄_𝑖0𝐴^𝑇_𝑖𝜎𝑖0000𝑄_𝑖0𝐴^𝑇_𝑖𝜎𝑖𝐸_𝑖^1/2𝐴00_{𝑇𝑖𝜂𝑖}00000𝐴^𝑇_𝑖𝜂𝑖𝐸_𝑖^1/2000000000−𝑋^𝑇_𝑖𝐵^𝑇_𝑖𝛿𝑖000000−𝑋^𝑇_𝑖𝐵^𝑇_𝑖𝛿𝑖𝐸_𝑖^1/2,Γ^𝑖₁₅=𝑄_𝑖0⋯𝑄_{𝑖0𝑁−1},Γ^𝑖₂₆=000000000000000−𝑄_𝑖0𝑁^𝑇_𝑖,𝐸_𝑖=^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝐴^𝑇_𝑖𝑗,Γ^𝑖₃₃=Ψ^𝑖₁₁⋯Ψ^𝑖_1𝑖−1Ψ^𝑖_1𝑖+1⋯Ψ^𝑖_1𝑁⋮⋱⋮⋮⋮⋮∗∗Ψ^𝑖_{𝑖−1𝑖−1}Ψ^𝑖_{𝑖−1𝑖+1}⋯Ψ^𝑖_𝑖−1𝑁∗∗∗Ψ^𝑖_𝑖+1𝑖+1⋯Ψ^𝑖_𝑖+1𝑁⋮⋮⋮⋮⋱⋮∗∗∗∗∗Ψ^𝑖_𝑁𝑁,Γ^𝑖₄₄=diag−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑆⁻¹_𝑖1,−𝑆⁻¹_𝑖2,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,Γ^𝑖₄₆=𝜀_𝑖1𝐵_𝑖𝑀_𝑖𝜀_𝑖1𝐵_𝑖𝛿𝑖𝑀_𝑖𝜀000000_𝑖1𝑀_𝑖𝜀0000000_𝑖1𝐸_𝑖^1/2𝐵_𝑖𝑀_𝑖000000000000𝜀_𝑖1𝐸_𝑖^1/2𝐵_𝑖𝛿𝑖𝑀_𝑖,𝑒00_𝑖1=11−𝑔_𝑖+𝑓²_𝑖0,𝑒_𝑖25=1+1−𝑔_𝑖+5𝑓²_𝑖0,Γ^𝑖₅₅=diag−1−𝑙₁𝑊_1𝑖,…,−1−𝑙_𝑖−1𝑊_𝑖−1𝑖,−1−𝑙_𝑖+1𝑊_𝑖+1𝑖,…,−1−𝑙_𝑁𝑊_𝑁𝑖,Γ^𝑖₆₆=diag−𝜀_𝑖1𝐼_𝑛𝑖,−𝜀_𝑖1𝐼_𝑛𝑖,−𝜀_𝑖1𝐼_𝑛𝑖,−𝜀_𝑖1𝐼_𝑛𝑖,Ψ^𝑖₁₁=𝑒_𝑖2𝐼_𝑛1−𝑊_𝑖1+𝑒_𝑖1𝐴^𝑇_𝑖1𝐴_𝑖1,Ψ^𝑖_1𝑖−1=𝑒_𝑖1𝐴^𝑇_𝑖1𝐴_𝑖𝑖−1,Ψ^𝑖_1𝑖+1=𝑒_𝑖1𝐴^𝑇_𝑖1𝐴_𝑖𝑖+1,Ψ^𝑖_1𝑁=𝑒_𝑖1𝐴^𝑇_𝑖1𝐴_𝑖𝑁,Ψ^𝑖_{𝑖−1𝑖−1}=𝑒_𝑖2𝐼_𝑛𝑖−1−𝑊_𝑖𝑖−1+𝑒_𝑖1𝐴^𝑇_𝑖𝑖−1𝐴_𝑖𝑖−1,Ψ^𝑖_{𝑖−1𝑖+1}=𝑒_𝑖1𝐴^𝑇_𝑖𝑖−1𝐴_𝑖𝑖+1,Ψ^𝑖_𝑖−1𝑁=𝑒_𝑖1𝐴^𝑇_𝑖𝑖−1𝐴_𝑖𝑁,Ψ^𝑖_𝑖+1𝑖+1=𝑒_𝑖2𝐼_𝑛𝑖+1−𝑊_𝑖𝑖+1+𝑒_𝑖1𝐴^𝑇_𝑖𝑖+1𝐴_𝑖𝑖+1,Ψ^𝑖_𝑖+1𝑁=𝑒_𝑖1𝐴^𝑇_𝑖𝑖+1𝐴_𝑖𝑁,Ψ^𝑖_𝑁𝑁=𝑒_𝑖2𝐼_𝑛𝑁−𝑊_𝑖𝑁+𝑒_𝑖1𝐴^𝑇_𝑖𝑁𝐴_𝑖𝑁,𝑊_𝑗𝑖=𝑊⁻¹_𝑗𝑖,𝑖,𝑗=1,2…,𝑁,𝑗≠𝑖. (12)

Proof. Choose 𝑃_𝑖0=𝑄⁻¹_𝑖0, 𝑃_𝑖1=𝑄⁻¹_𝑖0𝑄_𝑖1𝑄⁻¹_𝑖0, and 𝑃_𝑖2=𝑄⁻¹_𝑖0𝑄_𝑖2𝑄⁻¹_𝑖0, and construct the following Lyapunov functional: 𝑉(𝑥(𝑡))=,𝑡_𝑁𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖0𝑥_𝑖(𝑡)+^𝑡_{𝑡−𝜎𝑖(𝑡)}𝑥^𝑇_𝑖(𝑠)𝑃_𝑖1𝑥_𝑖(𝑠)+1𝑑𝑠1−𝑔_𝑖^𝑡_{𝑡−𝜂𝑖(𝑡)}𝑥̇^𝑇_𝑖(𝑠)𝑥̇_𝑖(𝑠)𝑑𝑠+𝑓_𝑖0^𝑡_{𝑡−𝑓𝑖0}𝑠−𝑡−𝑓_𝑖0𝑥̇^𝑇_𝑖(𝑠)𝑥̇_𝑖(𝑠)+𝑑𝑠_{𝑡𝑡−𝛿𝑖(𝑡)}𝑥^𝑇_𝑖(𝑠)𝑃_𝑖2𝑥_𝑖(𝑠)+1𝑑𝑠1−𝑙_𝑖^𝑁_{𝑗=1,𝑗≠𝑖}^𝑡_{𝑡−𝜏𝑖𝑗(𝑡)}𝑥^𝑇_𝑗(𝑠)𝑊_𝑖𝑗𝑥_𝑗(𝑠)𝑑𝑠. (13)
Obviously, 𝑉(𝑥(𝑡),𝑡)>0 for all 𝑥_𝑖(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 𝑉̇(𝑥(𝑡))≤,𝑡_𝑁𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖0𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖0𝑥_𝑖(𝑡)+2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖0𝐴_𝑖𝜎𝑖𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖0𝐴_𝑖𝜂𝑖𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)−2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖0𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖𝑥_𝑖(𝑡)−2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖0𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖×𝑥_𝑖𝑡−𝛿_𝑖(𝑡)+2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖0×^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑥^𝑇_𝑖(𝑡)𝑃_𝑖1𝑥_𝑖(𝑡)−1−𝑓_𝑖𝑥^𝑇_𝑖×𝑡−𝜎_𝑖(𝑡)𝑃_𝑖1𝑥_𝑖𝑡−𝜎_𝑖(𝑡+1)1−𝑔_𝑖+𝑓²_𝑖0×𝑥^𝑇_𝑖(𝑡)𝐴^𝑇_𝑖+𝑥^𝑇_𝑖𝑡−𝜎_𝑖(𝑡)𝐴^𝑇_𝑖𝜎𝑖+𝑥̇^𝑇_𝑖𝑡−𝜂_𝑖(𝑡)𝐴^𝑇_𝑖𝜂𝑖−𝑥^𝑇_𝑖(𝑡)𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖−𝑥^𝑇_𝑖×𝑡−𝛿_𝑖(𝑡)𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝛿𝑖+^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗𝑡−𝜏_𝑖𝑗(𝑡)𝐴^𝑇_𝑖𝑗×𝐴_𝑖𝑥_𝑖(𝑡)+𝐴_𝑖𝜎𝑖𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+𝐴_𝑖𝜂𝑖𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)−𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖𝑥_𝑖(𝑡)−𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖𝑥_𝑖𝑡−𝛿_𝑖(𝑡+_{𝑁𝑗=1,𝑗≠𝑖})𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑥̇^𝑇_𝑖𝑡−𝜂_𝑖(𝑡)𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)−^𝑡_{𝑡−𝜎𝑖(𝑡)}𝑥̇_𝑖(𝑠)𝑑𝑠^𝑇𝑡_{𝑡−𝜎𝑖(𝑡)}𝑥̇_𝑖(𝑠)𝑑𝑠+𝑥^𝑇_𝑖(𝑡)𝑃_𝑖2𝑥_𝑖(𝑡)−1−ℎ_𝑖𝑥^𝑇_𝑖×𝑡−𝛿_𝑖(𝑡)𝑃_𝑖2𝑥_𝑖𝑡−𝛿_𝑖(𝑡)+11−𝑙_𝑖^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗(𝑡)𝑊_𝑖𝑗𝑥_𝑗(𝑡)−^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗𝑡−𝜏_𝑖𝑗(𝑡)𝑊_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡). (14)
According to Lemma 1 and the following the fact: ^𝑁_𝑖=111−𝑙_𝑖^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗(𝑡)𝑊_𝑖𝑗𝑥_𝑗(𝑡)=^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝑊_𝑗𝑖𝑥_𝑖(𝑡), (15) one can obtain ^𝑁_𝑖=12𝑥^𝑇_𝑖(𝑡)𝑃^𝑁_{𝑖0𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+2𝑒_𝑖1×𝑥^𝑇_𝑖(𝑡)𝐴^𝑇_𝑖+𝑥^𝑇_𝑖𝑡−𝜎_𝑖(𝑡)𝐴^𝑇_𝑖𝜎𝑖+𝑥̇^𝑇_𝑖𝑡−𝜂_𝑖(𝑡)𝐴^𝑇_𝑖𝜂𝑖−𝑥^𝑇_𝑖(𝑡)×𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖−𝑥^𝑇_𝑖𝑡−𝛿_𝑖(𝑡)×𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝛿𝑖×^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑒_𝑖1×^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗𝑡−𝜏_𝑖𝑗(𝑡)𝐴^𝑇_𝑖𝑗×^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡+1)1−𝑙_𝑖^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗(𝑡)𝑊_𝑖𝑗𝑥_𝑗(𝑡)−^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗𝑡−𝜏_𝑖𝑗(𝑡)×𝑊_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)≤^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖0𝐸_𝑖𝑃_𝑖0+𝑒_𝑖1𝐴^𝑇_𝑖𝐸_𝑖𝐴_𝑖