Proper rational matrix diophantine equations and the exact model matching problem

Relying on the theory behind the Smith-MacMillan form of a rational matrix at s=??, a necessary and sufficient condition is derived for the solvability of matrix Diophantine equations of the form: A (s) X (s) + B (s) Y (s) = M (s), where A (s), B (s) and M (s) are given proper rational matrices and X (s) and Y (s) are unknown proper rational matrices. It is shown that the above result can be used in order to resolve in a new illuminating way the exact model matching problem (EMMP). If a solution to EMMP exists then the family of all solutions is parametrised.