Fuzzy Large-Scale Systems Stabilization with

Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October 2009

Fuzzy Large-Scale Systems Stabilization with Nonlinear State Feedback Controller Iman Zamani

Nasser Sadati

Electrical Engineering Department The Amirkabir University of Technology Tehran, Iran [email protected]

Electrical and Computer Engineering Department The University of British Columbia Vancouver, BC Canada [email protected]

Abstract— This paper investigates the stability and stabilization problem of fuzzy large-scale systems in which the system is composed of a number of Takagi-Sugeno fuzzy subsystems with interconnection. Instead of fuzzy parallel distributed compensation (PDC) design, nonlinear state feedback controllers are used in the stabilization of the overall large-scale system. Based on Lyapunov criterion, some conditions are derived under which the whole fuzzy large-scale system is stabilized asymptotically.

subsystem equations

୬

୧୪ ൌ

୧୪ —୧ ሺ–ሻ

‫š‡Š–۔‬ሶ ୧ ሺ–ሻ ൌ ൅ ൅ ෍ ˆ୧୨୪ ൫š୨ ሺ–ሻ൯ ۖ ୨ୀଵ ‫ە‬ ୨ஷ୧ ሺ‹ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ሻǡ ሺŒ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ሻǡ ሺŽ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ”୧ ሻሺͳሻ

୧୪ —୧ ሺ–ሻ ‫ א‬Թ୫౟

is the Žth rule of ୧ . is the control input of ୧ at time – with appropriate length. the State vector of the ‹th subsystem š୧ ሺ–ሻ ‫ א‬Թ୬౟ ୘ š୧ ሺ–ሻ ൌ ൣš୧ଵ ሺ–ሻǡ š୧ଶ ሺ–ሻǡ ‫ ڮ‬ǡ š୧୬౟ ሺ–ሻ൧ . the interconnection between ‹th and ˆ୧୨୪ ൫š୨ ሺ–ሻ൯ Œth subsystem in the Žth rule of ୧ . the number of rules in subsystem ୧ . ”୧ the Number of states in subsystem ୧ . ୧ ൫୪୧ ǡ ୧୪ ൯ the system matrices of rule Ž in  subsystem ୧ that are controllable. ୧୩ the grade of membership of š୧୩ ሺ–ሻǡ ሺ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ୧ ሻ. If we utilize the standard fuzzy inference method i.e., singleton, minimum or product fuzzy inference, and centralaverage deffuzzifier, (1) can be inferred as ୰౟

šሶ ୧ ሺ–ሻ ൌ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ሺ୪୧ š୧ ሺ–ሻ ൅ ୧୪ —୧ ሺ–ሻሻ ୪ୀଵ

Consider a fuzzy large-scale system ሺሻ which consists of interconnected subsystems ୧ ሺ‹ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ሻ . The ‹ th fuzzy

978-1-4244-2794-9/09/$25.00 ©2009 IEEE

5156

୎

୰౟

൅ ෍ ෍ ρ୪୧ ሺš୧ ሺ–ሻሻˆ୧୨୪ ൫š୨ ሺ–ሻ൯ ሺʹሻ ୨ୀଵ ୪ୀଵ ୨ஷ୧

where

PRELIMINARIES

978-1-4244-2794-9/09/$25.00 ©2009 IEEE

୎

୪୧ š୧ ሺ–ሻ

where

INTRODUCTION

Stability analysis and stabilization of fuzzy large-scale systems for both discrete and continuous cases were discussed by some researches[1]-[4]. In this work, a fuzzy large-scale system consisting of subsystems ୧ ሺ‹ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ሻ with interconnection is considered. Each subsystem is decomposed into a set of fuzzy regions, were in each region there is a local dynamic behavior of the subsystems described by a T-S fuzzy model. Motivated from the work in [5] for fuzzy systems, the stability analysis and controller design of continuous-time fuzzy large-scale system using Lyapunov function is considered in this paper. Based on Lyapunov and relevant theory such as Lasalle’s theorem, a nonlinear state feedback controller for each subsystem is designed such that the whole closed loop large-scale system under some sufficient conditions is stabilized asymptotically. These conditions are derived in the form of matrix inequalities. The main advantages of the proposed approach are low computation, optional feedback gains and some merits which are discussed in the next sections. The article is organized as follows; the considered system and problem formulation are discussed in Section II. In Section III, nonlinear state feedback controllers are designed and the main results are proposed. Section IV provides an example to illustrate the applicability of the proposed controller design. Finally, some concluding remarks are given in Section V. II.

‹ˆš୧ଵ ሺ–ሻ‹•୧ଵ ƒ† ‫šڮ‬୧୬౟ ሺ–ሻ‹•୧ ౟ 

‫ۓ‬ ۖ

Keywords— Fuzzy large-scale system, nonlinear controller, T-S fuzzy systems, stability analysis

I.

