System equivalence for AR-systems over rings - with an application to delay-differential systems

In this paper the notion of autoregressive systems over an integral domain R is introduced, as a generalization of AR-systems over the rings IR[s] and IR[s, s(-1)]. The interpretation of the dynamics represented by a matrix over R is fixed by the choice of a module M over R, consisting of all time-trajectories under consideration. In this setup the problem of system equivalence is studied: when do two different AR-representations characterize the same behavior? This problem is solved using a ring extension of R, that explicitly depends on the choice of the module M of all time-trajectories. In this way the usual divisibility conditions on the system defining matrices can be recovered. The results apply to the class of delay-differential systems with (in)commensurable delays. In this particular application, the ring extension of R is characterized explicitly.