Learning Robust Data-Based LQG Controllers From Noisy Data
Wenjie Liu, Gang Wang, Jian Sun, Francesco Bullo, Jie Chen
Abstract
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.