Learning Stabilizing Controllers for Unstable Linear Quadratic Regulators from a Single Trajectory

Lenart Treven, Sebastian Curi, Mojmír Mutný, Andreas Krause
Proceedings of the 3rd Conference on Learning for Dynamics and Control, PMLR 144:664-676, 2021.

Abstract

The principal task to control dynamical systems is to ensure their stability. When the system is unknown, robust approaches are promising since they aim to stabilize a large set of plausible systems simultaneously. We study linear controllers under quadratic costs model also known as linear quadratic regulators (LQR). We present two different semi-definite programs (SDP) which results in a controller that stabilizes all systems within an ellipsoid uncertainty set. We further show that the feasibility conditions of the proposed SDPs are \emph{equivalent}. Using the derived robust controller syntheses, we propose an efficient data dependent algorithm – \textsc{eXploration} – that with high probability quickly identifies a stabilizing controller. Our approach can be used to initialize existing algorithms that require a stabilizing controller as an input while adding constant to the regret. We further propose different heuristics which empirically reduce the number of steps taken by \textsc{eXploration} and reduce the suffered cost while searching for a stabilizing controller.

Cite this Paper

BibTeX
@InProceedings{pmlr-v144-treven21a, title = {Learning Stabilizing Controllers for Unstable Linear Quadratic Regulators from a Single Trajectory}, author = {Treven, Lenart and Curi, Sebastian and Mutn\'{y}, Mojm\'{i}r and Krause, Andreas}, booktitle = {Proceedings of the 3rd Conference on Learning for Dynamics and Control}, pages = {664--676}, year = {2021}, editor = {Jadbabaie, Ali and Lygeros, John and Pappas, George J. and A. Parrilo, Pablo and Recht, Benjamin and Tomlin, Claire J. and Zeilinger, Melanie N.}, volume = {144}, series = {Proceedings of Machine Learning Research}, month = {07 -- 08 June}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v144/treven21a/treven21a.pdf}, url = {https://proceedings.mlr.press/v144/treven21a.html}, abstract = {The principal task to control dynamical systems is to ensure their stability. When the system is unknown, robust approaches are promising since they aim to stabilize a large set of plausible systems simultaneously. We study linear controllers under quadratic costs model also known as linear quadratic regulators (LQR). We present two different semi-definite programs (SDP) which results in a controller that stabilizes all systems within an ellipsoid uncertainty set. We further show that the feasibility conditions of the proposed SDPs are \emph{equivalent}. Using the derived robust controller syntheses, we propose an efficient data dependent algorithm – \textsc{eXploration} – that with high probability quickly identifies a stabilizing controller. Our approach can be used to initialize existing algorithms that require a stabilizing controller as an input while adding constant to the regret. We further propose different heuristics which empirically reduce the number of steps taken by \textsc{eXploration} and reduce the suffered cost while searching for a stabilizing controller.} }
Endnote
%0 Conference Paper %T Learning Stabilizing Controllers for Unstable Linear Quadratic Regulators from a Single Trajectory %A Lenart Treven %A Sebastian Curi %A Mojmír Mutný %A Andreas Krause %B Proceedings of the 3rd Conference on Learning for Dynamics and Control %C Proceedings of Machine Learning Research %D 2021 %E Ali Jadbabaie %E John Lygeros %E George J. Pappas %E Pablo A. Parrilo %E Benjamin Recht %E Claire J. Tomlin %E Melanie N. Zeilinger %F pmlr-v144-treven21a %I PMLR %P 664--676 %U https://proceedings.mlr.press/v144/treven21a.html %V 144 %X The principal task to control dynamical systems is to ensure their stability. When the system is unknown, robust approaches are promising since they aim to stabilize a large set of plausible systems simultaneously. We study linear controllers under quadratic costs model also known as linear quadratic regulators (LQR). We present two different semi-definite programs (SDP) which results in a controller that stabilizes all systems within an ellipsoid uncertainty set. We further show that the feasibility conditions of the proposed SDPs are \emph{equivalent}. Using the derived robust controller syntheses, we propose an efficient data dependent algorithm – \textsc{eXploration} – that with high probability quickly identifies a stabilizing controller. Our approach can be used to initialize existing algorithms that require a stabilizing controller as an input while adding constant to the regret. We further propose different heuristics which empirically reduce the number of steps taken by \textsc{eXploration} and reduce the suffered cost while searching for a stabilizing controller.
APA
Treven, L., Curi, S., Mutný, M. & Krause, A.. (2021). Learning Stabilizing Controllers for Unstable Linear Quadratic Regulators from a Single Trajectory. Proceedings of the 3rd Conference on Learning for Dynamics and Control, in Proceedings of Machine Learning Research 144:664-676 Available from https://proceedings.mlr.press/v144/treven21a.html.

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