Contents -- List of Figures -- List of Symbols -- 1 Prologue -- I: SIMPLICIAL APPROXIMATION AND ALGORITHMS -- 2 Robust Multivariable Nyquist Criterion -- 2.1 Multivariable Nyquist Criterion -- 2.2 Robust Multivariable Nyquist Criterion -- 2.3 Uncertainty Space -- 2.4 "Punctured" Uncertainty Spaces -- 2.5 Compactification of Imaginary Axis -- 2.6 Horowitz Supertemplate Approach -- 2.7 Crossover -- 2.8 Mapping into Other Spaces -- 3 A Basic Topological Problem -- 3.1 The Boundary Problem -- 3.2 Topology for Boundary and Continuity -- 3.3 Mathematical Formulation of Boundary Problem -- 3.4 Example (Continuous Fraction Criterion) -- 3.5 Example (Kharitonov) -- 3.6 Example (Real Structured Singular Value) -- 3.7 Example (Brouwer Domain Invariance) -- 3.8 Example (Covering Map) -- 3.9 Example (Holomorphic Mapping) -- 3.10 Example (Proper Mapping) -- 3.11 Example (Conformal Mapping) -- 3.12 Examples (Horowitz) -- 3.13 Example (Functions on Polydisks) -- 3.14 Several Complex Variables -- 3.15 Example (Plurisubharmonic functions) -- 3.16 Example (Proper Holomorphic and Biholomorphic Maps) -- 3.17 Example (Whitney's Root System) -- 4 Simplicial Approximation -- 4.1 Simplexes, Complexes, and Polyhedra -- 4.2 Abstract Complexes -- 4.3 Alexandroff Theorem -- 4.4 Simplicial Approximation-Point Set Topology -- 4.5 Simplicial Map-Algebra -- 4.6 Computational Issues -- 4.7 Relative Simplicial Approximation -- 4.8 Cell Complexes and Cellular Maps -- 4.9 Historical Notes -- 5 Cartesian Product of Many Uncertainties -- 5.1 Prismatic Decomposition -- 5.2 Boundary of Cartesian Product -- 5.3 Simplicial Combinatorics of Cube -- 5.4 Q-Triangulation -- 5.5 Combinatorial Equivalence -- 5.6 Flatness -- 6 Computational Geometry -- 6.1 Delaunay Triangulation of Template -- 6.2 Simplicial Edge Mapping -- 6.3 The SimplicialVIEW Software.

6.4 Numerical Stability, Flatness, and Conditioning -- 6.5 Making Map (Locally) Simplicial -- 6.6 Procedure -- 7 Piecewise-Linear Nyquist Map -- 7.1 Piecewise-Linear Nyquist Map -- 7.2 From Piecewise-Linear to Simplicial Map -- 7.3 Strict Linear Complementarity -- 8 Game of the Hex Algorithm -- 8.1 2-D Hex Board -- 8.2 n-D Hex Board -- 8.3 Combinatorial Equivalence -- 8.4 Two-Dimensional Hex Game Algorithm -- 8.5 Three-Dimensional Hex Game Algorithm -- 8.6 Higher-Dimensional Hex Games -- 9 Simplicial Algorithms -- 9.1 Simplicial Algorithms Over 2-D Uncertainty Space -- 9.2 Simplicial Algorithms Over 3-D Uncertainty Space -- 9.3 Relative Uncertainty Complex -- 9.4 Simplicial Labeling Map -- 9.5 Algorithm-Integer Search -- 9.6 Algorithm-Vector Labeling Search -- II: HOMOLOGY OF ROBUST STABILITY -- 10 Homology of Uncertainty and Other Spaces -- 10.1 Simplicial Homology -- 10.2 Semisimplicial Homology -- 10.3 Homology of a Chain Complex -- 10.4 Homotopy Invariance -- 10.5 Homology of Product of Uncertainty -- 10.6 Uncertainty Manifold-Mayer-Vietoris Sequence -- 10.7 Relative Homology Sequence -- 10.8 More Sophisticated Homology Computation -- 11 Homology of Crossover -- 11.1 Combinatorial Homology of Crossover -- 11.2 Projecting the Crossover -- 12 Cohomology -- 12.1 Simplicial Cohomology -- 12.2 de Rham Cohomology -- 13 Twisted Cartesian Product of Uncertainties -- 13.1 Fiber Bundle -- 13.2 Semisimplicial Bundles -- 13.3 Nyquist Fibration -- 13.4 Semisimplicial Fibration -- 13.5 Summary -- 14 Spectral Sequence of Nyquist Map -- 14.1 Homology Spectral Sequence -- 14.2 Example (Spectral Sequence of a Matrix) -- 14.3 Spectral Sequence of Geometric System Theory -- 14.4 Cohomology Spectral Sequence -- 14.5 Dihomology Spectral Sequence -- 14.6 Leray-Serre Spectral Sequence -- 14.7 Semisimplicial Serre Spectral Sequence of Nyquist Map.