୧ is described by the following rule-based

ρ୧ ୪ ൫š୧ ሺ–ሻ൯ ൌ

୰౟

୬౟ ™୧ ୪ , ™ ୪ ൌ ς୩ୀଵ ୧୩ ሺš୧୩ ሺ–ሻሻ ൒ Ͳ ൘σ୰౟ ™୨ ୧ ୨ୀଵ ୧

୎

šሶ ୧ ሺ–ሻ ൌ ෍ ෍ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯୩୧ ሺ–ሻ ቆ ୪ୀଵ ୩ୀଵ ୨ୀଵ ୨ஷ୧

୰

౟ and σ୪ୀଵ ρ୧ ୪ ൫š୧ ሺ–ሻ൯ ൌ ͳ. ρ୧ ୪ ൫š୧ ሺ–ሻ൯ is the firing strength of Žth rule of the ‹th subsystem. It is assumed that the fuzzy sets are normalized, were two examples of normalized fuzzy sets are shown in Fig. 1. Also it is assumed that ˆ୧୨୪ ൫š୨ ሺ–ሻ൯ ൌ ୧୨୪ š୨ ሺ–ሻ, where ୧୨୪ is a constant matrix with appropriate dimension.

1

ͳ  ୪୩ š ሺ–ሻ

െͳ ୧ ୧

൅ ୧୨୪ š୨ ሺ–ሻቇሺ͸ሻ where ୧୪୩ ൌ ୪୧ െ ୧୪  ୩୧ . Now, our task is to design  ୩୧ in such a way that the whole fuzzy large-scale system is stabilized asymptotically and exponentially. The stability condition of fuzzy large-scale system (6), can be summarized by the following theorem. Theorem 3.1: The fuzzy large-scale system (2) is stabilized asymptotically by the nonlinear state feedback controller (3), if there exist symmetric positive matrices ୧ , positive scalars Ʉ୪୧ and the state gainsଵ୧ such that the following conditions are satisfied:

”‡‹•‡ȋ‘”…‘•‡“—‡–Ȍ˜ƒ”‹ƒ„Ž‡

Figure 1. Two types of normalized fuzzy sets; solid line shows the normalized triangular fuzzy sets and dash line shows the other normalized fuzzy sets

III. NONLINEAR STATE FEEDBACK CONTROLLERS In this section, we will consider the controller design for fuzzy large-scale system (3). For stabilization, a nonlinear state feedback controller is considered as follows ୡ౟

—୧ ሺ–ሻ ൌ െ ෍ ୩୧ ሺ–ሻ ୩୧ š୧ ሺ–ሻ ሺ͵ሻ ୩ୀଵ

in which

ୡ౟

ୡ౟

෍ ୩୧ ሺ–ሻ ൌ ͳƬͲ ൑ ୩୧ ሺ–ሻ ൑ ͳ ୩ୀଵ

ሺ‹ ൌ ͳǡʹǡ ‫ ڮ‬ǡ  ൌ ͳǡʹǡ ‫ ڮ‬ǡ …୧ ሻሺͶሻ and  ୩୧ is the state feedback gain with appropriate dimension. ୩୧ ሺ–ሻ is a nonlinear function defined by (5. a) and (5. b), and …୧ is an optional number for designing the controller in each subsystem. ((5. a) and (5. b) are shown at the end of this page and the next page). It should be noted that if the control law based on PDC was used, then the proof of stabilization by Lyapunov method had much difficulty in the derivation of ሶ ൏ Ͳ . Hence, by proposing the new approach, using combination of (2) and (3), the closed-loop fuzzy subsystem becomes

୪ െ୪ଵ ୧ ൑ െɄ୧ ୧ ሺ͹Ǥ ƒሻ െ ൏ Ͳሺ͹Ǥ „ሻ where ͳ Ͳ ͳ  , ሺŽ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ”୧ ሻǡ ሺ‹ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ሻ  ୧ ൌ ൦ ൪ ‫ڰ‬ Ͳ ͳ ୬౟ ൈ୬౟ ୘

, ୧୪୩ ୧ ൅ ୧ ୧୪୩ ൌ െ୪୩ , ሺ ൌ ͳǡʹǡ ‫ ڮ‬ǡ …୧ ሻ ୧ ୧ ‹•–Š‡—„‡”‘ˆ•–ƒ–‡•‘ˆ୧ , Ʉ୧ ൌ ‹୪ ሺɄ୪୧ ሻ ൐ Ͳ, ‹ൌŒ ଵ୧ ሺ–ሻɄ୧ , Ⱦ୧୨ ൌ ƒš୪ ൫ฮ୧୨୪ ฮ൯ ൫୧ ൅ ୨ ൯ ,  ൌ ൣ୧୨ ൧ ൌ ቐ ଵ െ୧ ሺ–ሻ”୧ ɉ୧ Ⱦ୧୨ ‹ ് Œ ɉ୧ ൌ ƒš൫‡‹‰ሺ୧ ሻ൯. Proof: Consider the Lyapunov function ሺ–ሻ for the fuzzy large-scale system, given by ୎

