• 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.

Подпись: Q(X,u>)

Figure 12.8. Small gain test for checking the absence of limit-cycles – extension to

parametric uncertainties.

• 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)

0!(A, U>)

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).