Communications in Information and Systems

Volume 2 (2002)

Number 3

Properties of the graph topology for scalar transfer functions

Pages: 217 – 244

DOI: http://dx.doi.org/10.4310/CIS.2002.v2.n3.a1

Authors

Thomas S. Brinsmead (Research School of Info. Sciences and Engineering, Australian National University, Australia)

Brian D. O. Anderson (Research School of Info. Sciences and Engineering, Australian National University, Australia)

Abstract

Necessary and sufficient conditions for the existence of a scalar homotopy in the Vinnicombe metric between two given transfer functions are presented. Previous results show that for multivariable transfer functions, the existence of a homotopy is related to a winding number condition on the homotopy end-points. Here, it is shown that for scalar transfer functions the existence of a homotopy is also related to a Cauchy index condition. Extensive use is made of the relationship, previously noted by Brockett, between the Cauchy index of a real function and the argument of a particular related complex function.

Keywords

Vinnicombe metric, $\nu$-gap metric, scalar homotopy, subunitary homotopy, Cauchy index

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