Learning Robust Data-based LQG Controllers from Noisy Data

TL;DR

This paper develops a data-driven approach to design robust LQG controllers for unknown linear systems using noisy data, employing semi-definite programming to ensure stability and robustness.

Contribution

It introduces a novel SDP-based method to compute the Kalman gain directly from noisy data, enabling the construction of robust data-based LQG controllers.

Findings

Achieves robust global exponential stability for state estimation.

Ensures input-to-state practical stability under standard conditions.

Validated through numerical tests demonstrating effectiveness.

Abstract

This paper 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 semi-definite programming (SDP) using some noise-free input-state-output data. Additionally, a data-based LQG controller is developed, which is shown to be equivalent to the model-based LQG controller. For cases where offline data are corrupted by noise, a robust data-based observer gain is constructed by tackling a relaxed SDP. The proposed controllers are proven to achieve robust global exponential stability (RGES) for state estimation and input-to-state practical stability (ISpS) under standard conditions. Finally, numerical tests are conducted to validate the proposed controllers' correctness and effectiveness.