Static Collocated Piecewise Fuzzy Control Design of Quasi-Linear Parabolic PDE Systems Subject to Periodic Boundary Conditions

Abstract

This paper presents a Lyapunov and partial differential equation (PDE)-based methodology to solve static collocated piecewise fuzzy control design of quasi-linear parabolic PDE systems subject to periodic boundary conditions. Two types of piecewise control, i.e., globally piecewise control and locally piecewise control are considered, respectively. A Takagi–Sugeno (T–S) fuzzy PDE model that is constructed via local sector nonlinearity method is first employed to accurately describe spatiotemporal dynamics of quasi-linear PDEs. Based on the T–S fuzzy PDE model, a static collocated piecewise fuzzy feedback controller is constructed to guarantee the locally exponential stability of the resulting closed-loop system. Sufficient conditions for the existence of such fuzzy controller are developed by applying vector-valued Poincare–Wirtinger inequality and its variants and a linear matrix inequality (LMI) relaxation technique. These sufficient conditions are presented in terms of standard LMIs. Finally, the performance of the suggested fuzzy controller is illustrated by numerical simulation results of a nonlinear PDE system described by quasi-linear FitzHugh–Nagumo equation with periodic boundary conditions.