Decentralized Control of Uncertain Fuzzy Large-Scale System with Time Delay and Optimization

Abstract

This paper studies the decentralized stabilization problem for an uncertain fuzzy large-scale system with time delays. The considered large-scale system is composed of several T-S fuzzy subsystems. The decentralized parallel distributed compensation (PDC) fuzzy control for each subsystem is designed to stabilize the whole system. Based on Lyapunov criterion, some sufficient conditions are proposed. Moreover, the positive definite matrices 𝑃_𝑖 and PDC gain 𝐾_𝑖𝑗 can be solved by linear matrix inequality (LMI) toolbox of Matlab. Then, the optimization design method for decentralized control is also considered with respect to a quadratic performance index. Finally, numerical examples are given and compared with those of Zhang et al., 2004 to illustrate the effectiveness and less conservativeness of our method.

1. Introduction

Many real-life problems, such as power system, economic systems, societal system, and nuclear system are frequently of high dimension. Such systems are regarded as large-scale system. They consist of a number of subsystems which serve particular functions, share resources, and are governed by a set of interrelated goals and constrains [1]. Over the past decades, many methods have been to investigate the stability and stabilization of large-scale system [2–10].

Fuzzy systems of Takagi-Sugeno (T-S) models [11] have become an effective method to represent complex nonlinear dynamic system by fuzzy sets and fuzzy reasoning. The method of T-S model is feasible since, in many situations, human experts can provide linguistic descriptions of local systems in terms of IF-THEN rules [12–14]. Reference [15] proposed a control concept “parallel distributed compensation” (PDC) for fuzzy controller design of T-S fuzzy model. Under some conditions, PDC can stabilize the closed-loop fuzzy system asymptotically. Linear matrix inequalities (LMIs) methods to find the common positive matrix 𝑃_𝑖 always play a key role work in PDC design [16].

Now let us consider the control problem of large-scale system. Accordingly, suppose a large-scale system is composed of a number of subsystems with interconnection. Each subsystem is described by a T-S fuzzy model. That is, each subsystem dynamic is captured by a set of fuzzy implications that characterize local relations in the state space. Then, the global model of large-scale system can be achieved by smoothly connecting the local linear model in each fuzzy subspace together via the membership functions. Recently, the fuzzy decentralized control design methods and stability condition are addressed [17–24]. Wang and Luoh [17] have studied a fuzzy decentralized control design method for a fuzzy large-scale system on the assumption that all variables are available. Tseng and Chen [18] dealt with the model reference tracking control problem by using 𝐻_∞ decentralized fuzzy control, which relaxed the condition that the state variables are measurable. Wang and Zhang [19] considered the robust decentralized controller design method for nonlinear large-scale descriptor system. Hsiao et al. [20] proposed fuzzy decentralized control design methods for fuzzy large-scale system with time delays and gave the analysis of the closed-loop fuzzy large-scale system. Robust decentralized 𝐻_∞ output feedback controller was designed in [21]. Tong and Zhao [22] proposed a stabilization criterion of continuous-time interconnected fuzzy systems without uncertainties. Su and Liu [23] dealt with decentralized stabilization problem for a large-scale system in which the system is composed of several T-S fuzzy subsystems with nonlinear interconnections. Wang et al. [24] addressed robust 𝐻_∞ fuzzy controller design to overcome the parametric uncertainties of fuzzy large-scale systems without time delay and get 𝐻_∞ performance. Zhang et al. [25] proposed the stabilization problem of fuzzy large-scale system without parametric uncertainties.

However, the proposed decentralized control designs and the sufficient conditions of closed-loop system did not consider the parametric uncertainties or time delays, which is important in both theory and real-world application. In this paper, the stabilization problem of the uncertain fuzzy large-scale systems with time delays is considered. Based on decentralized control concept, we like to synthesize a PDC controller for each subsystem so that the whole system can be stabilized asymptotically. When applied to degenerated cases, (without uncertainties or time delay), the stabilization criteria are better than existing ones. Stability is one of the most important performance indecies, but it is not enough for control systems. Linear quadratic performance can reflect a lot of performance requirements, so a quadratic performance index is considered in this paper. Based on Lyapunov function stability theory, the optimization design method for decentralized control with respect to the quadratic performance index is transformed into solving linear inequality matrix by PDC controller.

This paper is organized as follows. In Section 2, the considered systems are stated and some preliminaries are presented.The problem of stabilization for fuzzy large-scale system with uncertainty is proposed in Section 3. Section 4 provides examples to illustrate the correctness of our theoretic results. Finally, a conclusion is given in Section 5.

2. System Description and Preliminaries

Consider a fuzzy large-scale system 𝑆, with uncertainties both in isolated subsystem and in their interconnection, which consists of 𝐽 interconnected subsystems 𝑆_𝑖. The 𝑖th fuzzy subsystem is described by the following equations: 𝑥̇_𝑖(𝑡)=^𝑟𝑖_𝑗=1ℎ_𝑖𝑗(𝑡)𝐴_𝑖𝑗+Δ𝐴_𝑖𝑗(𝑡)𝑥_𝑖(𝑡)+𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑡)𝑢_𝑖(𝑡+_𝐽𝑘=1)𝐶_𝑘𝑖+Δ𝐶_𝑘𝑖(𝑡)𝑥_𝑘(𝑡)+𝐷_𝑘𝑖+Δ𝐷_𝑘𝑖(𝑡)𝑥_𝑘𝑡−𝜏_𝑘𝑖, (2.1) where Δ𝐴_𝑖𝑗(𝑡),Δ𝐵_𝑖𝑗(𝑡)=𝐸_𝑖𝑗𝐹_𝑖𝑗(𝑡)𝐺_𝑖𝑗,𝐻_𝑖𝑗,Δ𝐶_𝑘𝑖(𝑡),Δ𝐷_𝑘𝑖(𝑡)=𝐸_𝑘𝑖𝐹_𝑘𝑖(𝑡)𝐿_𝑘𝑖,𝑀_𝑘𝑖 (2.2) for for all 𝑡≥0,𝑗=1,2,…,𝑟_𝑖,𝑘=1,2,…,𝐽, 𝑥_𝑖(𝑡) is the state vector, 𝑢_𝑖(𝑡) is the control input, 𝐴_𝑖𝑗,𝐵_𝑖𝑗,𝐺_𝑖𝑗,𝐻_𝑖𝑗,𝐸_𝑖𝑗,𝐶_𝑘𝑖,𝐷_𝑘𝑖,𝐸_𝑘𝑖,𝑀_𝑘𝑖,𝐿_𝑘𝑖 are known constant matrices with appropriate dimensions. 𝜏_𝑘𝑖 is the time delay. ℎ_𝑖𝑗(𝑡) is the normalized weigh in (2.6). 𝐹_𝑖𝑗(𝑡) and 𝐹_𝑘𝑖(𝑡) are time-varying matrices with appropriate dimensions satisfying 𝐹_𝑖𝑗(𝑡)𝐹^𝑇_𝑖𝑗(𝑡)≤𝐼,𝐹_𝑘𝑖(𝑡)𝐹^𝑇_𝑘𝑖(𝑡)≤𝐼. (2.3) Each isolated subsystem 𝑆_𝑖 is represented by a T-S fuzzy model. The 𝑗th rule of this T-𝑆 fuzzy model is represented as follows.

