# Stability of Large-Scale Fuzzy Interconnected System

DOI: 10.4018/978-1-5225-2385-7.ch002
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## Abstract

This chapter studies the asymptotic stability of large-scale fuzzy interconnected systems. It firstly focused on the general stability analysis. Then, by using some bounding techniques, the fuzzy rules in interconnections to other subsystems are eliminated. Such condition leads to a reduced number of LMIs. Also, we will present the stability result for the discrete-time case. Finally, we give several examples to illustrate the use of corresponding results.
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## 2.2 General Stability Analysis

This section will derive the general stability conditions for large-scale T-S fuzzy interconnected systems.

### 2.2.1 Problem Formulation

Consider a continuous-time large-scale nonlinear system containing subsystems with interconnections, where the -th nonlinear subsystem is represented by the following T-S fuzzy model:

Plant Rule : IF is and is and and is , THEN

(1) where , is the number of the subsystems. For the -th subsystem, is the -th fuzzy inference rule; is the number of inference rules; are fuzzy sets; denotes the system state; are the measurable variables; is the -th local model; denotes the nonlinear interconnection of the -th and -th subsystems for the -th local model.

Define the inferred fuzzy set and normalized membership function , it yields

(2) where is the grade of membership of in . Here we will denote for brevity.

By fuzzy blending, the -th global T-S fuzzy dynamic model is obtained by

(3) where

(4)

Before moving on, we give the following lemma which will be used to derive the main results.

• Lemma 2.2.1 Given the interconnected matrix in the system (1), and symmetric positive definite matrix , the following inequality holds:

(5)

Proof. Note that

(6) which implies that

(7)

By taking the relations in (6) and (7), we have

(8)

Thus, this proof is completed.

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