Noise Sensitivity of the Semidefinite Programs for Direct Data-Driven LQR

TL;DR

This paper investigates the noise sensitivity of semidefinite programs used for data-driven LQR control, revealing that noise leads to trivial solutions even with regularization, challenging their robustness.

Contribution

It demonstrates the inherent noise sensitivity of SDP-based data-driven LQR methods and shows regularization cannot fully mitigate this issue.

Findings

SDP solutions become trivial with noisy data

Regularization does not prevent trivial solutions

Noise sensitivity persists despite robustness efforts

Abstract

In this paper, we study the noise sensitivity of the semidefinite program (SDP) proposed for direct data-driven infinite-horizon linear quadratic regulator (LQR) problem for discrete-time linear time-invariant systems. While this SDP is shown to find the true LQR controller in the noise-free setting, we show that it leads to a trivial solution with zero gain matrices when data is corrupted by noise, even when the noise is arbitrarily small. We then study a variant of the SDP that includes a robustness promoting regularization term and prove that regularization does not fully eliminate the sensitivity issue. In particular, the solution of the regularized SDP converges in probability also to a trivial solution.