Regularized nonparametric Volterra kernel estimation

Abstract

In this paper, the regularization approach introduced recently for nonparametric estimation of linear systems is extended to the estimation of nonlinear systems modelled as Volterra series. The kernels of order higher than one, representing higher dimensional impulse responses in the series, are considered to be realizations of multidimensional Gaussian processes. Based on this, prior information about the structure of the Volterra kernel is introduced via an appropriate penalization term in the least squares cost function. It is shown that the proposed method is able to deliver accurate estimates of the Volterra kernels even in the case of a small amount of data points.

Gaussian process regression for the estimation of generalized frequency response functions

“…For multidimensional Volterra kernels, covariance functions have already been constructed in the time domain [1], by applying a diagonal/correlated (DC) structure [3] along multiple perpendicular 'regularizing directions.' The resulting covariance matrices are guaranteed to be valid and produce stable kernel realizations; two properties which we desire in the frequency domain context as well.…”

“…The estimates are plotted on top of the true GFRF (transparent) for comparison, revealing a close match in each case. It is clear that tuning and transforming the time-domain covariance structures from [1] leads to satisfactory GFRF estimation, even when the problem is severely rank deficient. While all previous results in this paper relied on the assumption of transient-free measurements from a periodic input excitation, one major benefit of Gaussian process regression for linear FRFs was the ability to estimate and remove transient functions for the non-periodic input case.…”

“…Kernel-based modeling of dynamic systems has garnered a significant amount of attention in the system identification literature since its introduction to the field in [7]. While the method was originally applied to linear impulse response estimation in the time domain, the concepts have since been extended to the frequency domain for estimation of frequency response functions (FRFs) [6], as well as to the Volterra series in [1]. In the latter case, smoothness and stable decay was imposed along the hypersurfaces of the multidimensional impulse responses (referred to as 'Volterra kernels' in the sequel), allowing lower variance estimates than could be obtained in a simple least squares framework.…”

Efficient multidimensional regularization for Volterra series estimation

“…The properties of decaying and smoothness for the 2 nd order Volterra kernel are encoded into the matrix P = . The ( , )element, which corresponds to =,A =,F ∀ , where =,A , =,F denote two Volterra coefficients in = , is given by [18]:…”

“…The regularization methods introduced for FIR modelling are extended to the case of Volterra kernels estimation using the method proposed in [18]. The benefit of regularization in this case with respect to FIR modelling is even more evident given the larger number of parameters usually involved in the Volterra series.…”