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

arXiv: https://arxiv.org/abs/2002.01023 · HTML: https://mlsanity.com/paper/2002.01023/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Feb 2020.

## Built on by (most cited first, 8 of 8)

- [Stabilizing reinforcement learning control: A modular framework for optimizing over all stable behavior](https://mlsanity.com/paper/2310.14098/) (2023)
- [Fading memory and the convolution theorem](https://mlsanity.com/paper/2408.07386/) (2024)
- [A modular framework for stabilizing deep reinforcement learning control](https://mlsanity.com/paper/2304.03422/) (2023)
- [Data driven feedback linearization of nonlinear control systems via Lie derivatives and stacked regression approach](https://mlsanity.com/paper/2508.13241/) (2025)
- [Discrete fully probabilistic design: towards a control pipeline for the synthesis of policies from examples](https://mlsanity.com/paper/2112.11210/) (2021)
- [SAFE--MA--RRT: Multi-Agent Motion Planning with Data-Driven Safety Certificates](https://mlsanity.com/paper/2509.04413/) (2025)
- [A Behavioral Framework for Data-Driven Modeling of Nonlinear Systems in Vector-Valued Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2605.07052/) (2026)
- [Learning Neural Feedback Linearization for Data-driven Systems via Augmented Lagrangian](https://mlsanity.com/paper/2609.25163/) (2026)

## Abstract (arXiv)

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