Stability Analysis of Interconnected Fuzzy Systems Using the Fuzzy Lyapunov Method

Abstract

The fuzzy Lyapunov method is investigated for use with a class of interconnected fuzzy systems. The interconnected fuzzy systems consist of 𝐽 interconnected fuzzy subsystems, and the stability analysis is based on Lyapunov functions. Based on traditional Lyapunov stability theory, we further propose a fuzzy Lyapunov method for the stability analysis of interconnected fuzzy systems. The fuzzy Lyapunov function is defined in fuzzy blending quadratic Lyapunov functions. Some stability conditions are derived through the use of fuzzy Lyapunov functions to ensure that the interconnected fuzzy systems are asymptotically stable. Common solutions can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. Finally, simulations are performed in order to verify the effectiveness of the proposed stability conditions in this paper.

1. Introduction

In the recent years, a number of research activities have been conducted concerning stability analysis and the stabilization problems of large-scale systems, including electric power systems, nuclear reactors, aerospace systems, large electrical networks, economic systems, process control systems, chemical and petroleum industrial systems, different societal systems, ecological systems, and transportation systems. The interconnected models can be used to represent practical large-scale systems. Moreover, the field of interconnected systems is so broad as to cover the fundamental theory of modeling, optimization and control aspects, and such applications. Therefore, the methodologies used when dealing with interconnected models provide a viable technique whereby the manipulation of system structure can be used to overcome the increasing size and complexity of the relevant mathematical models. Additionally, the fields of analysis, design, and control theory relating to interconnected systems have attained a considerable level of maturity and sophistication. They are currently receiving increasing attention from theorists and practitioners because they are methodologically interesting and have important real-life applications. Such systems comprise numerous interdependent subsystems which serve particular functions, share resources, and are governed by a set of interrelated goals and constraints. Recently, various approaches have been employed to elucidate the stability and stabilization of interconnected systems, as proposed in the literature and the references therein [1–3].

Since Zadeh [4] and Takagi and Sugeno [5] proposed a new concept for a fuzzy inference system which combines the flexibility of fuzzy logic theory and rigorous mathematical analysis tools into a unified framework, the application of fuzzy models has attracted great interest from the engineering and management community (e.g., see [6–20] and the references therein). This kind of fuzzy model suggests an efficient method to represent complex nonlinear systems via fuzzy reasoning. This has enabled the stability issues of fuzzy systems to be extensively applied in system analysis (see [21–26] and the references therein). Similarly, the stability criteria and stabilization problems have been discussed for fuzzy large-scale systems in Wang and Luoh [27], and Hsiao and Hwang [28], where the fuzzy large-scale system consists of J subsystems.

In the aforementioned results for T-S fuzzy models, most of the stability criteria and controller design have usually been derived based on the usage of a single Lyapunov function. However, the main drawback associated with this method is that the single Lyapunov function must work for all linear models of the fuzzy control systems, which in general leads to a conservative controller design [29]. Recently, in order to relax this conservatism, the fuzzy Lyapunov function approach has been proposed in [24, 29–33]. To the best of my knowledge, the stability analyses of interconnected systems based on fuzzy Lyapunov functions have not been discussed yet. Therefore, some novel sufficient conditions are derived from fuzzy Lyapunov functions for stability guarantees by fuzzy for interconnected fuzzy systems in this work.

The organization of the paper is presented as follows. First, the T-S fuzzy modeling is briefly reviewed and the interconnected scheme is used to construct a fuzzy dynamic model. Then, the stability conditions for the fuzzy Lyapunov functions are proposed which guarantee the stability of the interconnected fuzzy systems. In this section, the stability problems can be reformulated into a problem for solving a linear matrix inequality (LMI).

