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.