A generalized Bezout equation in output feedback design

Assigning a desired closed-loop structure using output feedback is addressed from the point of view of polynomial equations. A necessary and sufficient condition is the existence of a solution with some constant submatrices to a generalized Bezout equation (GBE) the right-hand side of which contains the desired closed-loop polynomials. Conditions are given in terms of a state feedback with some restrictions, in terms of coupled Diophantine equations, and in terms of a bilinear Diophantine equation. It is shown that the solution of the outer-feedback problem is equivalent to the solution of these polynomial equations.