2.1 COMPUTATION OF uos BOUNDS
The methods of subsections 1.1 and 1.2 (chapter 5) are briefly extended to the computation of a bound of where M is a complex matrix
and Д = Дag( Д і, Д2 ) contains only real non repeated scalars. The idea is simply to note that vertices (resp. two-dimensional faces) are handled in the method of subsection 1.1 (resp. 1.2). Let 6 a vector of parametric uncertainties. Two cases are to be considered:
1. 6 is to be expanded inside its one-sided unit ball BAos.
2. <5 is to be maintained inside its unit ball (i. e. the unit hypercube D).
In the context of the method of subsection 1.1 (chapter 5), remember that all scalar components 5- of a vertex Sl take an extreme value. In the first case above, this means that S is chosen as = 0 or S — k. In the second case, Jj ± 1.
In the context of the method of subsection 1.2 (chapter 5), 8 belongs to a two dimensional face of the hypercube if all its components take an extreme value, except two ones. The extreme values are chosen in the same way as above. The two other components 5i and Sj are obtained as in subsection 1.2 (chapter 5). In the first case, 8 is an acceptable solution if Si and Sj belong to the interval [0, k] . In the second case, S is acceptable when Si and Sj are inside [-1, 1].
Remark: both methods above are exponential time, and they are applicable to problems with a few parametric uncertainties and delays, which is the case of the missile example of section 4.. See (Ferreres et al., 1996b) for the computation of a polynomial time upper bound of