2. System Descriptions and Preliminaries

A fuzzy dynamic model was proposed in the pioneering work of Takagi and Sugeno [5] where complex nonlinear systems could be represented using local linear input/output relations. The main feature of this model is that each locally fuzzy implication (rule) is a linear system and the overall system model is achieved through linear fuzzy blending. This fuzzy dynamic model is described by fuzzy IF-THEN rules and are utilized in this study to deal with the stability analysis issue of an interconnected fuzzy system S that is composed of J subsystems Si (𝑗=1,…,𝐽). The ith rule of the interconnected fuzzy model of the jth subsystem is proposed as having the following form.

Plant Rule i:

IF𝑥_1𝑗(𝑡)is𝑀_𝑖1𝑗,…,𝑥_𝑔𝑗(𝑡)is𝑀_𝑖𝑔𝑗,THEN𝑥̇_𝑗(𝑡)=𝐴_𝑖𝑗𝑥_𝑗(𝑡)+^𝐽_{𝑛=1𝑛≠𝑗}𝐴_𝑖𝑛𝑗𝑥_𝑛(𝑡),𝑖=1,2,…,𝑟_𝑗, (2.1) where 𝑟_𝑗 is the IF-THEN rule number; 𝐴_𝑖𝑗 and 𝐴_𝑖𝑛𝑗are constant matrices with appropriate dimensions; 𝑥_𝑗(𝑡) is the state vector of the jth subsystem; 𝑥_𝑛(𝑡) is the interconnection between the nth and jth subsystems; 𝑀_𝑖𝑝𝑗(𝑝=1,2,…,𝑔) are the fuzzy sets; 𝑥_1𝑗(𝑡)∼𝑥_𝑔𝑗(𝑡) are the premise variables. Through the use of “fuzzy blending,” the overall fuzzy model of the jth fuzzy subsystem can be inferred as follows [13]:

𝑥̇_𝑗∑(𝑡)=^{𝑟𝑗𝑖=1}𝑤_𝑖𝑗(𝑡)[𝐴_𝑖𝑗𝑥_𝑗(𝑡)+∑^𝐽_{𝑛=1,𝑛≠𝑗}𝐴_𝑖𝑛𝑗𝑥_𝑛(𝑡)]∑^𝑟𝑗_𝑖=1𝑤_𝑖𝑗(𝑡)=^𝑟𝑗_𝑖=1ℎ_𝑖𝑗(𝑡)[𝐴_𝑖𝑗𝑥_𝑗(𝑡)+^𝐽_{𝑛=1𝑛≠𝑗}𝐴_𝑖𝑛𝑗𝑥_𝑛(𝑡)], (2.2) with 𝑤_𝑖𝑗(𝑡)≡^𝑔_𝑝=1𝑀_𝑖𝑝𝑗(𝑥_𝑝𝑗(𝑡)),ℎ_𝑖𝑗𝑤(𝑡)≡^𝑖𝑗(𝑡)∑^𝑟𝑗_𝑖=1𝑤_𝑖𝑗(𝑡), (2.3) in which 𝑀_𝑖𝑝𝑗(𝑥_𝑝𝑗(𝑡)) is the grade of membership of 𝑥_𝑝𝑗(𝑡) in 𝑀_𝑖𝑝𝑗. In this study, it is assumed that 𝑤_𝑖𝑗(𝑡)≥0,𝑖=1,2,…,𝑟_𝑗; 𝑗=1,2,…,𝐽. Therefore, the normalized membership function ℎ_𝑖𝑗(𝑡) satisfies

ℎ_𝑖𝑗(𝑡)≥0,^𝑟𝑗_𝑖=1ℎ_𝑖𝑗(𝑡)=1,∀𝑡. (2.4)

In the following, we state the lemmas which are useful to prove the stability of the interconnected fuzzy system 𝐒 which consists of J closed-loop subsystems described in (2.1).

