Loop-Shaping Design for an Ill-Conditioned Plant

In this paper, we study a special case of the robust performance problem studied in the companion paper (4). When the weightings used to describe the uncertainty and performance specifications vary only with frequency, then it is possible to strengthen the results in that paper 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 in (4), and essentially yield necessary and sufficient conditions for the structured singular value to be small. We use this information to suggest a strategy for compensator design to achieve robust performance despite plant ill-conditioning. Applying our strategy to an example, we see how the design tradeoffs quantified by the Bode gain-phase relation manifest themselves in the robust performance problem. Finally, we compare our design with one obtained using the "¿-synthesis" approach.