Optimal systems root loci: Relation to the McMillan structure of the open-loop system

This note shows that the orders of the Butterworth patterns formed by the asymptotes of optimal closed-loop poles of a time-invariant linear feedback system, as the weight of the input in the performance criterion approaches zero, always correspond to twice the McMillan orders of the zeros at infinity of the open-loop transfer matrix. Furthermore, bounds are given, in terms of the Newton polygons, on the orders of the Butterworth patterns formed by the root loci at finite pole-zero locations. Finally, some examples are presented to show that in the latter case the branching behavior need not always correspond to the McMillan structure of the open-loop system.