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Data-driven stabilization of continuous-time systems with noisy input-output data

Masashi Wakaiki

TL;DR

This work develops a data-informed approach to stabilizing unknown continuous-time AR systems from noisy input-output data. It introduces an operator-based data embedding that characterizes all data-consistent systems and derives a necessary-and-sufficient LMI condition for data informativity for quadratic stabilization, from which a stabilizing dynamic output-feedback controller is constructed. A byproduct result characterizes data informativity for noise-free system identification via surjectivity of the data-embedded operator, with a rigorous spline-based treatment and constructive surjectivity conditions. An inverted pendulum example illustrates the method: data from multiple trajectories yield an LMIs-feasible problem that produces a stabilizing controller without explicit system identification, validating the practical viability of the approach. Overall, the paper bridges continuous-time data with discrete-time synthesis techniques through synthesis operators, enabling derivative-free, robust data-driven stabilization and identification insights.

Abstract

We study data-driven stabilization of continuous-time systems in autoregressive form when only noisy input-output data are available. First, we provide an operator-based characterization of the set of systems consistent with the data. Next, combining this characterization with behavioral theory, we derive a necessary and sufficient condition for the noisy data to be informative for quadratic stabilization. This condition is formulated as linear matrix inequalities, whose solution yields a stabilizing controller. Finally, we characterize data informativity for system identification in the noise-free setting.

Data-driven stabilization of continuous-time systems with noisy input-output data

TL;DR

This work develops a data-informed approach to stabilizing unknown continuous-time AR systems from noisy input-output data. It introduces an operator-based data embedding that characterizes all data-consistent systems and derives a necessary-and-sufficient LMI condition for data informativity for quadratic stabilization, from which a stabilizing dynamic output-feedback controller is constructed. A byproduct result characterizes data informativity for noise-free system identification via surjectivity of the data-embedded operator, with a rigorous spline-based treatment and constructive surjectivity conditions. An inverted pendulum example illustrates the method: data from multiple trajectories yield an LMIs-feasible problem that produces a stabilizing controller without explicit system identification, validating the practical viability of the approach. Overall, the paper bridges continuous-time data with discrete-time synthesis techniques through synthesis operators, enabling derivative-free, robust data-driven stabilization and identification insights.

Abstract

We study data-driven stabilization of continuous-time systems in autoregressive form when only noisy input-output data are available. First, we provide an operator-based characterization of the set of systems consistent with the data. Next, combining this characterization with behavioral theory, we derive a necessary and sufficient condition for the noisy data to be informative for quadratic stabilization. This condition is formulated as linear matrix inequalities, whose solution yields a stabilizing controller. Finally, we characterize data informativity for system identification in the noise-free setting.
Paper Structure (16 sections, 13 theorems, 111 equations)

This paper contains 16 sections, 13 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Operator-based data embedding
  3. Data and synthesis operators
  4. Characterization of the set of data-consistent systems
  5. Products of synthesis operators and their adjoints
  6. Data-driven stabilization
  7. Closed-loop system
  8. Informativity for quadratic stabilization
  9. Informativity of noise-free data for system identification
  10. Basic facts on B-splines
  11. Relation to surjectivity of data-embedded operators
  12. Example
  13. Concluding remarks
  14. Proof of Lemma \ref{['lem:QMI']}
  15. Background material on quadratic differential forms
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $\mathfrak{D} = (u_k,y_k)_{k=1}^{K}$ and let $\Theta \in \mathbb{S}^p$ satisfy $\Theta \geq 0$. Define $H_k \in \mathcal{L}(\mathrm{H}^{L}_{0}[0,\tau_k],\mathbb{R}^{qL})$ and $\Pi\in \mathbb{S}^{p+qL}$ by eq:Hi_def and eq:Pi_def, respectively. Then the following statements are equivalent for all

Theorems & Definitions (27)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 4.1
  • ...and 17 more