Koopman operator-based model reduction for switched-system control of PDEs
Peitz, Sebastian
Klus, Stefan
We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navierâ€“Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.
2019
info:eu-repo/semantics/article
doc-type:article
text
https://ris.uni-paderborn.de/record/10593
Peitz S, Klus S. Koopman operator-based model reduction for switched-system control of PDEs. <i>Automatica</i>. 2019;106:184-191. doi:<a href="https://doi.org/10.1016/j.automatica.2019.05.016">10.1016/j.automatica.2019.05.016</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.automatica.2019.05.016
info:eu-repo/semantics/altIdentifier/issn/0005-1098
info:eu-repo/semantics/closedAccess