Observer-Based <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$</tex-math> </inline-formula> Sampled-Data Fuzzy Control for a Class of Nonlinear Parabolic PDE Systems

Abstract

In this paper, an observer-based $H_{\infty }$ sampled-data fuzzy control problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems. With the aid of the modal decomposition technique, a nonlinear ordinary differential equation (ODE) model is initially derived to describe the dominant (slow) dynamics of the PDE system. Subsequently, the resulting nonlinear ODE model is accurately represented by the Takagi–Sugeno (T–S) fuzzy model. Then, based on the T–S fuzzy model, a finite-dimensional observer-based sampled-data fuzzy control design with $H_\infty$ performance is developed for the PDE system via employing a novel time-dependent functional. The outcome of the observer-based $H_{\infty }$ sampled-data fuzzy control problem can be formulated as a bilinear matrix inequality optimization problem. Moreover, an iterative optimization algorithm based on the linear matrix inequalities is given to obtain a suboptimal $H_\infty$ sampled-data fuzzy controller. Finally, simulation results on the Fisher equation and a temperature cooling fin of high-speed aerospace vehicle illustrate that the proposed design method is effective.