AN EXTENSION

Parametric uncertainties are introduced in the closed loop system of Figure 12.1: the transfer function G(s, S) now depends on LTI uncertain parameters Si. For the sake of convenience, the vector 6 of parametric uncertainties is assumed to belong to the unit hypercube D:

The necessary condition for the existence of a limit-cycle becomes:

Defining the generalized value set G(ju) in the complex plane as:

and assuming as in the previous subsection that N(X, w) = N(X), the issue is to find the intersection (s) of the generalized value set G(jw) with the plot of 1 N(X) in the complex plane.

The problem is thus to build the generalized value set G(ju>) as a function of из. This is not an easy problem, which is only solved for special structures of parametric uncertainties. This is especially the case of interval plants:

where uncertain coefficients ai and bj belong to intervals. See e. g. (Kheel and Bhattacharyya, 1994) and included references.