The internal model principle states that all robustly regulating controllers must contain a suitably reduplicated internal model of the signal to be regulated. Using frequency domain methods, we show that the number of the copies may be reduced if the class of perturbations in the problem is restricted. We present a two-stage design procedure for a simple controller containing a reduced-order internal model achieving robust regulation. The results are illustrated with an example of a five-tank laboratory process where a restricted class of perturbations arises naturally.