  This paper provides a tutorial introduction to disk margins. These are robust
stability measures that account for simultaneous gain and phase perturbations
in a feedback system. The paper first reviews the classical (gain-only and
phase-only) margins and their limitations. This motivates the use of disk
margins which are defined using a set of perturbations that have simultaneous
gain and phase variations. A necessary and sufficient condition is provided to
compute the disk margin for a single-input, single-output feedback system.
Frequency-dependent disk margins can also be computed yielding additional
insight. The paper concludes with a discussion of stability margins for
multiple-input, multiple output (MIMO) feedback systems. A typical approach is
to assess robust stability "loop-at-a-time" with a perturbation introduced into
a single channel and all other channels held at their nominal values. MIMO disk
margins provide a useful extension to consider simultaneous variations in
multiple channels. This multiple-loop analysis can provide a more accurate
robustness assessment as compared to the loop-at-a-time approach.
