Stability analysis for the polynomial fuzzy systems by utilizing equality constraints of sum-of-squares program

Abstract

This paper presents the stability analysis for polynomial fuzzy systems by utilizing the equality constraints of sum-of-squares (SOS) Program. Recently, T-S fuzzy systems have been extended to polynomial fuzzy systems, in which the consequent parts could be polynomial sub-systems. Moreover, stability and stabilization analyses for polynomial fuzzy systems have been done based on polynomial Lyapunov functions that contain quadratic Lyapunov functions as a special case. Since polynomials exist in the sub-systems and the positive definite matrices of the Lyapunov functions, the linear matrix inequality (LMI) tools are not valid for the polynomial analyses. Instead of LMI tools, SOSTOOLS of Matlab toolbox is applied to solve solutions for the polynomial analyses. The SOS program of SOSTOOLS can solve two kinds of constraints that are equality constraints and inequality constraints. The existing stability and stabilization analyses for polynomial fuzzy systems represent their conditions only in the inequality constraints. This paper analyzes the stability of the polynomial fuzzy systems by polynomial Lyapunov functions, and represents the new stability conditions not only in terms of the inequality constraints but also in terms of the equality constraints.