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Willems' fundamental lemma asserts that all trajectories of a lineartime-invariant system can be obtained from a finite number of measured ones,assuming that controllability and a persistency of excitation condition hold.We show that these two conditions can be relaxed. First, we prove that thecontrollability condition can be replaced by a condition on the controllablesubspace, unobservable subspace, and a certain subspace associated with themeasured trajectories. Second, we prove that the persistency of excitationrequirement can be relaxed if the degree of a certain minimal polynomial istightly bounded. Our results show that data-driven predictive control usingonline data is equivalent to model predictive control, even for uncontrollablesystems. Moreover, our results significantly reduce the amount of data neededin identifying homogeneous multi-agent systems.

On Controllability and Persistency of Excitation in Data-Driven Control: Extensions of Willems' Fundamental Lemma