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Bridging the Prediction Error Method and Subspace Identification: A Weighted Null Space Fitting Method

Jiabao He, S. Joe Qin, Håkan Hjalmarsson

TL;DR

This work addresses improving state-space identification by bridging PEM and SIMs. It introduces Weighted Null-Space Fitting for State-Space models (WNSFSS), which starts from a high-order HOARX estimate of Markov parameters and reduces to a canonical state-space form via multi-step LS with optimal weighting, achieving consistency and, under admissible parametrizations, asymptotic efficiency. The authors prove these properties for single- and multi-output cases, show equivalence to PEM in efficiency under the same canonical structure, and demonstrate competitive performance across SISO, SIMO, MIMO, and DaISy benchmarks. The approach provides a robust, computationally tractable alternative to PEM and traditional SIMs, with practical benefits for open- and closed-loop identification in diverse settings.

Abstract

Subspace identification methods (SIMs) have proven to be very useful and numerically robust for building state-space models. While most SIMs are consistent, few if any can achieve the efficiency of the maximum likelihood estimate (MLE). Conversely, the prediction error method (PEM) with a quadratic criteria is equivalent to MLE, but it comes with non-convex optimization problems and requires good initialization points. This contribution proposes a weighted null space fitting (WNSF) approach for estimating state-space models, combining some key advantages of the two aforementioned mainstream approaches. It starts with a least-squares estimate of a high-order ARX model, and then a multi-step least-squares procedure reduces the model to a state-space model on canoncial form. It is demonstrated through statistical analysis that when a canonical parameterization is admissible, the proposed method is consistent and asymptotically efficient, thereby making progress on the long-standing open problem about the existence of an asymptotically efficient SIM. Numerical and practical examples are provided to illustrate that the proposed method performs favorable in comparison with SIMs.

Bridging the Prediction Error Method and Subspace Identification: A Weighted Null Space Fitting Method

TL;DR

This work addresses improving state-space identification by bridging PEM and SIMs. It introduces Weighted Null-Space Fitting for State-Space models (WNSFSS), which starts from a high-order HOARX estimate of Markov parameters and reduces to a canonical state-space form via multi-step LS with optimal weighting, achieving consistency and, under admissible parametrizations, asymptotic efficiency. The authors prove these properties for single- and multi-output cases, show equivalence to PEM in efficiency under the same canonical structure, and demonstrate competitive performance across SISO, SIMO, MIMO, and DaISy benchmarks. The approach provides a robust, computationally tractable alternative to PEM and traditional SIMs, with practical benefits for open- and closed-loop identification in diverse settings.

Abstract

Subspace identification methods (SIMs) have proven to be very useful and numerically robust for building state-space models. While most SIMs are consistent, few if any can achieve the efficiency of the maximum likelihood estimate (MLE). Conversely, the prediction error method (PEM) with a quadratic criteria is equivalent to MLE, but it comes with non-convex optimization problems and requires good initialization points. This contribution proposes a weighted null space fitting (WNSF) approach for estimating state-space models, combining some key advantages of the two aforementioned mainstream approaches. It starts with a least-squares estimate of a high-order ARX model, and then a multi-step least-squares procedure reduces the model to a state-space model on canoncial form. It is demonstrated through statistical analysis that when a canonical parameterization is admissible, the proposed method is consistent and asymptotically efficient, thereby making progress on the long-standing open problem about the existence of an asymptotically efficient SIM. Numerical and practical examples are provided to illustrate that the proposed method performs favorable in comparison with SIMs.

Paper Structure

This paper contains 42 sections, 11 theorems, 197 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Structure
  5. Notations
  6. Preliminaries
  7. Weighted Null-Space Fitting
  8. Multi-Step Least-Squares
  9. Extension to MIMO Systems
  10. Canonical Parametrizations
  11. A Case Study
  12. Asymptotic Properties
  13. Simulations
  14. Illustration of Asymptotic Properties
  15. A SISO System
  16. ...and 27 more sections

Key Result

Lemma 1

The state-space model MIMO_canonical_form with a particular Kronecker index $\bar{\nu}$ can describe almost all $n_x$-dimensional stochastic LTI systems.

Figures (7)

  • Figure 1: Average MSE of $\hat{\bm{\theta}}$ from 1000 Monte Carlo trials (SISO system): Open-loop (OL) and Closed-loop (CL) Cases.
  • Figure 2: Average MSE of $\hat{\bm{\theta}}$ from 100 Monte Carlo trials (SIMO system): Open-loop (OL) and Closed-loop (CL) Cases.
  • Figure 3: FITs from 100 Monte Carlo trials: Open-loop.
  • Figure 4: FITs from 100 Monte Carlo trials: Closed-loop.
  • Figure 5: Average transfer functions of the identified MIMO systems from 50 Monte Carlo trials.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Lemma 1: Gevers1984uniquelyLjung1999system
  • Lemma 2: Wertz1982determinationLjung1999system
  • Remark 3: Overlapping Parametrizations
  • Remark 4
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Remark 5
  • ...and 8 more