# A SECOND ц BASED METHOD

The sufficient condition of non oscillation is introduced in the first subsection. The method is extended to the case of parametric uncertain­ties in the second subsection. The third subsection shows that a skewed H problem with a special structure is to be solved. The problem of using (skewed) /г bounds is discussed in the last subsection.

3.2 A SUFFICIENT CONDITION OF NON OSCILLATION

monic part e(£) of the signal u(t), at the output of the nonlinearity Ф. To this aim, the frequency response of the nonlinearity is rewritten as N(X, uj) + Д, where Д represents the error induced by the SIDF approx­imation.

• In the case of a SISO nonlinearity, Д is known by the relation:

where a( X, a( is a function of X and ш. With respect to Figure 12.1, a sufficient condition of non oscillation is then that the inverse of the frequency response X( X, a) X Д of the nonlinear element does not inter­sect the frequency response of the linear part G(s) of the closed loop. As in section 2.1, when N(X, w) and a{X, u>) do not depend on frequency u>, Figure 12.6 suggests a graphical method for checking the absence of limit-cycles.

• In the case of a MIMO nonlinearity, Д is known by the relation:

a(A)<a(X, u) (12.32)

 Figure 12.1 is transformed into Figure 12.7. a, by replacing the nonlinear­ity Ф by its frequency response N(X, u>) + A. Figure 12.7.a is then trans­formed into Figure 12.7.b, where Qn(X, w) is the transfer seen by Д in Figure 12.7.a Applying finally the small gain theorem to Figure 12.7.b, a sufficient condition of non oscillation is obtained as:

where Д satisfies equation (12.32). The sufficient condition of non oscil­lation thus becomes: