An Introduction to Disk Margins

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.