Extended system matrices-Transfer functions and system equivalence

We introduce an extension of the usual notion of an input/output map and for linear constant-parameter systems an extended transfer function, that has a (matrix) polynomial- or more generally a differential/difference operator inverse, containing as a submatrix Rosenbrock\´s system matrix. A new class of system equivalence - "maximally strict system equivalence" (m.s.s.e.) is introduced as a necessary and sufficient condition for two minimal systems to have the same transfer function. We give an outline of the extension of these results to non-minimal systems, where results are only known for state-space systems in this generality. In the conclusion we discuss the relation of the extended system matrix to other deterministic and stochastic problems.