On the generalized Nyquist stability criterion

Abstract

The purpose of this paper is to provide a self-contained proof of the generalized Nyquist stability criterion. The frequency-dependent eigenvalues of a square transfer-function matrix G(8) are defined directly from a matrix fraction decomposition of G(8), and a discussion of the fixed modes and fixed gains is given. The theory of algebraic functions and Riemann surfaces is employed in the proof, and a detailed consideration is given to these theoretical ideas from the point of view of the stability result. The question of the relationship between the poles and zeros of an algebraic function (in the complex-variable sense) and the poles and zeros (in the Smith-McMillan sense) is carefully examined, and a proof of the argument principle for a regular region on a Riemann surface is given.