Singularity Over Stratified Uncertainty Space

Abstract

Abstract Originally the Morse theory was developed for real-valued functions defined on compact, differentiable manifolds [Morse 1925], [Milnor 1963]. Unfortunately, this does not lead us further than the multivariable phase margin problem where, typically,[ineq]. In many other structured stability problems, the set of uncertainties D x n is more complicated than a mere compact differentiable manifold. It suffices that the uncertainty manifold has a boundary for the “global” results of the previous chapter to fail, although the “local” results remain valid modulo some revisions. Early generalizations of the Morse theory to include the case of manifolds-with-boundary are due to Morse himself. This effort and other early attempts to extend the Morse theory have, more recently, crystallized around the so-called the Stratified Morse Theory [Goresky and MacPherson 1988]. The stratified Morse theory extends the classical Morse theory to the socalled stratified spaces, that encompass practically all sets of uncertainties encountered in robust stability. The idea of the stratified Morse theory, which comprises the case of an uncertainty manifold with boundary, is to decompose, or “stratify,” the space in a disjoint union of submanifolds, or “strata,” of varying dimensions.