Minimax Adaptive Control for a Finite Set of Linear Systems

Anders Rantzer
Proceedings of the 3rd Conference on Learning for Dynamics and Control, PMLR 144:893-904, 2021.

Abstract

An adaptive controller is derived for linear time-invariant systems with uncertain parameters restricted to a finite set, such that the closed loop system including the non-linear learning procedure is stable and satisfies a pre-specified l2-gain bound from disturbance to error. As a result, robustness to unmodelled (linear and non-linear) dynamics follows from the small gain theorem. The approach is based on a dynamic zero-sum game formulation with quadratic cost. Explicit upper and lower bounds on the optimal value function are stated and a simple formula for an adaptive controller achieving the upper bound is given. The controller uses semi-definite programming for optimal trade-off between exploration and exploitation. Once the uncertain parameters have been sufficiently estimated, the controller behaves like standard H-infinity state feedback.

Cite this Paper

BibTeX
@InProceedings{pmlr-v144-rantzer21a, title = {Minimax Adaptive Control for a Finite Set of Linear Systems}, author = {Rantzer, Anders}, booktitle = {Proceedings of the 3rd Conference on Learning for Dynamics and Control}, pages = {893--904}, 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/rantzer21a/rantzer21a.pdf}, url = {https://proceedings.mlr.press/v144/rantzer21a.html}, abstract = {An adaptive controller is derived for linear time-invariant systems with uncertain parameters restricted to a finite set, such that the closed loop system including the non-linear learning procedure is stable and satisfies a pre-specified l2-gain bound from disturbance to error. As a result, robustness to unmodelled (linear and non-linear) dynamics follows from the small gain theorem. The approach is based on a dynamic zero-sum game formulation with quadratic cost. Explicit upper and lower bounds on the optimal value function are stated and a simple formula for an adaptive controller achieving the upper bound is given. The controller uses semi-definite programming for optimal trade-off between exploration and exploitation. Once the uncertain parameters have been sufficiently estimated, the controller behaves like standard H-infinity state feedback.} }
Endnote
%0 Conference Paper %T Minimax Adaptive Control for a Finite Set of Linear Systems %A Anders Rantzer %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-rantzer21a %I PMLR %P 893--904 %U https://proceedings.mlr.press/v144/rantzer21a.html %V 144 %X An adaptive controller is derived for linear time-invariant systems with uncertain parameters restricted to a finite set, such that the closed loop system including the non-linear learning procedure is stable and satisfies a pre-specified l2-gain bound from disturbance to error. As a result, robustness to unmodelled (linear and non-linear) dynamics follows from the small gain theorem. The approach is based on a dynamic zero-sum game formulation with quadratic cost. Explicit upper and lower bounds on the optimal value function are stated and a simple formula for an adaptive controller achieving the upper bound is given. The controller uses semi-definite programming for optimal trade-off between exploration and exploitation. Once the uncertain parameters have been sufficiently estimated, the controller behaves like standard H-infinity state feedback.
APA
Rantzer, A.. (2021). Minimax Adaptive Control for a Finite Set of Linear Systems. Proceedings of the 3rd Conference on Learning for Dynamics and Control, in Proceedings of Machine Learning Research 144:893-904 Available from https://proceedings.mlr.press/v144/rantzer21a.html.

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