Rule 𝑗. If 𝑧_1𝑖(𝑡) is 𝑀_𝑗1𝑖 and … and 𝑧_𝑝𝑖(𝑡) is 𝑀_𝑗𝑝𝑖, then 𝑥̇_𝑖(𝑡)=𝐴_𝑖𝑗+Δ𝐴_𝑖𝑗(𝑡)𝑥_𝑖(𝑡)+𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑡)𝑢_𝑖(𝑡), (2.4) where 𝑧_𝑖(𝑡)=[𝑧_1𝑖(𝑡),𝑧_2𝑖(𝑡),…,𝑧_𝑝𝑖(𝑡)]^𝑇,𝑧_1𝑖,…,𝑧_{𝑝𝑖(𝑡)} are premise variables, and 𝑀_𝑗𝑙𝑖(𝑙=1,2,…,𝑝) are fuzzy sets. By “fuzzy blending,” the final output of the 𝑖th fuzzy subsystem is described as follows: 𝑥̇_𝑖(𝑡)=∑^𝑟𝑖_𝑗=1𝜔_𝑖𝑗(𝑡)𝐴_𝑖𝑗+Δ𝐴_𝑖𝑗(𝑡)𝑥_𝑖(𝑡)+𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑡)𝑢_𝑖(𝑡)∑^𝑟𝑖_𝑗=1𝜔_𝑖𝑗(𝑡)=^𝑟𝑖_𝑗=1ℎ_𝑖𝑗(𝑡)𝐴_𝑖𝑗+Δ𝐴_𝑖𝑗(𝑡)𝑥_𝑖(𝑡)+𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑡)𝑢_𝑖(𝑡) (2.5) with 𝜔_𝑖𝑗(𝑡)=^𝑝_𝑙=1𝑀_𝑗𝑙𝑖𝑧_𝑙𝑖(𝑡),ℎ_𝑖𝑗(𝑡)=𝜔_𝑖𝑗(𝑡)∑^𝑟𝑖_𝑗=1𝜔_𝑖𝑗(𝑡), (2.6) where 𝑀_𝑗𝑖𝑙(𝑧_𝑖𝑙(𝑡)) is the grade of membership of 𝑧_𝑖𝑙(𝑡) in 𝑀_𝑗𝑖𝑙, and 𝑟_𝑖 is the number of fuzzy rules of subsystem 𝑆_𝑖. We assumed that 𝜔_𝑖𝑗(𝑡)≥0 for all 𝑡, 𝑗=1,2,…,𝑟_𝑖. Therefore ^𝑟𝑖_𝑗=1ℎ_𝑖𝑗(𝑡)=1. (2.7)

Define a quadratic performance index 𝐽=^𝐽_𝑖=1^∞₀𝑥^𝑇_𝑖(𝑡)𝑄_𝑖𝑥_𝑖(𝑡)+𝑢^𝑇_𝑖(𝑡)𝑅_𝑖𝑢_𝑖(𝑡)𝑑𝑡, (2.8) where 𝑄_𝑖 and 𝑅_𝑖 are positive matrices.

The main propose of this paper is to synthesize a decentralized PDC fuzzy controller 𝑢_𝑖(𝑡) for each subsystem such that the closed-loop large-scale T-S fuzzy systems (2.1) is asymptotically stable and the optimization design method for decentralized control respect to the quadratic performance index.

Before starting the main results, we need the following lemmas.

Lemma 2.1 (see [26]). Let 𝑄 be any of 𝑛×𝑛 matrix; one will have for any constant 𝑘≥0; and any positive matrix 𝑆>0 that 2𝑥^𝑇𝑄𝑦≤𝑘𝑥^𝑇𝑄𝑆⁻¹𝑄^𝑇1𝑥+𝑘𝑦^𝑇𝑆𝑦 (2.9) for all 𝑥,𝑦∈𝑅^𝑛.

Lemma 2.2 (see [26]). Let 𝐷,𝐸 be any constant matrices and 𝐹^𝑇𝐹≤𝐼, where 𝑎 is a positive constant, all matrices with appropriate dimensions; one will have for any constant 𝑘>0 such that 2𝑥^𝑇DEF𝑦≤𝑘𝑥^𝑇𝐷𝐷^𝑇𝑎𝑥+𝑘𝑦^𝑇𝐸^𝑇𝐸𝑦, (2.10) for all 𝑥,𝑦∈𝑅^𝑛.

Lemma 2.3 (see [27]). Let 𝑅_𝑖 be any positive matrix; one will have for any 0≤ℎ_𝑖𝑗(𝑡)≤1 such that ^𝑟𝑖_𝑗=1ℎ_𝑖𝑗(𝑡)𝐾_𝑖𝑗^𝑇𝑅_𝑖^𝑟𝑖_𝑗=1ℎ_𝑖𝑗(𝑡)𝐾_𝑖𝑗≤^𝑟𝑖_𝑗=1ℎ_𝑖𝑗(𝑡)𝐾^𝑇_𝑖𝑗𝑅_𝑖𝐾_𝑖𝑗 (2.11) for all 𝐾_𝑖𝑗∈𝑅^𝑛.

3. Stabilization and PDC Synthesis of Fuzzy Large-Scale System

In this section, the decentralized concept and PDC approach are applied to synthesize a local feedback controller for each local subsystem. Let the fuzzy controller be as the PDC form:

Rule 𝑗. If 𝑧_1𝑖(𝑡) is 𝑀_𝑗1𝑖 and... and 𝑧_𝑝𝑖(𝑡) is 𝑀_𝑗𝑝𝑖, then 𝑢_𝑖(𝑡)=−𝐾_𝑖𝑗𝑥_𝑖(𝑡), (3.1) where 𝑖=1,2,…,𝐽,𝑗=1,2,…,𝑟_𝑖. The overall state feedback fuzzy control law is represented by: 𝑢_𝑖(𝑡)∑=−^{𝑟𝑖𝑗=1}𝜔_𝑖𝑗(𝑡)𝐾_𝑖𝑗𝑥_𝑖(𝑡)∑^𝑟𝑖_𝑗=1𝜔_𝑖𝑗(𝑡)=−^𝑟𝑖_𝑗=1ℎ_𝑖𝑗(𝑡)𝐾_𝑖𝑗𝑥_𝑖(𝑡). (3.2)

Substituting (3.2) into (2.1), the closed-loop fuzzy subsystem becomes 𝑥̇_𝑖(𝑡)=^𝑟𝑖𝑟_𝑗=1^𝑖_𝑛=1ℎ_𝑖𝑗(𝑡)ℎ_𝑖𝑛(𝑡)𝐴_𝑖𝑗+Δ𝐴_𝑖𝑗−𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗𝐾_𝑖𝑛𝑥_𝑖(𝑡)+^𝐽_𝑘=1𝐶_𝑘𝑖+Δ𝐶_𝑘𝑖𝑥_𝑘(𝑡)+𝐷_𝑘𝑖+Δ𝐷_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖. (3.3)

Now, our work is to determine the local feedback gains 𝐾_𝑖𝑗 such that the whole fuzzy large-scale system (3.3) is asymptotically stable.

