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 (SED-MPK), 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.

Volterra级数近似于广泛的非线性系统。由于维度的诅咒，它们的识别具有挑战性：模型参数的数量随着输入输出响应的复杂性呈指数增长。这一事实限制了此类模型的适用性，并且最近刺激了对正则化解决方案的大量研究。沿着这条思路，我们提出了两种使用基于内核的方法的新策略。首先，我们介绍乘法多项式内核（MPK）。与标准多项式内核相比，MPK配备了更丰富的超参数集，从而在选择真正影响系统输出的单项式方程式时增加了灵活性。其次，我们介绍了光滑指数衰减的多项式核（SED-MPK），这是MPK的规范化版本，需要较少的超参数，因此也可以处理高阶Volterra系列。数值结果表明了两种方法的有效性。