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- Juditsky, A., et al.
(författare)
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Nonlinear black-box models in system identification: Mathematical foundations
- 1995
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Ingår i: Automatica. - Linköping : Elsevier BV. - 0005-1098 .- 1873-2836. ; 31:12, s. 1725-1750
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Tidskriftsartikel (refereegranskat)abstract
- We discuss several aspects of the mathematical foundations of the nonlinear black-box identification problem. We shall see that the quality of the identification procedure is always a result of a certain trade-off between the expressive power of the model we try to identify (the larger the number of parameters used to describe the model, the more flexible is the approximation), and the stochastic error (which is proportional to the number of parameters). A consequence of this trade-off is the simple fact that a good approximation technique can be the basis of a good identification algorithm. From this point of view, we consider different approximation methods, and pay special attention to spatially adaptive approximants. We introduce wavelet and 'neuron' approximations, and show that they are spatially adaptive. Then we apply the acquired approximation experience to estimation problems. Finally, we consider some implications of these theoretical developments for the practically implemented versions of the 'spatially adaptive' algorithms. Copyright © 1995 Elsevier Science Ltd All rights reserved.
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- Sjöberg, Jonas, et al.
(författare)
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Nonlinear black-box modeling in system identification: A unified overview
- 1995
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Ingår i: Automatica. - Linköping : Elsevier BV. - 0005-1098 .- 1873-2836. ; 31:12, s. 1691-1724
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Tidskriftsartikel (refereegranskat)abstract
- 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, wavelet 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 radial- or 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. Copyright © 1995 Elsevier Science Ltd All rights reserved.
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