Feedback stabilization over commutative rings with no right-/left-coprime factorizations

V. Anantharam (1985) showed the existence of a model in which some stabilizable plants do not have its right-/left-coprime factorizations. In this paper, we give a condition of the non-existence of the right/left-coprime factorizations of stabilizable plants as a generalization of Anantharam´s result. As examples of models which satisfy the condition, we present two models; one is Anantharam´s example and the other the discrete finite-time delay system which does not have a unit delay. We illustrate the construction of stabilizing controllers of stabilizable single-input single-output plants of such model. The method presented is an application of the result of the necessary and sufficient condition of the stabilizability over commutative rings, which has been developed by K. Mori and K. Abe (1998)