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

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