On the Sample Complexity of the Linear Quadratic Regulator

This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage procedure, called Coarse-ID control , that estimates a model from a few experimental trials, estimates the error in that model with respect to t...

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Published in	Foundations of computational mathematics Vol. 20; no. 4; pp. 633 - 679
Main Authors	Dean, Sarah, Mania, Horia, Matni, Nikolai, Recht, Benjamin, Tu, Stephen
Format	Journal Article
Language	English
Published	New York Springer US 01.08.2020 Springer Springer Nature B.V
Subjects	Analysis Applications of Mathematics Computational geometry Computer Science Control systems design Controllers Convexity Cost control Design optimization Economics Errors Linear and Multilinear Algebras Linear quadratic regulator Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Optimal control Robust control Synthesis Uncertainty Robust control System level synthesis 93D21 Optimal control 93E35 Reinforcement learning 93E12 93D09 49N05 Statistical learning theory System identification
Online Access	Get full text
ISSN	1615-3375 1615-3383
DOI	10.1007/s10208-019-09426-y

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Abstract	This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage procedure, called Coarse-ID control , that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a quasi-convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.
AbstractList	This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a quasi-convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system. This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage procedure, called Coarse-ID control , that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a quasi-convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.
Audience	Academic
Author	Recht, Benjamin Dean, Sarah Mania, Horia Matni, Nikolai Tu, Stephen
Author_xml	– sequence: 1 givenname: Sarah surname: Dean fullname: Dean, Sarah organization: Department of Electrical Engineering and Computer Sciences, University of California – sequence: 2 givenname: Horia surname: Mania fullname: Mania, Horia organization: Department of Electrical Engineering and Computer Sciences, University of California – sequence: 3 givenname: Nikolai surname: Matni fullname: Matni, Nikolai organization: Department of Computing and Mathematical Sciences, California Institute of Technology – sequence: 4 givenname: Benjamin surname: Recht fullname: Recht, Benjamin email: brecht@berkeley.edu organization: Department of Electrical Engineering and Computer Sciences, University of California – sequence: 5 givenname: Stephen surname: Tu fullname: Tu, Stephen organization: Department of Electrical Engineering and Computer Sciences, University of California
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Snippet	This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage...
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SubjectTerms	Analysis Applications of Mathematics Computational geometry Computer Science Control systems design Controllers Convexity Cost control Design optimization Economics Errors Linear and Multilinear Algebras Linear quadratic regulator Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Optimal control Robust control Synthesis Uncertainty
Title	On the Sample Complexity of the Linear Quadratic Regulator
URI	https://link.springer.com/article/10.1007/s10208-019-09426-y https://www.proquest.com/docview/2430113038
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