By Edmond A. Jonckheere
During this publication, probably unrelated fields -- algebraic topology and strong keep an eye on -- are introduced jointly. The ebook develops algebraic/differential topology from an application-oriented perspective. The e-book takes the reader on a course ranging from a well-motivated powerful balance challenge, exhibiting the relevance of the simplicial approximation theorem and the way it may be successfully applied utilizing computational geometry. The simplicial approximation theorem serves as a primer to extra critical topological matters comparable to the obstruction to extending the Nyquist map, K-theory of strong stabilization, and finally the differential topology of the Nyquist map, culminating within the rationalization of the inability of continuity of the soundness margin relative to rounding error. The ebook is appropriate for graduate scholars in engineering and/or utilized arithmetic, educational researchers and governmental laboratories.
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Extra info for Algebraic and Differential Topology of Robust Stability
2. The shaded areas are regions of instability while the white areas are regions of stability. Clearly, these stability/instability regions and their separating boundaries can get complicated. The aim of this book is to formulate these robust stability problems in precise topological terms, to develop fast algorithms for computing the "crossover," and finally to develop algebraic procedures for computing the "topography" of the problem without the need for numerical exploration. Part I SIMPLICIAL APPROXIMATION AND ALGORITHMS This page intentionally left blank 2 ROBUST MULTIVARIABLE NYQUIST CRITERION SUMMARY In this chapter, we briefly review multivariable closed-loop stability and the related robustness issue.
2. The unstable (shaded) areas in the cubical model of the 3-torus. 2. The shaded areas are regions of instability while the white areas are regions of stability. Clearly, these stability/instability regions and their separating boundaries can get complicated. The aim of this book is to formulate these robust stability problems in precise topological terms, to develop fast algorithms for computing the "crossover," and finally to develop algebraic procedures for computing the "topography" of the problem without the need for numerical exploration.
Proper subset of A (A B) A \ B set of elements of A not in B A the closure of the space A the complex conjugate of z C disjoint union ~ equivalence relation is identical to ... is isomorphic or homeomorphic to ... is homotopic to ... is approximately equal to ... facing relation of an abstract simplicial complex or stratification in general, denotes the number of critical points of a certain type of a smooth function, or the number of n-simplexes of a complex, and so on the complex conjugate of z C p* denotes a critical point in the space of p-parameters X Y mapping or morphism from X to Y X,A Y,B a mapping or morphism X Y that maps the subset A C X into the subset B Y identity mapping or morphism homeomorphic mapping or isomorphic morphism a mapping, the existence of which is in question collapse [A, Bi the set of homotopy classes of maps A B [A, B] the set of homotopy classes of base point preserving maps A, a B,b identity mapping or morphism on A wedge product of exterior differential forms direct sum of groups, vector spaces, or Hilbert spaces tensor product of Abelian groups tensor product of R-modules .
Algebraic and Differential Topology of Robust Stability by Edmond A. Jonckheere