The combined sensitivity and phase margin problem

Any two design performance indices used in control system design have the potential to conflict with each other and a good control system is often some kind of compromise which optimizes neither index but secures satisfactory values for both. The objective of this paper is to study the two indices of sensitivity and phase margin simultaneously and reveal how the indices affect each other. The combined sensitivity and phase margin problem is basically solved, and both upper and lower bounds are derived on the achievable values of one index subject to a constraint on the other. It is shown that for a minimum phase plant, the optimal sensitivity and the optimal phase margin can be achieved simultaneously in a closed-loop system. Another particularly important result is that for a nonminimum phase plant, phase margin maximization will lead to an arbitrarily large sensitivity while sensitivity minimization does not cause an arbitrarily small phase margin. All the results are confined to scalar linear time-invariant plants.