Homotopy Classification Of Nyquist Maps

Abstract

Abstract Since we have removed 0 + jO from the complex plane, the Nyquist map D × Ω (ℂ \ {0 + j0}, if it exists, could have a nontrivial homotopy class. The set of all possible homotopy classes of all such maps depends essentially on D. This chapter addresses the question as to what homotopy classes can be expected for a given uncertainty space. In case of such uncontractible uncertainty spaces as toris—multichannel phase margin problems—the set of all homotopy classes is quite large. The next question is to determine which of these homotopy classes yield closed-loop stability. Finally, given an uncertainty structure D, it is shown how the loop function L(s) induces a homotopy class of the map.