Some Remarks on Adaptive Neuro-Fuzzy Systems

In this brief note we make three remarks concerning adaptive implementations of neural networks and fuzzy systems. First, we bring to the readers attention the fact that the potential power of these systems as function approximators is lost when, as done in recently published work, the adjustable parameters are only the linear combination weights of the basis functions. Second, we show that the stability analysis in those papers in any way uses properties particular to neural nets or fuzzy systems, and follows inmediately from well established results in adaptive systems theory. The second fact is well known to people familiar with adaptive systems theory, but not necessarily so to the neuro-fuzzy community. On the other hand, the opposite seems to be the case for the rst remark. Finally, we present a simple version of a result on adaptive stabilization of nonlinearly parametrized nonlinear systems which might be useful for the stability analysis of adaptive neuro-fuzzy systems. This result, though well known in the Russian literature for a long time, has apparently been overlooked in \western" publications. 1 Bounds on function approximation The recent massive availability of data {either from extensive simulations or experiments{ has renewed the interest of some control theorist on non-model-based techniques for identiication and control of coarsely known systems. In particular, attention has centered on methods that approximate nonlinear functions with some basis functions. It has recently been established 7] that a necessary and suucient condition for a basis to provide a \good approximation" for an arbitrary continuous function f(x) : R n ! R is that the basis is non polynomial. Hence the particular choice of one basis instead of the other becomes a matter only of taste, at least in what concerns its approximation capabilities. The key question that arises inmediately when talking about nonlinear function approximation is the \size of the basis". Estimates on the required number of elements for a given approximation