A quantitative and constructive proof of Willems' Fundamental Lemma and its implications
Julian Berberich, Andrea Iannelli, Alberto Padoan, Jeremy Coulson, Florian Dörfler, Frank Allgöwer
TL;DR
The paper tackles data-driven trajectory parametrization for controllable LTI systems by providing a quantitative, constructive proof of Willems' Fundamental Lemma. It develops a directional persistence of excitation (PE) framework with matrix-valued lower bounds and links PE transfer from inputs to outputs via a Cayley–Hamilton representation, yielding explicit data-richness bounds. Key contributions include extending adaptive-control PE concepts to data-driven settings, exposing the role of system zeros in PE requirements, and delivering a robustness-aware bound that remains informative even when system details are imperfect. This work enhances the reliability of data-driven control under noise and uncertainty and lays the groundwork for generalizations to longer horizons (L>1) and relaxed assumptions.
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.