Minimal Order Recovery through Rank-adaptive Identification
Frédéric Zheng, Yassir Jedra, Alexandre Proutière
TL;DR
This work tackles finite-time identification of discrete-time LTI systems from noisy input-output trajectories without any prior knowledge of the state or system order. It introduces Thresholded Ho-Kalman, a two-step method that first obtains a least-squares estimate of a Hankel-like matrix $H_\tau$ and then applies universal singular value thresholding to obtain a low-rank approximation $\hat{H}_\tau(\xi)$, enabling order recovery and Ho-Kalman-based estimation of Markov parameters. The authors provide non-asymptotic Frobenius-norm error bounds and sample-complexity guarantees for both single and multiple trajectory settings, showing that the bounds match those of methods that assume known order. Numerical experiments on synthetic data corroborate that the method recovers the true order and achieves near-oracle performance even when the order is unknown, highlighting practical viability for partially observed LTI identification.
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.