Path-Following-Based Design for Guaranteed Cost Control of Polynomial Fuzzy Systems

Abstract

This paper presents a path-following-based design for guaranteed cost control of a class of nonlinear systems represented by polynomial fuzzy systems. First, this paper proposes a polynomial Lyapunov function approach to guaranteed cost control for the feedback system consisting of a polynomial fuzzy system and a polynomial fuzzy controller. In particular, we introduce a new type of polynomial fuzzy controllers based on an approximate solution for the Hamilton–Jacobi–Bellman inequality. To design a guaranteed cost polynomial fuzzy controller effectively, a path-following-based design algorithm is newly developed by formulating as a sum-of-squares (SOS) stabilization problem. Two new relaxations are provided by bringing a peculiar benefit of the SOS framework. One is an $${\mathcal{S}}$$ -procedure relaxation for the considered outmost Lyapunov function level set that is contractively invariant set. The other is an $${\mathcal{S}}$$ -procedure relaxation for design conditions obtained for polynomial membership functions redefined by variable replacements in considered ranges. Furthermore, this paper provides a practical and reasonable way for estimating lower upper-bounds of a given performance function by increasing the order of a considered polynomial function. Finally, a complicated nonlinear system design example is employed to illustrate the validity of the proposed design algorithm and the lower upper-bound estimation.