Many control design specifications, such as the IFAC 1993 benchmark, include stringent closed-loop performance requirements, which are required to be met in the face of uncertainties. Such problems can be effectively designed by a combination of the method of inequalities, which designs for explicit closed-loop performance, and analytical optimization techniques, from which robustness is obtained. This paper describes how to combine the method of inequalities with a variety of analytical optimization methods. An introduction to mu synthesis is also included. The design of robust controllers for the benchmark problem using these methods is described, and a simple adaptive gain scheme to improve the performance is suggested.