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Data-based system representations from irregularly measured data

Mohammad Alsalti, Ivan Markovsky, Victor G. Lopez, Matthias A. Müller

TL;DR

Data-driven, non-parametric representations of discrete-time LTI systems are developed from irregularly measured data by exploiting the Hankel kernel structure. The authors modify a kernel-identification algorithm to recover a kernel representation from missing data, enabling reconstruction of any complete finite-length behavior and, in the special case of periodic missing outputs, guarantees under input conditions. They extend the framework to noisy data and demonstrate efficiency and accuracy through simulations and a physiological case study, showing advantages over data-completion via nuclear-norm methods. The work advances data-driven control by enabling reliable non-parametric modeling from incomplete offline measurements with practical impact across engineering and biomedical applications.

Abstract

Non-parametric representations of dynamical systems based on the image of a Hankel matrix of data are extensively used for data-driven control. However, if samples of data are missing, obtaining such representations becomes a difficult task. By exploiting the kernel structure of Hankel matrices of irregularly measured data generated by a linear time-invariant system, we provide computational methods for which any complete finite-length behavior of the system can be obtained. For the special case of periodically missing outputs, we provide conditions on the input such that the former result is guaranteed. In the presence of noise in the data, our method returns an approximate finite-length behavior of the system. We illustrate our result with several examples, including its use for approximate data completion in real-world applications and compare it to alternative methods.

Data-based system representations from irregularly measured data

TL;DR

Data-driven, non-parametric representations of discrete-time LTI systems are developed from irregularly measured data by exploiting the Hankel kernel structure. The authors modify a kernel-identification algorithm to recover a kernel representation from missing data, enabling reconstruction of any complete finite-length behavior and, in the special case of periodic missing outputs, guarantees under input conditions. They extend the framework to noisy data and demonstrate efficiency and accuracy through simulations and a physiological case study, showing advantages over data-completion via nuclear-norm methods. The work advances data-driven control by enabling reliable non-parametric modeling from incomplete offline measurements with practical impact across engineering and biomedical applications.

Abstract

Non-parametric representations of dynamical systems based on the image of a Hankel matrix of data are extensively used for data-driven control. However, if samples of data are missing, obtaining such representations becomes a difficult task. By exploiting the kernel structure of Hankel matrices of irregularly measured data generated by a linear time-invariant system, we provide computational methods for which any complete finite-length behavior of the system can be obtained. For the special case of periodically missing outputs, we provide conditions on the input such that the former result is guaranteed. In the presence of noise in the data, our method returns an approximate finite-length behavior of the system. We illustrate our result with several examples, including its use for approximate data completion in real-world applications and compare it to alternative methods.
Paper Structure (16 sections, 11 theorems, 51 equations, 2 figures, 2 tables, 3 algorithms)

This paper contains 16 sections, 11 theorems, 51 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kernel representations from missing data
  4. Problem formulation
  5. Algorithm for computing the kernel representation
  6. Retrieving a non-parametric system representation
  7. Kernel structure of Hankel matrices
  8. Retrieving the restricted behavior of an LTI system
  9. Periodically missing outputs
  10. Handling noisy data
  11. Simulation examples
  12. Performance of Algorithm \ref{['alg_generalalgorithm']} under random patterns of missing data
  13. Data completion: comparison vs nuclear norm heuristic
  14. Case study: physiological systems
  15. Discussion and Conclusions
  16. ...and 1 more sections

Key Result

Theorem 1

Markovsky22 Let $w\in\mathscr{B}|_T$ with $\mathscr{B}~\in~\partial\mathscr{L}_{m,n,\ell}^{q}$. Then, for all $L\in\mathbb{Z}_{[\ell,L_{\textup{max}}]}$ where $L_{\textup{max}}\coloneqq\lfloor*\rfloor{\frac{T+1}{q+1}}$, the following holds Moreover, for $L\geq\ell$, $\mathrm{im}\left(\mathscr{H}_L(w)\right)=\mathscr{B}|_L$ if and only if

Figures (2)

  • Figure 1: Transition diagram from failure (black) to success (white) when the data is missing randomly. It can be seen that the success rate of Algorithm \ref{['alg_generalalgorithm']} increases as both the length of the sequence $T$ and as the fraction of given to missing samples increase.
  • Figure 2: An illustration of the true vs. estimated trajectories using the three methods: structured low-rank approximation method (ident), our proposed subspace method (SS) and the nuclear norm method (NN). Unlike ident, SS does not require prior model knowledge or initialization steps but still performs similarly to ident.

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • proof
  • Example 1
  • Lemma 1
  • proof
  • Remark 1
  • Definition 2: Heinig84
  • ...and 15 more