Constructive approximation of non-linear discrete-time systems

✍ Scribed by Sandberg, Irwin W.

• Publisher: John Wiley and Sons
• Year: 2000
• Tongue: English
• Weight: 117 KB
• Volume: 28
• Category: Article
• ISSN: 0098-9886
• DOI: 10.1002/(sici)1097-007x(200003/04)28:2<109::aid-cta96>3.0.co;2-2

✦ Synopsis

It is known that large classes of approximately-"nite-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 "nding the structure. Our setting is as follows. There is a large family G of causal time-invariant approximately-"nite-memory input-output maps G from a set S of real d-vector-valued discrete-time inputs (with d*1) to the set of 1-valued discrete-time outputs, with both the inputs and outputs de"ned on the non-negative integers Z > . We show that for each '0, any G3G can be uniformly approximated by a structure map H(G, ) ) to within tolerance , and we give analytical results and an example to illustrate how such a H(G, ) ) can be determined in principle.