Data-driven control via Petersen's lemma

TL;DR

This paper develops a data-driven control design method using Petersen's lemma to robustly stabilize systems directly from noisy input and state data, applicable to linear and polynomial systems.

Contribution

It introduces a novel approach employing matrix ellipsoids and Petersen's lemma for data-driven stabilization, providing necessary and sufficient conditions for linear systems and sufficient conditions for polynomial systems.

Findings

Necessary and sufficient LMIs for linear systems stabilization.

Sufficient sum-of-squares conditions for polynomial systems.

Numerical examples demonstrating the effectiveness of the approach.

Abstract

We address the problem of designing a stabilizing closed-loop control law directly from input and state measurements collected in an open-loop experiment. In the presence of noise in data, we have that a set of dynamics could have generated the collected data and we need the designed controller to stabilize such set of data-consistent dynamics robustly. For this problem of data-driven control with noisy data, we advocate the use of a popular tool from robust control, Petersen's lemma. In the cases of data generated by linear and polynomial systems, we conveniently express the uncertainty captured in the set of data-consistent dynamics through a matrix ellipsoid, and we show that a specific form of this matrix ellipsoid makes it possible to apply Petersen's lemma to all of the mentioned cases. In this way, we obtain necessary and sufficient conditions for data-driven stabilization of linear systems through a linear matrix inequality. The matrix ellipsoid representation enables insights and interpretations of the designed control laws. In the same way, we also obtain sufficient conditions for data-driven stabilization of polynomial systems through (convex) sum-of-squares programs. The findings are illustrated numerically.