୎

ሺ–ሻ ൌ ෍ ୧ ሺ–ሻ ൌ ෍ š୧୘ ሺ–ሻ୧ š୧ ሺ–ሻ ሺͺሻ ୧ୀଵ

୧ୀଵ

Now, by taking the derivative of ୧ ሺ–ሻ along the trajectories of ‹th subsystem, we get ሶ୧ ሺ–ሻ ൌ šሶ ୧୘ ሺ–ሻ୧ š୧ ሺ–ሻ ൅ š୧୘ ሺ–ሻ୧ šሶ ୧ ሺ–ሻሺͻሻ

ଵ୧ ሺ–ሻ

ൌ

ͳ ୰౟ ୡ౟ ୪ ‫ۓ‬ σ୎୨ୀଵ ρŽ‹ ൫š‹ ሺ–ሻ൯ ቀ ൭σ୩ୀଶ ൭σ୪ୀଵ š ሺ–ሻ୘ ୪୩ ୡ౟ ୰౟ ୎ ୧ š୧ ሺ–ሻ െ ୧୨ ሺ–ሻቁ൱൱

െͳ ୧ ۖ ͳ ୨ஷ୧ ୪ ۖ ͳെ ǡ ‹ˆ ෍ ෍ ෍ ฬρŽ‹ ൫š‹ ሺ–ሻ൯ ൬ š୧ ሺ–ሻ୘ ୪୦ ୧ š୧ ሺ–ሻ െ ୧୨ ሺ–ሻ൰ฬ ് Ͳ ͳ ۖ ୡ౟ ୰౟ ୎

െ ͳ ୪୦ ሺ–ሻ ୪ ୘ Ž σ σ σ ሺ ሻ൯ ሺ–ሻ ୧ š୧ െ ୧୨ ሺ–ሻቁቚ ୦ୀଵ ୪ୀଵ ୨ୀଵ ۖ ୦ୀଵ ୪ୀଵ ୨ୀଵ ቚρ‹ ൫š‹ – ቀ െ ͳ š୧ ୨ஷ୧

୨ஷ୧

‫۔‬ ୡ౟ ୰౟ ୎ ۖ ͳ ۖͳ ୪ Ž š ሺ–ሻ୘ ୪୦ ۖ ‹ˆ ෍ ෍ ෍ ฬρ‹ ൫š‹ ሺ–ሻ൯ ൬ ୧ š୧ ሺ–ሻ െ ୧୨ ሺ–ሻ൰ฬ ൌ Ͳ

െͳ ୧ ۖ…୧ ୦ୀଵ ୪ୀଵ ୨ୀଵ ‫ە‬ ୨ஷ୧

ሺͷǤ ƒሻ

5157

Now since ୘ š୨୘ ሺ–ሻ୧୨୪ ୧ š୧ ሺ–ሻ

୧୨୪ ሺ–ሻ

ൌ (9) can be rewritten as ሶ୧ ሺ–ሻ ୡ౟

୰౟

୎

ൌ െ ෍ ෍ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯୩୧ ሺ–ሻ ቆ ୪ୀଵ ୩ୀଵ ୨ୀଵ ୨ஷ୧

൅ š୧ ሺ–ሻ

୘

୧ ୧୨୪ š୨ ሺ–ሻሺͳͲሻ

ͳ š ሺ–ሻ୘ ୪୩ ୧ š ୧ ሺ–ሻ

െͳ ୧

୰

୎

౟ σ୨ୀଵ ρ୪୧ ൫š୧ ሺ–ሻ൯ ቀ ൭σ୪ୀଵ

ୡ౟

ͳ ୪ š ሺ–ሻ୘ ୪୩ ୧ š ୧ ሺ–ሻ െ ୧୨ ሺ–ሻቁ൱

െͳ ୧

ଶ

‫ۇ‬ ‫ۇ‬ ‫ۊۊ‬ ୧ஷ୨ ‫ۋۋ‬ െ ‫ۈ‬෍ ‫ۈ‬ ͳ ୰౟ ୎ ‫ۈ‬ ‫ ۈ‬ୡ౟ ୪ š୧ ሺ–ሻ୘ ୪୦ š୧ ሺ–ሻ െ ୧୨୪ ሺ–ሻቁቚ‫ۋۋ‬ ୩ୀଶ σ୦ୀଵ σ୪ୀଵ σ୨ୀଵ ቚρ୧ ൫š ୧ ሺ–ሻ൯ ቀ ୧