+𝑒_𝑖1𝐾_𝑖+Δ𝐾_𝑖^𝑇×𝐵^𝑇_𝑖𝐸_𝑖𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖+^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝑊_𝑗𝑖×𝑥_𝑖(𝑡)+𝑒_𝑖1𝑥^𝑇_𝑖𝑡−𝜎_𝑖(𝑡)×𝐴^𝑇_𝑖𝜎𝑖𝐸_𝑖𝐴_𝑖𝜎𝑖𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+𝑒_𝑖1𝑥^𝑇_𝑖𝑡−𝛿_𝑖(𝑡)𝐾_𝑖+Δ𝐾_𝑖^𝑇×𝐵^𝑇_𝑖𝛿𝑖𝐸_𝑖𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖𝑥_𝑖𝑡−𝛿_𝑖(𝑡)+𝑒_𝑖1𝑥̇^𝑇_𝑖𝑡−𝜂_𝑖(𝑡)𝐴^𝑇_𝑖𝜂𝑖𝐸_𝑖𝐴_𝑖𝜂𝑖𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗𝑡−𝜏_𝑖𝑗(𝑡)×𝑒_𝑖2𝐼−𝑊_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑒_𝑖1×^𝑁_{𝑗=1,𝑗≠𝑖}𝑥^𝑇_𝑗𝑡−𝜏_𝑖𝑗(𝑡)𝐴^𝑇_𝑖𝑗×^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡). (16)
Therefore, it follows from (14) and (16) that 𝑉̇(𝑥(𝑡),𝑡)+^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑆_𝑖1𝑥_𝑖(𝑡)+𝑥^𝑇_𝑖(𝑡)𝐾_𝑖+Δ𝐾_𝑖^𝑇×𝑆_𝑖2𝐾_𝑖+Δ𝐾_𝑖𝑥_𝑖(𝑡≤_𝑁𝑖=1)𝜉^𝑇_𝑖Υ_𝑖𝜉_𝑖, (17) where𝜉^𝑇_𝑖=𝑥^𝑇_𝑖(𝑡)𝑥^𝑇_𝑖𝑡−𝜎_𝑖(𝑡)𝑥̇^𝑇_𝑖𝑡−𝜂_𝑖(𝑡)∫^𝑡_{𝑡−𝜎𝑖(𝑡)}𝑥̇^𝑇_𝑖(𝑠)𝑑𝑠𝑥^𝑇_𝑖𝑡−𝛿_𝑖(𝑡)𝑥^𝑇₁𝑡−𝜏_𝑖1(𝑡)⋯𝑥^𝑇_𝑖−1𝑡−𝜏_𝑖𝑖−1(𝑡)𝑥^𝑇_𝑖+1𝑡−𝜏_𝑖𝑖+1(𝑡)⋯𝑥^𝑇_𝑁𝑡−𝜏_𝑖𝑁(𝑡),Υ_𝑖=Ω_𝑖0∗Γ^𝑖₃₃+𝐺_𝑖10𝑒_𝑖1𝐼_𝑛𝑖𝐺^𝑇_𝑖10+𝐺_𝑖20𝑆_𝑖1𝐺^𝑇_𝑖20+𝐺_𝑖30𝑆_𝑖2𝐺^𝑇_𝑖30,Ω_𝑖=Ω^𝑖₁₁Ω^𝑖₁₂Ω^𝑖₁₃0Ω^𝑖₁₅∗Ω^𝑖₂₂000∗∗Ω^𝑖₃₃00∗∗∗Ω^𝑖₄₄0∗∗∗∗Ω^𝑖₅₅,Ω^𝑖₁₂=𝑃_𝑖0𝐴_𝑖𝜎𝑖,Ω^𝑖₁₃=𝑃_𝑖0𝐴_𝑖𝜂𝑖,Ω^𝑖₄₄=−𝐼_𝑛𝑖,Ω^𝑖₁₁=𝑃_𝑖0𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖0−𝑃_𝑖0𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖−𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝑃_𝑖0+𝑃_𝑖1+𝑃_𝑖2+𝑃_𝑖0𝐸_𝑖𝑃_𝑖0+𝑒_𝑖1𝐴^𝑇_𝑖𝐸_𝑖𝐴_𝑖+𝑒_𝑖1𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝐸_𝑖𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖+^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝑊_𝑗𝑖,Ω^𝑖₁₅=−𝑃_𝑖0𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖,Ω^𝑖₂₂=−1−𝑓_𝑖𝑃_𝑖1+𝑒_𝑖1𝐴^𝑇_𝑖𝜎𝑖𝐸_𝑖𝐴_𝑖𝜎𝑖,Ω^𝑖₃₃=−𝐼_𝑛𝑖+𝑒_𝑖1𝐴^𝑇_𝑖𝜂𝑖𝐸_𝑖𝐴_𝑖𝜂𝑖,Ω^𝑖₅₅=−1−ℎ_𝑖𝑃_𝑖2+𝑒_𝑖1𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝛿𝑖𝐸_𝑖𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖,𝐺_𝑖1=𝐴_𝑖−𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖𝐴_𝑖𝜎𝑖𝐴_𝑖𝜂𝑖0−𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖^𝑇,𝐺_𝑖2=𝐼_𝑛𝑖0000^𝑇,𝐺_𝑖3=−(𝐾_𝑖+Δ𝐾_𝑖)0000^𝑇. (18)
Define Υ_𝑖=Ω_𝑖0Υ^𝑖₁₃∗Γ^𝑖₃₃0∗∗Υ^𝑖₃₃, (19) where Υ^𝑖₁₃=𝐺_𝑖1𝐺_𝑖2𝐺_𝑖3, Υ^𝑖₃₃=diag{−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑆⁻¹_𝑖1,−𝑆⁻¹_𝑖2}.
