Observer-Based Decentralized Control for Uncertain Interconnected Systems of Neutral Type

Abstract

The observer-based decentralized control problem is investigated for a class of uncertain interconnected systems of neutral type. Using the singular value decomposition approach, a full-order observer is designed to guarantee the asymptotic stability of the error dynamic system. A novel mathematical technique is developed to solve this design problem. Sufficient condition for uncertain interconnected systems of neutral type to be asymptotic stable is established based on the singular value decomposition method. Furthermore, the desired gains of observer and controller are obtained by the explicit expressions in terms of some free parameters. Finally, an illustrative example is used to demonstrate the proposed approach, and the corresponding simulation results are given to elucidate the effectiveness.

1. Introduction

Nowadays the systems have become more and more large, and for the interconnected systems decentralized control has obvious advantage that overcomes the limitations of the traditional centralized control requiring sufficiently large communication bandwidth to exchange information between the subsystems. Therefore, the decentralized control scheme [1–11] only making use of local information available is very popular among the researchers and the engineers. In [1], an algorithm formulated within the convex optimization framework is proposed to investigate the strict dissipativity of the linear interconnected systems. A decentralized structure of dissipative state-feedback controllers is designed. In [2, 3], a decentralized adaptive output-feedback stabilizer and a decentralized 𝐿₁ adaptive controller are proposed to stabilize a class of large-scale nonlinear systems, respectively. Different from the constant delays involved in the considered system without a priori knowledge of subsystem high-frequency-gain signs in [2], the interconnected nonlinearities and unmodeled dynamics are included in the considered systems in [3]. Based on the concepts of dynamic graphs and dynamic adjacency matrix, a modeling method of a complex dynamic interconnected system is considered in [4]. In [5], a decentralized dynamic output feedback based on linear controller is proposed to robust stabilize a class of nonlinear interconnected systems coupled by nonlinear interconnections that are unknown and quadratically bounded. However, neither constant delay nor time-varying delays are considered in [4, 5].

On the other hand, the neutral system is the general form of delay system that contains the same highest order derivatives for the state vector 𝑥(𝑡), at both time 𝑡 and past time(s) 𝑡_𝑠≤𝑡. Many models of practical systems can be described by functional differential equation of neutral type [12]. Physical examples for neutral system have distributed networks [13], population ecology [14], heat exchangers, robots in contact with rigid environments [15], and so forth. In recent years, the stability analysis and robust control problems of neutral delay systems have been considered extensively (see, e.g., [16–23]). In [16–18], the stability problems of neutral systems are investigated. The difference among them is that Balasubramaniam et al. [16] focus on the stability of the neutral systems with both constant and time-varying delays using the method of nonuniformly dividing the whole delay interval into multiple segments, Rakkiyappan et al. [17] study the stability of the neutral systems with interval time-varying delays and nonlinear perturbations using the method of a new Lyapunov functional approach, and Nian et al. [18] deal with the stability of neutral systems with only a constant delay using the method of state matrix decomposition. However, the design problems of the control law have not been considered in them.

Furthermore, Ma et al. [19] develop a control method for neutral systems with a single input and some restrictions on the system matrices using a differential-difference inequality and the transformation technique. Han et al. [20] utilize a discretized Lyapunov-Krasovskii functional approach to investigate the stability of the linear neutral systems with small and nonsmall discrete delays. But its discrete application to control design yields nonlinear conditions, which may not be easily computable. In [21], the robust adaptive stabilization problem is investigated for neutral time-delay systems with uncertainties, and an adaptive scheme is introduced to estimate the bounds on uncertainties. But the matched condition is required for the disturbance vector of the considered system. It is worth pointing out that many researches on the neutral systems are often restricted to the stability analysis without controller or static state feedback control schemes.

Nowadays, observer control is also an attractive topic [24–28]. In [24], two controllers based on state feedback and observer output feedback are designed for networked systems with discrete and distributed delays subject to quantization and packet dropout. Also, a compensation scheme is proposed to deal with the effect of random packet dropout through communication network. In [25], Su et al. introduce a new model transformation for considered discrete-time T-S fuzzy systems to realize the design of dynamic output-feedback controller. Utilizing an approximation for time-varying delay state, a new comparison model is proposed. Furthermore, the 𝑙₂--𝑙_∞ filtering problem for a class of discrete-time T-S fuzzy systems with time-varying delays is studied in [26]. The anticipated full- and reduced-order filter design is cast into a convex optimization problem, which can be efficiently solved by standard numerical algorithms. In [27], an improved fuzzy observer design that has some advantages, such as the less conservatism and the satisfactory multiple performance required by the fault detection, is presented. Moreover, Messaoud et al. [28] propose a new design of functional unknown input observer for nonlinear systems. In particular, necessary and sufficient conditions for the existence of the anticipated observer are presented.

To the best of the authors’ knowledge, the observer-based decentralized control problems of uncertain interconnected systems with neutral type have not yet been investigated, which motivates the present study. In this paper, the asymptotic stabilization of a class of uncertain neutral interconnected systems with time-varying delays in state, control input, and interconnections is made. In framework of linear matrix equalities, the design of the observer and controller is formulated. Sufficiently the influence of interconnections on the system performance is taken into account, and interconnections are dealt with by flexible techniques. These strategies allow one to obtain less conservative stabilization conditions. A numerical example and the corresponding simulation results are given to illustrate the effectiveness of the proposed method of decentralized controller based on observer.