െͳ ୨ஷ୧ ‫یی‬ ‫ۉ‬ ‫ۉ‬ ሺͳ͵ሻ Therefore, ୎

ሶሺ–ሻ ൌ ෍ ሶ୧ ሺ–ሻ

െ ୧୨୪ ሺ–ሻቇሺͳͳሻ

୧ୀଵ ୰౟

୎

୰౟

‫ۇ‬ ͳ ‫ۇ‬ ଵ ሺ–ሻ ‫ۈ‬෍ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ቆ ൌ െ‫ۈ‬ š ሺ–ሻ୘ ୪ଵ ୧ š ୧ ሺ–ሻ ‫ۈ‬୧

െͳ ୧ ‫ۉ‬

‫ۉ‬

୎

ͳ ‫ۇ‬ ൑ െ ෍ ଵ୧ ሺ–ሻ ‫ۈ‬෍ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ቆ š ሺ–ሻ୘ ୪ଵ ୧ š ୧ ሺ–ሻ

െͳ ୧

୎

୧ୀଵ

‫ۉ‬

୪ୀଵ ୨ୀଵ ୨ஷ୧

୪ୀଵ ୨ୀଵ ୨ஷ୧

െ ୧୨୪ ሺ–ሻቇ൲ ‫ۊ‬ െ ୧୨୪ ሺ–ሻቇ൲‫ۋ‬ ‫ۋ‬ ‫ی‬

୰౟

ୡ౟

୎

ൌ

୎

ͳ ‫ۇ‬ ‫ۇ‬ െ ‫ۈ‬෍ ୩୧ ሺ–ሻ ‫ۈ‬෍ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ൬ š ሺ–ሻ୘ ୪୩ ୧ š ୧ ሺ–ሻ

െͳ ୧ ୩ୀଶ

‫ۉ‬

‫ۉ‬ െ

୧ୀଵ

୪ୀଵ ୨ୀଵ ୨ஷ୧

୰౟

୪ୀଵ

‫ۉ‬

୎

୪ୀଵ ୨ୀଵ ୨ஷ୧

‫ۊ‬

‫ی‬

୰

౟ Since σ୪ୀଵ ì୪୧ ൫š୧ ሺ–ሻ൯ ൌ ͳ and Ʉ୧ ൌ ‹୪ ሺɄ୪୧ ሻ, we can conclude that

୰౟

୎

‫ۇ‬ ͳ ‫ۇ‬ ଵ ሺ–ሻ ‫ۈ‬෍ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ቆ š ሺ–ሻ୘ ୪ଵ ሶ୧ ሺ–ሻ ൌ െ ‫ۈ‬ ୧ š ୧ ሺ–ሻ ‫ۈ‬୧

െͳ ୧ ‫ۉ‬

୰౟

െ ෍ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ୧୨୪ ሺ–ሻ൲ ሺͳͶሻ

୧୨୪ ሺ–ሻ൰൲‫ ۋ‬ሺͳʹሻ

So,

‫ۇ‬

െ ෍ ଵ୧ ሺ–ሻ ‫ۈ‬෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ቀš୧ ሺ–ሻ୘ ୪ଵ ୧ š ୧ ሺ–ሻቁ

‫ۉ‬

୰౟

୰౟

െ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ቀš୧ ሺ–ሻ୘ ୪ଵ ୧ š ୧ ሺ–ሻቁ

୪ୀଵ ୨ୀଵ ୨ஷ୧

୪ୀଵ

‫ۊ‬ െ ୧୨୪ ሺ–ሻቇ൲‫ۋ‬ ‫ۋ‬

൑ െߟ௜ ԡš୧

ԡଶ

෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ୪ୀଵ

ൌ െߟ௜ ԡš୧ ԡଶ ሺͳͷሻ

also

‫ی‬ ୩୧ ሺ–ሻ

ͳ ୰౟ ୪ σ୪ୀଵ σ୎୨ୀଵ ρ୪୧ ൫š୧ ሺ–ሻ൯ ቀ š ሺ–ሻ୘ ୪୩ ୡ౟ ୰౟ ୎ ‫ۓ‬ ୧ š ୧ ሺ–ሻ െ ୧୨ ሺ–ሻቁ

െͳ ୧ ͳ ୨ஷ୧ ୪ ۖ ǡ‹ˆ ෍ ෍ ෍ ฬρ୪୧ ൫š୧ ሺ–ሻ൯ ൬ š୧ ሺ–ሻ୘ ୪୦ ୧ š ୧ ሺ–ሻ െ ୧୨ ሺ–ሻ൰ฬ ് Ͳ ͳ ۖ ୡ౟ ୰౟ ୎