Lemma 2.1 (see [13]). For any A, B ∈R^𝑛and for any symmetric positive definite matrix G ∈R^𝑛×𝑛, one has 2𝐴^𝑇𝐵≤𝐴^𝑇𝐺𝐴+𝐵^𝑇𝐺⁻¹𝐵. (2.5)

Lemma 2.2 (see [34]). (Schur complements) One has TheLMI𝑄(𝑥)𝑆(𝑥)𝑆(𝑥)𝑅(𝑥)>0, (2.6) where 𝑄(𝑥)=𝑄^𝑇(𝑥), 𝑅(𝑥)=𝑅^𝑇(𝑥), and 𝑆(𝑥) depends on x that is equivalent to 𝑅(𝑥)>0,𝑄(𝑥)−𝑆(𝑥)𝑅⁻¹(𝑥)𝑆^𝑇(𝑥)>0. (2.7) In other words, the set of nonlinear inequalities (2.7) can be represented as the LMI (2.6).

3. Stability Analysis by a Fuzzy Lyapunov Function

Here we define a fuzzy Lyapunov function and consider the stability conditions for the jth fuzzy subsystem (2.2).

Definition 3.1. Equation (3.1) is said to be a fuzzy Lyapunov function for the T-S fuzzy system (2.2) if the time derivative of 𝑉(𝑡) is always negative 𝑉(𝑡)=^𝐽_𝑗=1𝑣_𝑗(𝑡)=^𝐽_𝑗=1^𝑟𝑗_𝑙=1ℎ_𝑙𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)𝑃_𝑙𝑗𝑥_𝑗(𝑡), (3.1) where 𝑃_𝑙𝑗 is a positive definite matrix.

Because the fuzzy Lyapunov function shares the same membership functions with the T-S fuzzy model of a system, the time derivative of the fuzzy Lyapunov function contains the time derivative of the premise membership functions. Therefore, how to deal with the time derivative of the time derivative of the premise membership functions is an important consideration.

By taking the time derivative of (3.1), the following stability condition of open-loop system (2.7) will be obtained.

Theorem 3.2. The fuzzy system (2.7) is stable in the large if there exist common positive definite matrices 𝑃₁,𝑃₂,…,𝑃_𝑟 such that the following inequality is satisfied: ℎ̇^𝑟𝑗_{𝜌=1𝜌𝑗}(𝑡)𝑃_𝜌𝑗+^𝑟𝑗_𝑙=1^𝑟𝑗_𝑖=1ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡)[𝐴^𝑇_𝑖𝑗𝑃_𝑙𝑖+𝑃_𝑙𝑖𝐴_𝑖𝑗+𝛼(𝐽−1)𝐼+^𝐽_𝑛=1𝛼⁻¹𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗]<0. (3.2)

