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

TL;DR

This paper relaxes key conditions in Willems' fundamental lemma, broadening its applicability to data-driven control and reducing data requirements for multi-agent systems, thus enhancing control design flexibility.

Contribution

It introduces relaxed controllability and persistency of excitation conditions, enabling data-driven control for uncontrollable systems and reducing data needs in multi-agent system identification.

Findings

Data-driven predictive control is equivalent to model predictive control for uncontrollable systems.

Controllability condition can be replaced by subspace conditions.

Persistency of excitation requirement can be relaxed with bounded polynomial degree.

Abstract

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.