Pre- and postmultiplying the matrix Υ_𝑖 in (19) by 𝑈^𝑇_𝑖 and 𝑈_𝑖, where 𝑈_𝑖=diag{𝑄_𝑖0,𝑄_𝑖0,𝐼_𝑛𝑖,𝐼_𝑛𝑖,𝑄_𝑖0,𝐼_𝑛1,…,𝐼_𝑛𝑖−1,𝐼_𝑛𝑖+1,…,𝐼_𝑛𝑁,𝐼_𝑛𝑖,𝐼_𝑛𝑖,𝐼_𝑚𝑖}, the following matrix is obtained: Υ_𝑖=Ω_𝑖0Υ^𝑖₁₃∗Γ^𝑖₃₃0∗∗Υ^𝑖₃₃, (20) where Ω_𝑖=Ω^𝑖₁₁Ω^𝑖₁₂Ω^𝑖₁₃0Ω^𝑖₁₅∗Ω^𝑖₂₂000∗∗Ω^𝑖₃₃00∗∗∗Ω^𝑖₄₄0∗∗∗∗Ω^𝑖₅₅,Ω^𝑖₁₂=𝐴_𝑖𝜎𝑖𝑄_𝑖0,Ω^𝑖₁₃=𝐴_𝑖𝜂𝑖,Ω^𝑖₁₁=𝐴_𝑖𝑄_𝑖0+𝑄_𝑖0𝐴^𝑇_𝑖−𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖𝑄_𝑖0−𝑄_𝑖0𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖+𝑄_𝑖1+𝑄_𝑖2+𝐸_𝑖+𝑒_𝑖1𝑄_𝑖0𝐴^𝑇_𝑖𝐸_𝑖𝐴_𝑖𝑄_𝑖0+𝑒_𝑖1𝑄_𝑖0×𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝐸_𝑖𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖𝑄_𝑖0+𝑄^𝑁_{𝑖0𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝑊_𝑗𝑖𝑄_𝑖0,Ω^𝑖₁₅=−𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖𝑄_𝑖0,Ω^𝑖₂₂=−1−𝑓_𝑖𝑄_𝑖1+𝑒_𝑖1𝑄_𝑖0𝐴^𝑇_𝑖𝜎𝑖𝐸_𝑖𝐴_𝑖𝜎𝑖𝑄_𝑖0,Ω^𝑖₅₅=−1−ℎ_𝑖𝑄_𝑖2+𝑒_𝑖1𝑄_𝑖0×𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝛿𝑖𝐸_𝑖𝐵_𝑖𝛿𝑖𝐾_𝑖+Δ𝐾_𝑖𝑄_𝑖0,Υ^𝑖₁₃=𝑄_𝑖0𝐴^𝑇_𝑖−𝑄_𝑖0𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝑄_𝑖0−𝑄_𝑖0𝐾_𝑖+Δ𝐾_𝑖^𝑇𝑄_𝑖0𝐴^𝑇_𝑖𝜎𝑖𝐴00_{𝑇𝑖𝜂𝑖}00000−𝑄_𝑖0𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝛿𝑖00. (21)
Define Θ_𝑖=Ω_𝑖0Υ^𝑖₁₃Ξ_𝑖Θ^𝑖₁₅∗Γ^𝑖₃₃000∗∗Υ^𝑖₃₃00∗∗∗Θ^𝑖₄₄0∗∗∗∗Θ^𝑖₅₅, (22) where Ω_𝑖=Ω^𝑖₁₁Ω^𝑖₁₂Ω^𝑖₁₃0Ω^𝑖₁₅∗Ω^𝑖₂₂000∗∗Ω^𝑖₃₃00∗∗∗Ω^𝑖₄₄0∗∗∗∗Ω^𝑖₅₅,Ξ_𝑖=Ξ^𝑖₁₁Ξ^𝑖₁₂00000Ξ^𝑖₂₃00000Ξ^𝑖₃₄0000000000Ξ^𝑖₅₅,Θ^𝑖₁₅=Γ^𝑖₁₅0000,Θ^𝑖₄₄=diag−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,−𝑒⁻¹_𝑖1𝐼_𝑛𝑖,Θ^𝑖₅₅=Γ^𝑖₅₅,Ω^𝑖₁₁=𝐴_𝑖𝑄_𝑖0+𝑄_𝑖0𝐴^𝑇_𝑖−𝐵_𝑖𝐾_𝑖+Δ𝐾_𝑖𝑄_𝑖0−𝑄_𝑖0𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖+𝑄_𝑖1+𝑄_𝑖2+𝐸_𝑖,Ω^𝑖₂₂=−1−𝑓_𝑖𝑄_𝑖1,Ω^𝑖₃₃=−𝐼_𝑛𝑖,Ω^𝑖₅₅=−1−ℎ_𝑖𝑄_𝑖2,Ξ^𝑖₁₁=𝑄_𝑖0𝐴^𝑇_𝑖𝐸_𝑖^1/2,Ξ^𝑖₁₂=−𝑄_𝑖0𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝐸_𝑖^1/2,Ξ^𝑖₂₃=𝑄_𝑖0𝐴^𝑇_𝑖𝜎𝑖𝐸_𝑖^1/2,Ξ^𝑖₃₄=𝐴^𝑇_𝑖𝜂𝑖𝐸_𝑖^1/2,Ξ^𝑖₅₅=−𝑄_𝑖0𝐾_𝑖+Δ𝐾_𝑖^𝑇𝐵^𝑇_𝑖𝛿𝑖𝐸_𝑖^1/2. (23)
The following equality is obvious: Θ_𝑖=Γ^𝑖₁₁Γ^𝑖₁₂0Γ^𝑖₁₄Γ^𝑖₁₅∗Γ^𝑖₂₂0Γ^𝑖₂₄0∗∗Γ^𝑖₃₃00∗∗∗Γ^𝑖₄₄0∗∗∗∗Γ^𝑖₅₅+Λ^𝑇_𝑖𝑅_𝑖1$_𝑖+$^𝑇_𝑖𝑅^𝑇_𝑖1Λ_𝑖, (24) where Λ_𝑖=𝑀^𝑇_𝑖𝐵^𝑇_𝑖00Λ^𝑖₁₄0𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝛿𝑖00Λ^𝑖₂₄0,$_𝑖=−𝑁_𝑖𝑄_𝑖000000$^𝑖₂₂,𝑅000_𝑖1=diag𝐹_𝑖(𝑡),𝐹_𝑖(𝑡,Λ_𝑖14)=𝑀^𝑇_𝑖𝐵^𝑇_𝑖0𝑀^𝑇_𝑖0𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝐸_𝑖^1/2,Λ000_𝑖24=𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝛿𝑖000000𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝛿𝑖𝐸_𝑖^1/2,$^𝑖₂₂=000−𝑁_𝑖𝑄_𝑖0. (25)
