Design problems for sensitivity and complementary sensitivity

The problem of finding a time-invariant controller for a given SISO (single-input, single output) plant so that the closed-loop system satisfies two tolerances for sensitivity and for complementary sensitivity simultaneously is considered. The solvability condition of the problem is derived in terms of an explicit inequality. It is also discovered that for a multivariable plant, there exists a very simple one-to-one correspondence between the sensitivity problem for r > 1 and the complementary sensitivity problem for the same r. Namely, a solution to either of the two problems can be obtained simply by multiplying a solution to the other problem by some certain constant. This indicates that sensitivity and complementary sensitivity have the same minimum for a plant if either minimum is greater than 1