Necessary and sufficient conditions for the robust stabilizing control of linear plants with ellipsoidal parametric uncertainties

Necessary and sufficient conditions for the robust stability of linear time-invariant single-input single-output systems composed of an uncertain plant and a fixed compensator are presented. Nominal values for the parameters of the plant transfer function are assumed to be known. In addition, it is assumed that these parameters are confined to a known ellipsoidal uncertainty region centered at a nominal value, i.e., the uncertainties of individual parameters can possibly be correlated and are thus not assumed to be independent of each other. It is shown that the ellipsoidal nature of the uncertainty structure is preserved by feedback. The robust stability problem is solved using the principle of arguments and the concept of critical direction. An algorithm for analyzing robust stability which involves computing a simple analytical expression as a function of frequency is proposed and illustrated with an example