Two-Step Stability Analysis for General Polynomial-Fuzzy-Model-Based Control Systems

This paper investigates the stability of a polynomial-fuzzy-model-based (PFMB) control system formed by a nonlinear plant represented by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop. Three cases of polynomial fuzzy controllers are proposed for the control process with the consideration of a matched/mismatched number of rules and/or premise membership functions, which demonstrate different levels of controller complexity, design flexibility, and stability analysis results. A general polynomial Lyapunov function candidate is proposed to investigate the system stability. Unlike the published work, there is no constraint on the polynomial Lyapunov function candidate, which is independent of the form of the polynomial fuzzy model. Thus, it can be applied to a wider class of PFMB control systems and potentially produces more relaxed stability analysis results. Two-step stability conditions in terms of sum-of-squares (SOS) are obtained to numerically find a feasible solution. To facilitate the stability analysis and relax the stability analysis result, the boundary information of membership functions is taken into account in the stability analysis and incorporated into the SOS-based stability conditions. Simulation examples are given to illustrate the effectiveness of the proposed approach.