DAILEY’S METHOD

This method, which provides a real p lower bound, is based on a conjecture about the structure of the minimum norm parameter per­turbation (Dailey, 1990). Assume that all but two components Si and Sj of the minimum norm parameter perturbation achieve the maximal magnitude k:

6[ = ±k VI ф і and VI ф j (5.3)

Rather than solving the general equation with respect to the vector of parametric uncertainties:

det(I — diag(S)M) = 0 (5.4)

it suffices now to solve it with respect to the two free components Si and Sj, and this reduces to finding the real roots of a set of two quadratic
equations in Si and Sj: remember indeed that Mis complex, so that the above equation has generally a single solution in Si and Sj. If the abso­lute values of the obtained Si and Sj are less or equal to k, a destabilizing parameter perturbation 5 with norm к has been obtained.

Relation (5.3) defines a two dimensional face of the hypercube kD, so that the above operation has to be done on each such face of the hypercube. The method, which is thus exponential time because of the exponential growth of the number of two dimensional faces as a function of the number of parametric uncertainties, is nevertheless easy to imple­ment. The technique provides a lower bound of real /і, since the counter­examples in (Holohan and Safonov, 1993; Ackermann, 1992) prove that it can not be assumed in the general case that more than one component Si of the minimum norm parameter perturbation achieves the maximal magnitude.

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