The remainder of the paper is organized as follows. The observer-based decentralized control problem formulation is described in Section 2. In Section 3, the desired gains of observer and controller are obtained by the explicit expressions in terms of some free parameters. A numerical example and the corresponding simulation results 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)−𝐴_𝑖𝜂𝑖𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)=𝐴_𝑖+Δ𝐴_𝑖(𝑡)𝑥_𝑖(𝑡)+𝐴_𝑖𝜎𝑖+Δ𝐴_𝑖𝜎𝑖(𝑡)𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+𝐵_𝑖+Δ𝐵_𝑖(𝑡)𝑢_𝑖(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗+Δ𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑦_𝑖(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡)+𝐷_𝑖𝑢_𝑖(𝑡),𝑥_𝑖(𝑡)=𝜙_𝑖(𝑡),𝑡∈[−𝑙,0],𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖(𝑡)∈ℜ^𝑛𝑖, 𝑢_𝑖(𝑡)∈ℜ^𝑚𝑖, and 𝑦_𝑖(𝑡)∈ℜ^𝑝𝑖 are the state, control input, and measurement output of the 𝑖th subsystem, respectively. 𝐴_𝑖, 𝐴_𝑖𝜎𝑖, 𝐴_𝑖𝜂𝑖, 𝐵_𝑖, 𝐴_𝑖𝑗, 𝐶_𝑖, and 𝐷_𝑖 are known constant matrices of appropriate dimensions. 𝜙_𝑖(𝑡) is the initial condition. 𝜎_𝑖(𝑡), 𝜂_𝑖(𝑡), and 𝜏_𝑖𝑗(𝑡) are the time-varying delays. Assume that there exist constants 𝑓_𝑖0, 𝑔_𝑖0, 𝑙_𝑖0, 𝑓_𝑖, 𝑔_𝑖, 𝑙_𝑖, and 𝑙 satisfying 0≤𝜎_𝑖(𝑡)≤𝑓_𝑖0,0≤𝜂_𝑖(𝑡)≤𝑔_𝑖0,0≤𝜏_𝑖𝑗(𝑡)≤𝑙_𝑖0,𝜎̇_𝑖(𝑡)≤𝑓_𝑖<1,𝜂̇_𝑖(𝑡)≤𝑔_𝑖<1,𝜏̇_𝑖𝑗(𝑡)≤𝑙_𝑖𝑓<1,𝑙=max_𝑖0,𝑔_𝑖0,𝑙_𝑖0,𝑖,𝑗=1,2…,𝑁,𝑗≠𝑖. (2)

Time-varying parametric uncertainties Δ𝐴_𝑖(𝑡), Δ𝐴_𝑖𝜎𝑖(𝑡), Δ𝐵_𝑖(𝑡), and Δ𝐴_𝑖𝑗(𝑡) are assumed to satisfy Δ𝐴_𝑖(𝑡)Δ𝐴_𝑖𝜎𝑖(𝑡)Δ𝐵_𝑖(𝑡)Δ𝐴_𝑖𝑗(𝑡)=𝑀_𝑖𝐹_𝑖(𝑡)𝑁_𝑖1𝑁_𝑖𝜎𝑖𝑁_𝑖2𝐸_𝑖𝑗, (3) where matrices 𝑀_𝑖, 𝑁_𝑖1, 𝑁_𝑖𝜎𝑖, 𝑁_𝑖2, and 𝐸_𝑖𝑗 are constant matrices of appropriate dimensions and 𝐹_𝑖(𝑡) is the unknown matrix function satisfying 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼,∀𝑡≥0. (4)

The following assumptions are made on the considered system (1).

Assumption 1. Suppose that the matrix 𝐶_𝑖 has full row rank (i.e., rank(𝐶_𝑖)=𝑝_𝑖). Then the singular value decomposition of 𝐶_𝑖 presents as follows: 𝐶_𝑖=𝑈_𝑖𝑆_𝑖0𝑉^𝑇_𝑖, (5) where 𝑆_𝑖∈ℜ^{𝑝𝑖×𝑝𝑖} is a diagonal matrix with positive elements in a decreasing order, 0∈ℜ^{𝑝𝑖×(𝑛𝑖−𝑝𝑖)} is a zero matrix, and 𝑈_𝑖∈ℜ^{𝑝𝑖×𝑝𝑖} and 𝑉_𝑖∈ℜ^{𝑛𝑖×𝑛𝑖} are unitary matrices.

Assumption 2. The matrix 𝐴_𝑖𝜂𝑖≠0 and ‖𝐴_𝑖𝜂𝑖‖<1.

