### 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"
}

**Related paper**

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.

**Related paper**

Some properties of an upper bound of mu

Back to reports index.