On frequency weighted balanced truncation: Hankel singular values and error bounds

The concept of frequency weighted balancing proposed by Enns is a generalisation of internally balanced model truncation which is simple to apply and additionally attractive because of the existence of an upper H_∞ error bound that is a function of the neglected Hankel singular values. However, a generalisation of this error bound based on the frequency weighted Hankel singular values has not been reported. In this paper, it is shown that there does not exist a frequency weighted upper error bound that depends only on the neglected frequency weighted Hankel singular values. Based on this result, it is shown that truncation of the states corresponding to the lowest frequency weighted Hankel singular values does not always yield the lowest approximation error. It is explained that this is due to cross-terms that appear in the frequency weighted error bound and make the discussion on stability of the reduced order model more complex. These cross-terms are inherent in the frequency weighted balancing technique proposed by Enns.