Consider the following observer-based decentralized control for system (1): 𝑥̇_𝑖(𝑡)−𝐴_𝑖𝜂𝑖𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)=𝐴_𝑖𝑥_𝑖(𝑡)+𝐵_𝑖𝑢_𝑖(𝑡)+𝐴_𝑖𝜎𝑖𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐿_𝑖𝑦_𝑖(𝑡)−𝑦_𝑖(𝑡),𝑦_𝑖(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡)+𝐷_𝑖𝑢_𝑖(𝑡),𝑢_𝑖(𝑡)=−𝐾_𝑖𝑥_𝑖(𝑡), (6) where 𝑥_𝑖(𝑡)∈ℜ^𝑛𝑖 and 𝑦_𝑖(𝑡)∈ℜ^𝑝𝑖 are the state and output vectors of the observer and 𝐾_𝑖∈ℜ^{𝑚𝑖×𝑛𝑖} and 𝐿_𝑖∈ℜ^{𝑛𝑖×𝑝𝑖} are the controller gain and observer gain to be designed. System (1) with the observer-based control (6) can be rewritten as 𝑥̇_𝑖(𝑡)𝑒_𝑖̇(𝑡)=𝐴_𝑖−𝐵_𝑖𝐾_𝑖𝐿_𝑖𝐶_𝑖0𝐴_𝑖−𝐿_𝑖𝐶_𝑖𝑥_𝑖(𝑡)𝑒_𝑖(𝑡)+𝐴_𝑖𝜎𝑖00𝐴_𝑖𝜎𝑖𝑥_𝑖𝑡−𝜎_𝑖(𝑡𝑒_𝑖)𝑡−𝜎_𝑖(𝑡+𝐴_𝑖𝜂𝑖)00𝐴_𝑖𝜂𝑖𝑥̇_𝑖𝑡−𝜂_𝑖(𝑡)𝑒_𝑖̇𝑡−𝜂_𝑖(𝑡)+^𝑁_𝑗=1∑_,𝑗≠𝑖𝐴_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)_𝑁𝑗=1∑_,𝑗≠𝑖𝐴_𝑖𝑗𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+0𝑀_𝑖Δ_𝑖(𝑡), (7)where the signal 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡) is defined as the estimated error of system, Δ_𝑖(𝑡)=𝐹_𝑖(𝑡)[(𝑁_𝑖1−𝑁_𝑖2𝐾_𝑖)𝑥_𝑖(𝑡)+𝑁_𝑖1𝑒_𝑖+𝑁_𝑖𝜎𝑖𝑥_𝑖𝜎𝑖+𝑁_𝑖𝜎𝑖𝑒_𝑖𝜎𝑖+∑^𝑁_{𝑗=1,𝑗≠𝑖}𝐸_𝑖𝑗𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡))+∑^𝑁_{𝑗=1,𝑗≠𝑖}𝐸_𝑖𝑗e_𝑗(𝑡−𝜏_𝑖𝑗(𝑡))], and the uncertainty Δ_𝑖(𝑡) satisfies the following quadratic inequality:Δ^𝑇_𝑖(𝑡)Δ_𝑖(𝑡)≤𝜉^𝑇_𝑖(𝑡)Θ^𝑇_𝑖Θ_𝑖𝜉_𝑖(𝑡), (8) where 𝜉^𝑇_𝑖(𝑡)=𝑥^𝑇_𝑖(𝑡)𝑒^𝑇_𝑖(𝑡)𝑥^𝑇_𝑖𝑡−𝜎_𝑖(𝑡)𝑒^𝑇_𝑖𝑡−𝜎_𝑖(𝑡)𝑥̇^𝑇_𝑖𝑡−𝜂_𝑖(𝑡)𝑒^𝑇_𝑖̇𝑡−𝜂_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)𝑒^𝑇₁𝑡−𝜏_𝑖1(𝑡)⋯𝑒^𝑇_𝑖−1𝑡−𝜏_𝑖𝑖−1(𝑡)𝑒^𝑇_𝑖+1𝑡−𝜏_𝑖𝑖+1(𝑡)⋯𝑒^𝑇_𝑁𝑡−𝜏_𝑖𝑁(𝑡),Θ_𝑖=𝑁_𝑖1−𝑁_𝑖2𝐾_𝑖𝑁_𝑖1𝑁_𝑖𝜎𝑖𝑁_𝑖𝜎𝑖00𝐸_𝑖1⋯𝐸_𝑖𝑖−1𝐸_𝑖𝑖+1⋯𝐸_𝑖𝑁𝐸_𝑖1⋯𝐸_𝑖𝑖−1𝐸_𝑖𝑖+1⋯𝐸_𝑖𝑁. (9)

Definition 3. Consider system (1) with the observer-based control (6). System (1) is said to be robustly stabilizable by the observer-based control (6), if the closed-loop system (7) satisfying Assumptions 1 and 2 is asymptotically stable.

Lemma 4 (see [29]). For a given 𝐶_𝑖∈ℜ^{𝑝𝑖×𝑛𝑖} with rank(𝐶_𝑖)=𝑝_𝑖, assume that 𝑄_𝑖2∈ℜ^{𝑛𝑖×𝑛𝑖} is a symmetric matrix; then there exists a matrix 𝑄_𝑖2∈ℜ^{𝑝𝑖×𝑝𝑖} such that 𝐶_𝑖𝑄_𝑖2=𝑄_𝑖2𝐶_𝑖 if and only if𝑄_𝑖2=𝑉_𝑖𝑋_𝑖100𝑋_𝑖2𝑉^𝑇_𝑖, (10)
where 𝑋_𝑖1∈ℜ^{𝑝𝑖×𝑝𝑖} and 𝑋_𝑖2∈ℜ^{(𝑛𝑖−𝑝𝑖)×(𝑛𝑖−𝑝𝑖)}.

3. Main Result

Theorem 5. Consider uncertain neutral interconnected systems (1) with (2), (3), and (6). If there exists a solution 𝑄_𝑖𝑘>0(𝑘=1,3,4,5,6), 𝑋_𝑖1>0, 𝑋_𝑖2>0, 𝑊_𝑗𝑖>0, 𝐺_𝑗𝑖>0, 𝑊_𝑖𝑗>0, 𝐺_𝑖𝑗>0, 𝑅_𝑖1, and 𝑅_𝑖2 such that the following inequalities hold: Ω_𝑖<0, (11)
then system (1) is robustly stabilizable by the observer-based control (6) with 𝐾_𝑖=𝑅_𝑖1𝑄⁻¹_𝑖1 and 𝐿_𝑖=𝑅_𝑖2𝑈_𝑖𝑆_𝑖𝑋⁻¹_𝑖1𝑆_𝑖⁻¹𝑈^𝑇_𝑖, where

