Root locus near isolated pole-zero dipoles and a counterintuitive case of digital control compensation

A convenient characterization is given for root loci associated with open-loop pole-zero dipoles. A particular dipole-locus effect is shown to influence the design of a digital PID controller for an oscillatory plant; the design is a counterintuitive one in which zeros of the compensator are placed outside the unit circle in the neighborhood of the plant poles.