A dual of mixed μ and on the losslessness of (D, G)-scaling

Abstract

This paper studies the mixed structured singular value, /spl mu/, and the well-known (D,G)-scaling upper bound, /spl nu/. A complete characterization of the losslessness of /spl nu/ (i.e., /spl nu/ being equal to /spl mu/) is derived in terms of the numbers of different perturbation blocks. Specifically, it is shown that /spl nu/ is guaranteed to be lossless if and only if 2(m/sub r/+m/sub c/)+m/sub C//spl les/3, where m/sub r/, m/sub c/ and m/sub 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 /spl mu/ and /spl nu/ which intimately links /spl mu/ with /spl 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.