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 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⩾1) to the set of ℝ-valued discrete-time outputs, with both the inputs and outputs defined on the non-negative integers +. We show that for each ϵ>0, any Gϵ can be uniformly approximated by a structure map H(G, ·) to within tolerance ϵ, and we give analytical results and an example to illustrate how such a H(G, ·) can be determined in principle. Copyright © 2000 John Wiley & Sons, Ltd.

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