On the Local Quadratic Stability of T-S Fuzzy Systems in the Vicinity of the Origin

TL;DR

This paper develops less conservative local stability conditions for continuous-time T-S fuzzy systems near the origin using LMIs and quadratic Lyapunov functions, providing necessary and sufficient criteria.

Contribution

Introduces new local stability conditions for T-S fuzzy systems that are less conservative and offer necessary and sufficient criteria near the origin.

Findings

Proposed conditions are less conservative than existing methods.

Conditions are necessary and sufficient for local exponential stability.

Validated through comprehensive examples.

Abstract

The main goal of this paper is to introduce new local stability conditions for continuous-time Takagi-Sugeno (T-S) fuzzy systems. These stability conditions are based on linear matrix inequalities (LMIs) in combination with quadratic Lyapunov functions. Moreover, they integrate information on the membership functions at the origin and effectively leverage the linear structure of the underlying nonlinear system in the vicinity of the origin. As a result, the proposed conditions are proved to be less conservative compared to existing methods using fuzzy Lyapunov functions in the literature. Moreover, we establish that the proposed methods offer necessary and sufficient conditions for the local exponential stability of T-S fuzzy systems. The paper also includes discussions on the inherent limitations associated with fuzzy Lyapunov approaches. To demonstrate the theoretical results, we provide comprehensive examples that elucidate the core concepts and validate the efficacy of the proposed conditions.