# Realization and identification of autonomous Wiener systems via low-rank approximation

Wiener systems are nonlinear dynamical systems, consisting of a linear dynamical system and a static nonlinear system in a series connection. Existing results for analysis and identification of Wiener systems assume zero initial conditions. In this paper, we consider the response of a Wiener system to initial conditions only, , we consider autonomous Wiener systems. Our main result is a proof that an autonomous Wiener system with a polynomial nonlinearity is equivalent to a finite-dimensional linear system. The order of the equivalent linear system is (n + d)-choose-d — the number of combinations with repetitions of d elements out of n elements — where n is the order of the linear subsystem and d is the degree of the nonlinearity. The relation between the eigenvalues of the equivalent linear system and the linear subsystem is given by a rank-1 factorization of a symmetric d-way tensor. As an application of the result, we outline a procedure for identification of autonomous Wiener systems.