Transfer function conditions for stabilizability

For the linear time invariant system[ {yc(s) yₘ(s)}] = [{G(s)&H(s) M(s)&N(s)} ] [ {uc(s) uᵣ(s)}]a necessary and sufficient condition based on the proper rational transfer function matrices\{G, H, M, N\}is given for the existence of a proper stabilizing controlleruc(s) = C(s)yₘ(s). The condition states that the McMillan degrees ofM⁺(s)and[{G⁺(s)&H⁺(s) M{+}(s)&N{+}(s)}]must be equal, where\{G⁺(s), H⁺(s), M⁺(s), N⁺(s)\}represents the unstable components of the partial fraction expansions of\{G(s), H(s), M(s), N(s)\}.