State Space Formulas for a Suboptimal Rational Leech Problem I: Maximum Entropy Solution

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 \({\|X\|_\infty={\rm sup}_{|z| \leq 1}\| X(z )\| \leq 1}\), 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.