Sequential Linear Quadratic Control of Bilinear Parabolic PDEs based on POD Model Reduction
C. Xu, Y. Ou and E. Schuster
Automatica, vol. 47, no. 2, pp. 418-426, February 2011
Abstract
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We present a framework to solve a finite-time optimal control problem for parabolic partial differential equations (PDEs) with
diffusivity-interior actuators, which is motivated by the control of the current density profile in tokamak plasmas. The proposed
approach is based on reduced order modeling (ROM) and successive optimal control computation. First we either simulate the
parabolic PDE system or carry out experiments to generate data ensembles, from which we then extract the most energetic modes to
obtain a reduced order model based on the proper orthogonal decomposition (POD) method and Galerkin projection. The obtained
reduced order model corresponds to a bilinear control system. Based on quasi-linearization of the optimality conditions derived
from Pontryagin's maximum principle, and stated as a two-boundary-value problem, we propose an iterative scheme for suboptimal
closed-loop control. We take advantage of linear synthesis methods in each iteration step to construct a sequence of controllers.
The convergence of the controller sequence is proved in appropriate functional spaces. When compared with previous iterative
schemes for optimal control of bilinear systems, the proposed scheme avoids repeated numerical computation of the Riccati
equation and therefore reduces significantly the number of ODEs that must be solved at each iteration step. A numerical
simulation study shows the effectiveness of this approach.