On simultaneous minimizing of the H∞ sensitivity function and maximizing the disk stability margin

The authors formulate the problem of achieving sensitivity less than r and D(-a,p) disk stability margin, and derive a necessary and sufficient condition for the solvability of this problem. The notion of D(-a,p) disk margin introduced means that the Nyquist plot of the nominal loop transfer function must not enter the forbidden region: the disk centered at -a with radius p. This is equivalent to robust stability under complex gain perturbation in a certain related disk and is believed to be a reasonable uncertainty in cases where the nominal performance is considered. The maximum and the minimum for the (real) gain are estimated, and it is believed that the unmodeled dynamics plays some role, but no estimates are available