Algebraic and Differential Topology of Robust Stability by Edmond A. Jonckheere

By Edmond A. Jonckheere

During this publication, possible unrelated fields -- algebraic topology and strong regulate -- are introduced jointly. The publication develops algebraic/differential topology from an application-oriented viewpoint. The publication takes the reader on a course ranging from a well-motivated strong balance challenge, exhibiting the relevance of the simplicial approximation theorem and the way it may be successfully carried out utilizing computational geometry. The simplicial approximation theorem serves as a primer to extra severe topological concerns 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 shortcoming of continuity of the soundness margin relative to rounding mistakes. The booklet is acceptable for graduate scholars in engineering and/or utilized arithmetic, educational researchers and governmental laboratories.

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The identification of the remaining pair of opposite faces of the cube is equivalent to identification of the inner boundary and the outer boundary of the hollow sphere. Clearly, this cannot be done in threedimensional space. Therefore, the most intuitive low-dimensional model of the uncertainty space in case of three uncertain phase angles is a hollow sphere where the inner boundary points and the outer boundary points PROLOGUE 3 Fig. 1. Positive real feedback perturbed by diagonal phase uncertainty.

3. " This space is defined by X+ = X U { }, for some { } X; and the open sets of X+ are the open sets of X and the sets of the form X+ \K , where K is compact in X . Moreover, X+ is unique up to homeomorphism. Proof. 3]. The one-point compactification of (—j unit circle S1, see [Dugundji 1970], , +j ) is well-known to be the This unit circle can the taken to be the imaginary section through the Riemann sphere since the Riemann sphere is the one-point compactification of the complex plane, Since the one-point compactification is unique, we can use any recipe to map the line to S1 \{point}.

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.

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