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On the role of the signature transform in nonlinear systems and data-driven control

Anna Scampicchio, Melanie N. Zeilinger

TL;DR

The work investigates the signature transform as a rigorous, data-efficient tool for modeling and controlling unknown nonlinear continuous-time systems. By mapping input paths to state trajectories through a truncated signature and a time-homogeneous linear operator, it enables accurate trajectory prediction and an open-loop output-matching control strategy, with theoretical universal-approximation guarantees. Numerical experiments on irregularly sampled data and a Langevin-type system demonstrate high-fidelity prediction with modest signature orders and practical open-loop tracking via a CasADi-implemented nonlinear program. The approach offers a transparent alternative to black-box models and aligns with pathwise, behavioral perspectives for predictive control.

Abstract

Classic control techniques typically rely on a model of the system's response to external inputs, which is difficult to obtain from first principles especially if the unknown dynamics are nonlinear. In this paper, we address this issue by presenting an approach based on the so-called signature transform, a tool that is still largely unexplored in data-driven control. We first show that the signature provides rigorous and practically effective features to represent and predict system trajectories. Furthermore, we propose a novel use of this tool on an output-matching problem, paving the way for signature-based, data-driven predictive control.

On the role of the signature transform in nonlinear systems and data-driven control

TL;DR

The work investigates the signature transform as a rigorous, data-efficient tool for modeling and controlling unknown nonlinear continuous-time systems. By mapping input paths to state trajectories through a truncated signature and a time-homogeneous linear operator, it enables accurate trajectory prediction and an open-loop output-matching control strategy, with theoretical universal-approximation guarantees. Numerical experiments on irregularly sampled data and a Langevin-type system demonstrate high-fidelity prediction with modest signature orders and practical open-loop tracking via a CasADi-implemented nonlinear program. The approach offers a transparent alternative to black-box models and aligns with pathwise, behavioral perspectives for predictive control.

Abstract

Classic control techniques typically rely on a model of the system's response to external inputs, which is difficult to obtain from first principles especially if the unknown dynamics are nonlinear. In this paper, we address this issue by presenting an approach based on the so-called signature transform, a tool that is still largely unexplored in data-driven control. We first show that the signature provides rigorous and practically effective features to represent and predict system trajectories. Furthermore, we propose a novel use of this tool on an output-matching problem, paving the way for signature-based, data-driven predictive control.
Paper Structure (14 sections, 16 equations, 4 figures, 2 algorithms)

This paper contains 14 sections, 16 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Set-up and theoretical results
  4. Proposed methodology
  5. Problem \ref{['problem:prediction']}: Prediction
  6. Problem \ref{['problem:output_matching_control']}: Output-matching control
  7. Numerical experiments
  8. Prediction task
  9. Output-matching control task
  10. Conclusions and outlook
  11. Review of signature properties
  12. Main algebraic structures
  13. Analytical properties
  14. Universal approximation

Figures (4)

  • Figure 1: Visualization of the signature represented as a collection of tensors of increasing dimension.
  • Figure 2: Values of the fit score $\mathcal{F}$ in the Monte Carlo test described in Section \ref{['subsec:predict_test']} for different orders of signature truncation $M$.
  • Figure 3: Sample performance of the output-matching strategy proposed in Section \ref{['subsec:control']}. The attained fit score in this examples is $\mathcal{F} = 90.4\%$.
  • Figure 4: Results of the Monte Carlo test of Section \ref{['subsec:control_test']}, evaluating the fit performance on trajectory portions of increasing length.

Theorems & Definitions (2)

  • Remark 1
  • Remark 2