The 2-block superoptimalAAK problem

Abstract

This paper presents a solution to a superoptimal version of the 2-blockAAK problem: Given a rational and antistable matrix functionR(s)=[R11(s)R12(s)] and a nonnegative integerk, find all superoptimal approximationsQ(s), with no more thank poles in the right-half complex plane, that minimize the supremum, over the imaginary axis, of the singular values of the error functionE(s)=[R11(s) R12(s)+Q(s)], with respect to lexicographic ordering. Conditions are given for which the superoptimal approximation is unique. In addition, ana priori upper bound on the MacMillan degree of the approximation is provided. The algorithm may be stopped after minimizing a given number of the singular values. This premature termination carries with it a predictable reduction in the MacMillan degree of the approximation. The algorithm only requires standard linear algebraic computations, and is therefore easily implemented.