Proof. Consider the Lyapunov function candidate for the fuzzy system (2.2) 𝑉(𝑡)=^𝐽_𝑗=1𝑣_𝑗(𝑡)=^𝐽_𝑗=1^𝑟𝑗_𝑙=1ℎ_𝑙𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)𝑃_𝑙𝑗𝑥_𝑗(𝑡). (A1) The time derivative of 𝑉is 𝑉̇(𝑡)=^𝐽_𝑗=1^𝑟𝑗_𝜌=1ℎ̇_𝜌𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)𝑃_𝜌𝑗𝑥_𝑗(𝑡)+^𝐽_𝑗=1^𝑟𝑗_𝑙=1ℎ_𝑙𝑗(𝑡){𝑥̇^𝑇_𝑗(𝑡)𝑃_𝑙𝑗𝑥_𝑗(𝑡)+𝑥^𝑇_𝑗(𝑡)𝑃_𝑙𝑗𝑥̇_𝑗=(𝑡)}_{𝐽𝑗=1𝑟𝑗𝜌=1}ℎ̇_𝜌𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)𝑃_𝜌𝑗𝑥_𝑗(𝑡)+^𝐽_𝑗=1^𝑟𝑗_𝑙=1ℎ_𝑙𝑗(𝑡){[^𝑟𝑗_𝑖=1ℎ_𝑖𝑗(𝑡)(𝐴_𝑖𝑗𝑥_𝑗(𝑡)+^𝐽_{𝑛=1𝑛≠𝑗}𝐴_𝑖𝑛𝑗𝑥_𝑛(𝑡))]^𝑇𝑃_𝑙𝑖𝑥_𝑗(𝑡)+𝑥^𝑇_𝑗(𝑡)𝑃_𝑙𝑗[^𝑟𝑗_𝑖=1ℎ_𝑖𝑗(𝑡)(𝐴_𝑖𝑗𝑥_𝑗(𝑡)+^𝐽_{𝑛=1𝑛≠𝑗}𝐴_𝑖𝑛𝑗𝑥_𝑛(𝑡))]}=^𝐽_𝑗=1^𝑟𝑗_𝜌=1ℎ̇_𝜌𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)𝑃_𝜌𝑗𝑥_𝑗(𝑡)+^𝐽_𝑗=1^𝑟𝑗_𝑙=1^𝑟𝑗_𝑖=1ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)[𝐴^𝑇_𝑖𝑗𝑃_𝑙𝑖+𝑃_𝑙𝑖𝐴_𝑖𝑗]𝑥_𝑗(𝑡)+^𝐽_𝑗=1^𝑟𝑗_𝑙=1^𝑟𝑗𝐽_{𝑖=1𝑛=1𝑛≠𝑗}ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡)[𝑥^𝑇_𝑛(𝑡)𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗𝑥_𝑗(𝑡)+𝑥^𝑇_𝑗(𝑡)𝑃_𝑗𝐴_𝑖𝑛𝑗𝑥_𝑛(𝑡)]. (A2)(A3)(A4) Based on Lemma 2.1, we have ^𝐽_𝑗=1^𝑟𝑗_𝑙=1^𝑟𝑗𝐽_{𝑖=1𝑛=1𝑛≠𝑗}ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡){𝑥^𝑇_𝑛(𝑡)𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗𝑥_𝑗(𝑡)+𝑥^𝑇_𝑗(𝑡)𝑃_𝑗𝐴_𝑖𝑛𝑗𝑥_𝑛≤(𝑡)}_{𝐽𝑗=1𝑟𝑗𝑙=1𝑟𝑗𝐽𝑖=1𝑛=1𝑛≠𝑗}ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡){𝛼[𝑥^𝑇_𝑛(𝑡)𝑥_𝑛(𝑡)]+𝛼⁻¹[𝑥^𝑇_𝑗(𝑡)𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗𝑥_𝑗(𝑡)]}=^𝐽_𝑗=1^𝑟𝑗_𝑙=1^𝑟𝑗_𝑖=1^𝐽_𝑛=1ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗1(𝑡){𝛼[(1−𝐽)𝑥^𝑇_𝑗(𝑡)𝑥_𝑗(𝑡)]+𝛼⁻¹[𝑥^𝑇_𝑗(𝑡)𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗𝑥_𝑗1(𝑡)−𝐽𝑥^𝑇_𝑗(𝑡)𝑃_𝑗𝐴_𝑖𝑗𝑗𝐴^𝑇_𝑖𝑗𝑗𝑃_𝑗𝑥_𝑗(𝑡)]}. (A5)(A6)(Based on the concept of interconnection, the matrix 𝐴_𝑖𝑗𝑗is equal to zero.) From (A4) and (A6), we obtain 𝑉̇(𝑡)≤^𝐽_𝑗=1^𝑟𝑗_𝜌=1ℎ̇_𝜌𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)𝑃_𝜌𝑗𝑥_𝑗(𝑡)+^𝐽_𝑗=1^𝑟𝑗_𝑙=1^𝑟𝑗_𝑖=1ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)×[𝐴^𝑇_𝑖𝑗𝑃_𝑙𝑖+𝑃_𝑙𝑖𝐴_𝑖𝑗+𝛼(𝐽−1)𝐼+^𝐽_𝑛=1𝛼⁻¹𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗]𝑥_𝑗(𝑡). (A7) Therefore, 𝑉̇(𝑡)<0 if (3.2) holds.However, condition (3.2) cannot be easily solved numerically because we need to consider the term of the time derivative ℎ̇_𝜌𝑗(𝑡). Eq. (3.2) is thus transformed into numerically feasible conditions described in Theorem 3.3 and upper bounds of the time derivative are used in place of the ℎ_𝜌(𝑡).

