Abstract—This work discusses the identification of nonlinear systems structured in blocks. Presently, the proposed method is addressed to Wiener-Hammerstein models. Hammerstein and Wiener models are nonlinear representations of systems composed by connecting of a nonlinearity element f(.) and a linear subsystem G(s) in the form f(.)-G(s) and G(s)-f(.) respectively. The identification of nonlinearity blocks and linear subsystems is not a trivial problem, and has attracted a lot of research interest. The linear subsystems G i (s) and G o (s) are allowed to be nonparametric and of unknown structure. Presently, the system nonlinearity is static and may be noninvertible. Moreover, this latter is of unknown structure and is only supposed to be well approximated, within any subinterval belonging to the working interval, with a polynomial of unknown order and parameters. Then, using a frequency identification method, a two-phase algorithm is presented for identifying the linear subsystems G i (s) and G o (s) (the frequency complex gains) and the nonlinearity element f(.). The procedure is illustrated with simulated and experimental data. The proposed strategy involves simples input signals. Key-Words—Block-oriented models, Nonlinear systems, Wiener systems, Hammerstein systems, Frequency identification, nonparametric systems. I. INTRODUCTION Wiener-Hammerstein systems consist of a series connection including a nonlinear static element sandwiched with two linear subsystems (Fig. 1). Clearly, this model structure is a generalization of Hammerstein and Wiener models and so it is expected to feature a superior modelling capability. This has been confirmed by several practical applications e.g. paralyzed skeletal muscle dynamics (Bai et al., 2009). As a matter of fact, Wiener-Hammerstein (WH) systems are more difficult to identify than the simpler Hammerstein and Wiener models. The complexity of the former lies in the fact that these systems involve two internal signals not accessible to measurements, whereas the latter only involve one. Then, it is not surprising that only a few methods are available that deal with WH system identification. The available methods have been developed following three main approaches i.e. iterative nonlinear optimization procedures (e.g. Marconato et al., 2012), stochastic methods (Bershad et al., 2001; Pillonetto et al., 2011); frequency methods (Giri et al., 2013; Brouri et al., 2014). The proposed identification methods also differ by the type of assumptions, made on both the system dynamics and the input signals, and the nature of convergence analysis results. Roughly, the iterative methods necessitate a large amount of data, since computation time and memory usage drastically increase, and have local convergence properties * ENSAM, L2MC, Moulay Ismail University, Morocco. Adil Brouri is with the ENSAM, L2MC, AEEE Department, Moulay Ismail University, Meknes, Morocco (e-mail: a.brouri@ensam-umi.ac.ma & brouri_adil@ yahoo.fr ). which necessitates that a fairly accurate parameter estimates are available to initialize the search process. This prior knowledge is not required in stochastic methods but these are generally relied on specific assumption on the input signals (e.g. gaussianity, persistent excitation....) and on system model (e.g. MA linear subsystems, smooth nonlinearity). The frequency methods are generally applied to nonparametric systems under minimal assumptions and only require simple periodic excitations. But, they sometime necessitate several data generation experiments. In this paper, the problem of identifying WH systems is addressed, for simplicity, in the continuous-time. Unlike many previous works, the model structure of the two linear subsystems is entirely unknown. Furthermore, the static nonlinearity is also of unknown structure and is not required to be invertible. This is only supposed to be well approximated, within any subinterval belonging to the working interval, with a polynomial of unknown order and parameters. The order p and the parameters of the polynomial can vary from one subinterval to another. It turns out that the complexity of the identification problem lies in: (i) the fact that the internal signals i u and o u are not accessible to measurement (Fig. 1); (ii) the nonparametric and nonlinear nature of the system. Given the system nonparametric nature, the identification problem is presently dealt with by developing a two-stage frequency identification method, involving periodic inputs. First, a set of points of the nonlinearity is identified using simple experiments; the size of this set is arbitrarily chosen by the user. Then, the frequency responses of the two linear subsystems are estimated for a number of frequencies; in turn, this number can be made arbitrarily large. The frequency gain estimator design relies on input/output Fourier series expansions. Compared to previous works, all involved estimators are presently shown to be consistent. Furthermore, the input excitation signals are deterministic and no particular assumption is made on the external noise except for stationarity. This paper is organized as follows: the identification problem is formally described in Section 2; then, the identification of the nonlinearity is coped with in Section 3; the identification of the linear subsystems is dealt with in Section 4. Simulation examples are provided in Section 5 to illustrate the performances of the whole identification method. Figure 1. Wiener-Hammerstein System Model. Wiener-Hammerstein Models Identification A. Brouri * , INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 10, 2016 ISSN: 1998-0140 244