Nonparametric volterra kernel estimation using regularization

Modeling of nonlinear dynamic systems constitutes one of the most challenging topics in the field of system identifi- cation. One way to describe the nonlinear behavior of a process is by use of the nonparametric Volterra Series representation. The drawback of this method lies in the fact that the number of parameters to be estimated increases fast with the number of lags considered for the description of the several impulse responses. The result is that the estimated parameters admit a very large variance leading to a very uncertain description of the nonlinear system. In this paper, inspired from the regularization techniques that have been applied to one-dimensional (1-D) impulse responses for a linear time invariant (LTI) system, we present a method to estimate efficiently finite Volterra kernels. The latter is achieved by constraining the estimated parameters appropriately during the identification step in a way that prior knowledge about the to-be-estimated kernels is reflected on the resulting model. The enormous benefit for the identification of Volterra kernels due to the regularization is illustrated with a numerical example.