Theorem 3.1. The fuzzy large-scale system (2.1) can be asymptotically stabilized by the decentralized PDC fuzzy control (3.2), if there exist matrices 𝐹_𝑖𝑗, positive definite matrices 𝑋_𝑖 to satisfy the following LMIs: 𝑋_𝑖𝐴^𝑇_𝑖𝑗+𝐴_𝑖𝑗𝑋_𝑖−𝐹^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗−𝐵_𝑖𝑗𝐹_𝑖𝑛𝑋_𝑖𝐺^𝑇_𝑖𝑗−𝐹^𝑇_𝑖𝑛𝐻^𝑇_𝑖𝑗𝑋_𝑖𝑀^𝑇_𝑖𝐶_𝑖𝐸_𝑖𝑋_𝑖𝐺_𝑖𝑗𝑋_𝑖−𝐻_𝑖𝑗𝐹_𝑖𝑛𝑀−𝐼0000_𝑖𝑋_𝑖0−𝐼000𝐶^𝑇_𝑖00−𝐼00𝐸^𝑇_𝑖1000−2𝑋𝐼0_𝑖10000−𝐼2𝐽<0, (3.4) for 𝑖=1,2,…,𝐽, 𝑗=1,2,…,𝑟_𝑖,𝑛=1,2,…,𝑟_𝑖, where 𝑀^𝑇_𝑖=[𝑀_𝑖1,𝑀_𝑖2,…,𝑀_𝑖𝐽,𝐿_𝑖1,𝐿_𝑖2,…,𝐿_𝑖𝐽]^𝑇, 𝐶_𝑖=[𝐸_𝑖𝑗,𝐶_1𝑖,𝐶_2𝑖,…,𝐶_𝐽𝑖,𝐷_1𝑖,𝐷_2𝑖,…,𝐷_𝐽𝑖],𝐸_𝑖=[𝐸_1𝑖,𝐸_2𝑖,…,𝐸_𝐽𝑖], 𝑋_𝑖=𝑃_𝑖⁻¹, 𝐹_𝑖𝑛=𝐾_𝑖𝑛𝑃_𝑖⁻¹.