Ω_𝑖=Ω^𝑖₁₁Ω^𝑖₁₂Ω^𝑖₁₃0Ω^𝑖₁₅0Ω^𝑖₁₇00𝑄_𝑖10Ω^𝑖₁₁₂0Ω^𝑖₁₁₄∗Ω^𝑖₂₂0Ω^𝑖₂₄0Ω^𝑖₂₆0Ω^𝑖₂₈𝑀_𝑖0Ω^𝑖₂₁₁Ω^𝑖₂₁₂Ω^𝑖₂₁₃Ω^𝑖₂₁₄∗∗−𝑄_𝑖300000000Ω^𝑖₃₁₂0Ω^𝑖₃₁₄∗∗∗−𝑄_𝑖400000000Ω^𝑖₄₁₃Ω^𝑖₄₁₄∗∗∗∗−𝑄_𝑖5000000Ω^𝑖₅₁₂00∗∗∗∗∗−𝑄_𝑖6000000Ω^𝑖₆₁₃0∗∗∗∗∗∗Ω^𝑖₇₇0000Ω^𝑖₇₁₂0Ω^𝑖₇₁₄∗∗∗∗∗∗∗Ω^𝑖₈₈0000Ω^𝑖₈₁₃Ω^𝑖₈₁₄∗∗∗∗∗∗∗∗−𝐼000𝑀^𝑇_𝑖0∗∗∗∗∗∗∗∗∗Ω^𝑖₁₀₁₀0000∗∗∗∗∗∗∗∗∗∗Ω^𝑖₁₁₁₁000∗∗∗∗∗∗∗∗∗∗∗Ω^𝑖₁₂₁₂00∗∗∗∗∗∗∗∗∗∗∗∗Ω^𝑖₁₃₁₃0,Ω∗∗∗∗∗∗∗∗∗∗∗∗∗𝐼_𝑖11=𝐴_𝑖𝑄_𝑖1+𝑄_𝑖1𝐴^𝑇_𝑖−𝐵_𝑖𝑅_𝑖1−𝑅^𝑇_𝑖1𝐵^𝑇_𝑖+^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝑊_𝑗𝑖,Ω^𝑖₁₂=𝑅_𝑖2𝐶_𝑖,Ω^𝑖₁₃=𝐴_𝑖𝜎𝑖𝑄_𝑖3,Ω^𝑖₁₅=𝐴_𝑖𝜂𝑖Q_𝑖5,Ω^𝑖₁₁₂=𝑄_𝑖1𝐴^𝑇_𝑖−𝑅^𝑇_𝑖1𝐵^𝑇_𝑖,Ω^𝑖₁₇=𝐴_𝑖1⋯𝐴_𝑖𝑖−1𝐴_𝑖𝑖+1⋯𝐴_𝑖𝑁,Ω^𝑖₁₁₄=𝑄_𝑖1𝑁^𝑇_𝑖1−𝑅^𝑇_𝑖1𝑁^𝑇_𝑖2,Ω^𝑖₂₄=𝐴_𝑖𝜎𝑖𝑄_𝑖4,Ω^𝑖₂₂=𝐴_𝑖𝑄_𝑖2+𝑄_𝑖2𝐴^𝑇_𝑖−𝑅_𝑖2𝐶_𝑖−𝐶^𝑇_𝑖𝑅^𝑇_𝑖2+^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝐺_𝑗𝑖,Ω^𝑖₂₆=𝐴_𝑖𝜂𝑖𝑄_𝑖6,Ω^𝑖₂₈=Ω^𝑖₁₇,Ω^𝑖₂₁₁=𝑄_𝑖2,Ω^𝑖₂₁₂=𝐶^𝑇_𝑖𝑅^𝑇_𝑖2,Ω^𝑖₂₁₄=𝑄_𝑖2𝑁^𝑇_𝑖1,Ω^𝑖₂₁₃=𝑄_𝑖2𝐴^𝑇_𝑖−𝐶^𝑇_𝑖𝑅^𝑇_𝑖2,Ω^𝑖₃₁₂=𝑄_𝑖3𝐴^𝑇_𝑖𝜎𝑖,Ω^𝑖₃₁₄=𝑄_𝑖3𝑁^𝑇_𝑖𝜎𝑖,Ω^𝑖₄₁₃=𝑄_𝑖4𝐴^𝑇_𝑖𝜎𝑖,Ω^𝑖₆₁₃=𝑄_𝑖6𝐴^𝑇_𝑖𝜂𝑖,Ω^𝑖₄₁₄=𝑄_𝑖4𝑁^𝑇_𝑖𝜎𝑖,Ω^𝑖₅₁₂=𝑄_𝑖5𝐴^𝑇_𝑖𝜂𝑖,Ω^𝑖₇₇=diag−𝑊_𝑖1,…,−𝑊_𝑖𝑖−1,−𝑊_𝑖𝑖+1,…,−𝑊_𝑖𝑁,Ω^𝑖₇₁₂=Ω^𝑖₁₇^𝑇,Ω^𝑖₈₁₃=Ω^𝑖₇₁₂,Ω^𝑖₇₁₄=𝐸_𝑖1⋯𝐸_𝑖𝑖−1𝐸_𝑖𝑖+1⋯𝐸_𝑖𝑁^𝑇,Ω^𝑖₈₁₄=Ω^𝑖₇₁₄,Ω^𝑖₈₈=diag−𝐺_𝑖1,…,−𝐺_𝑖𝑖−1,−𝐺_𝑖𝑖+1,…,−𝐺_𝑖𝑁,Ω^𝑖₁₀₁₀=−1−𝑓_𝑖𝑄_𝑖3,Ω^𝑖₁₁₁₁=−1−𝑓_𝑖𝑄_𝑖4,Ω^𝑖₁₂₁₂=−1−𝑔_𝑖𝑄_𝑖5,Ω^𝑖₁₃₁₃=−1−𝑔_𝑖𝑄_𝑖6,𝑊_𝑗𝑖=𝑄_𝑖1𝑊_𝑗𝑖𝑄_𝑖1,𝐺_𝑗𝑖=𝑄_𝑖1𝐺_𝑗𝑖𝑄_𝑖1,𝑄_𝑖2=𝑉_𝑖𝑋_𝑖100𝑋_𝑖2𝑉^𝑇_𝑖,𝑖,𝑗=1,2…,𝑁,𝑗≠𝑖. (12)

