State space formulas for a suboptimal rational Leech problem I: Maximum entropy solution

TL;DR

This paper derives explicit state space formulas for the maximum entropy solution to a rational Leech problem, demonstrating its stability, rationality, and providing entropy computation methods.

Contribution

It introduces explicit state space formulas for the maximum entropy solution to the Leech problem, including stability, rationality, and entropy calculation.

Findings

The maximum entropy solution is a stable rational matrix function.

An explicit state space realization for the solution is provided.

The entropy of the solution can be explicitly computed.

Abstract

For the strictly positive case (the suboptimal case) the maximum entropy solution $X$ to the Leech problem $G(z)X(z)=K(z)$ and $\|X\|_\infty=\sup_{|z|\leq 1}\|X(z)\|\leq 1$, with $G$ and $K$ stable rational matrix functions, is proved to be a stable rational matrix function. An explicit state space realization for $X$ is given, and $\|X\|_\infty$ turns out to be strictly less than one. The matrices involved in this realization are computed from the matrices appearing in a state space realization of the data functions $G$ and $K$. A formula for the entropy of $X$ is also given.