Nonlinear black-box modeling in system identification: a unified overview

A nonlinear black-box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area, with structures based on neural networks, radial basis networks, waveiet networks and hinging hyperplanes, as well as wavelet-transform-based methods and models based on fuzzy sets and fuzzy rules. This paper describes all these approaches in a common framework, from a user’s perspective. It focuses on what are the common features in the different approaches, the choices that have to be made and what considerations are relevant for a successful system-identification application of these techniques. It is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space. These mappings are discussed separately. The latter mapping is usually formed as a basis function expansion. The basis functions are typically formed from one simple scalar function, which is modified in terms of scale and location. The expansion from the scalar argument to the regressor space is achieved by a radialor a ridge-type approach. Basic techniques for estimating the parameters in the structures are criterion minimization, as well as two-step procedures, where first the relevant basis functions are determined, using data, and then a linear least-squares step to determine the coordinates of the function approximation. A particular problem is to deal with the large number of potentially necessary parameters. This is handled by making the number of ‘used’ parameters considerably less than the number of ‘offered’ parameters, by regularization, shrinking, pruning or regressor selection. *Received 19 October 1994; revised 21 March 1995; received in final form 23 June 1995. The original version of this paper was presented as an invited survey paper at the 10th IFAC Symposium on System Identification, which was held in Copenhagen, Denmark during 4-6 July 1994. The Published Proceedings of this IFAC meeting may be ordered from: Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 IGB, U.K. This paper was recommended for publication in revised form by Guest Editors Torsten SSderstrGm and Karl Johan Astrom. Corresponding author: Dr Jonas Sjoberg. Tel. + 41 1 632 3111; Fax +41 1632 1222; E-mail [email protected]. t Department of Electrical Engineering, Linkoping University, S-581 83 LinkSping, Sweden. SIRISA/INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France. § INSA. 20 avenue des Buttes de Coesmes. 35045 Rennes Cedex, France.