Proof. Let the Lyapunov functional be 𝑉(𝑡)=^𝐽_𝑖=1𝑉_𝑖(𝑡)=^𝐽_𝑖=1𝑥^𝑇_𝑖𝑃_𝑖𝑥_𝑖+^𝐽_𝑘=1^𝑡_{𝑡−𝜏𝑘𝑖}𝑥^𝑇_𝑘(𝑠)𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑥_𝑘(𝑠)𝑑𝑠, (3.5) where 𝑃_𝑖>0 is to be selected. It is obviously that there exist 𝜎₁ and 𝜎₂ such that 𝜎₁𝑥_𝑖(𝑡)²≤𝑉_𝑖(𝑡)≤𝜎₂𝑥_𝑖(𝑡)². (3.6) Taking the derivative of the 𝑉_𝑖(𝑡) along the trajectories of (3.3), 𝑉̇_𝑖(𝑡)=𝑥̇^𝑇_𝑖(𝑡)𝑃_𝑖𝑥_𝑖(𝑡)+𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝑥̇^𝑇_𝑖(𝑡)+^𝐽_𝑘=1𝑥^𝑇_𝑘(𝑡)𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑥_𝑘(𝑡)−^𝐽_𝑘=1𝑥^𝑇_𝑘𝑡−𝜏_𝑘𝑖𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖=^𝑟𝑖𝑟_𝑗=1^𝑖_𝑛=1ℎ_𝑖𝑗(𝑡)ℎ_𝑖𝑛(𝑡)𝑥^𝑇_𝑖(𝑡)𝐴_𝑖𝑗+Δ𝐴_𝑖𝑗^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑗+Δ𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗^𝑇𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗𝐾_𝑖𝑛𝑥_𝑖(𝑡)+2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐶_𝑘𝑖+Δ𝐶_𝑘𝑖𝑥_𝑘(𝑡)+2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐷_𝑘𝑖+Δ𝐷_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖+^𝐽_𝑘=1𝑥^𝑇_𝑘(𝑡)𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑥_𝑘(𝑡)−^𝐽_𝑘=1𝑥^𝑇_𝑘𝑡−𝜏_𝑘𝑖𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖=^𝑟𝑖𝑟_𝑗=1^𝑖_𝑛=1ℎ_𝑖𝑗(𝑡)ℎ_𝑖𝑛(𝑡)𝑥^𝑇_𝑖(𝑡)𝑁_𝑖𝑗+Δ𝐴^𝑇_𝑖𝑗𝑃_𝑖+𝑃_𝑖Δ𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛Δ𝐵^𝑇_𝑖𝑗𝑃_𝑖−𝑃_𝑖Δ𝐵_𝑖𝑗𝐾_𝑖𝑛𝑥_𝑖(𝑡)+2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐶_𝑘𝑖𝑥_𝑘(𝑡)+2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1Δ𝐶_𝑘𝑖𝑥_𝑘(𝑡)+2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐷_𝑘𝑖𝑥_𝑘(𝑡)+2𝑥^𝑇_𝑖𝑡−𝜏_𝑘𝑖𝑃_𝑖^𝐽_𝑘=1Δ𝐷_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖+^𝐽_𝑘=1𝑥^𝑇_𝑘(𝑡)𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑥_𝑘(𝑡)−^𝐽_𝑘=1𝑥^𝑇_𝑘𝑡−𝜏_𝑘𝑖𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖, (3.7) where 𝑁_𝑖𝑗=𝐴^𝑇_𝑖𝑗𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑗𝐾_𝑖𝑛.
Using Lemma (2.2), we have 𝑥^𝑇_𝑖(𝑡)Δ𝐴^𝑇_𝑖𝑗𝑃_𝑖+𝑃_𝑖Δ𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛Δ𝐵^𝑇_𝑖𝑗𝑃_𝑖−𝑃_𝑖Δ𝐵_𝑖𝑗𝐾_𝑖𝑛𝑥_𝑖(𝑡)=𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝐸_𝑖𝑗𝐹_𝑖𝑗(𝑡)𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛+𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛^𝑇𝐹^𝑇_𝑖𝑗(𝑡)𝐸^𝑇_𝑖𝑗𝑃_𝑖𝑥_𝑖(𝑡)≤𝑥^𝑇_𝑖(𝑡)𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛^𝑇𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛𝑥_𝑖(𝑡)+𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝐸_𝑖𝑗𝐸^𝑇_𝑖𝑗𝑃_𝑖𝑥_𝑖(𝑡),2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1Δ𝐶_𝑘𝑖𝑥_𝑘(𝑡)=2^𝐽_𝑘=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝐸_𝑘𝑖𝐹_𝑘𝑖(𝑡)𝐿_𝑘𝑖𝑥_𝑘(𝑡)≤𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐸_𝑘𝑖𝐸^𝑇_𝑘𝑖𝑃_𝑖𝑥_𝑖(𝑡)+^𝐽_𝑘=1𝑥^𝑇_𝑘(𝑡)𝐿_𝑘𝑖𝐿^𝑇_𝑘𝑖𝑥_𝑘(𝑡),2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1Δ𝐷_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖=2^𝐽_𝑘=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝐸_𝑘𝑖𝐹_𝑘𝑖(𝑡)𝑀_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖≤𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐸_𝑘𝑖𝐸^𝑇_𝑘𝑖𝑃_𝑖𝑥_𝑖(𝑡)+^𝐽_𝑘=1𝑥^𝑇_𝑘𝑡−𝜏_𝑘𝑖𝑀_𝑘𝑖𝑀^𝑇_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖. (3.8) Using Lemma 2.1, we get 2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐶_𝑘𝑖𝑥_𝑘(𝑡)≤𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐶_𝑘𝑖𝐶^𝑇_𝑘𝑖𝑃_𝑖𝑥_𝑖(𝑡)+^𝐽_𝑘=1𝑥^𝑇_𝑘(𝑡)𝑥_𝑘(𝑡),2𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐷_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖≤𝑥^𝑇_𝑖(𝑡)𝑃_𝑖^𝐽_𝑘=1𝐷_𝑘𝑖𝐷^𝑇_𝑘𝑖𝑃_𝑖𝑥_𝑖(𝑡)+^𝐽_𝑘=1𝑥^𝑇_𝑘𝑡−𝜏_𝑘𝑖𝑥_𝑘𝑡−𝜏_𝑘𝑖, (3.9) Noticing that the facts as follows: ^𝐽𝐽_𝑖=1𝑘=1𝑥^𝑇_𝑘(𝑡)𝑥_𝑘(𝑡)=^𝐽𝐽_𝑖=1𝑘=1𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)=^𝐽_𝑖=1𝑥^𝑇_𝑖(𝑡)(𝐽𝐼)𝑥_𝑖(𝑡),_{𝐽𝐽𝑖=1𝑘=1}𝑥^𝑇_𝑘(𝑡)𝐿_𝑘𝑖𝐿^𝑇_𝑘𝑖𝑥_𝑘(𝑡)=^𝐽𝐽_𝑖=1𝑘=1𝑥^𝑇_𝑖(𝑡)𝐿_𝑖𝑘𝐿^𝑇_𝑖𝑘𝑥_𝑖(𝑡),_{𝐽𝐽𝑖=1𝑘=1}𝑥^𝑇_𝑘(𝑡)𝑀_𝑘𝑖𝑀^𝑇_𝑘𝑖𝑥_𝑘(𝑡)=^𝐽𝐽_𝑖=1𝑘=1𝑥^𝑇_𝑖(𝑡)𝑀_𝑖𝑘𝑀^𝑇_𝑖𝑘𝑥_𝑖(𝑡) (3.10) Based on (3.7)–(3.10), we have 𝑉̇(𝑡)=^𝐽_𝑖=1𝑉̇_𝑖(𝑡)≤^𝐽_𝑖=1^𝑟𝑖𝑟_𝑗=1^𝑖_𝑛=1ℎ_𝑖𝑗(𝑡)ℎ_𝑖𝑛(𝑡)𝑥^𝑇_𝑖(𝑡)×𝐴^𝑇_𝑖𝑗𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑗𝐾_𝑖𝑛+𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛^𝑇×𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛+𝑃_𝑖𝐸_𝑖𝑗𝐸^𝑇_𝑖𝑗𝑃_𝑖+2𝑃_𝑖^𝐽_𝑘=1𝐸_𝑘𝑖𝐸^𝑇_𝑘𝑖𝑃_𝑖+2𝐽𝐼+𝑃_𝑖^𝐽_𝑘=1𝐶_𝑘𝑖𝐶^𝑇_𝑘𝑖𝑃_𝑖+𝑃_𝑖^𝐽_𝑘=1𝐷_𝑘𝑖𝐷^𝑇_𝑘𝑖𝑃_𝑖+^𝐽_𝑘=1𝑀_𝑖𝑘𝑀^𝑇_𝑖𝑘+^𝐽_𝑘=1𝐿_𝑖𝑘𝐿^𝑇_𝑖𝑘𝑥_𝑖(𝑡). (3.11) From Schur complement, we know 𝐴^𝑇_𝑖𝑗𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑗𝐾_𝑖𝑛+𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛^𝑇𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛+𝑃_𝑖𝐸_𝑖𝑗𝐸^𝑇_𝑖𝑗𝑃_𝑖+2𝑃_𝑖^𝐽_𝑘=1𝐸_𝑘𝑖𝐸^𝑇_𝑘𝑖𝑃𝑖+2𝐽𝐼+𝑃_𝑖^𝐽_𝑘=1𝐶_𝑘𝑖𝐶^𝑇_𝑘𝑖𝑃𝑖+𝑃_𝑖^𝐽_𝑘=1𝐷_𝑘𝑖𝐷^𝑇_𝑘𝑖𝑃𝑖+^𝐽_𝑘=1𝑀_𝑖𝑘𝑀^𝑇_𝑖𝑘+^𝐽_𝑘=1𝐿_𝑖𝑘𝐿^𝑇_𝑖𝑘<0 (3.12) is equivalent to 𝐴^𝑇_𝑖𝑗𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑗𝐾_𝑖𝑛+2𝐽𝐼𝐺^𝑇_𝑖𝑗−𝐾^𝑇_𝑖𝑛𝐻_𝑖𝑗𝑀^𝑇_𝑖𝐶_𝑖𝐸_𝑖𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛𝑀−𝐼000_𝑖𝐶0−𝐼00_𝑇𝑖𝐸00−𝐼0_𝑇𝑖1000−2𝐼<0, (3.13) where 𝑀^𝑇_𝑖=[𝑀_𝑖1,𝑀_𝑖2,…,𝑀_𝑖𝐽,𝐿_𝑖1,𝐿_𝑖2,…,𝐿_𝑖𝐽]^𝑇, 𝐶_𝑖=𝑃_𝑖[𝐸_𝑖𝑗,𝐶_1𝑖,𝐶_2𝑖,…,𝐶_𝐽𝑖,𝐷_1𝑖,𝐷_2𝑖,…,𝐷_𝐽𝑖], 𝐸_𝑖=𝑃_𝑖[𝐸_1𝑖,𝐸_2𝑖,…,𝐸_𝐽𝑖].
So we have 𝑉̇(𝑡)<0 while 𝑥_𝑖(𝑡)(𝑖=1,2,…,𝐽) are not all zero vectors. Note that the matrix inequalities in (3.13) can be transformed into certain forms of linear matrix inequalities (LMIs). Therefore, multiplying both sides of matrix inequalities (3.13) by diag{𝑃_𝑖⁻¹,𝐼,𝐼,𝐼,𝐼} and applying the change of variables such that 𝑃_𝑖=𝑋_𝑖⁻¹, 𝐾_𝑖𝑛=𝐹_𝑖𝑛𝑋_𝑖⁻¹,(𝑖=1,2,…,𝐽,𝑛=1,2,…,𝑟_𝑖), then (3.4) is obtained.

