Eigenvalue and generalized eigenvalue formulations for Hankel norm reduction directly from polynomial data

Abstract

Using the results of Adamjan, Arov and Krein [1], we develop a new algorithm for computing the optimal Hankel-norm approximants for SISO continuous-time systems. Given a rational transfer function f(s) = n(s)/d(s), we can construct optimal Hankel-norm approximants of all orders from the eigenvectors of a certain matrix M. The specific feature of this new algorithm is that the matrix M has the form 1/2(X2 -1Y2 - X1 -1Y1), where X1 and X2 are rearranged versions of the Hurwitz matrix of d(s), and Y1 and Y2 are obtained by arranging the coefficients of n(s) in a certain pattern. Further, M is a certain representation of the Hankel operator induced by f. Finally, if f(s) has lightly damped poles, the computation of M may be ill-conditioned, in which case a generalized eigenvalue formulation with coefficient matrices Xi and Yi is proposed.