State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions

For the strictly positive case (the suboptimal case), given stable rational matrix functions $G$ and $K$, the set of all $H^{\mathrm{\infty }}$ solutions $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and ${\sup }_{\mathrm{\mid }z\mathrm{\mid }\le 1}\mathrm{\parallel }X(z)\mathrm{\parallel }\le 1$, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions $G$ and $K$. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.