Stability Analysis of Recurrent Fuzzy Systems: A Hybrid System and SOS Approach

This paper presents a new approach to the stability analysis of recurrent fuzzy systems (RFSs). RFSs are rule-based dynamic fuzzy systems that are usually obtained from heuristic or data-driven modeling. In the presented approach, the stability of both continuous-time and discrete-time RFS can be analyzed in a common framework. It is based on the representation of an RFS as a hybrid polynomial system. Due to the polynomial structure, the recently developed method of sum-of-squares (SOS) decomposition along with semidefinite programming can be employed to derive sufficient conditions for the stability of equilibrium points. We consider stability analysis for known equilibrium points as well as unknown equilibrium points. The latter results in a two-step procedure. In the first step, a polynomial is constructed. In the second verification step, this polynomial is proven to be a Lyapunov function for the RFS. The applicability of the approach is shown by two systems formulated as a rule-based RFS.

I. Introduction

One of the most-popular application areas of fuzzy-logic systems is modeling and control of complex dynamical systems. In fuzzy modeling and control [5] [9] [25], we roughly distinguish between rule-based fuzzy systems and Takagi–Sugeno–Kang (TSK) fuzzy systems. While rule-based fuzzy systems allow a design based on linguistic knowledge about the system, TSK fuzzy systems blend between multiple linear models and controllers in different operating regions. One key issue for both kinds of dynamic fuzzy systems is the analysis of their stability.