[Paper Review] Optimal model order reduction with the Steiglitz-McBride method for open-loop data

Niklas Everitt, Miguel Galrinho+1 more|arXiv (Cornell University)|Oct 26, 2016

Control Systems and Identification10 references3 citations

TL;DR

This paper proposes the Model Order Reduction Steiglitz-McBride (MORSM) method, which combines high-order ARX modeling with a novel application of the Steiglitz-McBride algorithm to achieve asymptotically efficient plant model estimation from open-loop data. By pre-filtering data using a non-parametric ARX estimate and applying a single iteration of the Steiglitz-McBride method, MORSM avoids non-convex optimization pitfalls while achieving efficiency comparable to PEM, with improved finite-sample convergence over existing alternatives.

ABSTRACT

In system identification, it is often difficult to find a physical intuition to choose a noise model structure. The importance of this choice is that, for the prediction error method (PEM) to provide asymptotically efficient estimates, the model orders must be chosen according to the true system. However, if only the plant estimates are of interest and the experiment is performed in open loop, the noise model may be over-parameterized without affecting the asymptotic properties of the plant. The limitation is that, as PEM suffers in general from non-convexity, estimating an unnecessarily large number of parameters will increase the chances of getting trapped in local minima. To avoid this, a high order ARX model can first be estimated by least squares, providing non-parametric estimates of the plant and noise model. Then, model order reduction can be used to obtain a parametric model of the plant only. We review existing methods to perform this, pointing out limitations and connections between them. Then, we propose a method that connects favorable properties from the previously reviewed approaches. We show that the proposed method provides asymptotically efficient estimates of the plant with open loop data. Finally, we perform a simulation study, which suggests that the proposed method is competitive with PEM and other similar methods.

Motivation & Objective

• To address the challenge of non-convex optimization in prediction error method (PEM) estimation for system identification.
• To eliminate the need for parametric noise model order selection, which is often arbitrary and affects convergence in PEM.
• To develop a method that achieves asymptotically efficient plant model estimation without relying on iterative non-linear optimization.
• To improve finite-sample performance over existing methods like BJSM and ASYM by combining their strengths.

Proposed method

• Estimate a high-order ARX model using least squares on open-loop data, providing a non-parametric, globally optimal initial estimate.
• Use the high-order ARX model to pre-filter input and output data, approximating a noise-whitening transformation.
• Apply a single iteration of the Steiglitz-McBride algorithm to the pre-filtered data to estimate a low-order parametric plant model.
• Formulate the model reduction step to implicitly minimize an approximate maximum likelihood criterion, ensuring asymptotic efficiency.
• Connect the ASYM and BJSM methods by embedding the pre-filtering idea of BJSM into the model reduction framework of ASYM.
• Ensure the method remains computationally efficient by avoiding iterative non-linear optimization while maintaining statistical efficiency.

Experimental results

Research questions

• RQ1 Can a model order reduction method be designed that achieves asymptotic efficiency in open-loop system identification without requiring iterative non-linear optimization?
• RQ2 How can the pre-filtering approach of BJSM be combined with the asymptotic efficiency of ASYM to improve finite-sample performance?
• RQ3 Is it possible to achieve asymptotic efficiency with only one iteration of the Steiglitz-McBride algorithm when using high-order ARX pre-estimation?
• RQ4 Does the proposed method outperform BJSM and PEM in terms of convergence robustness and finite-sample accuracy?
• RQ5 Can the method avoid the need for noise model order selection while preserving efficiency for plant model estimation?

Key findings

• The proposed MORSM method achieves asymptotically efficient plant model estimation with only one iteration of the Steiglitz-McBride algorithm.
• MORSM demonstrates superior finite-sample convergence properties compared to the BJSM method in simulation studies.
• The method avoids the non-convex optimization issues of PEM by using a globally optimal high-order ARX estimate as a starting point.
• The simulation results show that MORSM performs competitively with PEM in terms of estimation accuracy and efficiency.
• The method successfully decouples plant and noise model estimation, allowing for efficient, consistent plant model estimation without requiring a parametric noise model.