The generalized Nyquist stability criterion and multivariable root loci
作者:
A. G. J. MAcFARLANE,
I. POSTLETHWAITE,
期刊:
International Journal of Control
(Taylor Available online 1977)
卷期:
Volume 25,
issue 1
页码: 81-127
ISSN:0020-7179
年代: 1977
DOI:10.1080/00207177708922217
出版商: Taylor & Francis Group
数据来源: Taylor
摘要:
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.
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