A SECOND MIXED і UPPER BOUND
In the general case of mixed uncertainties, (Fan et al., 1991) first formulate the problem of computing the exact value of //as a smooth constrained optimization problem. The result is then used to derive a computable p upper bound, in which a maximal generalized eigenvalue is to be minimized with respect to the sets V and Q of scaling matrices (see the previous subsection).
An other approach for computing a /t upper bound is proposed by (Safonov and Lee, 1993). This method basically uses the positivity theorem: the interconnection structure M — A is first transformed, so that the new feedback uncertainty block becomes positive (using a bilinear transformation). The conservatism of the positivity theorem is then minimized with multipliers. The optimal value of these multipliers is here again obtained as the solution of an LMI problem.
Both methods in (Fan et al., 1991; Safonov and Lee, 1993) solve in a more efficient way the problem of (Doyle, 1985), in which a p upper bound was proposed for the case of non repeated real scalars. As noted in (Safonov and Lee, 1993), the two upper bounds in (Fan et al., 1991; Safonov and Lee, 1993) are equivalent for this case. The equivalence can be further characterized using a result in (Haddad et al., 1992) (Corollary 2, p. 2819).