Geometric Design of Lead/Lag Compensators Meeting a Hinf Specification

This note deals with the problem of characterizing a class of second-order three-parameter compensators [including lead/lag compensators and proportional-integral- derivative (PID) controllers] satisfying given Hinfin closed-loop specification. Design characterizations of similar form as in the work on PID control, are derived for a larger class of compensators using simple geometric considerations. The linear fractional transformation is used to give the transfer function the required shape. Specifically it is shown that, fixing the value of one parameter: i) the region of the plane defined by the other two parameters where the considered Hinfin constraint is satisfied, consists of the union of disjoint convex sets whose number can be bounded by means of the pie-cutting formula, and ii) the closed-loop pole distribution can be related to them. An example illustrates how the method can be applied to design a lead/lag compensator in the case of bounded sensitivity.