Orthonormal Basis and Radial Basis Functions in Modeling and Identification of Nonlinear Block-Oriented Systems

Abstract

Nonlinear block-oriented systems, including the Hammerstein, Wiener and feedbacknonlinear systems have attracted considerable research interest both from the industrial and academic environments (Bai, 1998), (Greblicki, 1989), (Latawiec, 2004), (Latawiec et al., 2003), (Latawiec et al., 2004), (Pearson & Pottman, 2000). It is well known that orthonormal basis functions (OBF) (Bokor et al., 1999) have proved to be useful in identification and control of dynamical systems, including nonlinear blockoriented systems (Gomez & Baeyens, 2004), (Latawiec, 2004), (Latawiec et al., 2003), (Latawiec et al., 2006), (Latawiec et al., 2004), (Stanislawski et al., 2006). In particular, an inverse OBF (IOBF) modeling approach has been effective in identification of a linear dynamic part of the feedback-nonlinear and Hammerstein systems (Latawiec, 2004), (Latawiec et al., 2004). On the other hand, regular OBF (ROBF) modeling approach has proved to be useful in identification of the Wiener system. The approaches provide the separability in estimation of linear and nonlinear submodels (Latawiec et al., 2004), thus eliminating the bilinearity issue detrimentally affecting e.g. the ARX-based modeling schemes (Latawiec, 2004), (Latawiec et al., 2003), (Latawiec et al., 2006), (Latawiec et al., 2004). The IOBF modeling approach is continued to be efficiently used here to model a linear dynamic part of the feedback-nonlinear and Hammerstein systems and regular OBF modeling approach is used to model a linear part of the Wiener system. The problem of modeling of a nonlinear static part of the nonlinear block-oriented system can be classically tackled using e.g. the polynomial expansion (Latawiec, 2004), (Latawiec et al., 2004) or (cubic) spline functions. Recently, a radial basis function network (RBFN) has been used to model a nonlinear static part of the Hammerstein and feedback-nonlinear systems and a very good identification performance has been obtained (Hachino et al., 2004), (Stanislawski, 2007), (Stanislawski et al., 2007). The concept is extended here to cover the Wiener system. This paper presents a new strategy for nonlinear block-oriented system identification, which is a combination of OBF modeling for a linear dynamic part and RBFN modeling for a nonlinear static element. The effective OBF approach is finally coupled with the RBFN modeling concept, giving rise to the introduction of a powerful method for identification of the nonlinear block-oriented system.