Model selection approaches for non-linear system identification: a review

Abstract

The identification of non-linear systems using only observed finite datasets has become a mature research area over the last two decades. A class of linear-in-the-parameter models with universal approximation capabilities have been intensively studied and widely used due to the availability of many linear-learning algorithms and their inherent convergence conditions. This article presents a systematic overview of basic research on model selection approaches for linear-in-the-parameter models. One of the fundamental problems in non-linear system identification is to find the minimal model with the best model generalisation performance from observational data only. The important concepts in achieving good model generalisation used in various non-linear system-identification algorithms are first reviewed, including Bayesian parameter regularisation and models selective criteria based on the cross validation and experimental design. A significant advance in machine learning has been the development of the support vector machine as a means for identifying kernel models based on the structural risk minimisation principle. The developments on the convex optimisation-based model construction algorithms including the support vector regression algorithms are outlined. Input selection algorithms and on-line system identification algorithms are also included in this review. Finally, some industrial applications of non-linear models are discussed.

Keywords:

• adaptive learning
• cross validation
• model selection
• model generalisation
• system identification
• control engineering

Acknowledgements

The authors thank the referees and the editors for their constructive comments to improve the quality of the article.

Notes

Note

1. Functions that satisfy Mercer's condition, i.e. for any g(x) such that ∫g(x)²dx is finite, then ∫k(x, y)g(x)g(y) ×dx dy}  ≥ 0.