Two results on optimal modeling with differential-delay systems

Let be a fixed causal filter. The first result of this paper shows that for any positive integer and any delay time there is an "optimal model" for out of the class of differential-delay systems of order (and utilizing delay ). The second and main result shows that unless corresponds exactly to a purely differential system of order then there always exists some delay time and corresponding differential-delay system of order that is a better model for than even the optimal purely differential system of order .