Stabilization of Large-Scale Fuzzy Interconnected System

Stabilization of Large-Scale Fuzzy Interconnected System

DOI: 10.4018/978-1-5225-2385-7.ch003

Abstract

The chapter addresses the stabilization problem for large-scale fuzzy interconnected systems. Our aim is to present the design results on both the state feedback and static-output feedback (SOF) stabilizing fuzzy controllers. Firstly, by using some bounding techniques, the reduced number of LMIs to the decentralized state feedback controller design will be derived. Then, by using some matrix transformation techniques and singular system approach, we will also derive some design results on decentralized SOF control in terms of LMIs. Moreover, the proposed design results on the decentralized control will be extended to address the distributed control problem. Finally, several examples are given to illustrate the use of corresponding results.
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3.2 Decentralized Stabilization Design

In this section, we will study decentralized stabilization for large-scale T-S fuzzy interconnected systems.

3.2.1 Problem Formulation

Consider a continuous-time large-scale nonlinear interconnected system containing 978-1-5225-2385-7.ch003.m01 subsystems with interconnections, where the 978-1-5225-2385-7.ch003.m02-th nonlinear subsystem is represented by the following T-S fuzzy model:

Plant Rule 978-1-5225-2385-7.ch003.m03: IF

978-1-5225-2385-7.ch003.m04
THEN
978-1-5225-2385-7.ch003.m05
(1) where 978-1-5225-2385-7.ch003.m06, 978-1-5225-2385-7.ch003.m07, 978-1-5225-2385-7.ch003.m08 is the number of the subsystems; 978-1-5225-2385-7.ch003.m09 is the fuzzy inference rule; 978-1-5225-2385-7.ch003.m10 is the number of inference rules; 978-1-5225-2385-7.ch003.m11 are fuzzy sets;978-1-5225-2385-7.ch003.m12 and 978-1-5225-2385-7.ch003.m13 denotes the system state and control input, respectively; 978-1-5225-2385-7.ch003.m14 are the measurable variables; 978-1-5225-2385-7.ch003.m15 is the 978-1-5225-2385-7.ch003.m16-th local model; 978-1-5225-2385-7.ch003.m17 denotes the nonlinear interconnection of the 978-1-5225-2385-7.ch003.m18-th and 978-1-5225-2385-7.ch003.m19-th subsystems.

Define the inferred fuzzy set 978-1-5225-2385-7.ch003.m20 and normalized membership function 978-1-5225-2385-7.ch003.m21, it yields

978-1-5225-2385-7.ch003.m22
(2) where we will denote 978-1-5225-2385-7.ch003.m23 for brevity, and 978-1-5225-2385-7.ch003.m24 is the grade of membership of 978-1-5225-2385-7.ch003.m25 in 978-1-5225-2385-7.ch003.m26.

By fuzzy blending, the 978-1-5225-2385-7.ch003.m27-th global T-S fuzzy dynamic model is obtained by

978-1-5225-2385-7.ch003.m28
(3) where

978-1-5225-2385-7.ch003.m29
(4)

A decentralized fuzzy controller is given by:

Plant Rule 978-1-5225-2385-7.ch003.m30: IF 978-1-5225-2385-7.ch003.m31 is 978-1-5225-2385-7.ch003.m32 and 978-1-5225-2385-7.ch003.m33 is 978-1-5225-2385-7.ch003.m34 and 978-1-5225-2385-7.ch003.m35 and 978-1-5225-2385-7.ch003.m36 is 978-1-5225-2385-7.ch003.m37, THEN

978-1-5225-2385-7.ch003.m38
(5) where 978-1-5225-2385-7.ch003.m39 is controller gains to be determined.

Similarly, the overall controller can be given by

978-1-5225-2385-7.ch003.m40
(6) where 978-1-5225-2385-7.ch003.m41

Combined with the fuzzy system in (3) and the fuzzy controller in (6), the closed-loop fuzzy control system can be given by

978-1-5225-2385-7.ch003.m42
(7)

In this section, our aim is to design a decentralized fuzzy controller (6), such that the closed-loop fuzzy control system is asymptotically stable.

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