Nonlinear System Identification with Dominating Output Noise - A Case Study on the Silverbox

In the overview paper on nonlinear system identification Schoukens and Ljung (2019), it was indicated that reliable expressions to calculate the variance of an estimated nonlinear model are lacking, especially if the disturbing noise is entering the nonlinear regressors. In this study, we provide a better view on the driving mechanisms of the variability of estimated nonlinear models that is due to noise on the output. To do so, we follow a double approach. Firstly, a basic insight on the impact of disturbing noise on the monomial yTh is studied. Next, these insights are used in a case study on the forced Duffing oscilator data, also called the Silver box (Schoukens and Noel, 2016). The following models are studied: Nonlinear autoregressive exogenous models (NARX) using a 2-layer Neural Net (NARX-NN) and a polynomial (NARX-poly) expansion; and polynomial nonlinear state space models (PNLSS). This limited study indicates that an output error criterion (PNLSS, and NARX used in a simulation mode) does better than minimizing the equation error (NARX-NN and NARX-poly in prediction mode). When the signal-to-noise ratio (SNR) drops below 20 dB, the reduction in the error is more than a factor 10. This is a strong indication that, just as for linear identification, it is very important to properly deal with the noise properties in the cost function whenever the SNR of the output measurements drops such that noise becomes more important than structural model errors.