On a Continuous-Time Version of Willems' Lemma

TL;DR

This paper extends Willems' lemma to continuous-time linear systems by sampling signals and constructing Hankel-like structures, enabling data-driven trajectory generation using persistently excited data.

Contribution

It introduces a continuous-time version of Willems' lemma, allowing representation and control of continuous-time systems from data, which was previously limited to discrete-time systems.

Findings

Provides a method to represent continuous-time trajectories from data.

Shows how to design time-varying parameters for trajectory generation.

Identifies classes of input signals satisfying persistence of excitation.

Abstract

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 result is obtained by sampling the continuous signals at regular intervals, and constructing Hankel-like structures that closely resemble their discrete-time counterparts. Then, it is shown how to use measured persistently excited data to design a time-varying vector of parameters that allows the generation of arbitrary piecewise differentiable trajectories. A class of input signals that satisfies the conditions for persistence of excitation is also provided.