With the similar proof of Theorem 3.1, the stabilization criterion of large-scale system (2.1) without uncertainties is also discussed. The result is presented as follows.

Corollary 3.2. The fuzzy large-scale system (2.1) without uncertainties can be asymptotically stabilized by the decentralized PDC fuzzy control (3.2), if there exist matrices 𝐹_𝑖𝑗, positive definite matrices 𝑋_𝑖 to satisfy the following LMIs: 𝑋_𝑖𝐴^𝑇_𝑖𝑗+𝐴_𝑖𝑗𝑋_𝑖−𝐹^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗−𝐵_𝑖𝑗𝐹_𝑖𝑛𝐶_𝑖𝑋_𝑖𝐶^𝑇_𝑖𝑋−𝐼0_𝑖10−𝐼2𝐽<0 (3.14) for 𝑖=1,2,…,𝐽, 𝑗=1,2,…,𝑟_𝑖, 𝑛=1,2,…,𝑟_𝑖, where 𝐶_𝑖=[𝐶_1𝑖,𝐶_2𝑖,…,𝐶_𝐽𝑖,𝐷_1𝑖,𝐷_2𝑖,…,𝐷_𝐽𝑖].

Proof. The proof is similar with that of Theorem 3.1; therefore details are omitted.

Remark 3.3. This corollary is similar with Theorem 1 in [10], so the theorem in this paper is more general.

Theorem 3.4. If there exist matrices 𝐹_𝑖𝑗, positive definite matrices 𝑋_𝑖 to satisfy the following LMIs: 𝑋_𝑖𝐴^𝑇_𝑖𝑗+𝐴_𝑖𝑗𝑋_𝑖−𝐹^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗−𝐵_𝑖𝑗𝐹_𝑖𝑛𝑋_𝑖𝐺^𝑇_𝑖𝑗−𝐹^𝑇_𝑖𝑛𝐻^𝑇_𝑖𝑗𝑋_𝑖𝑀^𝑇_𝑖𝐶_𝑖𝐸_𝑖𝑋_𝑖𝑋_𝑖𝐹^𝑇_𝑖𝑛𝐺_𝑖𝑗𝑋_𝑖−𝐻_𝑖𝑗𝐹_𝑖𝑛𝑀−𝐼000000_𝑖𝑋_𝑖0−𝐼00000𝐶^𝑇_𝑖00−𝐼0000𝐸^𝑇_𝑖𝑋000−𝐼000_𝑖10000−𝑋2𝐽𝐼00_𝑖00000−𝑄_𝑖⁻¹0𝐹_𝑖𝑛000000−𝑅_𝑖⁻¹<0, (3.15) the fuzzy large-scale system (2.1) can be asymptotically stabilized by the decentralized PDC fuzzy control (3.2), and the performance index (2.8) is satisfied the following inequality: 𝐽≤𝐽^∗=^𝐽_𝑖=1𝑥^𝑇_𝑖(0)𝑃_𝑖𝑥_𝑖(0)+^𝐽_𝑘=1⁰_{−𝜏𝑘𝑖}𝑥^𝑇_𝑘(𝑠)𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑥_𝑘(𝑠)𝑑𝑠, (3.16) for 𝑖=1,2,…,𝐽, 𝑗=1,2,…,𝑟_𝑖, 𝑛=1,2,…,𝑟_𝑖, where 𝑀^𝑇_𝑖=[𝑀_𝑖1,𝑀_𝑖2,…,𝑀_𝑖𝐽,𝐿_𝑖1,𝐿_𝑖2,…, 𝐿_𝑖𝐽]^𝑇, 𝐶_𝑖=[𝐸_𝑖𝑗,𝐶_1𝑖,𝐶_2𝑖,…,𝐶_𝐽𝑖,𝐷_1𝑖,𝐷_2𝑖,…,𝐷_𝐽𝑖], 𝐸_𝑖=[𝐸_1𝑖,𝐸_2𝑖,…,𝐸_𝐽𝑖], 𝑋_𝑖=𝑃_𝑖⁻¹, 𝐹_𝑖𝑛=𝐾_𝑖𝑛𝑃_𝑖⁻¹.

