Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations

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Abstract

In this paper, we present a discussion of the proper orthogonal decomposition (POD) as applied to simulation and feedback control of the one-dimensional heat equation. We provide two examples of input collections to which the POD process is applied. First, we apply POD directly to the finite element basis of linear B-splines. Next, we additionally include time snapshots. We show that although the second case provides better simulations, this POD basis is ill-suited for control problems. We provide a discussion of both the linear quadratic regulator (LQR) problem and the linear quadratic Gaussian (LQG) problem.

Keywords

Proper orthogonal decomposition
Principal component analysis
Karhunen-Loève expansion
Heat equation
Feedback control

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We wish to thank J. A. Burns and E. M. Cliff for helpful discussions throughout this work.

This research was supported in part by the Air Force Office of Scientific Research under Grant F49620-96-1-0329 and by DARPA under ONR Contract N00014-98-C-0318.