IEEE Transactions on Automatic Control, Vol.42, No.7, 1032-1036, 1997
A Dual Formulation of Mixed Mu and on the Losslessness of (D,G)-Scaling
This paper studies the mixed structured singular value, mu, and the well-known (D.G)-scaling upper bound, nu. A dual characterization of mu and nu is derives, which intimately links the two values. Using the duals it is shown that nu is guaranteed to be lossless (i.e., equal to mu) if and only if 2(m(r) + m(c)) + m(c) less than or equal to 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 losslessness result further leads to a variation of the well-known Kalman-Yakubovich-Popov lemma and Lyapunov inequalities.