Given a plant transfer matrix as a fraction of two polynomial matrices, we present a doubly coprime factorisation identity over the proper stable rational matrices of the plant transfer matrix, and apply it to compute a controller transfer matrix in H-2 optimal control of discrete-time systems. If there are no uncontrollable modes of the plant transfer matrix, the controller is given by a fraction of two polynomial matrices, where the order of the denominator is less than or equal to the McMillan degree of the plant.