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Finite-dimensional approximation and control of non-linear parabolic PDE systems. (English) Zbl 1001.93034
Authors’ abstract: This article proposes a rigorous and practical methodology for the derivation of accurate finite-dimensional approximations and the synthesis of nonlinear output feedback controllers for nonlinear parabolic PDE systems for which the manipulated inputs and the controlled and measured outputs are distributed in space. The method consists of three steps: first, 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’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 nonlinear 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 nonlinear 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 nonlinear controller is successfully tested through simulations and is shown to be superior to the one of a linear controller.

MSC:
93C20 Control/observation systems governed by partial differential equations
93B11 System structure simplification
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
93C95 Application models in control theory
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