  The structured singular value $\mu$ 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 (which we call
$\hat \mu$) 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 a certain enhancement of the
uncertainty structure (i.e., letting the uncertainty parameters be freely
noncommuting linear operators on an infinite-dimensional separable Hilbert
space) resulted in the $D$-scaling procedure leading to an exact evaluation of
$\mu_{\text{enhanced}}$ ($\mu_{\text{enhanced}} = \hat \mu$), at least for the
tractable special cases which were analyzed in complete detail. On the one hand
this enhanced uncertainty has some appeal from the physical point of view: one
can allow the uncertainty in the plant parameters to be time-varying, or more
generally, one can catch the uncertainty caused by the designer's decision not
to model the more complex (e.g. nonlinear) dynamics of the true plant. On the
other hand, the precise mathematical formulation of this enhanced uncertainty
structure makes contact with developments in the growing theory of analytic
functions in freely noncommuting arguments and associated formal power series
in freely noncommuting indeterminates. In this article we obtain the
$\widetilde \mu = \hat \mu$ theorem for a more satisfactory general setting.
