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An input-output continuous-time version of Willems' lemma

Victor G. Lopez, Matthias A. Müller, Paolo Rapisarda

TL;DR

The paper tackles data-driven representation of continuous-time linear systems using only input-output data, extending Willems' fundamental lemma to the continuous-time setting. It introduces jet-based and latent-variable machinery to characterize admissible trajectories without internal state information, yielding necessary and sufficient conditions for data-based trajectory generation. A notion of sufficient informativity for external trajectories is defined, enabling exact trajectory reconstruction and a practical data-driven simulation framework without requiring persistently exciting inputs. The approach connects to existing LM and REtAl23b results while avoiding orthogonal-basis truncation, offering exact trajectory representations and a practical path to continuous-time simulation with external measurements.

Abstract

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 from the system and no internal (state) information is required. Second, our system representation makes use of exact system trajectories, without resorting to orthogonal bases representations and consequent approximations. We first establish sufficient and necessary conditions for data-based generation of system trajectories in terms of suitable latent variables. Subsequently, we reformulate these conditions using measured input-output data and show how to span the full behavior of the system. Furthermore, we show how to use the developed framework to solve the data-based continuous-time simulation problem.

An input-output continuous-time version of Willems' lemma

TL;DR

The paper tackles data-driven representation of continuous-time linear systems using only input-output data, extending Willems' fundamental lemma to the continuous-time setting. It introduces jet-based and latent-variable machinery to characterize admissible trajectories without internal state information, yielding necessary and sufficient conditions for data-based trajectory generation. A notion of sufficient informativity for external trajectories is defined, enabling exact trajectory reconstruction and a practical data-driven simulation framework without requiring persistently exciting inputs. The approach connects to existing LM and REtAl23b results while avoiding orthogonal-basis truncation, offering exact trajectory representations and a practical path to continuous-time simulation with external measurements.

Abstract

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 from the system and no internal (state) information is required. Second, our system representation makes use of exact system trajectories, without resorting to orthogonal bases representations and consequent approximations. We first establish sufficient and necessary conditions for data-based generation of system trajectories in terms of suitable latent variables. Subsequently, we reformulate these conditions using measured input-output data and show how to span the full behavior of the system. Furthermore, we show how to use the developed framework to solve the data-based continuous-time simulation problem.
Paper Structure (9 sections, 4 theorems, 36 equations)

This paper contains 9 sections, 4 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Main results
  4. Data-based representation of controllable systems
  5. Sufficiently informative external trajectories
  6. From input-output data to admissible $L$-jets
  7. Relations with the results in LM
  8. The data-based simulation problem
  9. Conclusions

Key Result

Proposition 1

Let $\mathop{\mathrm{col}}\nolimits(\overline{u},\overline{y})$ be defined by eq:Hua-eq:Hya with $\alpha:\mathbb{R}\rightarrow \mathbb{R}^{M+1}$. Assume that the latent variable $\ell$ is observable from the external variables $u$, $y$. The following statements are equivalent:

Theorems & Definitions (13)

  • Proposition 1
  • proof
  • Remark 1
  • Definition 1
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • Remark 4
  • Proposition 3
  • ...and 3 more