A Gaussian Sum Smoothing algorithm for Hammerstein-Wiener State-Space Systems

In this paper, we develop a novel Bayesian smoothing method for obtaining the smoothed probability density functions of Hammerstein-Wiener state-space systems and the corresponding state estimation. The proposed smoother is designed using the two-filter approach, based on the Gaussian sum filtering algorithm and a backward filtering method. In this work, this backward filter is obtained using an approximation of the probability function of the non-linear output conditioned to the system state. Both the forward filter and the backward filter are used to obtain the Gaussian sum smoothing algorithm, which also includes the computation of the joint probability density function of the system state in two consecutive time instants. Numerical examples are presented to illustrate the benefits of our proposal.