On Frequency Weighted Balanced Truncation: Hankel Singular Values and Error Bounds

Tlie concept of frequencj iveighted balancing proposed b j Enns is a generalisation of internally balanced model rrur~cation irhich is sin~ple to app!, and additionally attracti~e because of the existence o fan upper H , error bolrrid rlirrt is a function of the neglected Hankelsingular values. However, a generalisatiol~ of this error bound basedon the frequency ireiglited Haf~kelsingular values has not been reported. 111 this paper, it is shown that there does nor exist a frequency weighted upper error borrrzd that depends o n k on the r~eglected jkequency iveighred Haf~ke l singular values. Based on tl~is result, it is shoien that trrtncarion of the states corresponding to the loirestf,equency weighted Hankelsingular values does not always yield the lowest approximation error. It is explained that this is due to cross-terms that appear in the frequency 117eigIited error bound and rnake the discussion on stability of the reduced order model more cornplex. These cross-terms are inherent in the frequency iveighted balancing technique proposed by Enns.