Structured singular values versus diagonal scaling: the noncommutative setting (I)

Abstract

The structured singular value (often referred to simply as mu) was introduced independently by Doyle and Safanov as a tool for analyzing robustness of system stability and performance in the presence of structured uncertainty in the system parameters. While the structured singular value provides a necessary and sufficient criterion for robustness with respect to a structured ball of uncertainty, it is notoriously difficult to actually compute. The method of diagonal (or simply "D") scaling, on the other hand, provides an easily computable upper bound for the structured singular value, but provides an exact evaluation of mu (or even a useful upper bound for mu) only in special cases. However it was discovered in the 1990s that enhancement of the uncertainly structure to allow what can be interpreted as time-varying uncertainty (equivalently, letting the uncertainty parameters be freely noncommuting operators on an infinite-dimensional separable Hilbert space) resulted in the D-scaling procedure leading to an exact evaluation of mu. This report discusses recent refinements of these results to allow repetitions of full blocks as well as nonsquare blocks in the uncertainty structure.