Discussion on: “Subspace-based Identification Algorithms for Hammerstein and Wiener Models”

Abstract

Hammerstein and Wiener models are special nonlinear models composed of a cascade connexion of a linear dynamical system and a static nonlinear map, the nonlinear part being respectively located at the input and at the output for Hammerstein and Wiener models. Among the methods proposed in the literature for identification of these models, both parts of the model are either identified simultaneously or separately. The method presented by Gomez and Baeyens is among the first ones. Thanks to the developement of subspace methods for identification of state-space models for multivariable systems [1,2], the simultaneous identification of the dynamic part and the nonlinear map of Hammerstein and Wiener models has been investigated since 1996 and the pioneering works by Verhaegen and Westwick [2,3]. The interest of the paper by Gomez and Baeyens is to present the identification problem of both Hammerstein and Wiener models in one paper. They clearly show that, as soon as an algorithm is available for estimating multivariable linear systems, it is easy to identify simultaneaously the nonlinear map of a Hammerstein or Wiener model, the nonlinear function being written as a sum of independent functions. To obtain those simple results, some assumptions must be done. For the Hammerstein model, the noise is assumed to be at the output of the system and can also affect the dynamical part. This framework is similar to Verhaegen and Westwick (1996) [3]. For the Wiener model, the inputs noise between the dynamic system and the nonlinear map; no noise is considered at the output of the system. The nonlinearity is assumed to be invertible and a decomposition of its inverse is considered. The framework is then different from Westwic and Verhaegen (1996) [4] where it is shown that identifiability issues can occur when the nonlinearity is directly considered. Notice that the idea of considering the inverse of the nonlinearity is not original and can be found in Greblicki (1994) [5]. In addition, the nonlinear functions are considered as square.