Assignment of infinite structure to an open-loop system

Given a controllable pair of matrices (A,B) Rosenbrock’s theorem on assignment of open-loop zeros states that a matrix C may be chosen so that the transfer function matrix G(s)=C(sIn−A)−1B has prescribed zeros, that is, prescribed numerator polynomials in its McMillan form. In the same way, Rosenbrock’s theorem on assignment of closed-loop poles states that a matrix C may be chosen so that the McMillan form of the transfer function matrix of the closed-loop system has prescribed denominator polynomials. Following Rosenbrock’s ideas we can prove that given any pair of matrices (A,B), matrices C and D may be chosen so that the transfer function matrix of the system, G(s)=D+C(sIn−A)−1B, has prescribed infinite structure.