By Lemma 1 and Schur complement formula, the condition Γ_𝑖<0 in (10) is equivalent to Θ_𝑖<0 in (24). By Schur complement formula with Θ_𝑖<0, one can obtain Υ_𝑖<0 in (20). The condition Υ_𝑖𝑖<0 is equivalent to Υ_𝑖<0. Again, by Schur complement formula with Υ_𝑖<0, one can obtain Υ_𝑖<0. From the condition Υ_𝑖<0 in (17), there exists a constant 𝜌_𝑖>0, such that 𝑉̇(𝑥(𝑡),𝑡)≤^𝑁_𝑖=1−𝜌_𝑖𝑥_𝑖(𝑡)². (26)
By conditions (13) and (26) and ‖𝐴_𝑖𝜂𝑖‖<1, one can conclude that system (9) with (2) and (4) is asymptotically stable. From (17) with Υ_𝑖<0, one can obtain 𝑉̇^∞₀(𝑥(𝑡),𝑡)𝑑𝑡=lim_𝑡→∞𝑉(𝑥(𝑡),𝑡)−𝑉(𝑥(0),0)≤−^𝑁_𝑖=1^∞₀𝑥^𝑇_𝑖(𝑡)𝑆_𝑖1𝑥_𝑖(𝑡)+𝑢^𝑇_𝑖(𝑡)𝑆_𝑖2𝑢_𝑖(𝑡)𝑑𝑡. (27)
Therefore, the following equalities hold: lim_𝑡→∞𝑉(𝑥(𝑡),𝑡)=0,_{𝑁𝑖=1∞0}𝑥^𝑇_𝑖(𝑡)𝑆_𝑖1𝑥_𝑖(𝑡)+𝑢^𝑇_𝑖(𝑡)𝑆_𝑖2𝑢_𝑖(𝑡(𝑥(0)))𝑑𝑡≤𝑉,0=𝐽_𝑢. (28)
This completes the proof.

Remark 4. It is obvious that for every subsystem, the corresponding Γ_𝑖 in (10) is an LMI with obtained matrices 𝑊_𝑗𝑖(𝑊_𝑗𝑖=𝑊⁻¹_𝑗𝑖) and 𝑊_𝑖𝑗 in the last inequality (i.e., the inequality Γ_𝑖−1<0). Hence, the decentralized nonfragile control (4) and the guaranteed cost 𝐽_𝑢 in (11) can be obtained by finding feasible set to Γ_𝑖<0 with 𝑓𝑒𝑎𝑠𝑝 in [22] one by one.

Remark 5. Obviously, the guaranteed cost 𝐽_𝑢 in (11) cannot be directly optimized by using the toolbox of 𝑚𝑖𝑛𝑐𝑥 in [22]. One reason is that inequalities (10) with variable matrices 𝑊_𝑖𝑗 and 𝑊_𝑗𝑖(𝑊_𝑗𝑖=𝑊⁻¹_𝑗𝑖) are not a group of LMIs but 𝑁 coupled nonlinear inequalities. Another reason is that 𝐽_𝑢 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 matrices 𝑊_𝑖𝑗>0 and 𝑊_𝑗𝑖>0 satisfying Ψ^𝑖_𝑗𝑗<0 in Γ_𝑖, where 𝑊_𝑗𝑖=𝑊⁻¹_𝑗𝑖, 𝑖,𝑗=1,2…,𝑁,𝑗≠𝑖.
It is needed to simultaneously select 𝑁×(𝑁−1) constant matrices 𝑊_𝑖𝑗>0 and 𝑊_𝑗𝑖>0(𝑊_𝑗𝑖=𝑊⁻¹_𝑗𝑖) satisfying Ψ^𝑖_𝑗𝑗<0. For simplicity, one can choose 𝑊_𝑖𝑗 and 𝑊_𝑗𝑖 to be positive definite diagonal matrices according to the eigenvalues of 𝑒_𝑖2𝐼_𝑛𝑗+𝑒_𝑖1𝐴^𝑇_𝑖𝑗𝐴_𝑖𝑗 due to Ψ^𝑖_𝑗𝑗=𝑒_𝑖2𝐼_𝑛𝑗−𝑊_𝑖𝑗+𝑒_𝑖1𝐴^𝑇_𝑖𝑗𝐴_𝑖𝑗. The chosen entries need to be as small as possible, because 11−𝑙_𝑖^𝑁_{𝑗=1,𝑗≠𝑖}⁰_{−𝜏𝑖𝑗(0)}𝑥^𝑇_𝑗(𝑠)𝑊_𝑖𝑗𝑥_𝑗(𝑠)𝑑𝑠 (29) is involved in 𝐽_𝑢. 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 cost 𝐽_𝑢, based on Algorithm 6, is presented.

