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A data-based image representation for continuous-time LTI systems

Amine Othmane, Philipp Schmitz, Karl Worthmann, Kathrin Flaßkamp

TL;DR

The paper addresses data-driven modeling of continuous-time LTI systems by formulating a numerically stable image representation based on a continuous-time Willems lemma. It introduces a unimodular embedding and staircase-form strategy to bypass solving differential-algebraic equations and uses algebraic differentiators to estimate required derivatives from noisy data, preserving persistency of excitation. The method yields a compact, data-driven predictor $w = M(s)\ell$ that relies on a reduced set of unknowns and remains robust under substantial measurement disturbances. This work enables reliable data-driven prediction and provides a foundation for potential direct continuous-time data-driven control, with extensions to multi-output systems and deliberate input design for accuracy.

Abstract

We derive a numerically stable method to obtain an image representation of an unknown linear system only from data, leveraging a continuous-time version of Willems et al.'s fundamental lemma. We propose a data-based representation that, unlike previous approaches, avoids solving differential-algebraic equations and uses derivatives approximated by algebraic differentiators. Our image-based formulation significantly reduces the complexity of the data-driven representation by eliminating redundant degrees of freedom and thus reducing the number of unknown quantities to be identified. Simulation results confirm the effectiveness of the proposed approach, even in the presence of severe measurement disturbances.

A data-based image representation for continuous-time LTI systems

TL;DR

The paper addresses data-driven modeling of continuous-time LTI systems by formulating a numerically stable image representation based on a continuous-time Willems lemma. It introduces a unimodular embedding and staircase-form strategy to bypass solving differential-algebraic equations and uses algebraic differentiators to estimate required derivatives from noisy data, preserving persistency of excitation. The method yields a compact, data-driven predictor that relies on a reduced set of unknowns and remains robust under substantial measurement disturbances. This work enables reliable data-driven prediction and provides a foundation for potential direct continuous-time data-driven control, with extensions to multi-output systems and deliberate input design for accuracy.

Abstract

We derive a numerically stable method to obtain an image representation of an unknown linear system only from data, leveraging a continuous-time version of Willems et al.'s fundamental lemma. We propose a data-based representation that, unlike previous approaches, avoids solving differential-algebraic equations and uses derivatives approximated by algebraic differentiators. Our image-based formulation significantly reduces the complexity of the data-driven representation by eliminating redundant degrees of freedom and thus reducing the number of unknown quantities to be identified. Simulation results confirm the effectiveness of the proposed approach, even in the presence of severe measurement disturbances.
Paper Structure (17 sections, 7 theorems, 31 equations, 5 figures)

This paper contains 17 sections, 7 theorems, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setting and problem description
  3. Linear time-invariant systems
  4. A data-driven representation
  5. Image representation from noisy data
  6. Unimodular embedding
  7. Staircase form
  8. Image representation
  9. Gramians from noisy data
  10. Numerical experiments
  11. Generation of random persistently exciting inputs
  12. Regularized pseudo-inverse via SVD
  13. Simulation results and discussion
  14. Conclusion and outlook
  15. Algebraic differentiators
  16. ...and 2 more sections

Key Result

Lemma 2

Suppose system eq:sys is controllable, and let $(\bar{u},\bar{y})\in\mathcal{B}$ be such that $\bar{u}$ is persistently exciting of order $L+n+1$ for some $L\geq \mathop{\mathrm{\mathfrak l}}\nolimits(\mathcal{B})$. Define the Gramian matrix for $\overline W = \mathop{\mathrm{\operatorname{col}}}\nolimits(\bar{u},\dots, \bar{u}^{(L-1)},\bar{y},\dots, \bar{y}^{(L-1)})$. Then, ${\mathop{\mathrm{\op

Figures (5)

  • Figure 1: Evolution of input $\bar{u}$ (), output $\bar{y}$ () and disturbed output $\bar{y}_\eta$ () with $\mathrm{SNR}= 20.37\dB$.
  • Figure 2: Time evolution of the data-based state prediction using the data from Fig. \ref{['fig:experiment_A']} with respect to the true state and the corresponding input trajectory.
  • Figure 3: Evolution of input $\bar{u}$ (), output $\bar{y}$ () and disturbed output $\bar{y}_\eta$ () with $\mathrm{SNR}=80.44\dB$.
  • Figure 4: Time evolution of the data-based state prediction using the data from Fig. \ref{['fig:experiment_B']} with respect to the true state for the input trajectory from Fig. \ref{['fig:results_experiment_A']}.
  • Figure 5: Relative error \ref{['eq:relative_error_state']} versus SNR \ref{['eq:SNR']} for 1500 experiments using randomly sampled interpolation knots for the input generation and different random noise sequences, colored by noise STD. Letters I and II correspond to results in Figs. \ref{['fig:results_experiment_A']} and \ref{['fig:results_experiment_B']}.

Theorems & Definitions (12)

  • Definition 1
  • Lemma 2: Fundamental lemma
  • Remark 3
  • Proposition 4: Beelen1988
  • Corollary 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • Lemma 8
  • ...and 2 more