Multivariable circle criteria : an approach based on sector conditions

Abstract

The definition and properties of sector conditions introduced by Zames for scalar systems can be extended to the multivariable case. The present paper makes use of appropriate forms of sector conditions in conjunction with the decomposition of a linear operator into two linear operators φ and G - φ, where φ has a normal transfer function matrix ; and proposes a useful stability criterion for a class of multivariable non-linear feedback systems. The relevant stability result is developed further by a suitable interpretation of sector conditions in the frequency domain. A particular choice for φ leads to a generalization of the circle criterion in which the Nyquist plot of the frequency response of scalar systems is replaced by bands swept by circles whose centres and radii are related in a direct manner to the numerical range of G(jω). The result has a simple graphical interpretation and lends itself to a computer implementation which can be shown to be numerically stable. The approach is also suitable for the study of stability in the face of additive perturbations in the system model. Further choices of φ lead to stability tests which are based on the direct and inverse Nyquist plots of the diagonal elements of G(jω).