Proof. According to Schur complement, inequality (11) is equivalent to the following inequality:Γ_𝑖1=Ω^𝑖₁₁Ω^𝑖₁₂Ω^𝑖₁₃0Ω^𝑖₁₅0Ω^𝑖₁₇00∗Ω^𝑖₂₂0Ω^𝑖₂₄0Ω^𝑖₂₆0Ω^𝑖₂₈𝑀_𝑖∗∗−𝑄_𝑖3000000∗∗∗−𝑄_𝑖400000∗∗∗∗−𝑄_𝑖50000∗∗∗∗∗−𝑄_𝑖6000∗∗∗∗∗∗Ω^𝑖₇₇00∗∗∗∗∗∗∗Ω^𝑖₈₈0∗∗∗∗∗∗∗∗−𝐼+Υ^𝑇_𝑖111−𝑓_𝑖𝑄⁻¹_𝑖3Υ_𝑖1+Υ^𝑇_𝑖211−𝑓_𝑖𝑄⁻¹_𝑖4Υ_𝑖2+Υ^𝑇_𝑖311−𝑔_𝑖𝑄⁻¹_𝑖5Υ_𝑖3+Υ^𝑇_𝑖411−𝑔_𝑖𝑄⁻¹_𝑖6Υ_𝑖4+Υ^𝑇_𝑖5Υ_𝑖5<0, (13)
whereΥ_𝑖1=𝑄_𝑖1^𝑇00000000,Υ_𝑖2=0Ω^𝑖₂₁₁^𝑇,Υ0000000_𝑖3=Ω^𝑖₁₁₂^𝑇Ω^𝑖₂₁₂^𝑇Ω^𝑖₃₁₂^𝑇0Ω^𝑖₅₁₂^𝑇0Ω^𝑖₇₁₂^𝑇,Υ00_𝑖4=0Ω^𝑖₂₁₃^𝑇0Ω^𝑖₄₁₃^𝑇0Ω^𝑖₆₁₃^𝑇0Ω^𝑖₈₁₃^𝑇𝑀_𝑖,Υ_𝑖5=Ω^𝑖₁₁₄^𝑇Ω^𝑖₂₁₄^𝑇Ω^𝑖₃₁₄^𝑇Ω^𝑖₄₁₄^𝑇00Ω^𝑖₇₁₄^𝑇Ω^𝑖₈₁₄^𝑇0. (14)
By Lemma 4, the conditions 𝑄_𝑖2=𝑉_𝑖𝑋_𝑖100𝑋_𝑖2𝑉^𝑇_𝑖 and 𝑄_𝑖2=𝑈_𝑖𝑆_𝑖𝑋_𝑖1𝑆_𝑖⁻¹𝑈^𝑇_𝑖 imply the condition 𝐶_𝑖𝑄_𝑖2=𝑄_𝑖2𝐶_𝑖. In view of 𝐾_𝑖=𝑅_𝑖1𝑄⁻¹_𝑖1, 𝐿_𝑖=𝑅_𝑖2𝑈_𝑖𝑆_𝑖𝑋⁻¹_𝑖1𝑆_𝑖⁻¹𝑈^𝑇_𝑖, 𝑊_𝑗𝑖=𝑄_𝑖1𝑊_𝑗𝑖𝑄_𝑖1, and 𝐺_𝑗𝑖=𝑄_𝑖1𝐺_𝑗𝑖𝑄_𝑖1, the matrix inequality (13) is equivalent to Γ_𝑖2=Ξ^𝑖₁₁Ξ^𝑖₁₂Ξ^𝑖₁₃0Ξ^𝑖₁₅0Ω^𝑖₁₇00∗Ξ^𝑖₂₂0Ξ^𝑖₂₄0Ξ^𝑖₂₆0Ω^𝑖₁₈𝑀_𝑖∗∗−𝑄_𝑖3000000∗∗∗−𝑄_𝑖400000∗∗∗∗−𝑄_𝑖50000∗∗∗∗∗−𝑄_𝑖6000∗∗∗∗∗∗Ω^𝑖₇₇00∗∗∗∗∗∗∗Ω^𝑖₈₈0∗∗∗∗∗∗∗∗−𝐼+Λ^𝑇_𝑖11−𝑔_𝑖𝑄⁻¹_𝑖5Λ_𝑖+Λ^𝑇_𝑖11−𝑔_𝑖𝑄⁻¹_𝑖6Λ_𝑖+Λ^𝑇_𝑖Λ_𝑖+𝑄_𝑖10000000011−𝑓_𝑖𝑄⁻¹_𝑖3𝑄_𝑖100000000^𝑇+0𝑄_𝑖2000000011−𝑓_𝑖𝑄⁻¹_𝑖40𝑄_𝑖20000000^𝑇<0, (15)
whereΛ_𝑖=𝐴_𝑖−𝐵_𝑖𝐾_𝑖𝑄_𝑖1𝐿_𝑖𝐶_𝑖𝑄_𝑖2𝐴_𝑖𝜎𝑖𝑄_𝑖30𝐴_𝑖𝜂𝑖𝑄_𝑖50Ω^𝑖₁₇,Λ00_𝑖=0𝐴_𝑖−𝐿_𝑖𝐶_𝑖𝑄_𝑖20𝐴_𝑖𝜎𝑖𝑄_𝑖40𝐴_𝑖𝜂𝑖𝑄_𝑖60Ω^𝑖₁₇𝑀_𝑖,Λ_𝑖=𝑁_𝑖1−𝑁_𝑖2𝐾_𝑖𝑄_𝑖1𝑁_𝑖1𝑄_𝑖2𝑁_𝑖𝜎𝑖𝑄_𝑖3𝑁_𝑖𝜎𝑖𝑄_𝑖400Ω^𝑖₇₁₄^𝑇Ω^𝑖₈₁₄^𝑇0,Ξ^𝑖₁₁=𝑄_𝑖1𝐴_𝑖−𝐵_𝑖𝐾_𝑖^𝑇+𝐴_𝑖−𝐵_𝑖𝐾_𝑖𝑄_𝑖1+^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝑊_𝑗𝑖,Ξ^𝑖₁₂=𝐿_𝑖𝐶_𝑖𝑄_𝑖2,Ξ^𝑖₁₃=𝐴_𝑖𝜎𝑖𝑄_𝑖3,Ξ^𝑖₁₅=𝐴_𝑖𝜂𝑖𝑄_𝑖5,Ξ^𝑖₂₂=𝑄_𝑖2A_𝑖−𝐿_𝑖𝐶_𝑖^𝑇+𝐴_𝑖−𝐿_𝑖𝐶_𝑖𝑄_𝑖2+^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝐺_𝑗𝑖,Ξ^𝑖₂₄=𝐴_𝑖𝜎𝑖𝑄_𝑖4,Ξ^𝑖₂₆=𝐴_𝑖𝜂𝑖𝑄_𝑖6. (16)
