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

1 Aug 2022  ·  Julian Berberich, Andrea Iannelli, Alberto Padoan, Jeremy Coulson, Florian Dörfler, Frank Allgöwer ·

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, in contrast to the existing proofs, is constructive. Our proof also provides a quantitative and directional PE notion, as opposed to binary rank-based PE conditions. Moreover, the proof shows that the PE requirements of the Fundamental Lemma are not only sufficient but also necessary when considering arbitrary controllable systems. 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.

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Optimization and Control Systems and Control Systems and Control