An efficient method for generalised Wiener series estimation of nonlinear systems using Gaussian processes

Abstract

System identification of dynamical systems aims to predict the output of a system for a given input by inferring model details from data. This is particularly challenging for nonlinear vibrating systems with long memory, which often have complex dynamics for which the physics is not fully understood. However, in many engineering applications, it is important to capture as much of the nonlinearity in the response as possible. The Wiener series is a classical data-driven technique that decomposes the system response into a set of orthogonal functionals of increasing order. Unlike standard black-box algorithms, such as neural networks, the series is highly interpretable and can offer insight into the nonlinearities present. However, current methods require vast quantities of data to calculate higher order terms in the Wiener series. In this paper, a novel generalised formulation of the Wiener series is developed by framing the series explicitly as a QR decomposition. Then by placing a Gaussian process prior over the kernels, their inherent structure is exploited to significantly reduce the quantity of data required for inference. Initial investigations were carried out using two systems: first pre-defined kernels and second an idealised nonlinear oscillatory system with unknown kernels. It is demonstrated that the Gaussian Process Generalised Wiener Series (GPGWS) enables the calculation of up to and including the third order kernel for systems with long memory and improves the robustness to noise.