Article,
All optimal Hankel-norm approximations of linear multivariable systems and their L , ∞ -error bounds\dag
International Journal of Control, 39 (6): 1115--1193 (Mar 27, 2007)
DOI: 10.1080/00207178408933239
Abstract
The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by ?(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions ?(s) + F(s) that minimize where ?(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.