െ ͳ ୪ ୪୦ ୪ ୘ ୦ୀଵ ୪ୀଵ ୨ୀଵ ۖ σ୦ୀଵ σ୪ୀଵ σ୨ୀଵ ቚρ୧ ൫š୧ ሺ–ሻ൯ ቀ െ ͳ š୧ ሺ–ሻ  ୧ š୧ ሺ–ሻ െ ୧୨ ሺ–ሻቁቚ

ൌ

୨ஷ୧

୨ஷ୧

‫۔‬ ୡ౟ ୰౟ ୎ ۖͳ ͳ ୪ ۖ ‹ˆ ෍ ෍ ෍ ቤρ୪୧ ൫š୧ ሺ–ሻ൯ ቆ š ሺ–ሻ୘ ୪୦ ୧ š ୧ ሺ–ሻ െ ୧୨ ሺ–ሻቇቤ ൌ Ͳ

െ ͳ ୧ … ۖ ୧ ୦ୀଵ ୪ୀଵ ୨ୀଵ ‫ە‬ ୨ஷ୧ ሺͷǤ „ሻ

5158

୰౟

Remark 3.1: Considering (7. b), we can use Sylvester criterion [7] for positive definite matrices. It is easy to show that these conditions are independent of ଵ୧ ሺ–ሻሺ‹ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ሻ and we can get Ʉ୧ ሺ‹ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ሻ easily. Exponential stability is another important subject in stability analysis, which is concerned with asymptotic stability and decay rate. To treat the exponential stability analysis, an approach is often adopted, namely exponential stability theorem [8].

෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ୧୨୪ ሺ–ሻ ୪ୀଵ

୰౟

୘

ൌ ෍ ρ୪୧ ൫š୧ ሺ–ሻ൯ ൬š୨୘ ሺ–ሻ୧୨୪ ୧ š୧ ሺ–ሻ ൅ š୧ ሺ–ሻ୘ ୧ ୧୨୪ š୨ ሺ–ሻ൰ ୪ୀଵ

൑ ”୧ ԡ୧ š୧ ԡฮ୧୨୪ š୨ ฮ൫୧ ൅ ୨ ൯ ୘

൑ ”୧ ට൫୧୨୪ š୨ ൯ ୧୨୪ š୨ ටɉ୫ୟ୶ ൫୧ ୘ ୧ ൯ԡš୧ ԡฮš୨ ฮ൫୧ ൅ ୨ ൯ ൑ ”୧ ԡš୧ ԡฮš୨ ฮԡ୧ ԡଶ ƒš൫ฮ୧୨୪ ฮ൯ ൫୧ ൅ ୨ ൯ሺͳ͸ሻ ୪

Now based on (15) and (16), it is clear that ሶሺ–ሻ ൑

From the proposed method and the theorem, it is easy to conclude that I. This theorem and the proposed method are innovative, usable and applicable with low amount of computation. This method does not include some restrictive conditions such as norm bounded, and also this is not a pre-designed method, i.e. it is not necessary to check the stability for pre-designed system that requires trial-anderror for controller design. ଵ ୩ II. ሺ͹Ǥ ƒሻ is checked only for ୪ଵ ୧ and  ୧ . Therefore  ୧ is optional ሺˆ‘” ് ͳሻ and doesn’t affect the stability. The number of state feedback gains which must be designed decreases drastically. Based on the authors experience,  ୩୧ ሺ ് ͳሻaffect the type of responses i.e., the amount of oscillation, damping, settling time, etc. Therefore, the designer has more flexibility in designing.

୎

‫ۇ‬ ෍ ଵ୧ ሺ–ሻ ‫ۈ‬െߟ௜ ԡš୧ ԡଶ ୧ୀଵ

‫ۉ‬

୎

൅ ෍ ”୧ ԡš୧ ԡฮš୨ ฮԡ୧ ԡଶ ƒš൫ฮ୧୨୪ ฮ൯ ൫୧ ୪

୨ୀଵ ୨ஷ୧

൅ ୨ ሻ‫ ۊ‬ሺͳ͹ሻ ‫ۋ‬ ‫ی‬

IV. ILLUSTRATIVE EXAMPLE

Therefore, ୎

௃

ሶሺ–ሻ ൑ ෍ ଵ୧ ሺ–ሻ ቌെߟ௜ ԡš୧ ԡଶ ൅ ෍ ‫ݎ‬௜ ԡ‫ݔ‬௜ ԡฮ‫ݔ‬௝ ฮߣ௜ ߚ௜௝ ቍ ୧ୀଵ

௝ୀଵ

ሺͳͺሻ where

Here, an example is presented to verify the results of the proposed stabilization procedure of fuzzy large-scale systems. We consider the stability of the following fuzzy large-scale system composed of two subsystems described by • Subsystem ଵ ǣ

