Homotopy Groups And Sequences

Abstract

Abstract In this chapter, we “officially” start the homotopy theory of robust stabilization. “Officially” because the concept of homotopy has already pervaded the previous parts, and in particular because semisimplicial bundles and fibrations are traditionally considered to be part of homotopy theory. However, serious homotopy theory starts with the concept of homotopy groups, which are introduced in this chapter. Homotopy groups provide yet another algebraic picture of topological spaces, in addition to (co)homology groups. Probably the most important motivation for homotopy groups in robust stability is that they provide the natural value groups for the obstructions to extending the Nyquist map to higher-and-higher-dimensional skeleta of the polyhedron of uncertainty. Homotopy groups also yield exact sequences that play a role similar to homology exact sequences. In particular, in this chapter, we derive an exact homotopy sequence linking the homotopy groups of the uncertainty, the crossover, and the template of a robust stability problem.