Efficient parameterisation of nonlinear system models: a comment on Nöel and Schoukens (2018)

Abstract

Nöel, J. P., & Schoukens, J. [2018. Grey-box state-space identification of nonlinear mechanical vibrations. International Journal of Control, 91, 1–22] discuss a methodology for the discrete-time state-space identification of nonlinear systems and apply this to experimental data from the well known Silverbox nonlinear circuit, producing a model characterised by 13 parameters. This model explains the data very well but the parameter estimates are not well defined in the optimisation results, with the very large confidence bounds suggesting that the model is over-parameterised. This comment shows that this is indeed the case and that the data can be explained equally well by an alternative continuous-time, State-Dependent Parameter (SDP) transfer function model with only 6 parameters, the estimates of which are well defined with very tight confidence bounds. The comment also raises questions about how the model form for nonlinear systems such as the Silverbox should be identified and suggests that the Data-Based Mechanistic (DBM) approach to modelling has some advantages in this regard.

Keywords:

• System identification
• silverbox system
• nonlinear modelling
• continuous-time model
• efficient parameterisation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This can be downloaded from the 'Publication Downloads’ section of http://captaintoolbox.co.uk/Captain_Toolbox.html/Captain_Toolbox.html: PCY Technical Report 2018 TN5

2 Note that the parameter estimates shown here are rounded to four significant figures and this degrades the $R_T^2$.

3 Note that the $b_2=0.03086$ is not optimised because it only controls the gain of the system and can be set arbitrarily to any value, here to a value that relates to the transfer function modelling described in Section 3.