Bilinear state space realization for polynomial stochastic systems

AbstractThe Wiener approach to the non-linear stochastic systems permits the representation of single-valued systems with memory for which a small perturbation of the input produces a small perturbation of the output. The Wiener functional series representation contains infinitely many transfer functions to describe entirely the input output connections. One of the most important class of stochastic systems, especially in the statistical point of view, is the case when all the transfer functions are determined by finitely many parameters. The question of realizability, i.e., to construct a state space model for a system given in terms of a set of transfer functions, is a central problem in many cases for example in engineering. There are several results in this direction for systems with deterministic inputs see [1,2]. The realization of linear systems with stochastic Gaussian input was treated by Akaike [3] and studied intensively afterwards, see [4,5]. The bilinear system seems to be the next step leading from the linear towards non-linear ones. The realization problem of bilinear systems appeared in the papers [6–8], mainly by the time series side.In this work, we follow the early Akaike work and the results for the deterministic systems by Rugh [2] and also Brillinger [9], who investigated the identification of finite degree functional series by spectral analysis. First the bilinear realization problem is considered together with a simplest non-linear system, i.e., the second Hermite degree bilinear process. In the second section, the transfer function system with its main properties is given by a recursive formula for the bilinear states. Next, we show that these properties are necessary and sufficient for the transfer functions to be bilinear realizable. In section four, the stationarity of the bilinear state space model is investigated. We are constructing the abstract bilinear minimal realization for a time invariant stationary degree-N non-linear system driven by Gaussian white noise.Concerning to the identification of the bilinear model with Hermite degree-2 explicit formula of the bispectrum is given for this family of models