Summary: The phase-sensitive structured singular value was introduced in~[1,2] as a tool for the analysis of robust stability when, in addition to the knowledge of a magnitude bound and possible block-diagonal structure, certain phase information is available concerning the uncertainty. Specifically, possibly after an appropriate frequency-dependent phase shift, the numerical range (field of values) of each uncertainty value Delta(jw) is assumed to be contained in a sector of given aperture 2 theta about the positive real axis. (For example, theta=pi/2 corresponds to the case when the uncertainty is known to be passive.) In the scalar case, a corresponding ``small-mu'' theorem holds. One open question is under what conditions a ``small-mu'' theorem holds in the matrix case. More specifically, while a bound on mu is sufficient for robust stability, is this bound also necessary? An upper bound to the phase-sensitive structured singular value, computable via convex optimization, can also be defined, and is equal to phase-sensitive structured singular value in the scalar case. Another open question, in the matrix case, is under what conditions is the upper bound equal to the phase-sensitive structured singular value?

References:

1. L. Lee, ``Robustness Study of Systems with Phase-Informed Uncertainty,'' Ph.D. Dissertation, Department of Electrical Engineering, University of Maryland, College Park, MD 20742, 1992.

2. A. L. Tits, V. Balakrishnan, and L. Lee, ``Robustness Under Bounded Uncertainty with Phase Information,'' IEEE Trans. Aut. Contr., AC-44(1):50--65, January 1999.