Recursive Identification of Wiener Systems with Two–segment Polynomial Nonlinearities

For the subclasses of nonlinear dynamic systems which can be considered as block oriented systems [8] there exist several identification methods using topologically identical models. One of the simplest nonlinear models of this category is the so-called Wiener model consisting of one linear dynamic block and one nonlinear static block. The Wiener models appear in many engineering applications [6], [20], [22], [24] and more approaches were proposed for their identification, eg, [1–4], [7], [9], [10–13], [15, 16], [19], [21], [23], [25, 26], [29]. In the parametric form, the linear dynamic blocks of Wiener models are typically described by their transfer functions or in some cases by the FIR models. The characteristics of nonlinear blocks are often approximated by polynomials of proper degree. To obtain an accurate polynomial fit of given static nonlinearity may cause problems in some situations. With lower-degree polynomials, the approximation error can be quite notable, while, with high degrees, the number of parameters increases [6]. This is the case when the characteristics are strongly asymmetric [14], eg, their outputs differ significantly for the positive and negative inputs, respectively, and only the polynomials of higher degree can approximate the nonlinear block characteristics adequately. Therefore it may be reasonable, for such considerably asymmetric nonlinear characteristics, to use descriptions with two distinct maps, ie, two-segment polynomial descriptions. In this paper a special form of the Wiener model, based on a decomposition technique [27], is considered where the nonlinear static block is characterized by a twosegment polynomial approximation [28]. This model is linear-in-parameters and is used in a recursive estimation framework. The proposed algorithm is a direct application of the known recursive least squares method [17, 18] extended with the estimation of internal variables. It enables the estimation of both the parameters of the linear block transfer function and the coefficients of the polynomials approximating nonlinear characteristics using the system inputs, outputs and estimated internal variables. Simulation studies of Wiener system recursive identification are included.