Near optimal finite time identification of arbitrary linear dynamical systems

TL;DR

This paper provides finite time error bounds for estimating arbitrary linear time-invariant systems from a single trajectory, covering stable, marginally stable, and explosive regimes, and highlights conditions for statistical inconsistency.

Contribution

It offers the first comprehensive analysis of finite time identification errors for LTI systems across all eigenvalue regimes, including new bounds and insights into estimator consistency.

Findings

Sharp upper bounds for stable, marginally stable, and explosive systems.

Identification error bounds are logarithmic in key parameters.

Least squares may be inconsistent even with high signal-to-noise ratio.

Abstract

We derive finite time error bounds for estimating general linear time-invariant (LTI) systems from a single observed trajectory using the method of least squares. We provide the first analysis of the general case when eigenvalues of the LTI system are arbitrarily distributed in three regimes: stable, marginally stable, and explosive. Our analysis yields sharp upper bounds for each of these cases separately. We observe that although the underlying process behaves quite differently in each of these three regimes, the systematic analysis of a self--normalized martingale difference term helps bound identification error up to logarithmic factors of the lower bound. On the other hand, we demonstrate that the least squares solution may be statistically inconsistent under certain conditions even when the signal-to-noise ratio is high.