Frequency-weighted optimal Hankel-norm approximation of stable transfer functions

The present paper presents a method of frequency shaping the error obtained by performing an optimal Hankel-norm approximation of a scalar, finite-dimensional, linear, time-invariant system. The optimal Hankel-norm approximation procedure finds a transfer function (or transfer function matrix) of prescribed order which approximates a given transfer function (or transfer function matrix) of greater order 11-31. One motivation for frequency weighting comes from the desire to implement a reduced-order approximating controller within a closed-loop control system. Suppose an LQG designed series compensator is to be used in a control system implementation. The compensator will have the same dimension as the plant model. For simplicity, it is gain, the detailed shape of the approximating compensator is not so important, however around the unity gain crossover frequency, it is desirable to resume accurate approximation. We show here how to modify the approximation method originally developed by Adamjan, Arov and Krein [I] to allow for frequency weighting. The means for introducing frequency weighting while preserving the closed form solvability of the optimal Hankel-norm approximation problem is not immediately apparent. A technique for doing this is described in the paper.