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

This paper studies the mixed structured singular value, μ, and the well-known (D,G)-scaling upper bound, ν. A complete characterization of the losslessness of ν (i.e., ν being equal to μ) is derived in terms of the numbers of different perturbation blocks. Specifically, it is shown that ν is guaranteed to be lossless if and only if 2(m_r+m_c)+m_C⩽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 μ and ν which intimately links μ with ν. Further, a special case of the aforementioned losslessness result leads to a variation of the well-known Kalman-Yakubovich-Popov lemma and Lyapunov inequalities