Algebraic and Differential Topology of Robust Stability

Preface
	Simplicial Versus Singular Homology
	Cubes or Simplexes?
	Style
	Organization
	How to read this book
	Prerequisites
	Bibliographical Guide
	Acknowledgements
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
			2.6.1 No Imaginary-Axis Open-Loop Poles
			2.6.2 Imaginary-Axis Open-Loop Poles
		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)
			3.11.1 Locally Connected Boundary
			3.11.2 General Boundary
		3.12 Examples (Horowitz)
			3.12.1 Uncertain Gain/Real Pole
			3.12.2 Uncertain Pole/Zero Pair
		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)
			3.17.1 No Degree Uncertainty
			3.17.2 Uncertain Degree
	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
			4.5.1 Simplicial Theory
			4.5.2 Semisimplicial Theory
		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
			5.1.1 Simplicial Uncertainty-Frequency Product
			5.1.2 Semisimplicial Conceptualization
		5.2 Boundary of Cartesian Product
			5.2.1 Simplicial Approach
			5.2.2 Semisimplicial Approach
		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
			6.3.1 Coarse Initial Refinement
			6.3.2 Voronoi Diagram and Delaunay Triangulation
			6.3.3 Refinement and Point Location
			6.3.4 Checking Simplicial Property
			6.3.5 Identifying Simplex Containing the Origin
			6.3.6 Inverse Image
		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
			7.1.1 A Linear Program
			7.1.2 Polyhedral Crossover
		7.2 From Piecewise-Linear to Simplicial Map
			7.2.1 Simplicial Approximation to Piecewise-Linear Map
			7.2.2 Making Piecewise-Linear Map Simplicial
		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
			8.4.1 Primal
			8.4.2 Dual
			8.4.3 Complexity
		8.5 Three-Dimensional Hex Game Algorithm
			8.5.1 Dual
			8.5.2 Primal
		8.6 Higher-Dimensional Hex Games
	9. Simplicial Algorithms
		9.1 Simplicial Algorithms Over 2-D Uncertainty Space
			9.1.1 Integer Labeling
			9.1.2 Searching—Fundamental Graph Lemma
			9.1.3 Grid Refinement and Sperner's Lemma
			9.1.4 Vector Labeling
			9.1.5 Textbook Example
		9.2 Simplicial Algorithms Over 3-D Uncertainty Space
			9.2.1 Algorithm
			9.2.2 A 2-Torus Example
			9.2.3 Example
			9.2.4 Algebraic Curve Interpretation
		9.3 Relative Uncertainty Complex
		9.4 Simplicial Labeling Map
			9.4.1 Abstract Label Complex
			9.4.2 (Strong) Deformation Retract of Template
		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
			10.1.1 Homology Groups
			10.1.2 Homology Group Homomorphism
			10.1.3 Computation
		10.2 Semisimplicial Homology
		10.3 Homology of a Chain Complex
		10.4 Homotopy Invariance
			10.4.1 Chain Homotopy
			10.4.2 Acyclic Carriers
			10.4.3 Invariance Under Homotopy
			10.4.4 Homotopy Equivalence
		10.5 Homology of Product of Uncertainty
			10.5.1 Eilenberg-Zilber Theorem
			10.5.2 Künneth Theorem
			10.5.3 Remark
			10.5.4 Application—Uncertainty-Frequency Product
			10.5.5 Application—Uncertainty Torus
		10.6 Uncertainty Manifold—Mayer -Vietoris Sequence
			10.6.1 Application—Homology of Special Unitary Uncertainty
		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
			12.1.1 Cohomology Groups
			12.1.2 Cohomology Group Homomorphism
			12.1.3 Cup Product
			12.1.4 Cohomology of Product Space
		12.2 de Rham Cohomology
			12.2.1 Cohomology of Differential Forms
			12.2.2 Pull-Back
			12.2.3 Wedge Product
	13. Twisted Cartesian Product of Uncertainties
		13.1 Fiber Bundle
			13.1.1 Basic Definitions and Concepts
			13.1.2 Bundle Morphisms
			13.1.3 Clutching
			13.1.4 Cross Section
			13.1.5 Bundle Interpretation of Kharitonov's Theorem
			13.1.6 Principal Bundle
			13.1.7 Tangent Bundle
			13.1.8 Elementary Homotopy Theory of Bundles
			13.1.9 Fiber Bundle Interpretation of Doležal's Theorem
		13.2 Semisimplicial Bundles and Twisted Cartesian Product
			13.2.1 Background
			13.2.2 Semisimplicial Bundle
			13.2.3 Twisted Cartesian Product
			13.2.4 Example
			13.2.5 Cross Section
			13.2.6 Commutativity
			13.2.7 Twisted Tensor Product
		13.3 Nyquist Fibration
		13.4 Semisimplicial Fibration
		13.5 Summary
	14. Spectral Sequence of Nyquist Map
		14.1 Homology Spectral Sequence
			14.1.1 Graduation and Filtration
			14.1.2 Filtration of Cycle and Boundary Groups
			14.1.3 Filtration of Homology Group
			14.1.4 Successive Approximation
			14.1.5 Initialization
			14.1.6 Convergence
		14.2 Example (Spectral Sequence of a Matrix)
		14.3 Spectral Sequence of Geometric System Theory
		14.4 Cohomology Spectral Sequence
		14.5 Zeeman Dihomology Spectral Sequence of Simplicial Nyquist Map
		14.6 Leray-Serre Spectral Sequence
		14.7 Semisimplicial Serre Spectral Sequence of Nyquist Map
			14.7.1 Twisted Cartesian Product Spectral Sequence
