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Leveraging Non-Steady-State Frequency-Domain Data in Willems' Fundamental Lemma

T. J. Meijer, M. Wind, V. S. Dolk, W. P. M. H. Heemels

Abstract

Willems' fundamental lemma enables data-driven analysis and control by characterizing an unknown system's behavior directly in terms of measured data. In this work, we extend a recent frequency-domain variant of this result--previously limited to steady-state data--to incorporate non-steady-state data including transient phenomena. This approach eliminates the need to wait for transients to decay during data collection, significantly reducing the experiment duration. Unlike existing frequency-domain system identification methods, our approach integrates transient data without preprocessing, making it well-suited for direct data-driven analysis and control. We demonstrate its effectiveness by isolating transients in the collected data and performing FRF evaluation at arbitrary frequencies in a numerical case study including noise.

Leveraging Non-Steady-State Frequency-Domain Data in Willems' Fundamental Lemma

Abstract

Willems' fundamental lemma enables data-driven analysis and control by characterizing an unknown system's behavior directly in terms of measured data. In this work, we extend a recent frequency-domain variant of this result--previously limited to steady-state data--to incorporate non-steady-state data including transient phenomena. This approach eliminates the need to wait for transients to decay during data collection, significantly reducing the experiment duration. Unlike existing frequency-domain system identification methods, our approach integrates transient data without preprocessing, making it well-suited for direct data-driven analysis and control. We demonstrate its effectiveness by isolating transients in the collected data and performing FRF evaluation at arbitrary frequencies in a numerical case study including noise.

Paper Structure

This paper contains 18 sections, 5 theorems, 37 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Persistence of excitation in frequency domain
  4. Frequency-domain Willems' fundamental lemma
  5. Problem statement
  6. Willems' fundamental lemma using non-steady-state frequency-domain data
  7. Augmented system
  8. Willems' fundamental lemma
  9. Frequency-response-function evaluation and transient separation
  10. Numerical case study
  11. Noise-free data
  12. Noisy data
  13. Conclusions
  14. Appendix
  15. Proof of Lemma \ref{['lem:augmented-system']}
  16. ...and 3 more sections

Key Result

Lemma 1

Let $(\hat{U}_{[0,M-1]},\hat{X}_{[0,M-1]},\hat{Y}_{[0,M-1]})$ be a steady-state input-state-output spectrum of $\Sigma$ in eq:system satisfying Assumption asm:ctrb. Suppose that $\hat{U}_{[0,M-1]}$ is PE of order $L+n_x$. Then, the following statements hold:

Figures (4)

  • Figure 1: Estimated (using Theorem \ref{['thm:sep']}) FRF $Y_z$ () and transient $T_z$ () of the system, for $z=e^{j\omega}$ with $\omega\in\left[0,\pi\right)$, based on the data $\hat{Y}_{[0,M-1]}$ split into odd () and even () frequencies. The true transfer function $H(z)$ and transient $T(z)$ () are also depicted.
  • Figure 2: Estimation errors $|H(z)-Y_z|$ () and $|T(z)-T_z|$ () when using noise-free data.
  • Figure 3: Estimated (using Theorem \ref{['thm:sep']}) FRF $Y_z$ () and transient $T_z$ () of the system, for $z=e^{j\omega}$ with $\omega\in\left[0,\pi\right)$, based on the noisy data $\hat{Y}_{[0,M-1]}$ split into odd () and even () frequencies. The true transfer function $H(z)$ and transient $T(z)$ () are also depicted.
  • Figure 4: Estimation errors $|H(z)-Y_z|$ () and $|T(z)-T_z|$ () when using noisy data.

Theorems & Definitions (11)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Lemma 2
  • Theorem 1
  • Remark 2
  • Theorem 2
  • ...and 1 more