Online Learning of Linear Quadratic Gaussian Controllers from Noisy Data

This paper addresses the joint state estimation and online control problems of unknown linear time-invariant systems subject to process and measurement noises. The proposal is to design a finite-horizon linear quadratic Gaussian (LQG) controller from noisy data. To achieve this, relaxed data-based semi-definite programs (SDPs) are constructed, upon solving which, a robust finite-horizon linear quadratic regulator (LQR) and a robust Kalman filter are developed, and consequently, a robust data-driven finite-horizon LQG controller is designed. It is shown that the proposed data-driven finite-horizon LQG controller ensures robust global exponential stability (RGES) of the observer and the input-to-state stability (ISS) of the closed-loop system under standard conditions. Finally, a numerical example is provided to demonstrate its effectiveness.