Sample Complexity of the Robust LQG Regulator with Coprime Factors Uncertainty

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-loopidentification of the nominal model of the true plant is performed by constructing a Hankel-likematrix from a single time-series of noisy finite length input-output data, using the ordinary least squares algorithm from Sarkar and Rakhlin (2019). Next, an H$\infty$ bound on the estimated model error is provided and the robust controller is designed via convex optimization, much in the spirit of Mania et al. (2019) and Zheng et al. (2020b), 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.