Theorem 3.3. The fuzzy system (2.7) is stable in the large if there exist common positive definite matrices 𝑃_1𝑗,𝑃_2𝑗,…,𝑃_𝑟𝑗 such that inequality |ℎ̇_𝜌𝑗(𝑡)|≤𝜙_𝜌𝑗 is satisfied and ^𝑟𝑗_𝜌=1𝜙_𝜌𝑗𝑃_𝜌𝑗+𝐴^𝑇_𝑖𝑗𝑃_𝑙𝑖+𝑃_𝑙𝑖𝐴_𝑖𝑗+𝛼(𝐽−1)𝐼+^𝐽_𝑛=1𝛼⁻¹𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗. (3.3)

Proof. 𝑉̇(𝑡)≤^𝐽_𝑗=1^𝑟𝑗_𝜌=1ℎ̇_𝜌𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)𝑃_𝜌𝑗𝑥_𝑗+(𝑡)_{𝐽𝑗=1𝑟𝑗𝑙=1𝑟𝑗𝑖=1}ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)[𝐴^𝑇_𝑖𝑗𝑃_𝑙𝑖+𝑃_𝑙𝑖𝐴_𝑖𝑗+𝛼(𝐽−1)𝐼+^𝐽_𝑛=1𝛼⁻¹𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗]𝑥_𝑗(𝑡)≤^𝐽_𝑗=1^𝑟𝑗_𝑙=1^𝑟𝑗_𝑖=1ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡){^𝑟𝑗_𝜌=1𝑥^𝑇_𝑗(𝑡)[𝜙_𝜌𝑗𝑃_𝜌𝑗]𝑥_𝑗(𝑡)}+^𝐽_𝑗=1^𝑟𝑗_𝑙=1^𝑟𝑗_𝑖=1ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)[𝐴^𝑇_𝑖𝑗𝑃_𝑙𝑖+𝑃_𝑙𝑖𝐴_𝑖𝑗+𝛼(𝐽−1)𝐼+^𝐽_𝑛=1𝛼⁻¹𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗]𝑥_𝑗(𝑡)=^𝐽_𝑗=1^𝑟𝑗_𝑙=1^𝑟𝑗_𝑖=1ℎ_𝑙𝑗(𝑡)ℎ_𝑖𝑗(𝑡)𝑥^𝑇_𝑗(𝑡)[^𝑟𝑗_𝜌=1𝜙_𝜌𝑗𝑃_𝜌𝑗+𝐴^𝑇_𝑖𝑗𝑃_𝑙𝑖+𝑃_𝑙𝑖𝐴_𝑖𝑗+𝛼(𝐽−1)𝐼+^𝐽_𝑛=1𝛼⁻¹𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗]𝑥_𝑗(𝑡), (A8)𝑉̇(𝑡)<0 if (3.3) holds.

Remark 3.4. A special case of sufficient conditions without 𝜙_𝜌𝑗 is proposed to guarantee the asymptotically stability of fuzzy large-scale system 𝐒. If there exist symmetric positive definite matrices 𝑃_𝑗 which satisfy that each isolated subsystem is asymptotically stable as described in (3.4), the trajectories of the interconnected system are stable: 𝐴^𝑇_𝑖𝑗𝑃_𝑗+𝑃_𝑗𝐴_𝑖𝑗+^𝐽_𝑛=1𝛼⁻¹𝑃_𝑗𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑃_𝑗+𝛼(𝐽−1)𝐼<0 (3.4) for 𝑖=1,2,…,𝑟_𝑗; 𝑗=1,2,…,𝐽.

