Range Space or Null Space: Least-Squares Methods for the Realization Problem

TL;DR

This work reexamines the classical approximate realization problem for stable, minimal LTI systems by exploiting Hankel-structure information in the first $n$ Markov parameters. It shows that range-space realization (RASBR) corresponds to a total least-squares (TLS) solution, while null-space realization (NUSBR) corresponds to ordinary least-squares (OLS), and analyzes their sensitivities and ill-conditioning. Recognizing suboptimality in both, the paper proposes a weighted null-space fitting (WLS) approach that achieves the smallest asymptotic variance, and provides a rigorous statistical treatment (consistency and asymptotic normality) under mild regularity. Collectively, the results offer a unified, least-squares framework to compare realization methods, explain case-dependent performance, and guide toward asymptotically efficient subspace identification, with extensions to weighted strategies and potential MIMO realizations.

Abstract

This contribution revisits the classical approximate realization problem, which involves determining matrices of a state-space model based on estimates of a truncated series of Markov parameters. A Hankel matrix built up by these Markov parameters plays a fundamental role in this problem, leveraging the fact that both its range space and left null space encode critical information about the state-space model. We examine two prototype realization algorithms based on the Hankel matrix: the classical range-space-based (SVD-based) method and the more recent null-space-based method. It is demonstrated that the range-space-based method corresponds to a total least-squares solution, whereas the null-space-based method corresponds to an ordinary least-squares solution. By analyzing the differences in sensitivity of the two algorithms, we determine the conditions when one or the other realization algorithm is to be preferred, and identify factors that contribute to an ill-conditioned realization problem. Furthermore, recognizing that both methods are suboptimal, we argue that the optimal realization is obtained through a weighted least-squares approach. A statistical analysis of these methods, including their consistency and asymptotic normality is also provided.

Figures (4)

• Figure 1: FITs of impulse responses from 200 Monte Carlo trials: System 1 ($\circ$) and System 2 ($\blacktriangle$), and the solid line is a bisector line.
• Figure 2: FITs of impulse responses for three experiment groups in Table \ref{['Table1a']}: A random system ($\blacktriangle$), and the solid line is a bisector line.
• Figure 3: FITs of impulse responses for six experiment groups in Table \ref{['Table1']}: A random system ($\blacktriangle$), and the solid line is a bisector line.
• Figure 4: FITs of impulse responses from 200 Monte Carlo trials: System 1 ($\circ$, OLS VS WLS) and System 2 ($\blacktriangle$, TLS VS WLS), and the solid line is a bisector line.

Theorems & Definitions (26)

• Remark 1
• Remark 2
• Theorem III.1
• proof
• Remark 3
• Definition III.1: $\sin\theta$ theory Wedin1972perturbation
• Definition III.2: Golub2013matrix
• Lemma 1
• proof
• Remark 4