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

Abstract

For the strictly positive case (the suboptimal case), given stable rational matrix functions G and K, the set of all H ∞ solutions X to the Leech problem associated with G and K, that is, G(z)X(z) = K(z) and sup∣z∣≤1X||(z)|| ≤1sup∣z∣≤1||X(z)|| ≤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.