Frequency weighted model reduction via optimal projection

The problem of order reduction with frequency weighting is considered. Necessary conditions characterizing the reduced-order model which minimizes a quadratic error criterion in the frequency domain, with either input or output frequency weighting, are given. The problem is transformed to the time domain where the weighting matrix is expressed as a shaping filter that generates colored noise at the input. When the weighting matrix is singular at infinity, the reduced-order model has a direct transmission term even if the original system is strictly proper. This additional degree of freedom can be used to obtain better approximation over a prespecified frequency range