By Edmond A. Jonckheere
During this publication, doubtless unrelated fields -- algebraic topology and strong keep an eye on -- are introduced jointly. The ebook develops algebraic/differential topology from an application-oriented standpoint. 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 equivalent to the obstruction to extending the Nyquist map, K-theory of sturdy stabilization, and finally the differential topology of the Nyquist map, culminating within the clarification of the shortcoming of continuity of the soundness margin relative to rounding error. The publication is appropriate for graduate scholars in engineering and/or utilized arithmetic, educational researchers and governmental laboratories.
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Extra resources for Algebraic and Differential Topology of Robust Stability
Therefore, absorbing exp(—j ) in the Nyquist map makes the domain of definition more trivial, but at the expense of making the Nyquist map more "singular"; that is, more points of the domain are mapped to the same point in C. Nevertheless, if the latter is not our concern, it is clearly more economical to use [0, 2 ] rather than M as domain of definition. The problem with such a domain of definition as [0, 2 ] is that it is not a manifold. As already argued, a domain of definition with manifold structure is essential for singularity analysis of the Nyquist map, itself related to the extremely important problem of continuity of robustness margin relative to "certain" parameters.
For example, combining a real parameter q together with a phase UNCERTAINTY SPACE 13 margin term yields the topological product, that is, a cylinder of uncertainty. In the SISO gain margin problem, the uncertainty is the half-interval [0, ), which is not compact. Leaving the domain noncompact raises some questions as to what is "happening" at infinity. To clarify the behavior of the map at infinity, it is convenient to compactify the domain. We defer the compactification issue to the next section, where we will do it explicitly on the imaginary axis (—j , +j ).
422 Fig. 7 Connection between the "swallow tail" in the Cerf critical value plot, the "kink" in the supertemplate, and the Thom-Boardman singularity sets. 431 Plot of critical values of f versus 9 for a = 1 (Example I). 435 Plot of critical values of f versus 9 for a = 2 (Example I). 436 Fig. 10 Plots of critical points of f versus 6 on a planar model of the 2-torus of uncertainty for a — 1; the horizontal axis is q; the vertical axis is (Example I). 437 Fig. 5 Fig. 8 Fig. 9 Fig. 11 Nyquist template reconstructed by mapping the critical point curves to the complex plane.
Algebraic and Differential Topology of Robust Stability by Edmond A. Jonckheere