Learning Robust Data-Based LQG Controllers From Noisy Data

This article addresses the joint state estimation and control problems for unknown linear time-invariant systems subject to both process and measurement noise. The aim is to redesign the linear quadratic Gaussian (LQG) controller-based solely on data. The LQG controller comprises a linear quadratic regulator (LQR) and a steady-state Kalman observer; while the data-based LQR design problem has been previously studied, constructing the Kalman gain and the LQG controller from noisy data presents a novel challenge. In this work, a data-based formulation for computing the steady-state Kalman gain is proposed based on semidefinite programming (SDP) using some noise-free input-state-output data. To compensate for the offline noise, a relaxed SDP is proposed, upon solving which, a robust observer gain is constructed. In addition, a robust LQG controller is designed based on the observer gain and a data-based LQR gain. The proposed controller is proven to achieve robust global exponential stability for the observer and input-to-state stability for the resultant closed-loop systems under standard conditions. Finally, numerical tests are conducted to validate the proposed controllers' correctness and effectiveness.