An SOS-Based Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems

This paper deals with a sum-of-squares (SOS)-based control Lyapunov function (CLF) design for polynomial fuzzy control of nonlinear systems. The design starts with exactly replacing (smooth) nonlinear systems dynamics with polynomial fuzzy models, which are known as universal approximators. Next, global stabilization conditions represented in terms of SOS are provided in the framework of the CLF design, i.e., a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying Sontag's control law, once a CLF for a given nonlinear system is constructed. Furthermore, semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions. Both global and semiglobal stabilization problems are formulated as SOS optimization problems, which reduce to numerical feasibility problems. Five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and SOS approaches.

I. Introduction

Nonlinear systems analysis and design using the Takagi–Sugeno (T–S) fuzzy model [1] based control methodology [2], [3] have received much attention as a powerful tool to deal with complex nonlinear control systems over the past two decades. The T–S fuzzy model provides a convenient platform that can represent any smooth nonlinear systems by fuzzily blending linear subsystems, and its stabilization conditions based on Lyapunov stability theory [4] can be represented in terms of linear matrix inequalities (LMIs) [2], [5]. Thus, the designs have been carried out using LMI optimization techniques [6]. In the T–S fuzzy model-based control, the parallel distributed compensation (PDC) concept [2], [5] based on a common quadratic Lyapunov function has been mainly employed to design a fuzzy controller for the system. Nowadays, there are a large number of research studies [7] –[21] conducted to obtain more relaxed stability (stabilization) conditions of nonlinear systems expressed as the T–S fuzzy model.