Algebraic and Differential Topology of Robust Stability

Hardcover | April 30, 1999

byEdmond A. Jonckheere

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In this book, two seemingly unrelated fields -- algebraic topology and robust control -- are brought together. The book develops algebraic/differential topology from an application-oriented point of view. The book takes the reader on a path starting from a well-motivated robust stabilityproblem, showing the relevance of the simplicial approximation theorem and how it can be efficiently implemented using computational geometry. The simplicial approximation theorem serves as a primer to more serious topological issues such as the obstruction to extending the Nyquist map, K-theory ofrobust stabilization, and eventually the differential topology of the Nyquist map, culminating in the explanation of the lack of continuity of the stability margin relative to rounding errors. The book is suitable for graduate students in engineering and/or applied mathematics, academic researchersand governmental laboratories.

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In this book, two seemingly unrelated fields -- algebraic topology and robust control -- are brought together. The book develops algebraic/differential topology from an application-oriented point of view. The book takes the reader on a path starting from a well-motivated robust stabilityproblem, showing the relevance of the simplicia...

Edmond A. Jonckheere is at University of Southern California.
Format:HardcoverDimensions:624 pages, 9.49 × 6.26 × 1.34 inPublished:April 30, 1999Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0195093011

ISBN - 13:9780195093018

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Table of Contents

1. ProloguePart I: Simplicial approximations of algorithms2. Robust multivariable Nyquist criterion3. A basic topological problem4. Simplicial approximation5. Cartesian product of many uncertainties6. Computational geometry7. Piece-wise Nyquist map8. Game of Hex algorithm9. Simplicial algorithmsPart II: Homology of robust stability10. Homology of uncertainty and other spaces11. Homology of crossover12. Cohomology13. Twisted Cartesian product of uncertainty14. Spectral sequence of Nyquist mapPart III: Homotopy of robust stability15. Homotopy groups and sequences16. Obstruction to extending the Nyquist map17. Homotopy classification of Nyquist maps18. Brouwer degree of Nyquist map19. Homotopy of matrix return difference map20. K-Theory of robust stabilizationPart IV: Differential topology of robust stability21. Compact differentiable uncertainty manifolds22. Singularity over stratified uncertainty space23. Structural stability of crossoverPart V: Algebraic geometry of crossover24. Geometry of crossover25. Geometry of stability boundaryPart VI: EpiloguePart VII: Appendices

Editorial Reviews

"The opening sentence of the monograph reads as follows: 'In this book, two seemingly unrelated fields of intellectual endeavor--algebraic/differential topology and robust control--are brought together.' Indeed, there are probably just a few control engineers who are familiar with the modernconcepts of algebraic and differential topology; others need strong motivation to learn these disciplines. The book under review attempts to convince experts in robustness analysis to do this and to provide the needed material to learn. . . . The book consists of 7 parts, which contain 26 chaptersand 4 appendices. . . . [T]his book is a serious attempt to develop new approaches to robust stability. It will inspire research in control based on modern mathematics."--Mathematical Reviews