A dual formulation 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 dual characterization of μ and ν is derived, which intimately links the two values. Using the duals it is shown that ν is guaranteed to be lossless (i.e. equal to μ) if and only if 2(mr+me)+mC ⩽3, where mr, mc; and mC are the numbers of repeated real scalar blocks, repeated complex scalar blocks, and full complex blocks, respectively. The losslessness result further leads to a variation of the well-known Kalman-Yakubovich-Popov lemma and Lyapunov inequalities.