Stability and Stabilization: An Introduction

List of Figures xi
Preface xiii

Chapter 1: Introduction 1
1.1 Open Loop Control 1
1.2 The Feedback Stabilization Problem 2
1.3 Chapter and Appendix Descriptions 5
1.4 Notes and References 11

Chapter 2: Mathematical Background 12
2.1 Analysis Preliminaries 12
2.2 Linear Algebra and Matrix Algebra 12
2.3 Matrix Analysis 17
2.4 Ordinary Differential Equations 30
2.4.1 Phase Plane Examples: Linear and Nonlinear 35
2.5 Exercises 44
2.6 Notes and References 48

Chapter 3: Linear Systems and Stability 49
3.1 The Matrix Exponential 49
3.2 The Primary Decomposition and Solutions of LTI Systems 53
3.3 Jordan Form and Matrix Exponentials 57
3.3.1 Jordan Form of Two-Dimensional Systems 58
3.3.2 Jordan Form of n-Dimensional Systems 61
3.4 The Cayley-Hamilton Theorem 67
3.5 Linear Time Varying Systems 68
3.6 The Stability Definitions 71
3.6.1 Motivations and Stability Definitions 71
3.6.2 Lyapunov Theory for Linear Systems 73
3.7 Exercises 77
3.8 Notes and References 81

Chapter 4: Controllability of Linear Time Invariant Systems 82
4.1 Introduction 82
4.2 Linear Equivalence of Linear Systems 84
4.3 Controllability with Scalar Input 88
4.4 Eigenvalue Placement with Single Input 92
4.5 Controllability with Vector Input 94
4.6 Eigenvalue Placement with Vector Input 96
4.7 The PBH Controllability Test 99
4.8 Linear Time Varying Systems: An Example 103
4.9 Exercises 105
4.10 Notes and References 108

Chapter 5: Observability and Duality 109
5.1 Observability, Duality, and a Normal Form 109
5.2 Lyapunov Equations and Hurwitz Matrices 117
5.3 The PBH Observability Test 118
5.4 Exercises 121
5.5 Notes and References 123

Chapter 6: Stabilizability of LTI Systems 124
6.1 Stabilizing Feedbacks for Controllable Systems 124
6.2 Limitations on Eigenvalue Placement 128
6.3 The PBH Stabilizability Test 133
6.4 Exercises 134
6.5 Notes and References 136

Chapter 7: Detectability and Duality 138
7.1 An Example of an Observer System 138
7.2 Detectability, the PBH Test, and Duality 142
7.3 Observer-Based Dynamic Stabilization 145
7.4 Linear Dynamic Controllers and Stabilizers 147
7.5 LQR and the Algebraic Riccati Equation 152
7.6 Exercises 156
7.7 Notes and References 159

Chapter 8: Stability Theory 161
8.1 Lyapunov Theorems and Linearization 161
8.1.1 Lyapunov Theorems 162
8.1.2 Stabilization from the Jacobian Linearization 171
8.1.3 Brockett's Necessary Condition 172
8.1.4 Examples of Critical Problems 173
8.2 The Invariance Theorem 176
8.3 Basin of Attraction 181
8.4 Converse Lyapunov Theorems 183
8.5 Exercises 183
8.6 Notes and References 187

Chapter 9: Cascade Systems 189
9.1 The Theorem on Total Stability 189
9.1.1 Lyapunov Stability in Cascade Systems 192
9.2 Asymptotic Stability in Cascades 193
9.2.1 Examples of Planar Systems 193
9.2.2 Boundedness of Driven Trajectories 196
9.2.3 Local Asymptotic Stability 199
9.2.4 Boundedness and Global Asymptotic Stability 202
9.3 Cascades by Aggregation 204
9.4 Appendix: The Poincar´e-Bendixson Theorem 207
9.5 Exercises 207
9.6 Notes and References 211

Chapter 10: Center Manifold Theory 212
10.1 Introduction 212
10.1.1 An Example 212
10.1.2 Invariant Manifolds 213
10.1.3 Special Coordinates for Critical Problems 214
10.2 The Main Theorems 215
10.2.1 Definition and Existence of Center Manifolds 215
10.2.2 The Reduced Dynamics 218
10.2.3 Approximation of a Center Manifold 222
10.3 Two Applications 225
10.3.1 Adding an Integrator for Stabilization 226
10.3.2 LAS in Special Cascades: Center Manifold Argument 228
10.4 Exercises 229
10.5 Notes and References 231

Chapter 11: Zero Dynamics 233
11.1 The Relative Degree and Normal Form 233
11.2 The Zero Dynamics Subsystem 244
11.3 Zero Dynamics and Stabilization 248
11.4 Vector Relative Degree of MIMO Systems 251
11.5 Two Applications 254
11.5.1 Designing a Center Manifold 254
11.5.2 Zero Dynamics for Linear SISO Systems 257
11.6 Exercises 263
11.7 Notes and References 267

Chapter 12: Feedback Linearization of Single-Input Nonlinear Systems 268
12.1 Introduction 268
12.2 Input-State Linearization 270
12.2.1 Relative Degree n 271
12.2.2 Feedback Linearization and Relative Degree n 272
12.3 The Geometric Criterion 275
12.4 Linearizing Transformations 282
12.5 Exercises 285
12.6 Notes and References 287

Chapter 13: An Introduction to Damping Control 289
13.1 Stabilization by Damping Control 289
13.2 Contrasts with Linear Systems: Brackets, Controllability,
Stabilizability 296
13.3 Exercises 299
13.4 Notes and References 300

Chapter 14: Passivity 302
14.1 Introduction to Passivity 302
14.1.1 Motivation and Examples 302
14.1.2 Definition of Passivity 304
14.2 The KYP Characterization of Passivity 306
14.3 Positive Definite Storage 309
14.4 Passivity and Feedback Stabilization 314
14.5 Feedback Passivity 318
14.5.1 Linear Systems 321
14.5.2 Nonlinear Systems 325
14.6 Exercises 327
14.7 Notes and References 330

Chapter 15: Partially Linear Cascade Systems 331
15.1 LAS from Partial-State Feedback 331
15.2 The Interconnection Term 333
15.3 Stabilization by Feedback Passivation 336
15.4 Integrator Backstepping 349
15.5 Exercises 355
15.6 Notes and References 357

Chapter 16: Input-to-State Stability 359
16.1 Preliminaries and Perspective 359
16.2 Stability Theorems via Comparison Functions 364
16.3 Input-to-State Stability 366
16.4 ISS in Cascade Systems 372
16.5 Exercises 374
16.6 Notes and References 376

Chapter 17: Some Further Reading 378

Appendix A: Notation: A Brief Key 381
Appendix B: Analysis in R and Rⁿ 383
B.1 Completeness and Compactness 386
B.2 Differentiability and Lipschitz Continuity 393
Appendix C: Ordinary Differential Equations 393
C.1 Existence and Uniqueness of Solutions 393
C.2 Extension of Solutions 396
C.3 Continuous Dependence 399
Appendix D: Manifolds and the Preimage Theorem; Distributions and the Frobenius Theorem 403
D.1 Manifolds and the Preimage Theorem 403
D.2 Distributions and the Frobenius Theorem 410
Appendix E: Comparison Functions and a Comparison Lemma 420
E.1 Definitions and Basic Properties 420
E.2 Differential Inequality and Comparison Lemma 424
Appendix F: Hints and Solutions for Selected Exercises 430

Bibliography 443
Index 451