Theorem 7. Consider unperturbed system (9) with cost function (6), and assume ‖𝐴_𝑖𝜂𝑖‖<1. If the following optimization problem: min^𝑁_𝑖=1𝛼_𝑖+Tr𝑈^𝑇_𝑖1Φ_𝑖1𝑈_𝑖1+𝑈^𝑇_𝑖2Φ_𝑖2𝑈_𝑖2 (30) subject to LMI (10) with Algorithm 6, and −𝛼_𝑖𝜙^𝑇_𝑖(0)∗−𝑄_𝑖0<0,−2𝑄_𝑖0+𝑄_𝑖1𝐼_𝑛𝑖∗−Φ_𝑖1<0,−2𝑄_𝑖0+𝑄_𝑖2𝐼_𝑛𝑖∗−Φ_𝑖2<0 (31) has a solution set (𝛼_𝑖>0,  𝜀_𝑖1>0,  𝑄_𝑖𝑘>0(𝑘=0,1,2),  Φ_𝑖1>0,  Φ_𝑖2>0,  𝑋_𝑖), where ∫⁰_{−𝜎𝑖(0)}𝜙_𝑖(𝑠)𝜙^𝑇_𝑖(𝑠)𝑑𝑠=𝑈_𝑖1𝑈^𝑇_𝑖1, ∫⁰_{−𝛿𝑖(0)}𝜙_𝑖(𝑠)𝜙^𝑇_𝑖(𝑠)𝑑𝑠=𝑈_𝑖2𝑈^𝑇_𝑖2, (1/(1−𝑔_𝑖))∫⁰_{−𝜂𝑖(0)}𝜙̇^𝑇_𝑖(𝑠)𝜙̇_𝑖(𝑠)𝑑𝑠=𝐿_𝑖1, 𝑓_𝑖0∫⁰_−𝑓𝑖0(𝑠+𝑓_𝑖0)𝜙̇^𝑇_𝑖(𝑠)𝜙̇_𝑖(𝑠)𝑑𝑠=𝐿_𝑖2, (1/(1−𝑙_𝑖))∑^𝑁_{𝑗=1,𝑗≠𝑖}∫⁰_{−𝜏𝑖𝑗(0)}𝜙^𝑇_𝑗(𝑠)𝑊_𝑖𝑗𝜙_𝑗(𝑠)𝑑𝑠=𝐿_𝑖3, then control (4) with 𝐾_𝑖=𝑋_𝑖𝑄⁻¹_𝑖0 is the decentralized nonfragile guaranteed cost control of unperturbed system (9) with the minimization of the guaranteed cost 𝐽_𝑢 as follows: 𝐽^∗=min^𝑁_𝑖=1𝛼_𝑖+Tr𝑈^𝑇_𝑖1Φ_𝑖1𝑈_𝑖1+𝑈^𝑇_𝑖2Φ_𝑖2𝑈_𝑖2+^𝑁_𝑖=1𝐿_𝑖1+𝐿_𝑖2+𝐿_𝑖3. (32)

Proof. Applying the Schur complement formula to LMIs (31) leads to 𝜙^𝑇_𝑖(0)𝑄⁻¹_𝑖0𝜙_𝑖(0)<𝛼_𝑖,  −2𝑄_𝑖0+𝑄_𝑖1+Φ⁻¹_𝑖1<0,  −2𝑄_𝑖0+𝑄_𝑖2+Φ⁻¹_𝑖2<0, respectively.
Noting that [8] 𝑄_𝑖0−Φ⁻¹_𝑖1Φ_𝑖1𝑄_𝑖0−Φ⁻¹_𝑖1=𝑄_𝑖0Φ_𝑖1𝑄_𝑖0−2𝑄_𝑖0+Φ⁻¹_𝑖1𝑄≥0,_𝑖0−Φ⁻¹_𝑖2Φ_𝑖2𝑄_𝑖0−Φ⁻¹_𝑖2=𝑄_𝑖0Φ_𝑖2𝑄_𝑖0−2𝑄_𝑖0+Φ⁻¹_𝑖2≥0, (33) the following inequalities are obtained 𝑃_𝑖1=𝑄⁻¹_𝑖0𝑄_𝑖1𝑄⁻¹_𝑖0<Φ_𝑖1,𝑃_𝑖2=𝑄⁻¹_𝑖0𝑄_𝑖2𝑄⁻¹_𝑖0<Φ_𝑖2. (34)