Pre- and postmultiplying matrix Γ_𝑖2 in (15) by Π^𝑇_𝑖 and Π_𝑖, where Π_𝑖=diag𝑄⁻¹_𝑖1,𝑄⁻¹_𝑖2,𝑄⁻¹_𝑖3,𝑄⁻¹_𝑖4,𝑄⁻¹_𝑖5,𝑄⁻¹_𝑖6𝑃,𝐼,𝐼,𝐼=diag_𝑖1,𝑃_𝑖2,𝑃_𝑖3,𝑃_𝑖4,𝑃_𝑖5,𝑃_𝑖6,𝐼,𝐼,𝐼>0, (17) we obtain Γ_𝑖3=Ψ^𝑖₁₁Ψ^𝑖₁₂Ψ^𝑖₁₃0Ψ^𝑖₁₅0Ψ^𝑖₁₇00∗Ψ^𝑖₂₂0Ψ^𝑖₂₄0Ψ^𝑖₂₆0Ψ^𝑖₂₈𝑃_𝑖2𝑀_𝑖∗∗−𝑄_𝑖3000000∗∗∗−𝑄_𝑖400000∗∗∗∗−𝑄_𝑖50000∗∗∗∗∗−𝑄_𝑖6000∗∗∗∗∗∗Ψ^𝑖₇₇00∗∗∗∗∗∗∗Ψ^𝑖₈₈0∗∗∗∗∗∗∗∗−𝐼+Φ^𝑇_𝑖11−𝑔_𝑖𝑃_𝑖5Φ_𝑖+Φ^𝑇_𝑖11−𝑔_𝑖𝑃_𝑖6Φ_𝑖+Φ^𝑇_𝑖Φ_𝑖<0, (18) where Φ_𝑖=𝐴_𝑖−𝐵_𝑖𝐾_𝑖𝐿_𝑖𝐶_𝑖𝐴_𝑖𝜎𝑖0𝐴_𝑖𝜂𝑖0Ω^𝑖₁₇,Φ00_𝑖=0𝐴_𝑖−𝐿_𝑖𝐶_𝑖0𝐴_𝑖𝜎𝑖0𝐴_𝑖𝜂𝑖0Ω^𝑖₁₇𝑀_𝑖,Φ_𝑖=𝑁_𝑖1−𝑁_𝑖2𝐾_𝑖𝑁_𝑖1𝑁_𝑖𝜎𝑖𝑁_𝑖𝜎𝑖00Φ^𝑖₇₁₄^𝑇Φ^𝑖₈₁₄^𝑇0,Ψ^𝑖₁₁=𝐴_𝑖−𝐵_𝑖𝐾_𝑖^𝑇𝑃_𝑖1+𝑃_𝑖1𝐴_𝑖−𝐵_𝑖𝐾_𝑖+^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝑊_𝑗𝑖+11−𝑓_𝑖𝑃_𝑖3,Ψ^𝑖₁₂=𝑃_𝑖1𝐿_𝑖𝐶_𝑖,Ψ^𝑖₁₃=𝑃_𝑖1𝐴_𝑖𝜎𝑖,Ψ^𝑖₁₅=𝑃_𝑖1𝐴_𝑖𝜂𝑖,Ψ^𝑖₁₇=𝑃_𝑖1Ω^𝑖₁₇,Ψ^𝑖₂₄=𝑃_𝑖2𝐴_𝑖𝜎𝑖,Ψ^𝑖₂₆=𝑃_𝑖2𝐴_𝑖𝜂𝑖,Ψ^𝑖₂₂=𝐴_𝑖−𝐿_𝑖𝐶_𝑖^𝑇𝑃_𝑖2+𝑃_𝑖2𝐴_𝑖−𝐿_𝑖𝐶_𝑖+^𝑁_{𝑗=1,𝑗≠𝑖}11−𝑙_𝑗𝐺_𝑗𝑖+11−𝑓_𝑖𝑃_𝑖4,Ψ^𝑖₂₈=𝑃_𝑖2Ω^𝑖₁₇. (19)
Construct the following Lyapunov functional candidate: 𝑉𝑥_𝑡,𝑒_𝑡=^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖1𝑥_𝑖(𝑡)+𝑒^𝑇_𝑖(𝑡)𝑃_𝑖2𝑒_𝑖(𝑡)+11−𝑓_𝑖×^𝑡_{𝑡−𝜎𝑖(𝑡)}𝑥^𝑇_𝑖(𝑠)𝑃_𝑖3𝑥_𝑖(𝑠)+𝑒^𝑇_𝑖(𝑠)𝑃_𝑖4𝑒_𝑖(𝑠+1)𝑑𝑠1−𝑔_𝑖×^𝑡_{𝑡−𝜂𝑖(𝑡)}𝑥̇^𝑇_𝑖(𝑠)𝑃_𝑖5𝑥̇_𝑖(𝑠)+𝑒^𝑇_𝑖̇(𝑠)𝑃_𝑖6𝑒_𝑖̇(𝑠+1)𝑑𝑠1−𝑙_𝑖×^𝑁_{𝑗=1,𝑗≠𝑖}^𝑡_{𝑡−𝜏𝑖𝑗(𝑡)}𝑥^𝑇_𝑗(𝑠)𝑊_𝑖𝑗𝑥_𝑗(s)+𝑒^𝑇_𝑗(𝑠)𝐺_𝑖𝑗𝑒_𝑗(𝑠)𝑑𝑠. (20)
For the system (1), the following structural identity holds: ^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). (21)
By some simple derivations, the time derivative of 𝑉(𝑥_𝑡,𝑒_𝑡) along the trajectories of (7) satisfies the following inequality: 𝑉̇𝑥_𝑡,𝑒_𝑡≤^𝑁_𝑖=1𝜁^𝑇_𝑖(𝑡)Γ_𝑖3𝜁_𝑖(𝑡), (22) where𝜁^𝑇_𝑖(𝑡)=𝜉^𝑇_𝑖(𝑡)Δ^𝑇_𝑖(𝑡). (23)
According to conditions (18) and (22), one can obtain that, for all 𝜁_𝑖≠0, 𝑉̇𝑥_𝑡,𝑒_𝑡<0. (24)
In addition, Assumption 2 guarantees system (1) is Lipschitzian in the term 𝑥̇_𝑖(𝑡−𝜂_𝑖(𝑡)) with Lipschitz constant less than 1 [12]. Therefore, by Definition 3 with conditions (20) and (24), system (7) is asymptotically stable and system (1) is robustly stabilizable by observer-based control (6). This completes the proof.

