Stability of Large-Scale Fuzzy Interconnected System

Stability of Large-Scale Fuzzy Interconnected System

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

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 978-1-5225-2385-7.ch002.m02 subsystems with interconnections, where the 978-1-5225-2385-7.ch002.m03-th nonlinear subsystem is represented by the following T-S fuzzy model:

Plant Rule 978-1-5225-2385-7.ch002.m04: IF 978-1-5225-2385-7.ch002.m05 is 978-1-5225-2385-7.ch002.m06and 978-1-5225-2385-7.ch002.m07 is 978-1-5225-2385-7.ch002.m08 and 978-1-5225-2385-7.ch002.m09 and 978-1-5225-2385-7.ch002.m10 is 978-1-5225-2385-7.ch002.m11, THEN

978-1-5225-2385-7.ch002.m12
(1) where 978-1-5225-2385-7.ch002.m13, 978-1-5225-2385-7.ch002.m14 is the number of the subsystems. For the 978-1-5225-2385-7.ch002.m15-th subsystem, 978-1-5225-2385-7.ch002.m16 is the 978-1-5225-2385-7.ch002.m17-th fuzzy inference rule; 978-1-5225-2385-7.ch002.m18 is the number of inference rules; 978-1-5225-2385-7.ch002.m19 are fuzzy sets;978-1-5225-2385-7.ch002.m20 denotes the system state;978-1-5225-2385-7.ch002.m21 are the measurable variables; 978-1-5225-2385-7.ch002.m22 is the 978-1-5225-2385-7.ch002.m23-th local model; 978-1-5225-2385-7.ch002.m24 denotes the nonlinear interconnection of the 978-1-5225-2385-7.ch002.m25-th and 978-1-5225-2385-7.ch002.m26-th subsystems for the 978-1-5225-2385-7.ch002.m27-th local model.

Define the inferred fuzzy set 978-1-5225-2385-7.ch002.m28 and normalized membership function 978-1-5225-2385-7.ch002.m29, it yields

978-1-5225-2385-7.ch002.m30
(2) where 978-1-5225-2385-7.ch002.m31 is the grade of membership of 978-1-5225-2385-7.ch002.m32 in 978-1-5225-2385-7.ch002.m33. Here we will denote 978-1-5225-2385-7.ch002.m34 for brevity.

By fuzzy blending, the 978-1-5225-2385-7.ch002.m35-th global T-S fuzzy dynamic model is obtained by

978-1-5225-2385-7.ch002.m36
(3) where

978-1-5225-2385-7.ch002.m37
(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 978-1-5225-2385-7.ch002.m38 in the system (1), and symmetric positive definite matrix 978-1-5225-2385-7.ch002.m39, the following inequality holds:

    978-1-5225-2385-7.ch002.m40
    (5)

Proof. Note that

978-1-5225-2385-7.ch002.m41
(6) which implies that

978-1-5225-2385-7.ch002.m42
(7)

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

978-1-5225-2385-7.ch002.m43
(8)

Thus, this proof is completed.

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