H$_{\infty}$ Fuzzy Observer-Based Control for a Class of Nonlinear Distributed Parameter Systems With Control Constraints

Abstract

An H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> fuzzy observer-based control design is proposed for a class of nonlinear parabolic partial differential equation (PDE) systems with control constraints, for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. In the proposed control scheme, Galerkin's method is initially applied to the PDE system to derive a nonlinear ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. The resulting nonlinear ODE system is subsequently represented by the Takagi-Sugeno (T-S) fuzzy model. Then, based on the T-S fuzzy model, a fuzzy observer-based controller is developed to stabilize the nonlinear PDE system and achieve an optimized H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> disturbance attenuation performance for the finite-dimensional slow system, while control constraints are respected. The outcome of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> fuzzy observer-based control problem is formulated as a bilinear matrix inequality (BMI) optimization problem. A local optimization algorithm that treats the BMI as a double linear matrix inequality is presented to solve this BMI optimization problem. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod to illustrate its effectiveness.