The problem of order reduction of a linear, time-invariant system with frequency weighting is considered. The resulting reduced-order model minimizes a quadratic error criterion in the frequency domain with either input or output frequency weighting. 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. The additional degree of freedom enables better approximation over a specified frequency range.
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