This book unites the seemingly unrelated fields of algebraic topology and robust control to provide new insights on problems in stability. It uses the simplicial approximation theorem and its implementation through computational geometry as a primer for deep topological issues in stability.

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

Edmond A. JonckheereDepartment of Electrical Engineering Systems, University of Southern California

"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 |k No