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

Willems' fundamental lemma asserts that all trajectories of a linear time-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 the controllability condition can be replaced by a condition on the controllable subspace, unobservable subspace, and a certain subspace associated with the measured trajectories. Second, we prove that the persistency of excitation requirement can be relaxed if the degree of a certain minimal polynomial is tightly bounded. Our results show that data-driven predictive control using online data is equivalent to model predictive control, even for uncontrollable systems. Moreover, our results significantly reduce the amount of data needed in identifying homogeneous multi-agent systems.