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

arXiv: https://arxiv.org/abs/2102.02953 · HTML: https://mlsanity.com/paper/2102.02953/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2021.

## Built on by (most cited first, 1 of 1)

- [Limits of Learning Linear Dynamics from Experiments](https://mlsanity.com/paper/2605.12010/) (2026)

## Abstract (arXiv)

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.