Minimal Differential Difference Realizations of Delay Differential, Differential Difference, and Neutral Delay Systems

Delay-Differential Equations (DDEs) are often used to represent control of and over large networks. However, the presence of delay makes the problems of analysis and control of such networks challenging. Recently, Differential Difference Equations (DDFs) have been proposed as a modelling framework which allows us to more efficiently represent the low-dimensional nature of delayed channels in a network or large-scale delayed system. Unfortunately, however, the standard conversion formulae from DDE to DDF do not account for this low-dimensional structure - hence any efficient DDF representation of a large delayed network or system must be hand-crafted. In this paper, we propose an algorithm for constructing DDF realizations of both DDE and DDF systems wherein the dimension of the delayed channels has been minimized. Furthermore, we provide a convenient PIETOOLS implementation of these algorithms and show that the algorithm significantly reduces the complexity of the model for several illustrative examples, including Neutral Delay Systems (NDSs).