A Weighted Least-Squares Method for Non-Asymptotic Identification of Markov Parameters from Multiple Trajectories

TL;DR

This paper introduces a weighted least-squares approach for non-asymptotic identification of Markov parameters from multiple trajectories, providing tighter error bounds and consistent estimation methods, with demonstrated improvements over ordinary least-squares.

Contribution

It develops a non-asymptotic analysis for weighted least-squares in system identification, including methods to estimate the optimal weighting matrix for stable and unstable systems.

Findings

Weighted least-squares yields tighter error bounds than ordinary least-squares.

Optimal weighting matrix improves estimation accuracy in finite samples.

Numerical experiments confirm the advantages of the proposed method.

Abstract

Markov parameters play a key role in system identification. There exists many algorithms where these parameters are estimated using least-squares in a first, pre-processing, step, including subspace identification and multi-step least-squares algorithms, such as Weighted Null-Space Fitting. Recently, there has been an increasing interest in non-asymptotic analysis of estimation algorithms. In this contribution we identify the Markov parameters using weighted least-squares and present non-asymptotic analysis for such estimator. To cover both stable and unstable systems, multiple trajectories are collected. We show that with the optimal weighting matrix, weighted least-squares gives a tighter error bound than ordinary least-squares for the case of non-uniformly distributed measurement errors. Moreover, as the optimal weighting matrix depends on the system's true parameters, we introduce two methods to consistently estimate the optimal weighting matrix, where the convergence rate of these estimates is also provided. Numerical experiments demonstrate improvements of weighted least-squares over ordinary least-squares in finite sample settings.