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Willems' Fundamental Lemma with Large Noisy Fragmented Dataset

Sahand Kiani, Constantino M. Lagoa

Abstract

Willems' Fundamental Lemma enables parameterizing all trajectories generated by a Linear Time-Invariant (LTI) system directly from data. However, this lemma relies on the assumption of noiseless measurements. In this paper, we provide an approach that enables the applicability of Willems' Fundamental Lemma with a large noisy-input, noisy-output fragmented dataset, without requiring prior knowledge of the noise distribution. We introduce a computationally tractable and lightweight algorithm that, despite processing a large dataset, executes in the order of seconds to estimate the invariants of the underlying system, which is obscured by noise. The simulation results demonstrate the effectiveness of the proposed method.

Willems' Fundamental Lemma with Large Noisy Fragmented Dataset

Abstract

Willems' Fundamental Lemma enables parameterizing all trajectories generated by a Linear Time-Invariant (LTI) system directly from data. However, this lemma relies on the assumption of noiseless measurements. In this paper, we provide an approach that enables the applicability of Willems' Fundamental Lemma with a large noisy-input, noisy-output fragmented dataset, without requiring prior knowledge of the noise distribution. We introduce a computationally tractable and lightweight algorithm that, despite processing a large dataset, executes in the order of seconds to estimate the invariants of the underlying system, which is obscured by noise. The simulation results demonstrate the effectiveness of the proposed method.

Paper Structure

This paper contains 9 sections, 3 theorems, 25 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main Contribution
  3. Willems' Fundamental Lemma in the Multi-Experiment Setting
  4. Problem Formulation
  5. Estimation of System Invariants from Large Noisy Fragmented Dataset
  6. Simulation Results
  7. Conclusion
  8. Proof of Lemma \ref{['lem:noiseless_kernel']}
  9. Proof of Theorem \ref{['thm:lln_convergence']}

Key Result

Theorem 1

berberich2020trajectory Suppose $\{u_{k}^{(i)},y_{k}^{(i)}\}_{k=0}^{N-1}$ is a trajectory of an LTI system $G$ for a given experiment $i$ under Assumption AssumpData. Then, a sequence $\{\overline{u}_{k},\overline{y}_{k}\}_{k=0}^{L-1}$ is a trajectory of $G$ if and only if there exists a vector $\al $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 3: Heatmap of the minimum singular value ($\sigma_{\min}$). The red dots denote feasible points where $\sigma_{\min} < \epsilon_\sigma = 0.001$, with the absolute minimum (magenta star) successfully approximating the noise moments (cyan star).
  • Figure 4: Heatmap of the numerical rank of $\hat{\mathbf{M}}_\mathcal{D}(m_1, m_2)$ across the search grid.
  • Figure 5: Logarithmic convergence of the maximum subspace error angle as the number of samples $N_t$ increases, demonstrating recovery of the underlying system invariants.

Theorems & Definitions (15)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Theorem 2
  • proof
  • ...and 5 more