Causal Behavior of Switched Delay Systems as Multi-Mode Multi-Dimensional Systems*

This paper is about minimal causal models for delay systems, where the delay is switched between fixed values. Deterministic and random switching are discussed. General properties of multi-mode multidimensional (M3D) systems are described. We digress about the constraint “delay derivative > 1” that needs to be imposed on any delay system. Its relevance stems from the fact that a causal minimal representation of a switched delay system must be cast as an M3D system. The backwards Kolmogorov equation for a delay system where switching occurs at Poisson times (PM3D) is derived. The multidimensional character adds a new twist to this problem: the backward equation is given as a set of coupled PDE's involving unknown functions with different number of arguments.