On the Sample Complexity of the Linear Quadratic Regulator

TL;DR

This paper introduces Coarse-ID control, a method for designing controllers for unknown linear systems using limited data, model estimation, and uncertainty quantification, achieving near-optimal bounds and robust stabilization.

Contribution

It presents a novel multi-stage approach combining system identification, error estimation, and robust control synthesis using System Level Synthesis, with theoretical guarantees.

Findings

Nearly optimal bounds on control cost error

Efficient stabilization with limited data

Robust control design outperforming simple schemes

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 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.