Proof. From the proof of Theorem 3.1, we know if 𝐺_𝑖𝑗=𝐴^𝑇_𝑖𝑗𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑗𝐾_𝑖𝑛+𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛^𝑇𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛+𝑃_𝑖𝐸_𝑖𝑗𝐸^𝑇_𝑖𝑗𝑃_𝑖+2𝑃_𝑖^𝐽_𝑘=1𝐸_𝑘𝑖𝐸^𝑇_𝑘𝑖𝑃𝑖+2𝐽𝐼+𝑃_𝑖^𝐽_𝑘=1𝐶_𝑘𝑖𝐶^𝑇_𝑘𝑖𝑃𝑖+𝑃_𝑖^𝐽_𝑘=1𝐷_𝑘𝑖𝐷^𝑇_𝑘𝑖𝑃𝑖+^𝐽_𝑘=1𝑀_𝑖𝑘𝑀^𝑇_𝑖𝑘+^𝐽_𝑘=1𝐿_𝑖𝑘𝐿^𝑇_𝑖𝑘<0 (3.17) then 𝑉̇(𝑡)<0. Noticing that 𝑄_𝑖 and 𝑅_𝑖 are positive matrices, if inequality 𝐴^𝑇_𝑖𝑗𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑗−𝐾^𝑇_𝑖𝑛𝐵^𝑇_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑗𝐾_𝑖𝑛+𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛^𝑇𝐺_𝑖𝑗−𝐻_𝑖𝑗𝐾_𝑖𝑛+𝑃_𝑖𝐸_𝑖𝑗𝐸^𝑇_𝑖𝑗𝑃_𝑖+2𝑃_𝑖^𝐽_𝑘=1𝐸_𝑘𝑖𝐸^𝑇_𝑘𝑖𝑃𝑖+2𝐽𝐼+𝑃_𝑖^𝐽_𝑘=1𝐶_𝑘𝑖𝐶^𝑇_𝑘𝑖𝑃𝑖+𝑃_𝑖^𝐽_𝑘=1𝐷_𝑘𝑖𝐷^𝑇_𝑘𝑖𝑃𝑖+^𝐽_𝑘=1𝑀_𝑖𝑘𝑀^𝑇_𝑖𝑘+^𝐽_𝑘=1𝐿_𝑖𝑘𝐿^𝑇_𝑖𝑘+𝑄_𝑖+𝐾^𝑇_𝑖𝑛𝑅_𝑖𝐾_𝑖𝑛<0, (3.18) holds, we obtain 𝑉̇(𝑡)<0: 𝐽=^𝐽_𝑖=1^∞₀𝑥^𝑇_𝑖(𝑡)𝑄_𝑖𝑥_𝑖(𝑡)+𝑢^𝑇_𝑖(𝑡)𝑅_𝑖𝑢_𝑖(𝑡)=𝑑𝑡_{𝐽𝑖=1∞0}𝑥^𝑇_𝑖(𝑡)𝑄_𝑖+^𝑟𝑖_𝑛=1ℎ_𝑖𝑛(𝑡)𝐾^𝑇_𝑖𝑛𝑅_𝑖^𝑟𝑖_𝑛=1ℎ_𝑖𝑛(𝑡)𝐾_𝑖𝑛𝑥_𝑖(𝑡)𝑑𝑡. (3.19) Using Lemma 2.3, we get 𝐽≤^𝐽_𝑖=1^∞₀𝑥^𝑇_𝑖(𝑡)𝑄_𝑖+^𝑟𝑖_𝑛=1ℎ_𝑖𝑛(𝑡)𝐾^𝑇_𝑖𝑛𝑅_𝑖𝐾_𝑖𝑛𝑥_𝑖(𝑡)𝑑𝑡. (3.20) If (3.18) holds, we have 𝐽≤−^𝐽𝑟_𝑖=1^𝑖𝑟_𝑗=1^𝑖_𝑛=1^∞₀ℎ_𝑖𝑗(𝑡)ℎ_𝑖𝑛(𝑡)𝑥^𝑇_𝑖(𝑡)𝐺_𝑖𝑗𝑥_𝑖(𝑡)𝑑𝑡≤−^𝐽_𝑖=1^∞₀𝑉̇_𝑖(𝑡)(0)(∞)=𝑑𝑡=𝑉−𝑉_𝐽𝑖=1𝑥^𝑇_𝑖(0)𝑃_𝑖𝑥_𝑖(0)+^𝐽_𝑘=1⁰_{−𝜏𝑘𝑖}𝐼+𝑀^𝑇_𝑘𝑖𝑀_𝑘𝑖𝑑𝑠. (3.21) Therefore, multiply both sides of (3.18) by 𝑃_𝑖⁻¹, and let 𝑋_𝑖=𝑃_𝑖⁻¹,𝐹_𝑖𝑛=𝐾_𝑖𝑛𝑃_𝑖⁻¹. From Schur complement, the proof is completed.

4. Numerical Examples

In this section, some numerical simulations for uncertain fuzzy large-scale system will be given to illustrate the effectiveness of the proposed stabilization criteria and also compared with the existing results.

Example 4.1. Consider an uncertain FLSS time delays 𝑆 composed of three fuzzy subsystems 𝑆_𝑖,𝑖=1,2,3 by the following equations: 𝑥̇₁(𝑡)=²_𝑗=1ℎ_1𝑗(𝑡)𝐴_1𝑗+Δ𝐴_1𝑗𝑥₁(𝑡)+𝐵_1𝑗+Δ𝐵_1𝑗𝑢₁(𝑡+_3𝑘=1)𝐶_𝑘1+Δ𝐶_𝑘1𝑥_𝑘(𝑡)+𝐷_𝑘1+Δ𝐷_𝑘1𝑥_𝑘𝑡−𝜏_𝑘1,𝑥̇₂(𝑡)=²_𝑗=1ℎ_2𝑗(𝑡)𝐴_2𝑗+Δ𝐴_2𝑗𝑥₂(𝑡)+𝐵_2𝑗+Δ𝐵_2𝑗𝑢₂(𝑡+_3𝑘=1)𝐶_𝑘2+Δ𝐶_𝑘2𝑥_𝑘(𝑡)+𝐷_𝑘2+Δ𝐷_𝑘2𝑥_𝑘𝑡−𝜏_𝑘2,𝑥̇₃(𝑡)=²_𝑗=1ℎ_3𝑗(𝑡)𝐴_3𝑗+Δ𝐴_3𝑗𝑥₃(𝑡)+𝐵_3𝑗+Δ𝐵_3𝑗𝑢₃(𝑡+_3𝑘=1)𝐶_𝑘3+Δ𝐶_𝑘3𝑥_𝑘(𝑡)+𝐷_𝑘3+Δ𝐷_𝑘3𝑥_𝑘𝑡−𝜏_𝑘3, (4.1) in which 𝑥^𝑇₁(𝑡)=[𝑥₁₁,𝑥₂₁]^𝑇, 𝑥^𝑇₂(𝑡)=[𝑥₁₂,𝑥₂₂]^𝑇, 𝑥^𝑇₁(𝑡)=[𝑥₁₃,𝑥₂₃]^𝑇Δ𝐴_1𝑗,Δ𝐵_1𝑗=𝐸_1𝑗𝐹_1𝑗𝐺_1𝑗,𝐻_1𝑗,Δ𝐴_2𝑗,Δ𝐵_2𝑗=𝐸_2𝑗𝐹_2𝑗𝐺_2𝑗,𝐻_2𝑗,Δ𝐴_3𝑗,Δ𝐵_3𝑗=𝐸_3𝑗𝐹_3𝑗𝐺_3𝑗,𝐻_3𝑗,Δ𝐶_𝑘1,Δ𝐷_𝑘1=𝐸_𝑘1𝐹_𝑘1𝐿_𝑘1,𝑀_𝑘1,Δ𝐶_𝑘2,Δ𝐷_𝑘2=𝐸_𝑘2𝐹_𝑘2𝐿_𝑘2,𝑀_𝑘2,Δ𝐶_𝑘3,Δ𝐷_𝑘3=𝐸_𝑘3𝐹_𝑘3𝐿_𝑘3,𝑀_𝑘3, (4.2) for 𝑖,𝑘=1,2,3, 𝑗=1,2. 𝐴₁₁=0.62.40.2091.9,𝐴₁₂=0.62.401.9,𝐴₂₁=,𝐴−1−0.513₂₂=0.62.4−0.2091.9,𝐴₃₁=0.61.20.2091.9,𝐴₃₂=,𝐵0.61.201.9₁₁=2001,𝐵₁₂=1002,𝐵₂₁=,𝐵3103₂₂=2032,𝐵₃₁=1001,𝐵₃₂=,𝐶1011₁₁=−0.10.100,𝐶₂₁=000.10.1,𝐶₃₁=,𝐶−0.20.150−1₁₂=000.10.1,𝐶₂₂=0000.1,𝐶₃₂=,𝐶−0.3−0.20.20.1₁₃=−0.1000,𝐶₂₃=0.1000.1,𝐶₃₃=,𝐷000.10₁₁=0.10.100,𝐷₂₁=0.1000.1,𝐷₃₁=,𝐷−1.080.360−0.72₁₂=0.10−0.10.1,𝐷₂₂=00.100,𝐷₃₂=,𝐷0.6−0.30.10₁₃=0000,𝐷₂₃=−0.1000,𝐷₃₃=,𝐸−0.100−0.1₁₁=−0.28−0.028−0.140.182,𝐸₁₂=−0.21−0.084−1.26−0.42,𝐸₂₁=,𝐸−0.50.251.350.75₂₂=−0.27075.00.81,𝐸₃₁=−0.280.280.140.182,𝐸₃₂=,0.210.841.26−0.42𝐸₁₁=,−0.280.028−0.140.182𝐸₂₁=,−0.21−840.0−1.26−0.42𝐸₃₁=,0.21−0.084−1.260.42𝐸₁₂=,−0.50.251.350.75𝐸₂₂=,−0.2700.810.81𝐸₃₂=,−0.2700.750.81𝐸₁₃=,−0.280.280.140.182𝐸₂₃=,0.210.841.26−0.42𝐸₃₃=,𝐹−0.210.841.260.42₁₁(𝑡)=1−sin⁴𝑡001−cos⁴𝑡,𝐹₁₁(𝑡)=sin𝑡²00cos𝑡²,𝐺₁₁=𝐻₁₁=𝐸₁₁,𝐺₁₂=𝐻₁₂=𝐸₁₂,𝐺₂₁=𝐻₂₁=𝐸₂₁,𝐺₂₂=𝐻₂₂=𝐸₂₂,𝐺₃₁=𝐻₃₁=𝐸₃₁,𝐺₃₂=𝐻₃₂=𝐸₃₂,𝐿₁₁=𝑀₁₁=𝐸₁₁,𝐿₂₁=𝑀₂₁=𝐸₂₁,𝐿₃₁=𝑀₃₁=𝐸₃₁,𝐿₁₂=𝑀₁₂=𝐸₁₂,𝐿₂₂=𝑀₂₂=𝐸₂₂,𝐿₃₂=𝑀₃₂=𝐸₃₂,𝐿₁₃=𝑀₁₃=𝐸₁₃,𝐿₂₃=𝑀₂₃=𝐸₂₃,𝐿₃₃=𝑀₃₃=𝐸₃₃,𝐹₁₁(𝑡)=𝐹₁₂(𝑡)=𝐹₂₁(𝑡)=𝐹₂₂(𝑡)=𝐹₃₁(𝑡)=𝐹₃₂(𝑡),𝐹₁₁(𝑡)=𝐹₂₁(𝑡)=𝐹₃₁(𝑡)=𝐹₁₂(𝑡)=𝐹₂₂(𝑡)=𝐹₃₂(𝑡)=𝐹₁₃(𝑡)=𝐹₂₃(𝑡)=𝐹₃₃(𝑡). (4.3)

