Finite Sample Analysis for a Class of Subspace Identification Methods

TL;DR

This paper provides a finite sample statistical analysis of subspace identification methods for MIMO systems, establishing convergence rates and proposing a causal SIM approach that improves analytical tractability.

Contribution

It introduces a causal subspace identification method that bypasses the non-causal projection step, enabling finite sample error bounds for system matrices.

Findings

Convergence rate of $\rac{1}{\sqrt{N}}$ for estimating system parameters.

A new SIM approach enforcing causality for better analysis.

Finite sample error bounds for ARX models and system matrices.

Abstract

While subspace identification methods (SIMs) are appealing due to their simple parameterization for MIMO systems and robust numerical realizations, a comprehensive statistical analysis of SIMs remains an open problem, especially in the non-asymptotic regime. In this work, we provide a finite sample analysis for a class of SIMs, which reveals that the convergence rates for estimating Markov parameters and system matrices are $\mathcal{O}(1/\sqrt{N})$, in line with classical asymptotic results. Based on the observation that the model format in classical SIMs becomes non-causal because of a projection step, we choose a parsimonious SIM that bypasses the projection step and strictly enforces a causal model to facilitate the analysis, where a bank of ARX models are estimated in parallel. Leveraging recent results from finite sample analysis of an individual ARX model, we obtain an overall error bound of an array of ARX models and proceed to derive error bounds for system matrices via robustness results for the singular value decomposition.