Remark 3.5. Equation (3.4) can be recast as an LMI problem based on Lemma 2.2. Therefore, new variables 𝑊_𝑗=𝑃_𝑗⁻¹ and 𝛼=𝛼⁻¹ are introduced and (3.4) is rewritten as 𝑊_𝑗𝐴^𝑇_𝑖𝑗+𝐴_𝑖𝑗𝑊_𝑗+^𝐽_𝑛=1𝛼𝐴_𝑖𝑛𝑗𝐴^𝑇_𝑖𝑛𝑗𝑊_𝑗𝑊_𝑗−1𝛼(𝐽−1)𝐼<0for𝑖=1,2,…,𝑟_𝑗. (3.5)

4. A Numerical Example

Consider a interconnected fuzzy system S which consists of two fuzzy subsystems described by (2.2) with one rule.

Subsystem 1:
𝑥̇₁(𝑡)=ℎ₁₁(𝑡)⌊𝐴₁₁𝑥₁(𝑡)+𝐴₁₂₁𝑥₂(𝑡)⌋,

Subsystem 2:
𝑥̇₂(𝑡)=ℎ₁₂(𝑡)⌊𝐴₁₂𝑥₂(𝑡)+𝐴₁₁₂𝑥₁(𝑡)⌋,

in which

𝑥^𝑇₁(𝑡)=[𝑥₁₁(𝑡)𝑥₂₁(𝑡)],𝑥^𝑇₂(𝑡)=[𝑥₁₂(𝑡)𝑥₂₂(𝑡)],𝐴₁₁=,𝐴−16.252−6.222−6.084−24.674₁₂₁=0.30.20.10.4,𝐴₁₂=,𝐴−21.800−7.710−7.150−23.210¹¹²=.0.30.30.30.3 (4.1)

At first, based on (3.5), we can get the common solutions 𝑊_𝑗 and 𝛼 via the Matlab LMI optimization toolbox

𝑊₁=0.811−0.2411−0.24110.342,𝑊₂=,2.94821.19521.19523.1873𝛼=1. (4.2)

Then, the following positive definite matrices 𝑃_𝑗(=𝑊_𝑗⁻¹) and 𝛼 can be obtained such that (3.4) is satisfied

𝑃₁=1.56011.10021.10023.7001,𝑃₂=0.4002−0.1511−0.15120.3703,𝛼=1. (4.3)

Therefore, based on the theorem, the interconnected fuzzy system S described in (4.1) is guaranteed to be asymptotically stable. From the simulation of Figures 1 and 2 and given the initial conditions, 𝑥₁₁(0)=2, 𝑥₂₁(0)=−2, 𝑥₁₂(0)=4, 𝑥₂₂(0)=−4, we observe that the interconnected fuzzy system S is asymptotically stable, because the trajectories of two subsystems starting from non-zero initial states both approach close to the origin.

Figure 1 

The state response of subsystem 1.

Figure 2 

The state response of subsystem 2.

5. Conclusions

For the class of continuous interconnected fuzzy system S, LMI-based stability conditions have been derived based on the new fuzzy Lyapunov function. Sufficient stability conditions were derived based on the asymptotically stability of the existence of a common positive definite matrix Pi which is able to satisfy the Lyapunov equation or the LMI for each subsystem Si. Finally, a numerical example was given to illustrate the effectiveness and ease of implementation of this approach.

Acknowledgment

This work was supported by the National Science Council of Republic of China under Grant nos. NSC 98-2221-E-237-009-, NSC 96-2628-E-366-004-MY2, and NSC 98-2221-E-366-006-MY2.