A Matrix Finsler's Lemma with Applications to Data-Driven Control

TL;DR

This paper introduces a matrix version of Finsler's lemma, enabling the derivation of control solutions from noisy data and broadening the theoretical foundation for data-driven control methods.

Contribution

It proves a matrix Finsler's lemma and applies it to enhance data-driven control design, including for systems with noisy data and Lur'e systems.

Findings

Provides a tractable condition linking quadratic equality and inequality solutions.

Extends data-driven control techniques to noisy data scenarios.

Demonstrates applications to Lur'e systems.

Abstract

In a recent paper it was shown how a matrix S-lemma can be applied to construct controllers from noisy data. The current paper complements these results by proving a matrix version of the classical Finsler's lemma. This matrix Finsler's lemma provides a tractable condition under which all matrix solutions to a quadratic equality also satisfy a quadratic inequality. We will apply this result to bridge known data-driven control design techniques for both exact and noisy data, thereby revealing a more general theory. The result is also applied to data-driven control of Lur'e systems.