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Willems et al.'s fundamental lemma asserts that all trajectories of a linearsystem can be obtained from a single given one, assuming that a persistency ofexcitation condition holds. This result has profound implications for systemidentification and data-driven control, and has seen a revival over the lastfew years. The purpose of this paper is to extend Willems' lemma to thesituation where multiple (possibly short) system trajectories are given insteadof a single long one. To this end, we introduce a notion of collectivepersistency of excitation. We will then show that all trajectories of a linearsystem can be obtained from a given finite number of trajectories, as long asthese are collectively persistently exciting. We will demonstrate that thisresult enables the identification of linear systems from data sets with missingdata samples. Additionally, we show that the result is of practicalsignificance in data-driven control of unstable systems.

Willems' Fundamental Lemma for State-space Systems and its Extension to Multiple Datasets