Output Feedback Control of Parabolic PDE Systems with Nonlinear Spatial Differential Operators

Abstract

This article focuses on nonlinear static (direct) output feedback control of parabolic partial differential equations (PDE) systems with nonlinear spatial differential operators with application to a rapid thermal chemical vapor deposition (RTCVD) process. Initially, a detailed mathematical model is presented for the RTCVD process, which consists of a nonlinear parabolic PDE that describes the spatiotemporal evolution of the wafer temperature, coupled with a set of nonlinear ordinary differential equations (ODEs) that describes the time evolution of the chamber temperature and the concentrations of the various species. Then, a systematic methodology is presented for the synthesis of nonlinear static output feedback controllers for parabolic PDE systems with nonlinear spatial differential operators. Initially, the Karhunen−Loéve expansion is used to derive empirical eigenfunctions of the nonlinear parabolic PDE system, then the empirical eigenfunctions are used as basis functions within a Galerkin 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 nonlinear static output feedback controllers that guarantee stability and enforce output tracking in the closed-loop system. The proposed control method is employed to synthesize a nonlinear easy-to-implement controller for the RTCVD process that uses measurements of wafer temperature at five locations to manipulate the power of the top lamps in order to achieve uniform temperature and, thus, uniform deposition of a thin film on the wafer over the entire process cycle. The performance of the developed nonlinear output feedback controller is successfully tested through simulations and is shown to be superior to the one of a linear control scheme.