Constructive approximation of non-linear discrete-time systems

It is known that large classes of approximately-finite-memory maps can be uniformly approximated arbitrarily well by the maps of certain non-linear structures. As an application, it was proved that time-delay networks can be used to uniformly approximate arbitrarily well the members of a large class of causal nonlinear dynamic discrete-time input-output maps. However, the proof is non-constructive and provides no information concerning the determination of a structure that corresponds to a prescribed bound on the approximation error. Here we give some general results concerning the problem of finding the structure. Our setting is as follows. There is a large family G of causal time-invariant approximately-finite-memory input-output maps G from a set S of real d-vector-valued discrete-time inputs (with d greater than or equal to 1) to the set of [W-valued discrete-time outputs, with both the inputs and outputs defined on the non-negative integers L+. We show that for each epsilon > 0, any G is an element of G can be uniformly approximated by a structure map H(G,.) to within tolerance epsilon, and we give analytical results and an example to illustrate how such a H(G, .) can be determined in principle. Copyright (C) 2000 John Wiley & Sons, Ltd.