Identification of Wiener systems with internal noises

In this paper, we study the identification problem for the Wiener systems with internal noises. If the internal noise is iid Gaussian, then the Wiener system can be transformed to the one without internal noise by the Weierstrass transform. The linear subsystems for both the original and transformed systems are the same. Recursive algorithms are proposed to identify the transformed Wiener system. The strong consistency of the estimates is proved under reasonable conditions. Nonparametric estimate for the nonlinear block of the original Wiener system is given trough the inverse Weierstrass transform of the nonlinear function of the transformed Wiener system, and it is proved that the estimate is consistent if the nonlinearity of the original Wiener system is a polynomial. A numerical example is presented showing that the simulation result is consistent with theoretical analysis.