A study is made of a special case of the robust performance problem given by Freudenberg (1989). When the weightings used to describe the uncertainty and performance specifications vary only with frequency, then it is possible to strengthen the results in the above-mentioned work by deriving both upper and lower bounds upon the structured singular value. Both sets of bounds are stated in terms of the coupling coefficients introduced by Freudenberg (1989), and essentially yield necessary and sufficient conditions for the structured singular value to be small. This information is used to suggest a strategy for compensator design to achieve robust performance despite plant ill-conditioning. Applying this strategy to an example, it can be seen how the design trade-offs quantified by the Bode gain-phase relation manifest themselves in the robust performance problem. Finally, the design is compared with one obtained using the'µ-synthesis' approach.