By Edmond A. Jonckheere
During this booklet, likely unrelated fields -- algebraic topology and strong keep watch over -- are introduced jointly. The e-book develops algebraic/differential topology from an application-oriented viewpoint. The booklet 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 carried out utilizing computational geometry. The simplicial approximation theorem serves as a primer to extra critical topological concerns comparable to the obstruction to extending the Nyquist map, K-theory of sturdy stabilization, and at last 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 info for Algebraic and Differential Topology of Robust Stability
The next step is to reduce the RHP zero test on det(D + N) to a test involving the usual Nyquist plot of det(I + L 0). By the principle of the arguments, det(D(s) + N ( s ) ) has no RHP zeros iff where the integral with respect to s is taken from -j to +j followed by a large semicircle enclosing the right half-plane. The above integral is usually referred to as winding number or topological index of det(D(s) + N ( s ) ) relative to 0 + j0. To convert the above into a criterion involving the Nyquist plot of det(I+ L(jw) 0), observe that Therefore, the winding number of the plot of det(D + N) around 0 + j0 is At this stage, let us agree to call number of open-loop right half-plane poles the number, multiplicity counted, of right half-plane zeros of det(D(s)).
111 Fig. 3. 112 Fig. 1 Illustrative example of making a piecewise-linear map simplicial in the case where the convex hull of the sample image vertices is a triangle. 122 Fig. 2 Illustrative example of making a piecewise-linear map simplicial in the case where the convex hull of the sample image vertices is a quadrilatere. 122 Fig. 1 The game of Hex versus the traditional tic-tac-toe game. 125 Fig. 2 8 8 Hex board and its dual graph. 126 Fig. 3 Finding winning path of the Hex game. 129 Fig. 4 The 2-D Hex game played on the dual Hex board.
1 Nyquist mapping formulation of Kharitonov's theorem. 27 Fig. 2 Example of a Schwarz-Christoffel conformal mapping exhibiting some pathological boundary behavior. 39 Fig. 3 A domain with a boundary that is not locally connected. 40 Fig. 1 The standard 2-simplex. 63 Fig. 2 Simplicial cylinder. 64 Fig. 3 Simplicial 2-torus. 64 Fig. 4 The simplicial Mobius strip. 65 Fig. 5 Illustration of the nerve of a covering. 67 Fig. 6 The barycentric and Q-subdivisions of the simplex. 69 Fig. 7 The stellar subdivision of the simplex.
Algebraic and Differential Topology of Robust Stability by Edmond A. Jonckheere