A State-Space Algorithm for the Solution of the 2-Block Superoptimal Distance Problem

Abstract

A state-space algorithm for computing the solution of the 2-block superoptimal distance problem (SODP) is presented. Given a rational and antistable matrix function $R(s) = [R_{11} (s)R_{12} (s)]$, find all stable approximations $Q(s)$ that lexicographically minimize the singular values of the error function $E(s) = [R_{11} (s)R_{12} (s) + Q(s)]$. Conditions are given for which the superoptimal approximation is unique. In addition, an a priori upper bound on the MacMillan degree of the approximation is given. The algorithm may be stopped after minimizing a given number of singular values. This premature termination of the algorithm carries with it an expected saving in the computational effort and a predictable reduction in the MacMillan degree of the approximation. The algorithm only requires standard linear algebraic computations and is, therefore, easily implemented.