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The nu-gap metric was originally developed to evaluate robustness of a plant-controller feedback loop and it has many attractive properties. For example, performance of the closed loop changes continuously and stability is robust with respect to small perturbations of the plant (or the controller) in the nu-gap metric. In light of these properties, one can state that the nu-gap metric provides a good measure of distance between systems in a closed loop setting. This is very useful in model approximation, which is the focus of this technical note. The presented nu-gap approximation method is based on semidefinite programming and frequency response matching, which allows to extend the method to account for frequency-dependent weights in the objective function. The frequency-weighted extension is the major advantage of the presented method in comparison with other nu-gap model reduction methods. This extension is applied to approximation of controllers obtained by loop shaping and illustrated on a numerical example.