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Finite Sample Analysis for a Class of Subspace Identification Methods

Jiabao He, Ingvar Ziemann, Cristian R. Rojas, Håkan Hjalmarsson

TL;DR

The paper addresses non-asymptotic guarantees for SIMs by analyzing a PARSIM-based bank of parallel ARX models that enforces causality and avoids projection-induced noncausality. It proves that estimation errors for Markov parameters and state-space matrices decay as $\mathcal{O}(1/\sqrt{N})$ under mild assumptions, with high-probability bounds that propagate through a balanced realization via SVD robustness. The analysis hinges on a martingale-based treatment of the stochastic ARX error and a decoupled treatment of truncation bias, aided by persistence-of-excitation conditions achieved with a logarithmic past horizon $p=\beta\log N$. The results provide a principled finite-sample perspective on SIMs with inputs, and establish a pathway to compare and refine SIMs via ARX-bank formulations and SVD-based realizations. Practical implications include guidance on horizon choices, PE considerations, and the robustness of the realization step to ARX-model errors.

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.

Finite Sample Analysis for a Class of Subspace Identification Methods

TL;DR

The paper addresses non-asymptotic guarantees for SIMs by analyzing a PARSIM-based bank of parallel ARX models that enforces causality and avoids projection-induced noncausality. It proves that estimation errors for Markov parameters and state-space matrices decay as under mild assumptions, with high-probability bounds that propagate through a balanced realization via SVD robustness. The analysis hinges on a martingale-based treatment of the stochastic ARX error and a decoupled treatment of truncation bias, aided by persistence-of-excitation conditions achieved with a logarithmic past horizon . The results provide a principled finite-sample perspective on SIMs with inputs, and establish a pathway to compare and refine SIMs via ARX-bank formulations and SVD-based realizations. Practical implications include guidance on horizon choices, PE considerations, and the robustness of the realization step to ARX-model errors.

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 , 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.
Paper Structure (13 sections, 7 theorems, 57 equations)

This paper contains 13 sections, 7 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model and Assumptions
  4. A Recap of PARSIM
  5. Problem and Roadmap Ahead
  6. Finite Sample Analysis of each ARX Model
  7. Persistence of Excitation
  8. Stochastic Error
  9. Truncation Bias Term
  10. Robustness of Balanced Realization
  11. Overall Bound
  12. Bounds on System Matrices
  13. Discussion and Future Work

Key Result

Lemma 1

Fix a failure probability $0<\delta<1$, if $N\geq N_{pe}(\delta,\beta,i)$, then with probability at least $1-\delta$, we have where $\lambda_{\text{min}}(\Sigma_{i,\tau_i}) > 0$.

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Definition III.1
  • Remark 3
  • Lemma 1
  • proof
  • Remark 4
  • Lemma 2
  • proof
  • Lemma 3: Lemma E.5 in Ziemann2023tutorial
  • ...and 5 more