Conditions for the Existence and Uniqueness of Optimal
Matrix Scalings
Conditions for the Existence and Uniqueness of Optimal
Matrix Scalings
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 problem of finding a similarity scaling of a prespecifed
structure to minimize the scaled singular value of a matrix
arises frequently in the robustness analysis of control
systems. Two questions that arise in this context are: (i)
When is the set of optimal scalings nonempty? (ii) When is
the optimal scaling unique? Sufficient conditions that
guarantee an affirmative answer to questions (i) and (ii)
are available, for the case when the scaling
matrices are diagonal; a complete answer to these questions
remains an open issue. The case when the scaling matrices
are block-diagonal is essentially open.
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