Algebraic and Differential Topology of Robust Stability

"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<br>

I. SIMPLICIAL APPROXIMATION AND ALGORITHMS; 1. Prologue; 2. Robust Multivariable Nyquist Criterion; 3. A Basic Topological Problem; 4. Simplicial Approximation; 5. Cartesian Product of Uncertainties; 6. Computational Geometry; 7. Piecewise Linear Nyquist Map; 8. Game of Hex Algorithm; 9. Simplicial Algorithms; II. HOMOLOGY OF ROBUST STABILITY; 10. Homology of Uncertainty and Other Spaces; 11. Homology of Crossover; 12. Cohomology; 13. Twisted Cartesian Product of Uncertainties; 14. Spectral Sequence of Nyquist Map; III. HOMOTOPY OF ROBUST STABILITY; 15. Homotopy Groups and Sequences; 16. Obstruction to Extending 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; IV. DIFFERENTIAL TOPOLOGY OF ROBUST STABILITY; 21. Compact Differentiable Uncertainty Manifolds; 22. Singularity Over Stratified Uncertainty Space; 23. Structural Stability of Crossover; V. ALGEBRAIC GEOMETRY OF CROSSOVER; 24. Geometry of Crossover; 25. Geopmetry of Stability Boundary; Epilogue; Appendices