Finite-dimensional approximation and control of non-linear parabolic PDE systems

Abstract

This article proposes a rigorous and practical methodology for the derivation of accurate finite-dimensional approximations and the synthesis of non-linear output feedback controllers for non-linear parabolic PDE systems for which the manipulated inputs, the controlled and measured outputs are distributed in space. The method consists of three steps: first, the Karhunen-Loeve expansion is used to derive empirical eigenfunctions of the non-linear parabolic PDE system, then the empirical eigenfunctions are used as basis functions within a Galerkin's and approximate inertial manifold model reduction framework to derive low-order ODE systems that accurately describe the dominant dynamics of the PDE system, and finally, these ODE systems are used for the synthesis of non-linear output feedback controllers that guarantee stability and enforce output tracking in the closed-loop system. The proposed method is used to perform model reduction and synthesize a non-linear dynamic output feedback controller for a rapid thermal chemical vapour deposition process. The controller uses measurements of wafer temperature at five locations to manipulate the power of the top lamps in order to achieve spatially uniform temperature, and thus, uniform deposition of the thin film on the wafer over the entire process cycle. The performance of the non-linear controller is successfully tested through simulations and is shown to be superior to the one of a linear controller.