14.8 Eilenberg-Moore Spectral Sequence -- III: HOMOTOPY OF ROBUST STABILITY -- 15 Homotopy Groups and Sequences -- 15.1 Homotopy Groups -- 15.2 Homotopy Group Homomorphism -- 15.3 Homotopy Groups of Spheres -- 15.4 Basic Obstruction Result -- 15.5 Homotopy Sequence of Nyquist Fibration -- 15.6 Corollaries of Exact Homotopy Sequence -- 15.7 Historical Notes -- 16 Obstruction to Extending the Nyquist Map -- 16.1 Statement of Nyquist Extension Problem -- 16.2 Obstruction to Extending a General Map -- 16.3 Obstruction to Extending Nyquist Map -- 16.4 Weak Converse -- 16.5 Computation of Homotopy Class -- 16.6 Homotopy Extension -- 16.7 Homotopy Extension and Edge Tests -- 16.8 Appendix-Obstruction to Cross Sectioning -- 17 Homotopy Classification of Nyquist Maps -- 17.1 Fundamental Classification Result -- 17.2 Classification of Maps to Spheres -- 17.3 Elementary Proof of Main Result -- 17.4 Cohomology of Product of Uncertainty -- 17.5 Formal Classification -- 18 Brouwer Degree of Nyquist Map -- 18.1 Orientation -- 18.2 Combinatorial Degree -- 18.3 Analytical Degree -- 18.4 Homological Degree of Maps Between Spheres -- 18.5 Simple Examples -- 18.6 Application (Index of Vector Field) -- 18.7 Cohomological Degree of Maps to Spheres -- 18.8 Degree Proof of Superstrong Sperner Lemma -- 18.9 Degree of Maps Between Pseudomanifolds -- 18.10 Homotopy Collapse of Template -- 18.11 Continuation or Embedding Methods -- 18.12 Historical Notes -- 19 Homotopy of Matrix Return Difference Map -- 19.1 Matrix Return Difference -- 19.2 General Linear versus Unitary Groups -- 19.3 Homotopy Groups of GL -- 19.4 Degree (Stable Homotopy Case) -- 19.5 Differential Degree -- 19.6 Example (the Principle of Argument) -- 19.7 Example (Mapping into SO) -- 19.8 Example (Brouwer Degree) -- 19.9 Example (McMillan Degree) -- 19.10 Obstruction to Extending GL-Valued Nyquist Map.

