A necessary and sufficient condition for feedback stabilization in a factorial ring

The necessity for a coprime fractional representation of a plant, in order to completely parametrize all stabilizing compensators by the Youla theory, is examined. It is shown that the coprimeness is necessary for systems in a factorial ring, a class inclusive of multidimensional scalar systems. Some observations on partial-state transfer functions in the feedback configuration are found to be useful in showing the necessity.