Model structure selection for nonlinear system identification using feedforward neural networks IVAN PETROVIĆ, MATO BAOTIĆ, NEDJELJKO PERIĆ University of Zagreb, Faculty of Electrical Engineering and Computing, Department of Control and Computer Engineering in Automation, Unska 3, HR-10 000 Zagreb, Croatia, E-mail: ivan.petrovic@fer.hr Abstract - A nonlinear black-box structure for a dynamic system is a model structure that is prepared to describe virtually any nonlinear dynamics. The majority of nonlinear models based on neural networks are of the black-box structure. A nonlinear system can be nonlinear in many different ways, thus the nonlinear black-box model structure must be very flexible. This means that it must have many parameters. A model offering many parameters usually creates problems, and the variance contribution to the error might be high. For a particular identification problem, only a subset of the parameters might be necessary and the main topic in nonlinear system identification is how to select a model structure that describes the system dynamics with the minimum number of parameters. This paper discusses nonlinear input-output models that are suitable for implementation of feedforward neural networks. The proposed model structures were tested and compared using the identification procedure of a pH process. The results indicated that it would be worthwhile using the simplest model structure that can satisfactorily represent the investigated process. 1. INTRODUCTION One of the goals of system identification is to provide a good predictor model [1], [2]. This means that the model output () yk represents the expected value of the true system output y(k): () () () yk yk k = E ϕ , (1) where ϕ(k) is the regression vector, which contains information from the system available at time k. Let us now consider the parametrized candidate model structure f N (ϕ(k),Θ), where Θ is the vector of model parameters. The goal of parameter estimation is to find an estimate Θ such that the system can be described according to yk f k k N () ( ( ), ) () = + ϕ Θ ξ , (2) where ξ(k) is a white noise sequence whose variance is minimized with respect to a given criterion. To propose a candidate model like (2), there are two design questions which have to be addressed: 1) Which variables, constructed from the observed past data, should be chosen as regressors, i.e., as components of the regression vector ϕ(k)? 2) How should the nonlinear map f N (⋅,⋅) from the regressor space to the output space be chosen? The first question also arises in linear black-box identification. However, the linear model is completely specified by the choice of the regressor ϕ(k), because f N (⋅,⋅) is linear in ϕ(k). For nonlinear models, however, there are many possible choices for f N (⋅,⋅). The brute-force approach would be to plug the chosen ϕ(k) into a general black-box model, e.g., neural network. This would provide the most general model structure possible with this particular regressor. The latter approach is the most often employed one in nonlinear system identification using neural networks [3], [4]. It may be advantageous in some cases to consider restrictions of the general black-box model structure obtained by placing constraints on the structure of the nonlinear map f N (⋅,⋅). The restricted model obtained is less flexible than the