Sample Complexity of the Robust LQG Regulator with Coprime Factors Uncertainty

TL;DR

This paper establishes sample complexity bounds for learning the robust LQG controller in unknown LTI systems by combining system identification with convex optimization to handle model uncertainty.

Contribution

It introduces a method for end-to-end learning of robust LQG controllers using Hankel-like matrices and bounded uncertainty on coprime factors, extending prior results.

Findings

Provides a sample complexity bound for system identification.

Designs robust controllers via convex optimization considering model uncertainty.

Achieves results consistent with previous LQG and LQR learning studies.

Abstract

This paper addresses the end-to-end sample complexity bound for learning the H2 optimal controller (the Linear Quadratic Gaussian (LQG) problem) with unknown dynamics, for potentially unstable Linear Time Invariant (LTI) systems. The robust LQG synthesis procedure is performed by considering bounded additive model uncertainty on the coprime factors of the plant. The closed-loop identification of the nominal model of the true plant is performed by constructing a Hankel-like matrix from a single time-series of noisy finite length input-output data, using the ordinary least squares algorithm from Sarkar et al. (2020). Next, an H-infinity bound on the estimated model error is provided and the robust controller is designed via convex optimization, much in the spirit of Boczar et al. (2018) and Zheng et al. (2020a), while allowing for bounded additive uncertainty on the coprime factors of the model. Our conclusions are consistent with previous results on learning the LQG and LQR controllers.