On frequency weighted balanced truncation: A constructive counterexample to Enns' conjecture
2001 European Control Conference (ECC)
However, the generalization of the a priori balanced model reduction upper error bound in terms of the so-called frequency weighted Hankel singular values, has not been found The concept of weighted as proposed yet [I, 3, 7, 14, 15, 171. In this paper we explain that cer-Enns, is a generalization of internally balanced model truncatain types of generalizations of the upper bound, which we will tion. Internally balanced truncation is simple to apply and adcall Enns, Conjecture 131 in Section 2,
... annot serve as an upditionally attractive because of the existence of an upper H, per error bound. It can be shown that this is due to a cross error bound that is a function of the neglected Hankel singular term that can become unbounded in terms of the frequency values. A conjecture on the generalization of this upper error weighted Hankel singular values. This cross term is inherent in bound for the case of weighted frequency balancing. We give numerical was by Enns' but the proof has been found' In amples to Enns' conjecture using a constructive algorithm that this paper, Enns' conjecture is refuted and it is shown that there generates counterexamples. does not exist a frequency weighted upper error bound that depends only on neglected frequency weighted Hankel singular This paper is organized as follows. Frequency weighted balvalues. It is explained that this is due to cross terms which ap-ancing and Enns' conjecture are reviewed in Section 2. The pear in the frequency weighted error bound. However, these conjecture is refuted in Section 3. This paper is a companion cross terms are inherent in the frequency weighted balancing paper of  . technique proposed by Enns.