The geometry of the multivariable phase margin

A collection of multiplicative perturbations with a given structure can be viewed as a nontrivial n-dimensional manifold. This manifold can be divided into regions of stability and instability where the unperturbed identity matrix is a point on the manifold. The stability margin for a collection of such structured perturbations can be defined as the smallest distance on the manifold from the identity matrix to the region of instability. The class of unitary perturbations which has been shown to define the multivariable phase margin is examined. Specifically, it is seen that the margin is the smallest arc length on the three-sphere S³ between the north pole and the region of instability