# THE USE OF д BOUNDS

A necessary condition of oscillation is used in the above method. With reference to the end of section 3.1, two different cases are to be considered:

■ If no limit-cycle is obtained for the nominal closed loop system, the aim is to find the minimal amount of parametric uncertainties, for which a limit-cycle is obtained. It is interesting in this context to

compute both upper and lower bounds, since the flower bound (resp. the / upper bound) gives a size of the parametric uncertainties, for which the necessary condition of oscillation is satisfied (resp. not satisfied). The ц lower bound moreover gives a model perturbation Д[14], for which the necessary condition of oscillation is satisfied. It is then interesting to apply Д* to the closed loop, in order to check whether the first harmonic approximation is valid and whether the corresponding limit-cycle is stable or unstable (see section 5.).

■ If a limit-cycle is already obtained for the nominal closed loop system, the idea is rather to visualize the movement of this limit-cycle (i. e. the variation of its amplitude and frequency) as a function of the parametric uncertainties. In this new context, it is more interesting to obtain a model perturbation Д*, which satisfies indeed the necessary condition of oscillation. A /x lower bound is thus more interesting than a Ц upper bound.

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