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

TL;DR

This paper provides a state space parametrization of all solutions to a suboptimal rational Leech problem using linear fractional transformations, with coefficients derived from state space realizations of the data functions.

Contribution

It introduces a new parametrization method for all solutions to the suboptimal Leech problem using state space formulas and Riccati equations.

Findings

Explicit state space formulas for solution parametrization

Solutions expressed via linear fractional transformations

Matrices computed from data function realizations

Abstract

For the strictly positive case (the suboptimal case), given stable rational matrix functions $G$ and $K$, the set of all $H^\infty$ solutions $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and $\sup_{|z|\leq 1}\|X(z)\|\leq 1$, 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.