The object of this paper is the identification of Hammerstein systems, which are dynamic systems consisting of a static nonlinearity and a linear time-invariant dynamic system in cascade. We assume that the nonlinear function can be described as a linear combination of p basis functions. We model the system dynamics by means of an np-dimensional vector. This vector, usually referred to as overparameterized vector, contains all the combinations between the nonlinearity coefficients and the first n samples of the impulse response of the linear block. The estimation of the overparameterized vector is performed with a new regularized kernel-based approach. To this end, we introduce a novel kernel tailored for overparameterized models, which yields estimates that can be uniquely decomposed as the combination of an impulse response and p coefficients of the static nonlinearity. As part of the work, we establish a clear connection between the proposed identification scheme and our recently developed nonparametric method based on the stable spline kernel.
Risuleo, R. S., Bottegal, G., & Hjalmarsson, H. (2015). A new kernel-based approach to overparameterized Hammerstein system identification. In Proceedings of the IEEE Conference on Decision and Control (Vol. 54rd IEEE Conference on Decision and Control,CDC 2015, pp. 115–120). Institute of Electrical and Electronics Engineers Inc. https://doi.org/10.1109/CDC.2015.7402095