Tight bounds for robust sensitivity reduction from functional analysis

This paper deals with the so-called robust disturbance attenuation problem. This problem is to find the controller which robustly minimizes the infinity norm of the weighted sensitivity function. So this is a robust performance problem. This problem has not been solved at an analytical level. However, one can try to approach the optimum using, for instance, μ-synthesis or the convex optimization paradigm of Boyd and Barratt. The controllers obtained from these approaches might not be optimal. How can a designer check if these controllers are optimal or not? How can a designer determine rigorously how far these controllers are from the true optimal controller? This paper shows how to obtain both upper and lower bounds on how near a given controller´s performance is to the performance of the true optimal (but unknown) controller. These bounds are shown to approach each other as the controller approaches optimality. These results are established by functional analysis techniques, namely duality theory for optimization in function spaces.