ԡܲ௜ ԡଶ ൌ ߣ௜ ൌ ߣ௠௔௫ ሺܲ௜ ሻǡߚ௜௝ ൌ ݉ܽ‫ݔ‬௟ ൫ฮ‫ܥ‬௜௝௟ ฮ൯ ൫݊௜ ൅ ݊௝ ൯ Following [6], the right-hand side of (18) is quadratic in terms of ൛ԡšଵ ԡ ԡšଶ ԡ ‫ ڮ‬ฮš୎ ฮൟ, which can be rewritten as ”‹‰Š– െ Šƒ†•‹†‡‘ˆሺͳͺሻ ԡš ԡ ‫ ۍ‬ଵ ‫ې‬ ԡšଶ ԡ ‫ ۑ‬ሺͳͻሻ ൌ െൣԡšଵ ԡ ԡšଶ ԡ ‫ ڮ‬ฮš୎ ฮ൧  ൈ  ൈ ‫ێ‬ ‫ۑ ڭ ێ‬ ‫ ۏ‬ฮš୎ ฮ ‫ے‬ Now if (15) and (18) hold, then we get ሶሺ–ሻ ൏ Ͳ. Now, if ଵ ୩୧ ൫š୧ ሺ–ሻ൯ ൌ , then ୡ౟ ͳ ୪ ρ୪୧ ൫š୧ ሺ–ሻ൯ ൬ š ሺ–ሻ୘ ୪୩ ୧ š ୧ ሺ–ሻ െ ୧୨ ሺ–ሻ൰ ൌ Ͳ

െͳ ୧ As it can be seen, the above procedure is more simpler and it can be easily resulted that ሶሺ–ሻ ൑ Ͳ. With using the Laslle’s principle, since the limit set includes only the trivial trajectory šሺ–ሻ ‫Ͳ ؠ‬, and since the limit set only includes the origin, the origin is asymptotically stable. Thus, the proof is completed.

—Ž‡ͳǣ‹ˆšଵଵ ሺ–ሻ‹•ଵଶ ƒ†šଵଶ ሺ–ሻ‹•ଵଵ –Š‡šሶ ଵ ሺ–ሻ ൌ

ଵଵ šଵ ሺ–ሻ

൅

ଵଵ —ଵ ሺ–ሻ

ଶ ଵ ൅ ෍ ଵ୨ š୨ ሺ–ሻ ୨ୀଵ ୨ஷଵ

—Ž‡ʹǣ‹ˆšଵଵ ሺ–ሻ‹•ଵଶ ƒ†šଵଶ ሺ–ሻ‹•ଵଶ

ଶ

ଶ –Š‡šሶ ଵ ሺ–ሻ ൌ ଶଵ šଵ ሺ–ሻ ൅ ଵଶ —ଵ ሺ–ሻ ൅ ෍ ଵ୨ š୨ ሺ–ሻ ୨ୀଵ ୨ஷଵ

—Ž‡͵ǣ‹ˆšଵଵ ሺ–ሻ‹•ଵଵ ƒ†šଵଶ ሺ–ሻ‹•ଵଵ –Š‡šሶ ଵ ሺ–ሻ ൌ

ଷଵ šଵ ሺ–ሻ

൅

ଵଷ —ଵ ሺ–ሻ

ଶ ଷ ൅ ෍ ଵ୨ š୨ ሺ–ሻ ୨ୀଵ ୨ஷଵ

Ͳ ͳ ଶ Ͳ ͳ in which ଵଵ ൌ ቂ ቃ ǡ ଵ ൌ ቂ ቃǡ ଷଵ ൌ െͳ Ͳ െͳǤʹ ͳ Ͳ ͵ ͲǤͳ ͳǤ͸ Ͳ ቂ ቃ , ଵଵ ൌ ቂ ቃ ǡ ଵଶ ൌ ቂ ቃ ǡ ଵଷ ൌ ቂ ቃǤ Moreover, the ͳ Ͳ Ͳ Ͳ ͳǤʹ interconnection matrices among the two subsystems are given

5159

ͲǤͳ Ͳ ͲǤͲͳ ͲǤͲͳ ଵ ଷ by ଵଶ ൌቂ ቃ ǡ ଶ ൌ ቂ ቃ , ଵଶ ൌ െͲǤ͸ Ͳ ଵଶ െͲǤ͸ Ͳ െͲǤͳ ͲǤʹͳ ୪ ቂ ቃ. Also Ⱦଵଶ ൌ ƒš୪ ൫ฮଵଶ ฮ൯ሺଵ ൅ ଶ ሻ െͲǤ͸ Ͳ ଵ ԡǡ ԡ ଶ ԡǡ ԡ ଷ ԡሽ ൈ ሺଵ ൅ ଶ ሻ ൌ ͲǤ͸ͲͻͶ , ൌ ƒšሼԡଵଶ ଵଶ ଵଶ šଵଵ ሺ–ሻ ”ଵ ൌ ͵ and šଵ ሺ–ሻ ൌ ൤ ൨ . The normalized membership šଵଶ ሺ–ሻ functions of subsystem 1 are shown in Fig. 2. •

