A state-space solution of bilateral Diophantine equations over

This paper studies a class of real-rational matrix bilateral Diophantine equations (BDE) arising in numerous control problems. A necessary and sufficient solvability condition is derived in terms of state-space realizations of rational matrices involved in the equation. This condition is given in terms of a constrained matrix Sylvester equation and is numerically tractable. An explicit state-space parametrization of all solutions is also derived. This parameterization effectively includes two parameters: one is a “standard” RH ∞  parameter and another one arises if the Sylvester equation is non-uniquely solvable. A condition, in terms of zeros of rational matrices involved in the BDE, is found under which the Sylvester equation has a unique solution and, hence, the parametrization is affine in a single RH ∞  parameter.