Remark 6. One of the distinctive features of this paper is that the 𝑁 inequalities are included in (11), and only one is LMI due to the existence of the interconnected matrix variables 𝑊_𝑗𝑖=𝑄_𝑖1𝑊_𝑗𝑖𝑄_𝑖1, 𝑊_𝑖𝑗, 𝐺_𝑗𝑖=𝑄_𝑖2𝐺_𝑗𝑖𝑄_𝑖2, and 𝐺_𝑖𝑗 among 𝑁 inequalities. When 𝑖=1, the corresponding equality Ω₁<0 in (11) is an LMI. Under the solvable condition of Ω₁<0, we can apply Schur complement formula to the second inequality Ω₂<0 in order to obtain the solvable LMI. The process is repeated until the last inequality. Thus, the controller gain 𝐾_𝑖 and observer gain 𝐿_𝑖 can be obtained by finding feasible set to Ω_𝑖<0 with feasp in [30].

4. Illustrative Example

Consider system (1) composed of two three-order subsystems with the following parameters: 𝐴₁=,𝐴−7.11.22.5−3.8−2.10.42.1−3.1−5.4_1𝜎1=,𝐵−0.2−0.1−0.20.20.10.10.1−0.2−0.1₁=−2.9−1.3−0.32.52.7−2.2,𝐴_1𝜂1=,𝑀−0.20.10.20.2−0.30.20.1−0.2−0.1₁=−0.2−0.1−0.40.6−0.20.2−0.1−0.30.3,𝑁₁₂=,𝐴−0.10.30.2−0.50.10.2₁₂=−2.2−1.51.1−1.11.30.20.31.1−0.5,𝑁₁₁=,𝐷0.20.70.30.2−0.5−0.20.5−0.1−0.1₁=−1.40.41.1−1.1,𝑁_1𝜎1=,𝐸0.1−0.1−0.30.2−0.8−0.2−0.50.10.1₁₂=0.30.40.2−0.3−0.3−0.1−0.10.20.1,𝜎₁(𝑡)((𝑡,𝐶=0.12+sin))₁=−2.33.91.20.22.3−2.1,𝜂₁(𝑡)((𝑡,𝜏=0.21+cos))₁₂(𝑡)=0.1(1+cos(𝑡)),𝐴₂=,𝐴−7.62.11.3−2.1−1.5−2.11.2−0.5−5.3_2𝜎2=0.1−0.1−0.2−0.20.30.10.20.30.2,𝐵₂=,𝐴−2.11.8−4.10.22.1−2.1_2𝜂2=0.10.1−0.10.30.20.40.10.20.1,𝐴₂₁=,𝑀−0.10.3−0.10.1−0.20.20.1−0.10.1₂=0.5−0.10.3−0.2−0.10.1−0.20.30.1,𝑁₂₁=,𝑁−0.10.20.1−0.1−0.30.5−0.20.1−0.5_2𝜎2=0.2−0.10.10.30.20.2−0.20.1−0.1,𝑁₂₂=,𝐸0.30.1−0.2−0.20.10.3₂₁=−0.20.2−0.10.10.10.2−0.1−0.20.1,𝐶₂=,𝐷−4.50.10.10.10.3−0.2₂=0.170.31−0.120.23,𝜎₂(𝑡)((𝑡,𝜂=0.21+cos))₂(𝑡)=0.1(2+cos(𝑡)),𝜏₂₁(𝑡)=0.2(2+sin(𝑡)). (25)

