Frequency domain estimation of parallel Hammerstein systems using Gaussian process regression

There exists a number of nonparametric model structures specifically developed for frequency domain modelling of nonlinear systems. Here we consider the nonlinear output frequency response function (NOFRF) structure, which is a series of input-dependent one-dimensional functions representing each nonlinear order present in the system. When used to model parallel Hammerstein systems, the NOFRFs lose their input dependence and become ‘linear’ in structure. In this paper, we extend a linear Gaussian process regression method to the nonlinear setting, where the pseudo-linear form of Hammerstein NOFRFs can be exploited by applying standard covariance structures from the linear theory. Compared to the traditional method of NOFRF estimation, the proposed method can be performed using simple experimental conditions and shows a significant improvement in estimation accuracy in the presence of measurement noise. The proposed method can also be adapted to estimate and remove the effect of transients in the case of non-periodic excitation. Numerical results are presented which show the veracity of the proposed algorithms for systems with polynomial nonlinearities of known degree.