Willems' Fundamental Lemma with Large Noisy Fragmented Dataset

TL;DR

This paper extends Willems' Fundamental Lemma to noisy, fragmented datasets, providing a fast, practical algorithm for system invariants estimation without noise distribution assumptions.

Contribution

It introduces a novel, computationally efficient method to apply Willems' Lemma to large noisy datasets without prior noise knowledge.

Findings

Algorithm executes in seconds on large datasets

Effective estimation of system invariants from noisy data

Simulation confirms method's robustness and practicality

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.