CHOICE OF WEIGHTS ON THE MODEL UNCERTAINTIES

An important practical issue is the choice of weights on model un­certainties. It was seen in subsection 1.2 that a template is to be de­termined for each block of neglected dynamics. Consider now the case of parametric_uncertainties 6{. Each is assumed to belong to an in­terval [—ві, ві]. The normalized parametric uncertainties 8{ are then introduced as

Let kmax the robustness margin: the parametrically uncertain closed loop remains asymptotically stable for each 8{ Є [—kmax, kmax], or equi­valently for_each ві Є [—kmax ві, kmax ві]. The choice of the initial intervals [—ві, ві] is critical, as illustrated by the example of Figure 1.8, with only two parametric uncertainties ві and в2. The zero point cor­responds to the nominal values of the uncertain parameters: remember that the nominal closed loop system is asymptotically stable. The space of parametric uncertainties ві and 62 can then be split into two subdo­mains. In the first one, which contains the zero point, the parametrically
uncertain closed loop remains asymptotically stable. In the second one, this closed loop is unstable. At the limit between the two subdomains, the closed loop is marginally stable.

Generally speaking, the ц approach provides the largest hypercube in the space of the normalized parametric uncertainties Si, inside which the closed loop stability is guaranteed. This hypercube becomes a box in the space of the initial parametric uncertainties 0*. What illustrates Figure 1.8 is that various stability boxes are obtained depending on the choice of the initial intervals [—&i, 0i.

Obviously, the above discussion can be extended to the general case of a structured model perturbation containing parametric uncertainties and neglected dynamics: various stability domains are obtained depending on the choice of the weights on the parametric uncertainties and neglected dynamics. We will come back to the study of this important issue in the following.