Willems' Fundamental Lemma for State-Space Systems and Its Extension to Multiple Datasets

Henk J. van Waarde*, Claudio De Persis, M. Kanat Camlibel, Pietro Tesi

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

101 Citations (Scopus)
215 Downloads (Pure)


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 and a controllability condition hold. 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 letter 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 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 samples. Additionally, we show that the result is of practical significance in data-driven control of unstable systems.

Original languageEnglish
Article number9062331
Pages (from-to)602-607
Number of pages6
JournalIEEE Control Systems Letters
Issue number3
Publication statusPublished - Jul-2020
Event59th IEEE Conference on Decision and Control - Jeju Island, Jeju Island, Korea, Republic of
Duration: 14-Dec-202018-Dec-2020


  • Trajectory
  • Linear systems
  • Kernel
  • Controllability
  • Computational modeling
  • Instruments
  • Identification for control
  • linear systems

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