Reducing Data Requirements for Wiener Series Identification With Gaussian Process Priors

Abstract

Abstract System identification of nonlinear dynamical systems aims to predict the output of a system for a given input. In many engineering applications, the underlying physics are not fully understood and so there is no analytical solution. 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. To date, in order to calculate higher order terms in the Wiener series, vast quantities of data are needed. In this paper, a novel formulation of the Wiener series is developed in the frequency domain which applies to general stochastic inputs with an arbitrary spectrum. It is enhanced by placing Gaussian process priors over the Wiener kernels to enforce prior knowledge of their structure. This significantly reduces the quantity of data required for inference and has the benefit of enabling the calculation of the third order kernel for systems with long memory. The benefits were demonstrated in initial investigations using an idealised nonlinear oscillatory system. Decomposition of the system response into Wiener functionals also sheds light on the learnability of nonlinear dynamical systems, which could be used to assess the value of collecting additional data.