On Tikhonov Regularization for Direct and Indirect Data-Driven LQR Control

TL;DR

This paper introduces a new regularization technique for direct data-driven LQR control that improves robustness in ill-conditioned cases and extends to nonlinear systems via Koopman embeddings.

Contribution

It proposes a regularized covariance parameterization method that enhances data-driven LQR control and demonstrates its equivalence to Tikhonov-regularized certainty-equivalence.

Findings

Effective in handling ill-conditioned data matrices

Equivalent to Tikhonov-regularized certainty-equivalence LQR

Validated through simulation results

Abstract

In recent years, the so-called `direct data-driven control' has been a topic of intense research, and it is expected that it will become prominent in future complex dynamical systems control. Within this framework, regularization not only implicitly enforces system identification, but also plays a crucial role in ensuring reliable closed-loop behavior. To further enhance the performance of data-driven controllers, we propose a new regularization method for direct data-driven LQR control of unknown LTI systems, based on a regularized covariance parameterization. Unlike existing data-driven techniques, the proposed method remains effective in handling ill-conditioned cases, such as when the data matrix has a large condition number. Then, we demonstrate that our method is equivalent to the indirect certainty-equivalence LQR combined with Tikhonov regularization. Furthermore, we extend our method to the design of controllers for unknown nonlinear systems using Koopman linear embedding. Finally, the simulation results validate the effectiveness and advantages of the proposed regularization method.