NonSysId: Nonlinear System Identification with Improved Model Term Selection for NARMAX Models

An open-source MATLAB package for system identification of ARX, NARX and (N)ARMAX models, featuring improved term selection and robust long-term simulation capabilities.

Authors: Rajintha Gunawardena¹, Zi-Qiang Lang², Fei He¹

1. Centre for Computational Science and Mathematical Modelling, Coventry University, Coventry CV15FB, UK.
2. School of Electrical and Electronic Engineering, The University of Sheffield, Western Bank, Sheffield S10 2TN, UK.

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Overview 📖

NonSysId is a MATLAB package designed for the identification of nonlinear dynamic systems using (N)AR(MA)X models. It incorporates an enhanced Orthogonal Forward Regression (OFR) algorithm, iterative-OFR (iOFR), and PRESS-statistic based criterion to improve model term selection and ensure robust long-term predictions. The package is particularly suited for applications where separate validation datasets are difficult to obtain, such as real-time fault diagnosis and electrophysiological studies.

Features

• Iterative OFR (iOFR): Improves term selection by iterating through multiple orthogonalisation paths to produce parsimonious models.
• Simulation-based Model Selection: Ensures simulation stability and enhances long-term prediction accuracy.
• PRESS-statistic Integration: Includes a PRESS-statistic based term selection criterion that aims to minimise the leave-one-out cross-validation error. Therefore, the model can be validated without requiring separate validation datasets.
• Reduced Computational Time (RCT): Optimized procedures to accelerate model term selection for complex NARX models.
• Custom Householder QR Orthogonalisation: Replaces the original Gram–Schmidt orthogonalisation in the FROLS/i-FRO procedure with a Householder orthogonalisation. The Householder orthogonalisation is more stable when considering a larger number of regressors. The customisation reconstructs each normalised candidate orthogonal regressor in the original sample coordinates, ensuring that the sample-wise PRESS-statistic updates remain mathematically equivalent to those in the original formulation. The original back-substitution method is retained for parameter estimation, thereby avoiding matrix inversion within the FROLS/i-FRO algorithm.
• NonSysID-i: This is a dedicated variant of NonSysID for identifying linear and nonlinear input-only (N)ARX models. Its candidate regressors are constructed exclusively from lagged inputs and their nonlinear combinations, with no lagged output terms included. Consequently, recursive model simulation does not depend on previously predicted outputs, making model simulation equivalent to one-step-ahead prediction. By eliminating the repeated recursive simulation required by conventional NonSysID models during model selection, NonSysID-i provides substantially faster system identification for input-only model structures.

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Getting Started 🚀

Prerequisites

• MATLAB R2017a or later.
• Required MATLAB Toolboxes:
  • Signal Processing Toolbox (required if using earlier than Matlab 2019a, for correlation analysis).
  • Parallel Computing Toolbox (required for accelerating system identification procedures).

Installation

1. Clone the repository:

   git clone https://github.com/raj-gun/NonSysId.git

   or manually download the folder 'NonSysId'.

2. In Matlab, either;

   • add the folder 'NonSysId' to the Matlab path permanently using the 'pathtool' (https://uk.mathworks.com/help/matlab/ref/pathtool.html).
   • or use the 'addpath' command in the Matlab script to add the folder 'NonSysId' and use the functions within (https://uk.mathworks.com/help/matlab/ref/addpath.html).

Documentation

Brief documentation explaining the main functions and a code structure for identifying a model, simulating and validating an identified model is given in the doc folder.

Examples

• Identifying a SISO NARX model from real data, see the example in Examples/Electro-mechanical system.
• Identifying a MISO NARX model is shown in Examples/Hysteresis_model_MISO.

Paper

If you are using the NonSysId package for academic purposes, kindly reference our paper as follows:

NonSysId: Nonlinear System Identification with Improved Model Term Selection for NARMAX Models

Rajintha Gunawardena, Zi-Qiang Lang, Fei He

DOI: 10.21105/joss.08028

@article{Gunawardena2025,
doi = {10.21105/joss.08028},
url = {https://doi.org/10.21105/joss.08028},
year = {2025},
publisher = {The Open Journal},
volume = {10},
number = {114},
pages = {8028},
author = {Gunawardena, Rajintha and Lang, Zi-Qiang and He, Fei},
title = {NonSysId: Nonlinear System Identification with Improved Model Term Selection for NARMAX Models},
journal = {Journal of Open Source Software}
}

References

[1] M. Korenberg, S. Billings, Y. Liu, and P. McIlroy, “Orthogonal parameter estimation algorithm for non-linear stochastic systems,” Int. J. Control, vol. 48, no. 1, pp. 193–210„ 1988.

[2] S. Chen, S. Billings, and W. Luo, “Orthogonal least squares methods and their application to non-linear system identification,” Int. J. Control, vol. 50, no. 5, pp. 1873–1896„ 1989.

[3] S. Billings, Nonlinear System Identification: NARMAX Methods In The Time, Frequency, And Spatio-Temporal Domains, vol. 13. Chichester, UK: John Wiley & Sons, Ltd, 2013.

[4] S. B. Yuzhu Guo, L.Z. Guo and H.-L. Wei, “An iterative orthogonal forward regression algorithm,” International Journal of Systems Science, vol. 46, no. 5, pp. 776–789, 2015.

[5] X. Hong, P. Sharkey, and K. Warwick, “Automatic nonlinear predictive model-construction algorithm using forward regression and the press statistic,” IEE Proceedings-Control Theory and Applications, vol. 150, no. 3, pp. 245–254, 2003.

[6] L. Ljung, System identification. Springer, 1998.