Characterization of PID and lead/lag compensators satisfying given H∞ specifications

This note deals with the problem of characterizing a class of second-order three-parameter controllers [including proportional-integral-derivative (PID) and lead/lag compensators] satisfying given H_∞ closed-loop specifications. Design characterizations of similar form as in the recent work on PID control, are derived for a larger class of compensators using simple geometric considerations. Specifically it is shown that, given the value of one parameter: i) the region of the plane defined by the other two parameters where the considered H_∞ constraint is satisfied, consists of the union of disjoint convex sets whose number can be bounded by means of the pancake-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 PID controller in the case of bounded sensitivity.