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

By Edmond A. Jonckheere

During this e-book, likely unrelated fields -- algebraic topology and powerful regulate -- are introduced jointly. The e-book 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, displaying 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 concerns reminiscent of 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 inability of continuity of the steadiness 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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More generally, observe that the following fractions, do not appear in the diagonal entries of the Smith-McMillan form, but do appear in det(D(s) + N ( s ) ) where some cancellation might occur. It follows that there could be a discrepancy between the zeros of det(D + N) and the poles of (D + N ) - l ; to be more specific, ROBUST MULTIVARIABLE NYQUIST CRITERION 9 However, if any cancellation occurs, this could only be a LHP pole/zero cancellation because the bi(s)'s are Hurwitz. Therefore, { RHP poles of (D+N)-1} = { RHP zeros of det (D+N)} At this stage we have reached the following result: The multivariable closed-loop matrix (/ + L 0)-1L 0 is stable iff det(D(s) + N(s)) has no right half-plane zeros.

129 Fig. 4 The 2-D Hex game played on the dual Hex board. 130 Fig. 5 Rhombic dodecahedron. 133 Fig. 6 FCC sphere packing I. 134 Fig. 7 FCC sphere packing II. 135 Fig. 1 Illustration of the labeling. The label of a vertex depends on where its image bi falls relative to the 120degree butterfly in the complex plane. 139 Fig. 2 The intuitive idea behind the algorithms for chasing completely labeled simplexes. 140 Fig. 3 Constructive proof of Sperner's lemma. 143 Fig. 4 The "variable grid refinement" concept, illustrated on a simple phase margin example.

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. on the same radial are identified. The resulting geometric structure is the 3-torus T3. Another procedure for removing the singularity of the mapping is to make ( 1, 2, 3) local Riemann coordinates by covering { } with open sets O, such that the restriction of ( 1, 2, 3) to each open set Oi is a homeomorphism hi.

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