The generalized Nyquist stability criterion and multivariable root loci

Abstract A comprehensive discussion is given of the background to the generalized Nyquist stability criterion for linear multivariable feedback systems. This leads to a proof based on the use of the Principle of the Argument applied to an algebraic function defined on an appropriate Riemann surface. It is shown how the matrix-valued functions of a complex variable which define the loop transmittance, return-ratio and return-difference matrices of feedback systems analysis may be associated with a set of characteristic algebraic functions, each associated with a Riemann surface. These characteristic functions enable the characteristic loci, which featured in previous heuristic treatments of the generalized Nyquist stability criterion, to be put on a sound basis. The relationship between the algebraic structure of the matrix-valued functions and the appropriate complex-variable theory is carefully discussed. These extensions of the complex-variable concepts underlying the Nyquist criterion are then related to an appropriate generalization of the root locus concept. It is shown that multivariable root loci are the 180° phase loci of the characteristic functions of a return-ratio matrix on an appropriate Riemann surface, plus some possibly degenerate loci each consisting of a single point.