Algebraic and Differential Topology of Robust Stability

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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 stability problem, 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 of robust 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 researchers and governmental laboratories.

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Editorial Reviews

From the Publisher
"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 modern concepts 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 chapters and 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
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Product Details

  • ISBN-13: 9780195093018
  • Publisher: Oxford University Press, USA
  • Publication date: 5/28/1997
  • Pages: 624
  • Product dimensions: 6.25 (w) x 9.50 (h) x 1.32 (d)

Table of Contents

1. Prologue
Part I: Simplicial approximations of algorithms
2. Robust multivariable Nyquist criterion
3. A basic topological problem
4. Simplicial approximation
5. Cartesian product of many uncertainties
6. Computational geometry
7. Piece-wise Nyquist map
8. Game of Hex algorithm
9. Simplicial algorithms
Part II: Homology of robust stability
10. Homology of uncertainty and other spaces
11. Homology of crossover
12. Cohomology
13. Twisted Cartesian product of uncertainty
14. Spectral sequence of Nyquist map
Part III: Homotopy of robust stability
15. Homotopy groups and sequences
16. Obstruction to extending the Nyquist map
17. Homotopy classification of Nyquist maps
18. Brouwer degree of Nyquist map
19. Homotopy of matrix return difference map
20. K-Theory of robust stabilization
Part IV: Differential topology of robust stability
21. Compact differentiable uncertainty manifolds
22. Singularity over stratified uncertainty space
23. Structural stability of crossover
Part V: Algebraic geometry of crossover
24. Geometry of crossover
25. Geometry of stability boundary
Part VI: Epilogue
Part VII: Appendices

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