On frequency weighted balanced truncation: A constructive counterexample to Enns´ conjecture

The concept of frequency weighted balancing, as proposed by Enns, is a generalization of internally balanced model truncation. Internally balanced truncation is simple to apply and additionally attractive because of the existence of an upper _ff_œ error bound that is a function of the neglected Hankel singular values. A conjecture on the generalization of this upper error bound for the case of frequency weighted balanced truncation was formulated by Enns, but the proof has not been found. In this paper, Enns´ conjecture is refuted and it is shown that there does not exist a frequency weighted upper error bound that depends only on neglected frequency weighted Hankel singular values. It is explained that this is due to cross terms which appear in the frequency weighted error bound. However, these cross terms are inherent in the frequency weighted balancing technique proposed by Enns.