20 K-Theory of Robust Stabilization -- 20.1 Return Difference Operator -- 20.2 Index of Fredholm Operators -- 20.3 Index of Fredholm Toeplitz Operators -- 20.4 Index of Fredholm Family -- 20.5 K-Group -- 20.6 Index of Uncertain Return Difference Operator -- 20.7 Open-Loop Unstable Return Difference Operator -- 20.8 Reduced K-Groups -- 20.9 Unitary Approach to K-Theory -- 20.10 Higher K-Groups and Bott Periodicity -- 20.11 Index for Fredholm Toeplitz Family -- 20.12 Atiyah-Hirzebruch Spectral Sequence -- 20.13 KO-Theory of Real Perturbation -- 20.14 KR-Theory of Real Perturbation -- 20.15 Connection with Algebraic K-Theory -- IV: DIFFERENTIAL TOPOLOGY OF ROBUST STABILITY -- 21 Compact Differentiable Uncertainty Manifolds -- 21.1 Compact Differentiable Uncertainty Manifold -- 21.2 Singularity Analysis of Nyquist Map -- 21.3 Nyquist Curve as Critical Value Plot -- 21.4 Nash Functions -- 21.5 Sard's Theorem -- 21.6 Critical Values Plots -- 21.7 Loops of Critical Points -- 21.8 Degree Approach to Critical Points -- 21.9 Vector Field Approach to Critical Points -- 21.10 Quadratic Differential of Nyquist Map -- 21.11 Thom-Boardman Singularity Sets -- 21.12 The Case of Two Uncertain Parameters -- 21.13 Template Boundary Revisited -- 21.14 Example I -- 21.15 Example II -- 21.16 Cell Decomposition -- 22 Singularity Over Stratified Uncertainty Space -- 22.1 (Whitney) Stratified Uncertainty Space -- 22.2 Stratified Morse Theory -- 22.3 Boundary Singularity -- 22.4 Application to Mapping Theorems -- 23 Structural Stability of Crossover -- 23.1 Jet Space -- 23.2 Whitney Topology -- 23.3 (Elementary) Transversality -- 23.4 Singularity Sets Revisited -- 23.5 Universal Singularity Sets -- 23.6 Stability of Nyquist Map -- 23.7 Infinitesimal Stability -- 23.8 Local Infinitesimal Stability -- 23.9 Stability of Whitney Fold and Cusp -- 23.10 Example (Simple).

23.11 Example (Whitney Fold) -- 23.12 Example (Phase Margin) -- 23.13 Example (Uncertain Degree) -- 23.14 Example (Pole/Zero Cancellation) -- 23.15 Structural Stability of Crossover -- 23.16 The Counter-Example -- V: ALGEBRAIC GEOMETRY OF CROSSOVER -- 24 Geometry of Crossover -- 24.1 Crossover as a Real Algebraic Set -- 24.2 Triangulation of Real Algebraic Sets -- 24.3 Local Euler Characteristic of Real Algebraic Crossover Set -- 24.4 Betti Numbers of Real Algebraic Crossover Set -- 24.5 Algebraic Crossover Curve -- 24.6 Example -- 25 Geometry of Stability Boundary -- 25.1 Tarski-Seidenberg Elimination -- 25.2 Complexity -- 25.3 Example -- VI: EPILOGUE -- 26 Epilogue -- VII: APPENDICES -- A: Homological Algebra of Groups -- A.1 Abelian Groups and Homomorphisms -- A.2 Chain Complexes -- A.3 Tensor Product -- A.4 Categories and Functors -- A.5 Exact Sequences -- A.6 Free Resolution -- A.7 Connecting Morphism -- A.8 Torsion Product -- A.9 Universal Coefficient Theorem -- A. 10 Künneth Formula -- B: Matrix Analysis of Integral Homology Groups -- B.1 Matrix Computation of Homology Groups -- B.2 Hopf Trace Theorem -- C: Homological Algebra of Modules -- C.1 Modules -- C.2 Algebra -- C.3 Differential Graded Modules -- D: Algebraic Singularity Theory -- D.1 Weierstrass Preparation Theorem -- D.2 Malgrange Preparation Theorem -- D.3 Jets and Germs -- D.4 Rings and Ideals of Functions -- D.5 Formal Inverse Function Theorem -- D.6 Local Ring of a Map -- D.7 Modules Over Rings of Functions -- D.8 Generalized Malgrange Preparation Theorem -- D.9 Jacobi Ideal, Codimension, and Determinacy -- D.10 Universal Unfolding -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W.