Decoupling non-polynomial functions: A neural network example

The decoupling algorithm proposed by Dreesen et al. (SIAM J. Matrix Anal. Appl., 2015) was developed for MIMO polynomials. Polynomial based models are commonly used in nonlinear system identification despite their poor extrapolation behaviour and the often poor numerical conditioning of the resulting estimation problems. Several alternatives have been used in the identification literature including: various spline representations, neural networks, and support vector machines. In this contribution, we consider the approximation of non-polynomial nonlinearities using a decoupled structure. The derivation of the decoupling algorithm is repeated, but without assuming a polynomial structure at any stage of the development. As an example, a NARX model employing a feedforward sigmoidal neural network as its nonlinearity is fitted to data from the Silverbox benchmark system. The resulting nonlinearity is then decoupled using the approach proposed in this contribution. The non-parametric filtered CPD is used to impose smoothness on the solution without imposing a polynomial constraint.