NONLINEAR BLACK-BOX MODELS IN SYSTEM-IDENTIFICATION - MATHEMATICAL FOUNDATIONS

We discuss several aspects of the mathematical foundations of the nonl inear 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 f lexible is the approximation), and the stochastic error (which is prop ortional to the number of parameters). A consequence of this trade-off is the simple fact that a good approximation technique can be the bas is of a good identification algorithm. From this point of view, we con sider different approximation methods, and pay special attention to sp atially adaptive approximants. We introduce wavelet and 'neuron' appro ximations, and show that they are spatially adaptive. Then we apply th e acquired approximation experience to estimation problems. Finally, w e consider some implications of these theoretical developments for the practically implemented versions of the 'spatially adaptive' algorith ms.