By Edmond A. Jonckheere

During this publication, doubtless unrelated fields -- algebraic topology and powerful regulate -- are introduced jointly. The ebook develops algebraic/differential topology from an application-oriented perspective. The ebook takes the reader on a direction ranging from a well-motivated strong 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 resembling the obstruction to extending the Nyquist map, K-theory of sturdy stabilization, and finally the differential topology of the Nyquist map, culminating within the rationalization of the shortcoming of continuity of the steadiness margin relative to rounding mistakes. The e-book is appropriate for graduate scholars in engineering and/or utilized arithmetic, educational researchers and governmental laboratories.

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**Example text**

Fig. 8 Top view of stability crossover curve, equivalent to projecting the crossover curve on the space of parameters D and getting the boundary between the regions in which the number of LHP closed-loop poles is constant. 155 Fig. 9 Number of LHP closed-loop poles diagram. The horizontal axis is q1 ; the vertical axis is q2. 156 Fig. 1 Simplicial hole in the supertemplate. 173 Fig. 1 Cross section in Kharitonov's cube. 206 ; the vertical axis is 154 FIGURES Fig. 2 xxxvii Semisimplicial version of the principal bundle of the Mobius strip.

Indeed, we could take the phase angle 6 as parameter (running in R or [0,2 ]) and absorb exp(-j ) in the Nyquist map. The problem is that the mapping exp(—j ) is many-to-one (it is actually a covering map if 0 runs in R). 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.

155 Fig. 9 Number of LHP closed-loop poles diagram. The horizontal axis is q1 ; the vertical axis is q2. 156 Fig. 1 Simplicial hole in the supertemplate. 173 Fig. 1 Cross section in Kharitonov's cube. 206 ; the vertical axis is 154 FIGURES Fig. 2 xxxvii Semisimplicial version of the principal bundle of the Mobius strip. Thick lines and dark vertices correspond to the vertex f° of the fiber complex; thin lines and circled vertices correspond to vertex fl of fiber complex F {f0,f1}. 220 Fig. 3 Illustration of lack of fibration property of the orthogonal projection of a circle on a line.