Modelling unknown nonlinear systems defined on a unbounded set via neural networks

This paper presents a general approach to the modelling of unknown non-linear systems represented by NARMA models, where the unknown nonlinear function is defined on a non-compact set. Since neural networks modelling requires that the unknown nonlinear function be defined on a compact set, a continuous, monotonic and invertible one-to-one mapping is used to transfer the non-compact definition domain of the nonlinear unknown function into a bounded open set, which can be further covered by a bounded closed set (compactness). As a result, the original nonlinear function can be regarded as a new function defined on the bounded closed set where a B-spline neural network can be directly applied. Due to the one-to-one mapping, the weights in B-splines neural networks are no longer the linear combination of the model output. Training algorithm are therefore developed and shown to exhibit local convergence. A pH process is studied to demonstrate the applicability of the method and desired modelling results are obtained.