Proper stable transfer matrix factorizations and internal system descriptions

The exact relations between coprime proper stable factorizations of
P(s)
and coprime polynomial matrix factorizations are derived, and they directly lead to relations with internal descriptions of the plant in differential operator or state-space form. It is shown that obtaining any right or left proper stable coprime factorization is equivalent to state-feedback stabilization or to designing a full-order full-state observer, respectively. Solving the Diophantine equation is shown to be equivalent to designing a full or reduced-order observer of a linear functional of the state and to designing a stable inverse system; and this suggests new computational methods to solve the Diophantine.