An Introduction
Buch, Englisch, 474 Seiten, Format (B × H): 183 mm x 260 mm, Gewicht: 1079 g
ISBN: 978-0-691-13444-4
Verlag: Princeton University Press
Stability and Stabilization is the first intermediate-level textbook that covers stability and stabilization of equilibria for both linear and nonlinear time-invariant systems of ordinary differential equations. Designed for advanced undergraduates and beginning graduate students in the sciences, engineering, and mathematics, the book takes a unique modern approach that bridges the gap between linear and nonlinear systems. Presenting stability and stabilization of equilibria as a core problem of mathematical control theory, the book emphasizes the subject's mathematical coherence and unity, and it introduces and develops many of the core concepts of systems and control theory. There are five chapters on linear systems and nine chapters on nonlinear systems; an introductory chapter; a mathematical background chapter; a short final chapter on further reading; and appendixes on basic analysis, ordinary differential equations, manifolds and the Frobenius theorem, and comparison functions and their use in differential equations. The introduction to linear system theory presents the full framework of basic state-space theory, providing just enough detail to prepare students for the material on nonlinear systems.Focuses on stability and feedback stabilization Bridges the gap between linear and nonlinear systems for advanced undergraduates and beginning graduate students Balances coverage of linear and nonlinear systems Covers cascade systems Includes many examples and exercises
Autoren/Hrsg.
Fachgebiete
- Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
- Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik
- Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
Weitere Infos & Material
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 Sta
