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 Rs] and Rs;s ?1 ]. Unlike the behavioral approach, the signal space is considered as a module M over the ring R. In this setup the problem of system equivalence is studied: when do two diierent AR-representations characterize the same behavior? This problem is solved using a ring extension of R, that explicitly depends on the choice of the signal space M. In this way the usual divisibility conditions on the system-deening matrices can be recovered. The results apply to the class of delay-diierential systems with (in)commensurable delays. In this particular application, the ring extension of R is characterized explicitly.