Further, one can obtain ⁰_{−𝜎𝑖(0)}𝜙^𝑇_𝑖(𝑠)𝑃_𝑖1𝜙_𝑖(𝑠)𝑑𝑠=Tr⁰_{−𝜎𝑖(0)}𝜙_𝑖(𝑠)𝜙^𝑇_𝑖(𝑠)𝑃_𝑖1𝑈𝑑𝑠=Tr^𝑇_𝑖1𝑃_𝑖1𝑈_𝑖1≤Tr𝑈^𝑇_𝑖1Φ_𝑖1𝑈_𝑖1,_{0−𝛿𝑖(0)}𝜙^𝑇_𝑖(𝑠)𝑃_𝑖2𝜙_𝑖(𝑠)𝑑𝑠=Tr⁰_{−𝛿𝑖(0)}𝜙_𝑖(𝑠)𝜙^𝑇_𝑖(𝑠)𝑃_𝑖2𝑈𝑑𝑠=Tr^𝑇_𝑖2𝑃_𝑖2𝑈_𝑖2≤Tr𝑈^𝑇_𝑖2Φ_𝑖2𝑈_𝑖2,11−𝑔_𝑖⁰_{−𝜂𝑖(0)}𝜙̇^𝑇_𝑖(𝑠)𝜙̇_𝑖(𝑠)𝑑𝑠=𝐿_𝑖1,𝑓_𝑖0⁰_−𝑓𝑖0𝑠+𝑓_𝑖0𝜙̇^𝑇_𝑖(𝑠)𝜙̇_𝑖(𝑠)𝑑𝑠=𝐿_𝑖2,11−𝑙_𝑖^𝑁_{𝑗=1,𝑗≠𝑖}⁰_{−𝜏𝑖𝑗(0)}𝜙^𝑇_𝑗(𝑠)𝑊_𝑖𝑗𝜙_𝑗(𝑠)𝑑𝑠=𝐿_𝑖3. (35)
Therefore, it follows from (11) that 𝐽_𝑢≤^𝑁_𝑖=1𝛼_𝑖+Tr𝑈^𝑇_𝑖1Φ_𝑖1𝑈_𝑖1+𝑈^𝑇_𝑖2Φ_𝑖2𝑈_𝑖2+^𝑁_𝑖=1𝐿_𝑖1+𝐿_𝑖2+𝐿_𝑖3. (36)
The minimization of the right hand of inequality (36) implies the minimization of the guaranteed cost 𝐽_𝑢 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 𝜀_𝑖1>0, 𝜀_𝑖2>0, and 𝜀_𝑖3>0, some symmetric positive definite matrices 𝑄_𝑖𝑘(𝑘=0,1,2), 𝑊_𝑗𝑖, 𝑊_𝑖𝑗, and matrix 𝑋_𝑖 such that the following inequalities hold: Γ_𝑖=Γ_𝑖Γ_𝑖∗Γ_𝑖<0,−𝐼_𝑛𝑖+𝜀_𝑖3𝐷^𝑇_𝑖𝜂𝑖𝐷_𝑖𝜂𝑖𝐴^𝑇_𝑖𝜂𝑖0∗−𝐼_𝑛𝑖𝐶_𝑖∗∗−𝜀_𝑖3𝐼_𝑛𝑖<0, (37)(38) then control (4) with 𝐾_𝑖=𝑋_𝑖𝑄⁻¹_𝑖0 is the decentralized nonfragile guaranteed cost control of uncertain neutral interconnected systems (1) with the guaranteed cost in (11), where Γ_𝑖=𝜀_𝑖2Π_𝑖1Π_𝑖2, Π_𝑖1=Γ^𝑖₁₁00Γ^𝑖₄₁00,Π_𝑖2=Γ^𝑖₁₂Γ^𝑖₂₂000Γ^𝑖₆₂,Γ^𝑖₂₂=00𝑄_𝑖0𝐷^𝑇_𝑖𝜎𝑖000000𝐷^𝑇_𝑖𝜂𝑖000000000000−𝑋^𝑇_𝑖𝐷^𝑇_𝑖𝛿𝑖0,Γ^𝑖₁₁=𝐶_𝑖𝐶_𝑖𝐶_𝑖𝐶_𝑖𝐶_𝑖𝐶_𝑖,Γ^𝑖₁₂=𝑄_𝑖0𝐷^𝑇_𝑖1−𝑋^𝑇_𝑖𝐷^𝑇_𝑖20000,Γ^𝑖₄₁=𝐶_𝑖𝐶_𝑖𝐶_𝑖𝐶_𝑖𝐶_𝑖𝐶_𝑖𝐸000000000000_𝑖1/2𝐶_𝑖000000𝐸_𝑖^1/2𝐶_𝑖000𝐸_𝑖^1/2𝐶_𝑖00𝐸_𝑖^1/2𝐶_𝑖000000𝐸_𝑖^1/2𝐶_𝑖000000𝐸_𝑖^1/2𝐶_𝑖0,Γ^𝑖₆₂=00000𝜀_𝑖1𝑀^𝑇_𝑖𝐷^𝑇_𝑖20000𝜀_𝑖1𝑀^𝑇_𝑖𝐷^𝑇_𝑖𝛿𝑖0,Γ000000000000_𝑖=diag−𝜀_𝑖2𝐼_𝑛𝑖,…,−𝜀_𝑖2𝐼_𝑛𝑖12. (39)

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: Σ_𝑖=Γ_𝑖+Π_𝑖1𝑅_𝑖2Π^𝑇_𝑖2+Π_𝑖2𝑅^𝑇_𝑖2Π^𝑇_𝑖1<0, (40) where 𝑅_𝑖2=diag{𝐹_𝑖(𝑡),𝐹_𝑖(𝑡),𝐹_𝑖(𝑡),𝐹_𝑖(𝑡),𝐹_𝑖(𝑡),𝐹_𝑖(𝑡)}.