			14.7.2 Twisted Tensor Product Spectral Sequence
		14.8 Eilenberg-Moore Spectral Sequence
			14.8.1 Contractible Template—Open-Loop Stable Case
			14.8.2 Uncontractible Template
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
			15.5.1 Exact Homotopy Sequence
		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
			16.2.1 Path Connectedness of Range
			16.2.2 Absolute Obstruction to Extension
			16.2.3 Relative Obstruction to Extension
		16.3 Obstruction to Extending Nyquist Map
			16.3.1 Absolute Results
			16.3.2 Relative Results
		16.4 Weak Converse
		16.5 Computation of Homotopy Class
			16.5.1 Piecewise-Linear Nyquist Map
			16.5.2 Comparison with Part I
			16.5.3 Relative Results
		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
			17.4.1 Multivariable Phase Margin
			17.4.2 Special Orthogonal Perturbation
		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
			18.5.1 Degree of a Linear Map
			18.5.2 Degree of a Holomorphic Function
			18.5.3 Degree of Real Polynomial Map
			18.5.4 Application to Robust Stability
		18.6 Application (Index of Vector Field)
		18.7 (Co)homological Degree of Maps to Spheres
			18.7.1 Degree of Nyquist-Related Map
			18.7.2 Degree of Maps from Manifolds to Spheres
			18.7.3 A Counterexample
		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
			19.3.1 Stable Homotopy (2n_l >= n)
			19.3.2 Unstable Homotopy (2n_l < n)
		19.4 Degree (Stable Homotopy Case)
			19.4.1 2n_l = n
			19.4.2 2n_l > n
		19.5 Differential Degree (Stable and Unstable Homotopy)
			19.5.1 Cohomology of General Linear Group
			19.5.2 de Rham Cohomology of Differential Forms
			19.5.3 Invariant Differential Forms on GL
			19.5.4 Invariant Differential Forms on U
			19.5.5 Example (SO Group)
			19.5.6 Pull-Back
			19.5.7 Degree
			19.5.8 Connection with Analytical Degree
		19.6 Example (the Principle of Argument)
		19.7 Example (Mapping into SO)
			19.7.1 Degree 1 Map
			19.7.2 Degree 2 Map
		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
			20.1.1 Open-Loop Stable, Discrete-Time Systems
			20.1.2 Toeplitz Operators
			20.1.3 Open-Loop Unstable Systems
			20.1.4 Closed-Loop Stability
		20.2 Index of Fredholm Operators
		20.3 Index of Fredholm Toeplitz Operators
		20.4 Index of Fredholm Family
			20.4.1 Constant Cokernel Dimension
			20.4.2 Vector Bundle Formulation
		20.5 K-Group
			20.5.1 Complex Bundle over Uncertainty Space
			20.5.2 Equivalent Bundles
			20.5.3 trivial bundle
			20.5.4 Whitney Sum
			20.5.5 Grothendieck Construction
			20.5.6 K-Group
			20.5.7 K-Group Homomorphism
		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
			20.9.1 Chern Classes and Character
		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. Singularity over Compact Differentiable Uncertainty Manifolds
		21.1 Compact Differentiable Uncertainty Manifold
		21.2 Singularity Analysis of Nyquist Map
			21.2.1 Variational Interpretation of Template Boundary
			21.2.2 Basic Facts of Morse Theory
			21.2.3 Degeneracy Phenomena
			21.2.4 Three Approaches to Singularity Analysis
		21.3 Nyquist Curve as Critical Value Plot
		21.4 Nash Functions
		21.5 Sard's Theorem
		21.6 Critical Values Plots
			21.6.1 The Problem
			21.6.2 Classification of Critical Points by Their Codimensions
			21.6.3 Isotopy
			21.6.4 Stratification of Space of Differentiable Functions
			21.6.5 Stratification of the Space of Morse Functions
			21.6.6 Local Properties of Family
			21.6.7 Global Properties of Family
			21.6.8 Effect of Variation of "Certain" Parameters
		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
			23.2.1 C^0 Case
			23.2.2 C^k and C^\infty Cases
		23.3 (Elementary) Transversality
		23.4 Singularity Sets Revisited
			23.4.1 Iterated Jacobi Extensions
			23.4.2 Jacobi Extension Definition of Singularity Sets
			23.4.3 Jacobian of a Set of Functions
		23.5 Universal Singularity Sets
			23.5.1 Total Jacobi Extension
			23.5.2 Universal Singularity Sets
			23.5.3 (Strong) Transversality
		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.1.1 Modules and Projective Modules
			C.1.2 Projective Resolution
			C.1.3 Tensor Product
			C.1.4 Higher Torsion Products
		C.2 Algebra
		C.3 Differential Graded Modules
			C.3.1 dg Modules
			C.3.2 dg Algebra
			C.3.3 dg Module Over dg Algebra
			C.3.4 Tensor Product
			C.3.5 Torsion Product
	D. Algebraic Singularity Theory
		D.1 Weierstrass Preparation Theorem
			D.1.1 Example (Root-Locus Breakaway Point)
			D.1.2 Proofs
		D.2 Malgrange Preparation Theorem
			D.2.1 Proofs
		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
	1-15
	16-33
	34-51
	52-69
	70-84
	85-101
	102-119
	120-138
	139-154
	155-171
	172-189
	190-207
	208-225
	226-239
Index