A quantitative and constructive proof of Willems' Fundamental Lemma and its implications

TL;DR

This paper offers a new, constructive proof of Willems' Fundamental Lemma, introducing a quantitative PE notion and explicit bounds, with implications for data-driven control of linear systems.

Contribution

It provides a novel, constructive proof of Willems' Fundamental Lemma using a quantitative PE concept and generalizes PE results by highlighting the role of system zeros.

Findings

Introduces a quantitative directional PE notion.

Derives explicit PE lower bounds for data.

Generalizes PE results from adaptive control literature.

Abstract

Willems' Fundamental Lemma provides a powerful data-driven parametrization of all trajectories of a controllable linear time-invariant system based on one trajectory with persistently exciting (PE) input. In this paper, we present a novel proof of this result which is inspired by the classical adaptive control literature and differs from existing proofs in multiple aspects. The proof involves a quantitative and directional PE notion, allowing to characterize robust PE properties via singular value bounds, as opposed to binary rank-based PE conditions. Further, the proof is constructive, i.e., we derive an explicit PE lower bound for the generated data. As a contribution of independent interest, we generalize existing PE results from the adaptive control literature and reveal a crucial role of the system's zeros.