By Lemma 1 and Schur complement formula, the condition Γ_𝑖<0 in (37) is equivalent to Σ_𝑖<0 in (40). For the same reason, (38) is equivalent to 𝐴_𝑖𝜂𝑖+Δ𝐴_𝑖𝜂𝑖(𝑡)^𝑇𝐴_𝑖𝜂𝑖+Δ𝐴_𝑖𝜂𝑖(𝑡)<𝐼_𝑛𝑖. (41)
This implies that uncertain neutral interconnected systems (1) are Lipschitz in the term 𝑥̇_𝑖(𝑡−𝜂_𝑖(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 𝐽_𝑢 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: min^𝑁_𝑖=1𝛼_𝑖+Tr𝑈^𝑇_𝑖1Φ_𝑖1𝑈_𝑖1+𝑈^𝑇_𝑖2Φ_𝑖2𝑈_𝑖2 (42) is subject to LMI (37) with Algorithm 6, (38), and (31) has a solution set (𝛼_𝑖>0, 𝜀_𝑖1>0, 𝜀_𝑖2>0, 𝜀_𝑖3>0, 𝑄_𝑖𝑘>0(𝑘=0,1,2), Φ_𝑖1>0, Φ_𝑖2>0, 𝑋_𝑖), then control (4) with 𝐾_𝑖=𝑋_𝑖𝑄⁻¹_𝑖0 is the decentralized nonfragile guaranteed cost control for uncertain neutral interconnected systems (1) with the minimization 𝐽^∗ of the guaranteed cost 𝐽_𝑢 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: 𝐴₁=,𝐴1.122170.1414−5.12474.1437−1.12033.12432.4589−0.5671−2.2548_1𝜎1=,𝐴−0.03210.0012−0.01230.1325−0.0321−0.02460.03480.00230.0236_1𝜂1=,𝐵0.2236−0.2011−0.03210.21340.0271−0.12820.01230.5621−0.0124₁=,𝐵−2.1231−4.0126−1.12453.47250.1243−9.3417_1𝛿1=,𝐴0.1012−0.02190.1427−0.0537−0.05310.05324₁₂=,𝐶−0.08980.0161−0.0682−0.03590.0205−0.0542−0.02050.01760.0814₁=,𝐷0.0680−0.06550.0283−0.0086−0.03810.08890.04220.00880.0366₁₁=,𝐷−0.00510.04290.04640.0792−0.0749−0.0321−0.05790.09000.0946_1𝜎1=,𝐷−0.02120.0481−0.0933−0.09910.08960.0941−0.0347−0.0204−0.0837₁₂=,𝐷0.0928−0.0609−0.03900.0897−0.0874−0.0064_1𝜂1=,𝑀−0.05010.0860−0.0565−0.0311−0.09130.01850.00680.0621−0.0214₁=,𝐷0.010.010.010.010.010.01_1𝛿1=,𝑁−0.01750.00860.01720.06210.0142−0.0739₁{},𝜎=diag1,1,1₁(𝑡)((𝑡,𝜂=0.1∗2+sin))₁(𝑡)((𝑡,𝛿=0.2∗1+cos))₁(𝑡)((𝑡,𝜏=0.3∗1+sin))₁₂(𝑡)((𝑡,𝐴=0.1∗1+cos))₂=,𝐴8.19060.45712.5678−0.4724−4.45401.45270.4561−2.4561−5.9568_2𝜎2=,𝐴0.06140.0973−0.0627−0.0819−0.0535−0.0848−0.0844−0.08950.0602_2𝜂2=,𝐵0.1147−0.02180.0157−0.12540.0282−0.05150.0919−0.0763−0.2169₂=,𝐵9.7954−1.3341−7.5894−1.0482−0.0893−0.3494_2𝛿2=,𝐴−0.0349−0.0189−0.0854−0.02310.03120.0993₂₁=,𝐶−0.28980.3161−0.0682−0.03590.0205−0.0542−0.02050.31760.3814₂=,𝐷0.03870.07380.0668−0.04410.0863−0.0823−0.09090.07500.0727₂₁=,𝐷0.04170.03330.04830.0805−0.08880.09140.0205−0.06770.0988_2𝜎2=,𝐷−0.0149−0.0275−0.0303−0.04880.0497−0.0892−0.06140.0711−0.0632₂₂=,𝐷−0.0860−0.00730.03800.06730.04620.0578_2𝜂2=,𝐷−0.0103−0.0529−0.03890.01360.0132−0.00280.0579−0.0099−0.0987_2𝛿2=,𝑀0.05660.0541−0.0304−0.03350.08940.0378₂=𝑀₁,𝑁₂=𝑁₁,𝜎₂(𝑡)((𝑡,𝜂=0.12∗1+cos))₂(𝑡)((𝑡,𝛿=0.1∗2+sin))₂(𝑡)((𝑡,𝜏=0.2∗1+cos))₂₁(𝑡)=0.2∗(2+sin(𝑡)). (43)
Let 𝑆₁₁{},𝑆=diag0.3,0.3,0.3₁₂{},𝑆=diag0.5,0.5₂₁{},𝑆=diag0.1,0.1,0.1₂₂=diag{1,1}, (44) and give the following initial condition: 𝜙₁(𝑡)=−0.1𝑒^2𝑡0.150.2𝑡+0.05^𝑇,𝜙₂(𝑡)=0.2𝑒^𝑡−0.06+0.2𝑡0.06^𝑇. (45)
According to Algorithm 6, 𝑊₁₂ and 𝑊₂₁ are chosen as follows: 𝑊₁₂{},𝑊=diag7.7378,7.7123,7.7253₂₁=diag{6.9514,9.0184,7.2122}. (46)
Solving the optimization problem (42) subject to condition LMI (37) with Algorithm 6, (38), and (31), one can obtain 𝐾₁=,𝐾−3.9594−1.2404−1.2278−0.6353−0.0710−0.6896₂=,𝐽1.03130.05510.1222−4.53970.3862−1.8594_∗=1.3053. (47) 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 𝐴_1𝜂1=0,𝐷_1𝜂1𝑀=0,₁=0,𝑁₁𝜎=0,₁(𝑡)=0.3,𝜂₁(𝑡)𝛿=0,₁(𝑡)=0.6,𝜏₁₂(𝑡)𝐴=0,_2𝜂2=0,𝐷_2𝜂2𝑀=0,₂=0,𝑁₂=0,𝜎₂(𝑡)𝜂=0.24,₂(𝑡)=0,𝛿₂(𝑡)=0.3,𝜏₂₁(𝑡)=0, (48) 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 𝐽_𝑢 is given by 𝐽^∗=0.9429. (49)
Since system (13a) in [18] is a nonlinear large-scale system, the authors choose the nonlinear vector function 𝑔_𝑖𝑗=𝑊_𝑖𝑗𝑥_𝑗 satisfying the assumptions in [18] and 𝐺_𝑖𝑗=diag{1,1,1}, 𝐷_𝑖𝑗=𝐸_𝑖𝑗=0. By Theorem  4.2 in [18], one can obtain the minimization 𝐽^∗ as follows: 𝐽^∗=2.2627. (50)

Remark 11. It is clear from Example 2 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.

Acknowledgments

This research is supported by Natural Science Foundation of China (no. 61104106), Science Foundation of Department of Education of Liaoning Province (no. L2012422), and Start-up Fund for Doctors of Shenyang University (no. 1120212340).