Here, the T-S fuzzy models of the isolated subsystem are of the following:

Subsystem 1
Rule 1. If 𝑥₁₁(𝑡) is 𝑀₁₁₁ then 𝑥̇₁(𝑡)=𝐴₁₁+Δ𝐴₁₁(𝑡)𝑥₁(𝑡)+𝐵₁₁+Δ𝐵₁₁(𝑡)𝑢₁(𝑡). (4.4)Rule 2. If 𝑥₁₁(𝑡) is 𝑀₂₁₁ then 𝑥̇₁(𝑡)=𝐴₁₂+Δ𝐴₁₂(𝑡)𝑥₁(𝑡)+𝐵₁₂+Δ𝐵₁₂(𝑡)𝑢₁(𝑡), (4.5) and the membership functions for Rules 1 and 2 are, respectively, 𝑀₁₁₁(𝑥₁₁(𝑡))=1/(1+exp(−3𝑥₁₁(𝑡))),𝑀₂₁₁(𝑥₁₁(𝑡))=1−𝑀₁₁₁(𝑥₁₁(𝑡)).

Subsystem 2
Rule 1. If 𝑥₁₂(𝑡) is 𝑀₁₁₂ Then 𝑥̇₂(𝑡)=𝐴₂₁+Δ𝐴₂₁(𝑡)𝑥₂(𝑡)+𝐵₂₁+Δ𝐵₂₁(𝑡)𝑢₂(𝑡). (4.6)Rule 2. If 𝑥₁₂(𝑡) is 𝑀₂₁₂ then 𝑥̇₂(𝑡)=𝐴₂₂+Δ𝐴₂₂(𝑡)𝑥₂(𝑡)+𝐵₂₂+Δ𝐵₂₂(𝑡)𝑢₂(𝑡), (4.7) and the membership functions for Rules 1 and 2 are, respectively, 𝑀₁₁₂(𝑥₁₂(𝑡))=exp(−2𝑥²₁₂(𝑡)), 𝑀₂₁₂(𝑥₁₂(𝑡))=1−𝑀₁₁₂(𝑥₁₂(𝑡)).

Subsystem 3
Rule 1. If 𝑥₁₃(𝑡) is 𝑀₁₁₃ then 𝑥̇₃(𝑡)=𝐴₃₁+Δ𝐴₃₁(𝑡)𝑥₃(𝑡)+𝐵₃₁+Δ𝐵₃₁(𝑡)𝑢₃(𝑡). (4.8)Rule 2. If 𝑥₁₃(𝑡) is 𝑀₂₁₃ then 𝑥̇₃(𝑡)=𝐴₃₂+Δ𝐴₃₂(𝑡)𝑥₃(𝑡)+𝐵₃₂+Δ𝐵₃₂(𝑡)𝑢₃(𝑡), (4.9) and the membership functions for Rules 1 and 2 are, respectively, 𝑀₁₁₃(𝑥₁₃(𝑡))=1/(1+exp(−1.5𝑥₁₃(𝑡))), 𝑀₂₁₃(𝑥₁₃(𝑡))=1−𝑀₁₁₃(𝑥₁₃(𝑡)).
It is noted that the above large-scale system without control (𝑢_𝑖(𝑡)=0) has unstable responses with initial conditions 𝑥₁(𝑡)=[−3,3]^𝑇, 𝑥₂(𝑡)=[−2,2]^𝑇, 𝑥₃(𝑡)=[1,−1]^𝑇 as shown in Figure 1.

(a)

(b)

(c)

Figure 1 

The state response with 𝑢𝑖(𝑡)=0.

