On the realization theory of polynomial matrices and the algebraic structure of pure generalized state space systems

We review the realization theory of polynomial (transfer function) matrices via “pure” generalized state space models. The concept of an irreducible at infinity generalized state space realization of a polynomial matrix is defined and the mechanism of the “cancellations” of “decoupling zeros at infinity” is closely examined. The difference between the concepts of irreducibility and minimality of generalized state space realizations of polynomial (transfer function) matrices is pointed out and the associated concepts of dynamic and non dynamic variables appearing in generalized state space realizations are also examined. Finally the isomorphism between the zeros at infinity of the “infinite pole pencil” and the Rosenbrock system matrix of an irreducible at infinity generalized state space realization of a polynomial matrix and the pole and zero structure at infinity of such a polynomial matrix is examined.