A polynomial characterization of (A,B)-invariant and reachability subspaces

Based on the state space model of P. Fuhrmann, a link is laid between the geometric approach to linear system theory, as developed by W. M. Wonham and A. S. Morse, and the approach based on polynomial matrices. In particular polynomial characterizations of $(A,B)$-invariant and reachability subspaces are given. These characterizations are used to prove the equivalence of the disturbance decoupling problem and the exact model matching problem and also to connect the polynomial matrix and the geometric approach to the construction of observers. Finally, constructive procedures and conditions are given for computing the supremal $(A,B)$-invariant subspace and reachability space and for checking the solvability of the exact model matching problem.