In order to stabilize the large-scale fuzzy system, three decentralized PDC fuzzy controllers are designed in the following.

Fuzzy Controller 𝐶₁
Rule 1. If 𝑥₁₁(𝑡) is 𝑀₁₁₁ then 𝑢₁(𝑡)=−𝐾₁₁𝑥₁(𝑡). (4.10)Rule 2. If 𝑥₁₁(𝑡) is 𝑀₂₁₁ then 𝑢₁(𝑡)=−𝐾₁₂𝑥₁(𝑡). (4.11)

Fuzzy Controller 𝐶₂
Rule 1. If 𝑥₁₂(𝑡) is 𝑀₁₁₂ then 𝑢₂(𝑡)=−𝐾₂₁𝑥₂(𝑡). (4.12)Rule 2. If 𝑥₁₂(𝑡) is 𝑀₂₁₂ then 𝑢₂(𝑡)=−𝐾₂₂𝑥₂(𝑡). (4.13)

Fuzzy Controller 𝐶₃
Rule 1. If 𝑥₁₃(𝑡) is 𝑀₁₁₃ then 𝑢₃(𝑡)=−𝐾₃₁𝑥₃(𝑡). (4.14)Rule 2. If 𝑥₁₃(𝑡) is 𝑀₂₁₃ Then 𝑢₃(𝑡)=−𝐾₃₂𝑥₃(𝑡). (4.15)

By using the approaches of Theorem 3.1, we obtain the matrices 𝐾_𝑖𝑗 for subsystems 𝑆₁,𝑆₂, and 𝑆₃: 𝐾₁₁=1.56810.47330.52963.0430,𝐾₁₂=2.22610.26800.58011.7275,𝐾₂₁=,𝐾0.87170.7769−0.48561.2398₂₂=1.48310.51790.20981.7072,𝐾₃₁=1.65480.72810.41222.9992,𝐾₃₂=.1.57690.49580.58232.0029 (4.16)

The complete simulation results with initial conditions 𝑥₁(𝑡)=[−3,3]^𝑇, 𝑥₂(𝑡)=[−2,2]^𝑇, 𝑥₃(𝑡)=[1,−1]^𝑇 are shown in Figure 2. It is obvious that they are stabilized asymptotically.

(a)

(b)

(c)

Figure 2 

The state response with fuzzy control by Theorem 3.1.

Example 4.2. Consider the fuzzy large-scale system 𝑆 composed of two subsystems 𝑆_𝑖 as follows [25]: 𝑥̇₁(𝑡)=²_𝑗=1ℎ_1𝑗(𝑡)𝐴_1𝑗𝑥₁(𝑡)+𝐵_1𝑗𝑢₁(𝑡)+²_ℎ=1𝐶_ℎ1𝑥_ℎ(𝑡),𝑥̇₂(𝑡)=²_𝑗=1ℎ_2𝑗(𝑡)𝐴_2𝑗𝑥₂(𝑡)+𝐵_2𝑗𝑢₂(𝑡)+²_ℎ=1𝐶_ℎ2𝑥_ℎ(𝑡), (4.17) where 𝑥₁(𝑡)=[𝑥₁₁(𝑡),𝑥₁₂(𝑡)]^𝑇,𝑥₂(𝑡)=[𝑥₂₁(𝑡),𝑥₂₂(𝑡)]^𝑇𝐴₁₁=510−4,𝐴₁₂=−402−8,𝐵₁₁=10,𝐵₁₂=01,𝐶₁₁=𝐶₂₂=0000,𝐴₂₁=,𝐴−304−2₂₂=−4−51−3,𝐵₂₁=1−2,𝐵₂₂=1,𝐶−1₂₁=5100,𝐶₁₂=1010 (4.18) and the membership functions are, respectively, 𝑀₁₁₁(𝑥₁₁(𝑡))=exp(−𝑥²₁₁(𝑡)), 𝑀₂₁₁(𝑥₁₁(𝑡))=1−𝑀₁₁₁(𝑥₁₁(𝑡)), 𝑀₁₁₂(𝑥₂₁(𝑡))=1/(1+exp(−𝑥₂₁(𝑡))), 𝑀₂₁₂(𝑥₂₁(𝑡))=1−𝑀₁₁₂(𝑥₂₁(𝑡)).
In order to stabilize the large-sale fuzzy system, three decentralized PDC fuzzy controllers are designed in the following.

Fuzzy Controller 𝐶₁
Rule 1. If 𝑥₁₁(𝑡) is 𝑀₁₁₁ then 𝑢₁(𝑡)=−𝐾₁₁𝑥₁(𝑡). (4.19)Rule 2. If 𝑥₁₁(𝑡) is 𝑀₂₁₁ then 𝑢₁(𝑡)=−𝐾₁₂𝑥₁(𝑡). (4.20)

Fuzzy Controller 𝐶₂
Rule 1. If 𝑥₂₁(𝑡) is 𝑀₁₁₂ then 𝑢₂(𝑡)=−𝐾₂₁𝑥₂(𝑡). (4.21)Rule 2. If 𝑥₂₁(𝑡) is 𝑀₂₁₂ then 𝑢₂(𝑡)=−𝐾₂₂𝑥₂(𝑡). (4.22)

We let 𝑄₁=𝑄₂=⁵⁰₀₁, 𝑅₁=1, 𝑅₂=2. By using the approaches of Theorem 3.4, we obtain 𝑃₁=5.50841.63291.63291.5120,𝑃₂=0.9692−0.0406−0.04061.0681. (4.23)

The control gain 𝐾_𝑖𝑗 for subsystem 𝑆₁,𝑆₂ is compared with [24] in Table 1. The complete simulation results with initial conditions 𝑥₁(𝑡)=[1,−1]^𝑇,𝑥₂(𝑡)=[2,−2]^𝑇 are shown in Figure 3.

Table 1 

Comparison of fuzzy controller, for example, with Zhang et al. [25].

(a)

(b)

Figure 3 

The state response with fuzzy control by Theorem 3.4.

Remark 4.3. From Figure 3, we can see that the system can be stabilized through appropriate decentralized control. Obviously, in Table 1, we can see the method of this paper has smaller gain matrices and performance index, so it has better control effectiveness.

5. Conclusions

In this paper we explore the stabilization problems for uncertain fuzzy large-scale system with time delays. The decentralized PDC fuzzy controller has been designed under some conditions such that the whole closed-loop large-scale fuzzy system is asymptotically stable. Then, the optimization design method for decentralized control is also considered with respect to a quadratic performance index. Finally, numerical examples are provided to demonstrate the correctness and less conservativeness of the theoretical results. However, there are still some other problems to be addressed, such that time-varying delays and delay-dependent stability and stabilization of fuzzy large-scale system and the results developed in the paper can be extended to the case that the underlying systems are invovled with any switching dynamics.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (60774045, 61075065).