Finite Representations of Block Hankel Operators and Balanced Realizations

Abstract

The block Hankel operator Гg corresponding to a rational matrix function g, analytic in D and of McMillan degree d, has rank d. Its non-trivial part, acting from (Ker Гg)⊥ to Range Гg, can therefore in principle be represented by a d × d matrix with respect to a pair of orthonormal bases. We show how to obtain such a representation using polynomial methods: that is, we work with the coefficients of the numerator and denominator polynomials and do not require the solution of any polynomial equations. We use this representation to derive an algorithm for the construction of balanced realizations of rational transfer functions.