Applying Matlab toolbox to solving inequality (11), we obtain the following results: 𝑄₁₁=,𝑄1.48650.79550.42310.79552.2754−1.51400.4231−1.51405.2053₁₃=,𝑄1.43240.26460.86370.26461.1728−1.35990.8637−1.35995.0693₁₄=,𝑄1.42850.24130.65130.24131.0183−1.17820.6513−1.17824.2542₁₅=,𝑄47.92356.6463−2.81486.646344.10281.5746−2.81481.574643.3414₁₆=,𝐾40.19029.0430−2.84929.043039.54472.1458−2.84922.145841.1207₁=,𝑄1.4531−2.1903−0.7468−1.45702.45540.6993₁₂=,𝑋0.57470.2102−0.74203.6344₁₁=,𝑋0.8267−0.7426−0.74263.3824¹²=2.0431,𝑊₂₁=,𝑊6.62560.8468−2.81660.84683.7842−1.9443−2.8166−1.94437.7792₂₁=,8.0860−4.9568−2.7975−4.95683.91971.9259−2.79751.92591.2428𝐺₂₁=,𝐺3.63291.1769−1.72761.17693.4642−1.9414−1.7276−1.94147.6914₂₁=,𝑅2.2006−2.2199−1.0134−2.21994.20631.6203−1.01341.62031.4176₁₁=,𝑅−5.33503.3676−0.9899−1.58434.02580.3958₁₂=,𝑊−1.2497−0.63241.44330.67110.6835−0.8005₁₂=,𝐺32.5716−0.15600.7872−0.156033.06801.22480.78721.224832.6264₁₂=,𝐿33.16990.4384−0.01190.438433.46290.6883−0.01190.688333.2668₁=,𝑄−2.2325−0.04492.55880.03660.8420−0.2690₂₁=,𝑄0.2476−0.0193−0.1246−0.01930.1848−0.0060−0.1246−0.00600.2346₂₂=,X0.1239−0.0000−0.00000.1239₂₁=,𝑄0.1239−0.0005−0.00050.1197₂₃=,𝑅1.36380.0541−2.65500.05412.8900−0.9487−2.6550−0.948711.4203₂₂=,𝑄−0.03361.59120.07975.74630.0145−2.5460₂₄=,𝐾1.80541.0651−3.05581.06511.7394−3.5831−3.0558−3.58319.2685₂=,𝑄0.5773−8.9195−3.60703.8724−6.5449−12.5895₂₅=,𝑋416.8897−302.9290−158.6713−302.9290231.3825118.1184−158.6713118.118479.4590₂₂𝑄=0.1013,₂₆=,𝑅139.5613−92.2506−33.9039−92.250666.792618.6639−33.903918.663912.5276₂₁=,𝐿0.7642−1.6381−0.86472.6539−1.2087−3.3972₂=.−0.271212.84240.643446.37660.1172−20.5476 (26)

Under the following initial condition 𝜙₁(𝑡)=−0.2𝑒^2𝑡0.10.4𝑡−0.05,𝜙₂(𝑡)=0.2𝑒^𝑡0.2𝑡−0.10.05, (27)the simulation results are shown in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. With the observer-based control applied, the state responses of the closed-loop system are depicted in Figures 1–2. The control signals based on the observer are rather smooth in Figures 3 and 4. The output signals of the closed-loop system are shown in Figures 5 and 6. The state responses and the output signals of the observer system (6) are presented in Figures 7–10, respectively. The state responds of the error dynamic system (7) are shown in Figures 11 and 12.

Figure 1 

State responses of the first closed-loop subsystem.

Figure 2 

State responses of the second closed-loop subsystem.

Figure 3 

The control signals of the first subsystem.

Figure 4 

The control signals of the second subsystem.

Figure 5 

Output responses of the first subsystem.

Figure 6 

Output responses of the second subsystem.

Figure 7 

State responses of the first observer subsystem.

Figure 8 

State responses of the second observer subsystem.

Figure 9 

Output responses of the first observer subsystem.

Figure 10 

Output responses of the second observer subsystem.

Figure 11 

State responses of the first error subsystem.

Figure 12 

State responses of the second error subsystem.

The simulations results indicate that the designed observer can stabilize the error dynamic system and estimate the states of the interconnected systems of neutral type.

Remark 7. For this example, 𝑁=2. When 𝑖=1, we can obtain 𝑄₁₁, 𝑄₁₂, 𝑊₂₁, 𝑊₁₂, 𝐺₂₁, and 𝐺₁₂ by solving LMI Ω₁<0 in (11). Furthermore, according to 𝑊₂₁=𝑄₁₁𝑊₂₁𝑄₁₁ and 𝐺₂₁=𝑄₁₂𝐺₂₁𝑄₁₂, we have 𝑊₂₁ and 𝐺₂₁. When 𝑖=2, inequality Ω₂<0 in (11) is not LMI because of the existence of the interconnection matrix 𝑊₁₂=𝑄₂₁𝑊₁₂𝑄₂₁ and 𝐺₁₂=𝑄₂₂𝐺₁₂𝑄₂₂ (here, 𝑊₁₂ and 𝐺₁₂ are constant matrices coming from Ω₁<0). However, the transformed inequality is an LMI after applying Schur complement to Ω₂<0. Hence, the controller gain 𝐾₂ and observer gain 𝐿₂ can be obtained by finding feasible set to the transformed inequality of Ω₂<0.

5. Conclusion

The observer-based decentralized control problem of uncertain interconnected systems of neutral type is complex and challenging. In framework of Lyapunov stability theory, a novel mathematical technique to deal with the parametric disturbances is developed to obtain the sufficient conditions of existing anticipated controller and observer. The sufficient conditions are the coupled LMIs and depend on not only the sizes of delays, but also the information of derivatives. The numerical example and the corresponding simulation results elucidate that the results obtained in this paper are effective.

Acknowledgments

This research is supported by the Natural Science Foundation of China (no. 61104106), the Science Foundation of Department of Education of Liaoning Province (no. L2012422), and the Doctorial Science Foundation of Shenyang University (no. 20212340).