Minimum return difference as a compensator design tool

Minimum return difference (RDmin) is a single robustness measure that, when large, guarantees that both gain margin (GM) and phase margin (PM) are large. In this paper, a lag and lead compensator design procedure based on RD is proposed, derived, and compared with the commonly used Bode PM-based design methods that undergraduates are taught in the first course on controls and that continue to appear in new textbooks. To introduce students to modern compensator design concepts while avoiding the complexities of optimal control theory, a cost function is minimized over a search focused on compensators that potentially may yield a large RDmin. An approximate relation between RDmin and {GM, PM} suggests a lower bound on RDmin for robust system stability. An efficient procedure for exact calculation of RDmin is presented and is a valuable component of the compensator design algorithm. The compensator parameter search is conducted over a domain that approximately enforces/exceeds the lower bound requirement on RDmin. All designs violating the requirement are rejected. The settling time and overshoot of the step response are usually both reduced relative to the traditional design methods, sometimes substantially, and the RD method often succeeds when the traditional methods fail. The undergraduate instructor is thus given a fresh, modern, and successful alternative for teaching first-order compensation synthesis without getting into advanced graduate-level techniques. The procedure also helps students understand system robustness and provides an easy transition to optimization methods, linking them with familiar concepts.