In the above algorithm, if the aim is just to compute an upper bound of Umax, it is unnecessary to apply the method of subsection 3.1 to all intervals [tup Wi+і]. Assume indeed that a lower bound 7 of цтах was a priori computed (i. e. before applying the algorithm of the previous subsection). If the s. s.v. is proved to be less than7 on afrequency
interval [<u, tu], the critical frequency (which corresponds to the maximal s. s.v. mmax over the frequency range) does not belong to this interval. It is thus useless to apply the algorithm of the previous subsection to this interval.
The issue is to find a computationally cheap method for checking whether the s. s.v.
A solution is to proceed as follows (see also (Magni et al., 1999) for an alternative method): ■ Let a frequency point tao, which may belong to the gridding
in the algorithm of the previous subsection. Find (sub)optimal values of І), G scaling matrices, which minimize the singular value:
with F = (I + G2) 1^4. Remember that 7 is fixed.
If the singular value above is found to be less than unity, apply the method of subsection inside which the s. s.v. desire to minimize the singular value in equation (10.38) should be understood as an heuristic way to maximize the size of the interval [w, її].
The issue is thus to minimize to some extent the singular value in equation (10.38). In the aircraft example of the following section, the minimization was done as follows (see also subsection 2.4 of chapter 8):
■ Computation of a suboptimal diagonal І) scaling matrix, which minimizes to a large extent —), with the Perron eigenvector
approach: see (Safonov, 1982). The method is computationally efficient and yet accurate. A routine is moreover available in the Robust Control Toolbox of Matlab.
■ Computation of an initial value of the G scaling matrix with the idea of (Young et al., 1995). Loosely speaking, the idea is simply to cancel
with G the skewed hermitian part of the blocks of —, which
correspond to the real parametric uncertainties.
■ A simple gradient method further minimizes the quantity in equation (10.38) with respect to G. Note that this quantity is not necessarily differentiable, and that the optimization problem is seemingly non convex with respect to G. It is thus important to have a good initial guess for G.
Despite its roughness, the method above gave quite good results in the example. Nevertheless, more sophisticated methods could be investigated, with the constraint to remain computationally cheap. Otherwise, the best solution would be to apply the basic algorithm of the previous
134 A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS subsection.
Remark: the idea of reducing the computational burden by eliminating frequency intervals, which can not contain the critical frequency, can be traced back at least to (Ferreres et al., 1996b). Nevertheless, this reference uses the augmented fi problem, in which the frequency is treated as an additional uncertainty. The approach proposed here is computationally much cheaper.