State space formulas for stable rational matrix solutions of a Leech problem

TL;DR

This paper presents a method to compute stable rational matrix solutions for the Leech problem using state space formulas, enabling explicit solutions based on given data functions.

Contribution

It introduces a procedure to derive state space formulas for solutions of the Leech problem from data functions' realizations, advancing control theory methods.

Findings

Provides explicit state space formulas for solutions

Enables computation of solutions from data functions' realizations

Ensures solutions are stable and rational

Abstract

Given stable rational matrix functions $G$ and $K$, a procedure is presented to compute a stable rational matrix solution $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and $\sup_{|z|\leq 1}\|X(z)\|\leq 1$. The solution is given in the form of a state space realization, where the matrices involved in this realization are computed from state space realizations of the data functions $G$ and $K$.