Many control design specifications, such as the IFAC 1993 benchmark, i
nclude stringent closed-loop performance requirements, which are requi
red to be met in the face of uncertainties. Such problems can be effec
tively designed by a combination of the method of inequalities, which
designs for explicit closed-loop performance, and analytical optimizat
ion techniques, from which robustness is obtained. This paper describe
s how to combine the method of inequalities with a variety of analytic
al optimization methods. An introduction to mu synthesis is also inclu
ded. The design of robust controllers for the benchmark problem using
these methods is described, and a simple adaptive gain scheme to impro
ve the performance is suggested.