Characterization of standard controllers satisfying given H∞ specifications

The paper deals with the problem of characterizing a class of second-order three-parameter controllers (including PID and lag/lead) satisfying given H_∞ closed-loop specifications. Simple geometric considerations allow us to easily prove and extend to a larger class of compensators some recent results on PID control [A. Datta et al., 2000], [M.T. Ho, December 2001]. 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 convex sets, (ii) the closed-loop pole distribution is the same for all point inside each set and, (iii) a bound on the number of these sets is supplied by the pancake-cutting formula. An example illustrates how the method can be applied to design a PID controller in the case of bounded sensitivity.