Convex Stability Analysis of Mamdani-Like Fuzzy Systems With Singleton Consequents

Abstract

We study the stability of a class of discrete-time fuzzy systems with singleton consequents, called Mamdani-like fuzzy systems. The parametric expressions, specific to this class of fuzzy systems, are leveraged to derive stability analysis conditions via Finsler's lemma and Lyapunov stability tools. This allows avoiding the major challenge in dealing with high-dimensional cases, encountered in the related literature when using the classical state-space representation. Moreover, the information of the piecewise region partition can be fully taken into account in the stability analysis with the well-known <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$S-$</tex-math></inline-formula> procedure to further reduce the stability conservatism. The stability of Mamdani-like fuzzy systems can be checked by solving a set of linear matrix inequalities (LMIs), that is numerically tractable with a suitable semidefinite programming software. Several numerical and physically motivated examples are provided to illustrate the effectiveness of the proposed stability analysis results.