Kernel-based methods for Volterra series identification

Volterra series approximate a broad range of nonlinear systems. Their identification is challenging due to the curse of dimensionality: the number of model parameters grows exponentially with the complexity of the input–output response. This fact limits the applicability of such models and has stimulated recently much research on regularized solutions. Along this line, we propose two new strategies that use kernel-based methods. First, we introduce the multiplicative polynomial kernel (MPK). Compared to the standard polynomial kernel, the MPK is equipped with a richer set of hyperparameters, increasing flexibility in selecting the monomials that really influence the system output. Second, we introduce the smooth exponentially decaying multiplicative polynomial kernel (SEDMPK), that is a regularized version of MPK which requires less hyperparameters, allowing to handle also high-order Volterra series. Numerical results show the effectiveness of the two approaches.