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Data-driven control of continuous-time systems: A synthesis-operator approach

Masashi Wakaiki

TL;DR

This work develops a derivative-free data-driven control framework for continuous-time systems by embedding state and input trajectories into synthesis operators on $H^{1}_{0}([0,\tau])$. It provides necessary and sufficient informativity conditions for system identification and stabilization, including a robust linear matrix inequality (LMI) criterion for quadratic stabilization under process noise via adjoint representations and operator-based matrix inequalities. The approach avoids sampling and derivative estimation, enabling analysis and controller design directly from continuous-time data and furnishing both theoretical guarantees (necessity and sufficiency) and practical computational tools. A numerical batch-reactor example demonstrates the method's ability to certify informativity under noise and to synthesize stabilizing gains, highlighting practical relevance for continuous-time data-driven control.

Abstract

This paper addresses data-driven control of continuous-time systems. We develop a framework based on synthesis operators associated with input and state trajectories. A key advantage of the proposed method is that it does not require the state derivative and uses continuous-time data directly without sampling or filtering. First, systems compatible with given data are described by the synthesis operators into which data trajectories are embedded. Next, we characterize data informativity properties for system identification and for stabilization. Finally, we establish a necessary and sufficient condition for informativity for quadratic stabilization in the presence of process noise. This condition is formulated as linear matrix inequalities by exploiting the finite-rank structure of the synthesis operators.

Data-driven control of continuous-time systems: A synthesis-operator approach

TL;DR

This work develops a derivative-free data-driven control framework for continuous-time systems by embedding state and input trajectories into synthesis operators on . It provides necessary and sufficient informativity conditions for system identification and stabilization, including a robust linear matrix inequality (LMI) criterion for quadratic stabilization under process noise via adjoint representations and operator-based matrix inequalities. The approach avoids sampling and derivative estimation, enabling analysis and controller design directly from continuous-time data and furnishing both theoretical guarantees (necessity and sufficiency) and practical computational tools. A numerical batch-reactor example demonstrates the method's ability to certify informativity under noise and to synthesize stabilizing gains, highlighting practical relevance for continuous-time data-driven control.

Abstract

This paper addresses data-driven control of continuous-time systems. We develop a framework based on synthesis operators associated with input and state trajectories. A key advantage of the proposed method is that it does not require the state derivative and uses continuous-time data directly without sampling or filtering. First, systems compatible with given data are described by the synthesis operators into which data trajectories are embedded. Next, we characterize data informativity properties for system identification and for stabilization. Finally, we establish a necessary and sufficient condition for informativity for quadratic stabilization in the presence of process noise. This condition is formulated as linear matrix inequalities by exploiting the finite-rank structure of the synthesis operators.

Paper Structure

This paper contains 12 sections, 13 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Data and synthesis operators
  3. Data informativity for system identification
  4. Data informativity for stabilization
  5. Data informativity for quadratic stabilization in the presence of noise
  6. Adjoints of synthesis operators
  7. Integral representations
  8. Norm
  9. Noisy data
  10. Characterization via the matrix $S$-lemma
  11. Numerical example
  12. Conclusion

Key Result

Lemma 2.2

Let $\Xi$ and $\Xi_{\mathrm{d}}$ be the synthesis operator and the differentiated synthesis operator associated with $x \in \mathrm{H}^{1}[0,\tau]$, and let $\Upsilon$ be the synthesis operator associated with $u \in \mathrm{L}^{2}[0,\tau]$. Let $\mathfrak{D} \coloneqq (x,u,\tau)$. Then for all $A \

Theorems & Definitions (27)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 4.1
  • ...and 17 more