Phase-sensitive structured singular value

A. Tits and V. Balakrishnan

In Workshop on Open Problems in Mathematical Systems Theory and Control
Institute of Mathematics, University of Liege, Belgium, June 30, 1997

Also in Proc. IEEE Conf. on Decision and Control, Tampa, Florida, December 1998

Abstract The phase-sensitive structured singular value was introduced in [1,2] as a tool for the analysis of robust stability when, in addition to 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 (fields of values) of each uncertainty value Delta(j omega) 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 it holds in the matrix case; sufficiency always does. A convex upper bound \hat(mu)_theta(M) to the phase-sensitive structured singular value mu_theta(M) can also be defined, and is equal to mu_theta(M) in the scalar case. Another open question is under what conditions equality holds in the matrix case.

References:

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.
Robustness under Bounded Uncertainty with Phase Information (Tits, Balakrishnan and Lee. IEEE TAC, 1999)

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