Model Predictive Control of Parabolic PDE Systems with Dirichlet Boundary Conditions via Galerkin Model Reduction
Y. Ou and E. Schuster
American Control Conference
St. Louis, Missouri, USA, June 10-12, 2009
Abstract
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We propose a framework to solve a closed-loop, optimal tracking
control problem for a parabolic partial differential equation (PDE)
via diffusivity, interior, and boundary actuation. The approach is
based on model reduction via proper orthogonal decomposition (POD) and
Galerkin projection methods. A conventional integration-by-parts
approach during the Galerkin projection fails to effectively
incorporate the considered Dirichlet boundary control into the
reduced-order model (ROM). To overcome this limitation we use a
spatial discretization of the interior product during the Galerkin
projection. The obtained low dimensional dynamical model is bilinear
as the result of the presence of the diffusivity control term in the
nonlinear parabolic PDE system. We design a closed-loop optimal
controller based on a nonlinear model predictive control (MPC) scheme
aimed at bating the effect of disturbances with the ultimate goal of
tracking a nominal trajectory. A quasi-linear approximation approach
is used to solve on-line the quadratic optimal control problem subject
to the bilinear reduced-order model. Based on the convergence
properties of the quasi-linear approximation algorithm, the
asymptotical stability of the closed-loop nonlinear MPC scheme is
discussed. Finally, the proposed approach is applied to the current
profile control problem in tokamak plasmas and its effectiveness is
demonstrated in simulations.