Exact Nyquist-like stability results for ellipsoidal uncertainties

In this paper we develop a stability criterion for systems with uncertainties which are manifested in the frequency domain by simply-connected and closed, arbitrary uncertainty regions which satisfy a mild convexity constraint. In particular, well-known stability results for the case of disk-bounded frequency domain uncertainties are recovered as a special case of the proposed approach. The main results hinge on the definition of the critical direction as the direction of the line joining the -1+j0 point to the the nominal frequency response at a particular frequency. It is argued that the worst case uncertainties must lie along this line and this idea is exploited to yield a general stability criterion. An example arising from system and uncertainty identification is presented to illustrate the ideas developed in the paper. An application of the results of this paper yields exact and explicit formulae for the robust stability of systems with ellipsoidal parametric uncertainties.