Minimal Order Recovery through Rank-adaptive Identification

TL;DR

This paper introduces a rank-adaptive algorithm for identifying linear systems from noisy data, accurately recovering system order and parameters with optimal sample complexity bounds.

Contribution

It proposes Thresholded Ho-Kalman, a novel rank-adaptive method that improves system identification by balancing singular value estimation and approximation errors.

Findings

Finite-sample Frobenius norm error bounds established

System order and Markov parameters recovered accurately

Sample complexity bounds match state-of-the-art algorithms

Abstract

This paper addresses the problem of identifying linear systems from noisy input-output trajectories. We introduce Thresholded Ho-Kalman, an algorithm that leverages a rank-adaptive procedure to estimate a Hankel-like matrix associated with the system. This approach optimally balances the trade-off between accurately inferring key singular values and minimizing approximation errors for the rest. We establish finite-sample Frobenius norm error bounds for the estimated Hankel matrix. Our algorithm further recovers both the system order and its Markov parameters, and we provide upper bounds for the sample complexity required to identify the system order and finite-time error bounds for estimating the Markov parameters. Interestingly, these bounds match those achieved by state-of-the-art algorithms that assume prior knowledge of the system order.