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The Willems' fundamental lemma, which characterizes linear dynamics with measured trajectories, has found successful applications in controller design and signal processing, which has driven a broad research interest in its extension to…

Optimization and Control · Mathematics 2021-06-01 Yingzhao Lian , Colin N. Jones

Data-driven control uses a past signal trajectory to characterise the input-output behaviour of a system. Willems' lemma provides a data-based prediction model allowing a control designer to bypass the step of identifying a state-space or…

Systems and Control · Electrical Eng. & Systems 2024-03-25 Roy S. Smith , Mohamed Abdalmoaty , Mingzhou Yin

The fundamental lemma by Willems and coauthors facilitates a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays an important role in data-driven simulation and…

Optimization and Control · Mathematics 2022-05-16 Jeremy Coulson , Henk van Waarde , Florian Dörfler

Recently, data-driven predictive control of linear systems has received wide-spread research attention. It hinges on the fundamental lemma by Willems et al. In a previous paper, we have shown how this framework can be applied to predictive…

Systems and Control · Electrical Eng. & Systems 2023-01-02 Philipp Schmitz , Alexander Engelmann , Timm Faulwasser , Karl Worthmann

This paper presents a data-driven receding horizon control framework for discrete-time linear systems that guarantees robust performance in the presence of bounded disturbances. Unlike the majority of existing data-driven predictive control…

Optimization and Control · Mathematics 2025-10-08 Jian Zheng , Sahand Kiani , Mario Sznaier , Constantino Lagoa

Willems' fundamental lemma enables a trajectory-based characterization of linear systems through data-based Hankel matrices. However, in the presence of measurement noise, we ask: Is this noisy Hankel-based model expressive enough to…

Systems and Control · Electrical Eng. & Systems 2024-04-25 Nathan P. Lawrence , Philip D. Loewen , Shuyuan Wang , Michael G. Forbes , R. Bhushan Gopaluni

We illustrate a novel version of Willems' lemma for data-based representation of continuous-time systems. The main novelties compared to previous works are two. First, the proposed framework relies only on measured input-output trajectories…

Systems and Control · Electrical Eng. & Systems 2024-05-27 Victor G. Lopez , Matthias A. Müller , Paolo Rapisarda

Willems' fundamental lemma enables data-driven analysis and control by characterizing an unknown system's behavior directly in terms of measured data. In this work, we extend a recent frequency-domain variant of this result--previously…

Systems and Control · Electrical Eng. & Systems 2025-04-10 T. J. Meijer , M. Wind , V. S. Dolk , W. P. M. H. Heemels

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…

Systems and Control · Electrical Eng. & Systems 2021-04-13 Yue Yu , Shahriar Talebi , Henk J. van Waarde , Ufuk Topcu , Mehran Mesbahi , Behçet Açıkmeşe

Willems' fundamental lemma has recently received an impressive amount of attention from the data-driven control community. In this paper, we formulate a version of this celebrated result based on frequency-domain data. In doing so, we…

Optimization and Control · Mathematics 2026-02-09 T. J. Meijer , K. J. A. Scheres , S. A. N. Nouwens , V. S. Dolk , W. P. M. H. Heemels

This survey presents recent research on determining control-theoretic properties and designing controllers with rigorous guarantees using semidefinite programming and for nonlinear systems for which no mathematical models but measured…

Optimization and Control · Mathematics 2023-11-06 Tim Martin , Thomas B. Schön , Frank Allgöwer

Despite growing interest in data-driven analysis and control of linear systems, descriptor systems--which are essential for modeling complex engineered systems with algebraic constraints like power and water networks--have received…

Systems and Control · Electrical Eng. & Systems 2025-08-25 Yuan Zhang , Yu Wang , Jun Shang , Jinhui Zhang

We present an extension of Willems' Fundamental Lemma to the class of multi-input multi-output discrete-time feedback linearizable nonlinear systems, thus providing a data-based representation of their input-output trajectories. Two sources…

Optimization and Control · Mathematics 2023-03-17 Mohammad Alsalti , Victor G. Lopez , Julian Berberich , Frank Allgöwer , Matthias A. Müller

This letter presents a robust data-driven receding-horizon control framework for the discrete time linear quadratic regulator (LQR) with input constraints. Unlike existing data-driven approaches that design a controller from initial data…

Optimization and Control · Mathematics 2025-10-08 Jian Zheng , Mario Sznaier

In this paper, a method to represent every input-output trajectory of a continuous-time linear system in terms of previously collected data is presented. This corresponds to a continuous-time version of the well-known Willems' lemma. The…

Systems and Control · Electrical Eng. & Systems 2023-04-03 Victor G. Lopez , Matthias A. Müller

Based on the Fundamental Lemma by Willems et al., the entire behaviour of a Linear Time-Invariant (LTI) system can be characterised by a single data sequence of the system as long the input is persistently exciting. This is an essential…

Systems and Control · Electrical Eng. & Systems 2022-03-02 Chris Verhoek , Roland Tóth , Sofie Haesaert , Anne Koch

Willems' Fundamental Lemma provides a powerful data-driven parametrization of all trajectories of a controllable linear time-invariant system based on one trajectory with persistently exciting (PE) input. In this paper, we present a novel…

Optimization and Control · Mathematics 2024-12-04 Julian Berberich , Andrea Iannelli , Alberto Padoan , Jeremy Coulson , Florian Dörfler , Frank Allgöwer

Data-driven control of discrete-time and continuous-time systems is of tremendous research interest. In this paper, we explore data-driven optimal control of continuous-time linear systems using input-output data. Based on a density result,…

Optimization and Control · Mathematics 2024-07-18 Philipp Schmitz , Timm Faulwasser , Paolo Rapisarda , Karl Worthmann

We present a stochastic constrained output-feedback data-driven predictive control scheme for linear time-invariant systems subject to bounded additive disturbances. The approach uses data-driven predictors based on an extension of Willems'…

Systems and Control · Electrical Eng. & Systems 2025-10-07 Johannes Teutsch , Sebastian Kerz , Dirk Wollherr , Marion Leibold

The fundamental lemma by Jan C. Willems and co-authors enables the representation of all input-output trajectories of a linear time-invariant system by measured input-output data. This result has proven to be pivotal for data-driven…

Systems and Control · Electrical Eng. & Systems 2024-11-06 Guanru Pan , Ruchuan Ou , Timm Faulwasser
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