Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories

TL;DR

This paper develops finite-time, non-asymptotic methods for identifying linear systems from multiple trajectories, applicable to both stable and unstable systems, and highlights the impact of process noise on estimation difficulty.

Contribution

It introduces a finite-time analysis for Markov parameter estimation using multiple trajectories, extending previous work to unstable systems and analyzing the effect of process noise.

Findings

Estimation error bounds are independent of spectral radius without process noise.

Unstable systems with larger spectral radius are harder to estimate in the presence of process noise.

Numerical experiments confirm the effectiveness of the proposed OLS estimator.

Abstract

This paper considers the problem of linear time-invariant (LTI) system identification using input/output data. Recent work has provided non-asymptotic results on partially observed LTI system identification using a single trajectory but is only suitable for stable systems. We provide finite-time analysis for learning Markov parameters based on the ordinary least-squares (OLS) estimator using multiple trajectories, which covers both stable and unstable systems. For unstable systems, our results suggest that the Markov parameters are harder to estimate in the presence of process noise. Without process noise, our upper bound on the estimation error is independent of the spectral radius of system dynamics with high probability. These two features are different from fully observed LTI systems for which recent work has shown that unstable systems with a bigger spectral radius are easier to estimate. Extensive numerical experiments demonstrate the performance of our OLS estimator.