Relaxed Stability and Performance Conditions for Takagi–Sugeno Fuzzy Systems With Knowledge on Membership Function Overlap

Abstract

This correspondence presents a relaxation of some earlier linear matrix inequality (LMI) conditions, which allow setting up less conservative stability or performance conditions for Takagi-Sugeno fuzzy models. Unlike the previous literature, this correspondence takes into account the knowledge of the membership functions' shape by considering bounds on them and their cross products (interpreted as an overlap measure), introducing auxiliary LMI variables. Numerical examples illustrate the achieved improvements.

Relaxed stability conditions for polynomial‐fuzzy‐model‐based control system with membership function information

“…The early work to tackle this problem can be found in [36], in which the correlation between membership functions was taken into account in the stability analysis. The membership function information was further represented as a group of affine inequalities in [37] and a group of inequalities of the membership function overlap terms in [38], and the authors of [39] proposed the analysis approach to represent the membership function information in a more general form, i.e. a group of second-order polynomial inequalities.…”

Membership-Function-Dependent Stability Analysis and Control Synthesis of Guaranteed Cost Fuzzy-Model-Based Control Systems

“…When we consider continuous membership functions, the number of LMIs will reach infinity so it is impractical to apply numerical techniques to solve the solution to the stability conditions. In order to include the information of membership functions into the analysis, methods trying to add some constrains on the membership functions can be found in [20,21]. Besides, approximation of membership functions is also one of the methods to circumvent this difficulty by approximating the infinite number of stability conditions with finite ones.…”

Relaxed LMI conditions for control of nonlinear Takagi-Sugeno models

“…However, LMI conditions in current literature do not take into account that fact. These contributions have been published in (Sala & Ariño, 2007a). …”

“…From (Sala & Ariño, 2007a), expression (3.23) holds if there exist matrices X i j = X T ji nonlinear control 43…”

“…If a membership µ 1 is "temperature is low and pressure is low", and other membership µ 2 is "temperature is high and pressure is high", then µ 1 µ 4 ≤ 0.25 2 . Details and examples of these situations appear in (Sala & Ariño, 2007a).…”