State-space algorithm for the solution of the 2-block superoptimal distance problem

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) = [R₁₁(s)R₁₂(s)], find all stable approximations Q(s) that lexicographically minimize the singular values of the error function E(s) = [R₁₁(s)R₁₂(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.