A dual formulation of mixed mu and the losslessness of (D,G)-scaling

Gjerrit Meinsma, Yash Shrivastava and Minyue Fu.
Abstract
This paper studies the mixed structured singular value ($\mu$) and the well-known $(D,G)$-scaling upper bound ($\nu$). A complete characterization of the losslessness of $\nu$ (i.e., $\nu$ being equal to $\mu$) is derived in terms of the numbers of different perturbation blocks. Specifically, it is shown that $\nu$ is guaranteed to belossless if and only if $2(m_r+m_c)+m_C\leq 3$, where $m_r$, $m_c$, and $m_C$ are the numbers of repeated real scalar blocks, repeated complex scalar blocks and full complex blocks, respectively. The results hinge on a dual characterization of $\mu$ and $\nu$, which intimately links $\mu$ with $\nu$. Further, a special case of the aforementioned losslessness result leads to a variation of the well-known Kalman-Yakubovich-Popov lemma and Lyapunov inequalities.

Keywords: Mixed structured singular values, Kalman-Yakubovich-Popov lemma, Duality, Linear matrix inequalities.
Postscript file: mu-report.ps.gz (12 pages, 97 Kb, 600 dpi, gzip compressed).
BibTex entry
@Article{MSF95b, author = "G. Meinsma and Y. Shrivastava and Minyue Fu", title = "A dual formulation of mixed {$\mu$} and on the losslessness of {$(D,G)$}-scaling", year = "1997", journal = "IEEE Trans. Aut. Control", volume = "42", number = "7", pages = "1032--1036" }
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The CDC96 version: mu2-cdc96.ps.gz (6 pages, 129 Kb, 600 dpi, gzip compressed). This is almost the same, but better layout and some some proofs missing, and another little lemma added.
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