becomes stable. For each subsystem, let us set …ଵ ൌ …ଶ ൌ ʹ. By using the above procedure, we can have ଵଵ ൌ ሾʹǤͲ͹͵Ͷǡ െͲǤͳͶͷʹሿ, ͷǤͷͶʹͲ ͲǤ͵ͻ͵͹ “ൌቂ ቃ ൌ  ିଵ ൐ Ͳ ͲǤ͵ͻ͵͹ ʹǤͺͳ͵ͺ which implies Ʉଵ ൌ ሺͲǤͳ͹ͺ͸ሻଶ ൌ ሺɉ୫୧୬ ሺሻሻଶ ൌ ͲǤͲ͵ͳͻǡ ɉ୫ୟ୶ ሺሻ ൌ ͲǤ͵͸ʹ͸. Also ଶ

Subsystem ଶ ǣ

—Ž‡ͳǣ‹ˆšଶଵ ሺ–ሻ‹•ଶଵ ƒ†šଶଶ ሺ–ሻ‹•ଶଵ 

ଶ

ଵ –Š‡šሶ ଶ ሺ–ሻ ൌ ଵଶ šଶ ሺ–ሻ ൅ ଶଵ —ଶ ሺ–ሻ ൅ ෍ ଶ୨ š୨ ሺ–ሻ ୨ୀଵ ୨ஷଶ

—Ž‡ʹǣ‹ˆšଶଵ ሺ–ሻ‹•ଶଶ ƒ†šଶଶ ሺ–ሻ‹•ଶଵ 

ଶ

ଶ –Š‡šሶ ଶ ሺ–ሻ ൌ ଶଶ šଶ ሺ–ሻ ൅ ଶଶ —ଶ ሺ–ሻ ൅ ෍ ଶ୨ š୨ ሺ–ሻ ୨ୀଵ ୨ஷଶ

Ͳ ͳ Ͳ Ͳ ͳǤͷ ଶ in which ଵଶ ൌ ቂ ቃ ǡ ଶ ൌ ቂ ቃ , ଶଵ ൌ ቂ ቃ ǡ ଶଶ ൌ ʹ Ͳ ͲǤͷ͸ Ͷ Ͳ ͲǤͳ ቂ ቃ. Moreover, the interconnection matrices among the two Ͳ subsystems are given as ͲǤͳ ͲǤͳ ଵ Ͳ Ͳ ଶ ଶଵ ቂ ቃǡ ଶଵ ൌቂ ቃ Ͳ Ͳ ͲǤͳ ͲǤͳ also Ⱦଶଵ ୪ ଵ ԡǡ ԡ ଶ ԡሽሺ ൌ ƒš൫ฮଶଵ ฮ൯ ሺଶ ൅ ଵ ሻƒšሼԡଶଵ ଶ ൅ ଵ ሻ ଶଵ ୪

š ሺ–ሻ ൌ ͲǤʹͲͲͲ, ”ଶ ൌ ʹ and šଶ ሺ–ሻ ൌ ൤ ଶଵ ൨. šଶଶ ሺ–ሻ The normalized membership functions of subsystem 2 are shown in Fig. 3. First, we design a nonlinear state feedback controller for each subsystem so to make the fuzzy large-scale system

൫ɉ ሺ ሻ൯ െ”ଵ ɉȾଵଶ Ʉଶ ൌ ʹͶǤͲͻͷͻ ฺ  ቤ ୫୧୬ ୧ ቤ െ”ଶ ɉȾଶଵ Ʉଶ ͲǤͲ͵ͳͻ െ͵ ൈ ͲǤ͵͸ʹ͸ ൈ ͲǤ͸ͲͻͶ ฬ൐Ͳ ൌฬ െʹ ൈ ͲǤ͵͸ʹ͸ ൈ ͲǤʹͲͲͲ Ʉଶ and the state feedback gain is obtained as ଵଶ ൌ ሾͷǤͳ͸ͺʹͶ͵Ǥͷͻ͵Ͳሿ . Since ଵଶ ƒ† ଶଶ  are optional, by choosingଵଶ ൌ ሾͳǡͳሿ and  ଶଶ ൌ ሾͲǤͲͳǡͲǤͲͳሿ, the procedure is completed. Hence, the overall closed-loop fuzzy large-scale system is stabilized asymptotically. The complete simulation results with initial condition šଵ ሺͲሻ ൌ ሾͲǤ͵ǡ െͲǤͷሿ୘ , šଶ ሺͲሻ ൌ ሾͲǤͺǡ െͲǤͻሿ୘ are shown in Figs. 4-7, respectively. Also for comparison, the responses without controllers (—୧ ሺ–ሻ ‫Ͳ ؠ‬ሺ‹ ൌ ͳǡʹሻ) are shown.

