Deep Hankel matrices with random elements

TL;DR

This paper investigates the expressive power of noisy Hankel matrices in data-driven linear system modeling, revealing how data quantity and matrix depth influence output prediction accuracy and control design.

Contribution

It introduces a novel analysis connecting Hankel matrix depth and data size to model self-consistency and prediction accuracy in noisy environments.

Findings

Deeper Hankel matrices improve prediction accuracy.

Asymptotic relationship between data size and model self-consistency.

Application to optimal control with improved trajectory modeling.

Abstract

Willems' fundamental lemma enables a trajectory-based characterization of linear systems through data-based Hankel matrices. However, in the presence of measurement noise, we ask: Is this noisy Hankel-based model expressive enough to re-identify itself? In other words, we study the output prediction accuracy from recursively applying the same persistently exciting input sequence to the model. We find an asymptotic connection to this self-consistency question in terms of the amount of data. More importantly, we also connect this question to the depth (number of rows) of the Hankel model, showing the simple act of reconfiguring a finite dataset significantly improves accuracy. We apply these insights to find a parsimonious depth for LQR problems over the trajectory space.