International Journal of Sciences and Techniques of Automatic control & computer engineering IJ-STA, Volume 3, N° 1, July 2009, pp. 926−941. This paper was recommended for publication in revised form by the editor Staff. Edition: CPU of Tunis, Tunisia, ISSN: 1737-7749 Identification of the Structure and the Parameters of Volterra models using crosscumulants Kamel Abderrahim 1 , Houda Mathlouthi 1 , Faouzi Msahli 2 and Gérard Favier 3 1 National School of Engineers of Gabès, Route de Medenine 6029 Gabès, Tunisie 2 National School of Engineers of Monastir, Avenue Ibn Jazzar, 5019, Monastir, Tunisia. 3 I3S Laboratory, UNSA/CNRS, 2000 Route des Lucioles, Sophia-Antipolis, Biot, France. Abstract. In this paper, we address the problem of structure and parameter identification of Volterra models driven by symmetric input with four levels. It consists in estimating the model order, the memory length of each kernel and the parameters. The proposed approaches are based on the crosscumulants between the input and the output using the statistics proprieties of the input sequence. The structure identification method consists in estimating the order of the Volterra model that will be used to identify the length of each kernel. A closed form solu- tion has been developed to estimate the parameters of the Volterra models. Simu- lations are presented to illustrate the performance of the proposed methods. keywords. Structure Identification, Parameter Estimation, Cross-cumulant, Vol- terra system. 1. Introduction The truncated Volterra model has been the most popular since it can represent any nonlinear system time invariant with fading memory [1]-[5]. Moreover, the parameters of this model are linearly related to the output which allows the extension of some results of linear systems to nonlinear ones [12], [19]. For these reasons, the truncated Volterra model has found applications in many fields such as signal processing and control [1], [12], [15], [17], [19], [22]. Two main problems must be taken into account for the identification of truncated Volterra model: one is the identification of the model kernels and two is the identifica- tion of the model structure defined by the model order and the kernel memory lengths. Several methods have been proposed in the literature for the identification of the ker- nels of Volterra models [6]-[12]. They can be classified into two great families. The