Optimizing simultaneously over the numerator and denominator polynomials in the Youla-Kucˇera parametrization

Traditionally, when approaching controller design with the Youla-Kucˇera parametrization of all stabilizing controllers, the denominator of the rational parameter is fixed to a given stable polynomial, and optimization is carried out over the numerator polynomial. In this note, we revisit this design technique, allowing to optimize simultaneously over the numerator and denominator polynomials. Stability of the denominator polynomial, as well as fixed-order controller design with H_∞ performance are ensured via the notion of a central polynomial and linear matrix inequality (LMI) conditions for polynomial positivity.