Transmission networks

Proposed herein is a new method for solving arbitrary networks of nonuniform transmission ladders and lines. Two cases are considered, the first occurring when all circuit elements are purely resistive and in general non‐linear and the second occurring when they are linear but in general have both resistive and reactive components. The first case arises for example when the DC operating point of a non‐linear interconnection network is to be determined, and the second arises for example when the transient behaviour of a network of linear, possibly nonuniform, RLCG transmission lines and ladders is desired. Moreover, our method eliminates the voltages at internal nodes of a ladder and only treats nodes at which three or more ladders or lines meet or just one ladder or line terminates. The internal node voltages are obtained at the end of our procedure with very little additional computation. This provides a computational advantage, which may save orders of magnitude in computation time. A similar facility accrues to cascades of distributed lines and ladders. Copyright © 1999 John Wiley & Sons, Ltd.     

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⩾1) to the set of ℝ‐valued discrete‐time outputs, with both the inputs and outputs defined on the non‐negative integers 𝒵₊. We show that for each ϵ>0, any Gϵ𝒢 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.