Properties of the graph topology for scalar transfer functions (Q817389)

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.