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

A parameterization of stabilizing controllers over commutative rings

“…Directly from this, we have (7). The remaining relations (8) and (9) are obtained directly from (1).…”

“…They have introduced the notion of the generalized elementary factor, which is a generalization of the elementary factor introduced by Sule [3], and have given the necessary and sufficient condition of the feedback stabilizability.Since the stabilizing controllers are not unique in general, the choice of the stabilizing controllers is important for the resulting closed loop. In the classical case, that is, in the case where there exist the right-/left-coprime factorizations of the given plant, the stabilizing controllers can be parameterized by the method called "Youla-Kučera parameterization" [1,2,7,8]. However, it is known that there exist models in which some stabilizable transfer matrices do not have their right-/left-coprime factorizations [9].…”

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

“…It is obvious that a ∈ Λ pI1 . On the other hand, b is a member of Λ pI1 since the ratio aa ′−1 in (1) can be rewritten as In particular, if the plant is of the single-input single-output and if A is a unique factorization domain, then Raman and Liu in [6] gave the result that the plant is stabilizable if and only if it has a doubly coprime factorization. We now give two examples which satisfy the condition of Proposition 3.1.…”