EXTENSION TO THE CASE OF PARAMETRIC UNCERTAINTIES
• As a preliminary, it is straightforward to extend the graphical method
in the previous subsection (see also Figure 12.6) to the case of a transfer function G(s,6), which depends on some parametric uncertainties Sp see section 2.2.
Figure 12.8. Small gain test for checking the absence of limit-cycles – extension to
• The general case of a MIMO nonlinearity is considered. As in section 3.1, parametric uncertainties are introduced in the closed loop of Figure 12.1 byrewriting C7(s,£)as an LFT F)(P(s), Д2), where Дгів areal model perturbation (see Figure 12.3). Replacing then Ф by its frequency response N(X, ш) + А, Figure 12.3 is transformed into figures 12.8.a and 12.8.b. The sufficient condition of non oscillation is:
m(Q(X, u>), Д2)) < -7^-r (12.35)
The robustness margin is the maximal size of parametric uncer
tainties, for which the sufficient condition above is satisfied.
r(X, w) = min (k ЭД2 e kD-2 s. t. a(F,(Q(X, u), Д2)) > * – Л (12.S6)
r(X, w) = 0 if the nominal closed loop (obtained with A2 = 0) does not satisfy equation (12.34).
Remark: in the case of a diagonal MIMO nonlinearity, diagonal scaling matrices V can be introduced in the small gain test, in order to reduce the conservatism of the sufficient condition of non oscillation (i. e. to consider a(T>Fi(Q(X, u>),A2)V~1) instead of a(Fi(Q(X, to), Д2)) in definition 4..1).