On the Sample Complexity of the Linear Quadratic Regulator
This addresses the challenge of optimal control under uncertainty for systems with unknown dynamics, offering a robust solution that stabilizes systems where simpler methods fail.
The paper tackles the Linear Quadratic Regulator problem with unknown dynamics by proposing Coarse-ID control, a multi-stage method that estimates a model and its error to design a robust controller, achieving nearly optimal error bounds in control cost relative to system parameters.
This paper addresses the optimal control problem known as the Linear Quadratic Regulator in the case when the dynamics are unknown. We propose a multi-stage 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 convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are nearly 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.