Piecewise polynomial lyapunov functions based stability analysis for polynomial fuzzy systems

This paper proposes two stability criteria for polynomial fuzzy systems by applying minimum-type and maximum-type piecewise polynomial Lyapunov functions (PPLFs) respectively. Piecewise Lyapunov functions and polynomial Lyapunov functions (PLFs) have been utilized to the stability analysis for fuzzy-model-based (FMB) control systems to obtain relaxed results in literature. However, the minimum-type and maximum-type PPLFs have not been employed to analyze the stability of FMB control systems. Therefore, this paper applies the minimum-type and maximum-type PPLFs to the stability analysis of polynomial FMB control systems. Two relaxed stability criteria represented in terms of bilinear sum-of-squares (SOS) conditions are proposed. The proposed stability criteria are represented in terms of bilinear SOS conditions that cannot be directly solved by the mathematical tools of solving SOS optimization problem (e.g. SOSTOOLS and SOSOPT). Therefore, the path-following method that has been shown to be effective for the nonconvex bilinear matrix inequality problem is employed for solving the bilinear SOS problem of the proposed stability criteria. A numerical example is provided to demonstrate the relaxation of the proposed stability criteria.