V. CONCLUSION In this paper, a new approach is used to stabilize the fuzzy large-scale systems in sense of Lyapunov Stability. Under some sufficient conditions, the nonlinear state feedback controllers are developed to stabilize the fuzzy large-scale system exponentially. Since in the real world, almost all systems are complex, the extracted results in this paper can be widely used in the practical applications. An example was also presented to demonstrate the applicability and effectiveness of the proposed approach.

1

1

0.9

0.9

‫ܯ‬ଵଵ

0.8 0.7

‡Ǥ ˜ƒŽ—‡•

0.7

‡Ǥ ˜ƒŽ—‡•

0.6 0.5

0.6 0.5

0.4

0.4

0.3

0.3

‫ܯ‬ଵଶ

0.2

‫ܯ‬ଶଶ

0.2

0.1 0 -2

‫ܯ‬ଶଵ

0.8

0.1 -1.5

-1

-0.5

0

0.5

1

1.5

0

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Figure 3. Membership function of subsystem ࡿ૛

Figure 2. Membership functions of subsystem ࡿ૚

5160

2

3

5

2.5 4

2 1.5

3

‫ݔ‬ଶଶ ሺ‫ݐ‬ሻ

‫ݔ‬ଶଵ ሺ‫ݐ‬ሻ

1 0.5

2

0 1

-0.5 -1

0 -1.5 -1

-2 0

10

20

30

40

50

60

70

80

0

90

10

20

30

40

50

60

70

‹‡Ǧ•–‡’

‹‡Ǧ•–‡’ Figure 4. Response ‫ݔ‬ଵଵ ሺ‫ݐ‬ሻ of subsystem1 when ‫ݑ‬ሺ‫ݐ‬ሻ ‫( Ͳ ؠ‬dash-line) and with controller (solid-line)‫ݔ‬ଵଵ ሺͲሻ ൌ ͲǤ͵

Figure 5. Response šଵଶ ሺ–ሻ of subsystem1 when —ሺ–ሻ ‫( Ͳ ؠ‬dash-line) and with controller (solid-line) šଵଶ ሺͲሻ ൌ െͲǤͷ

1

1

0.5

0.5

0

‫ݔ‬ଵଵ ሺ‫ݐ‬ሻ

0

‫ݔ‬ଵଶ ሺ‫ݐ‬ሻ -0.5

-0.5 -1

-1 -1.5

-1.5

0

500

1000

1500

Figure 6. Response ‫ݔ‬ଶଵ ሺ‫ݐ‬ሻ of subsystem1 when‫ݑ‬ሺ‫ݐ‬ሻ ‫( Ͳ ؠ‬dash-line) and with controller (solid-line)‫ݔ‬ଶଵ ሺͲሻ ൌ ͲǤͺ

REFERENCES

[2] [3] [4]

[5]

[6] [7] [8]

0

50

100

150

‹‡Ǧ•–‡’

‹‡Ǧ•–‡’

[1]

-2

W. J. Wang and W. W. Lin “Decentralized PDC for large-scale T–S fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 13, no. 6, Dec 2005. W. J. Wang and L. Luoh “Stability and stabilization of fuzzy large-scale systems “IEEE Trans. Fuzzy Syst., vol. 12, no. 3, June 2004. F. H. Hsiao and J. D. Hwang “Stability analysis of fuzzy large-scale systems,” IEEE Trans. Syst., Man, Cybern. vol. 32, no. 1, Feb 2001. H. Zhang, C. Li, and X. Liao “Stability Analysis and H’ Controller Design of Fuzzy Large-Scale Systems Based on Piecewise Lyapunov Functions,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 36, no. 3, June 2006. H. K. Lam, F. H. Leung, and P. K. S. Tam,˝ Nonlinear state feedback controller for nonlinear systems: stability analysis and design based on fuzzy plant model,˝ IEEE Trans. Fuzzy Syst., vol. 9, no. 4, Aug 2001. H. K. Khalil, Nonlinear Systems, 3rd ed. Upper saddle River, NJ: Prentice-Hall, 2002. K. Ogata, Discrete Time Control Systems. Published 1995, Prentice Hall. M. Bodson., Stability, Convergence, and Robustness of Adaptive Systems, Doctoral thesis, College of Engineering, University of California, Berkeley (1986).

5161

Figure 7. Response ‫ݔ‬ଶଶ ሺ‫ݐ‬ሻ of subsystem1 when ‫ݑ‬ሺ‫ݐ‬ሻ ‫( Ͳ ؠ‬dash-line) and with controller (solid-line)‫ݔ‬ଶଶ ሺͲሻ ൌ െͲǤͻ