The Simultaneous Optimization Problem for Sensitivity and Gain Margin

In this note, the combined sensitivity and gain margin problem for SISO linear systems is formulated and solved using a complex function interpolation technique. It is proved that this problem always has a real rational solution provided it is solvable in the complex irrational sense. The sensitivity minimization problem subject to a gain margin constraint and its dual problem are also considered. In addition, the range of the gain margin constraint is given subject to which the optimal constrained sensitivity is identical with the optimal unconstrained sensitivity. Finally, it is shown that, not unexpectedly, the gain margin maximization conflicts with the sensitivity minimization for a nonminimum phase plant.