Nonlinear Systems Analysis / Edition 2

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When M. Vidyasagar wrote the first edition of Nonlinear Systems Analysis, most control theorists considered the subject of nonlinear systems a mystery. Since then, advances in the application of differential geometric methods to nonlinear analysis have matured to a stage where every control theorist needs to possess knowledge of the basic techniques because virtually all physical systems are nonlinear in nature. The second edition, now republished in SIAM's Classics in Applied Mathematics series, provides a rigorous mathematical analysis of the behavior of nonlinear control systems under a variety of situations. It develops nonlinear generalizations of a large number of techniques and methods widely used in linear control theory. The book contains three extensive chapters devoted to the key topics of Lyapunov stability, input-output stability, and the treatment of differential geometric control theory. In addition, it includes valuable reference material in these chapters that is unavailable elsewhere. The text also features a large number of problems that allow readers to test their understanding of the subject matter and self-contained sections and chapters that allow readers to focus easily on a particular topic.
About the Authors

M. Vidyasagar is Executive Vice President of Tata Consultancy Services in Hyderabad, India. He has held numerous academic visiting positions with universities such as MIT, the University of California, and Tokyo Institute of Technology and has also held various consultancy positions within the field. In addition, he has published many articles in linear, nonlinear, and robust control theory; robotics; and statistical learning theory. He has authored or co-authored more than a half dozen books on these topics.
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Product Details

  • ISBN-13: 9780136234630
  • Publisher: Prentice Hall Professional Technical Reference
  • Publication date: 7/28/1992
  • Edition description: 2nd ed
  • Edition number: 2
  • Pages: 498

Table of Contents

Note to the Reader
1 Introduction 1
2 Nonlinear Differential Equations 6
2.1 Mathematical Preliminaries 6
2.2 Induced Norms and Matrix Measures 19
2.3 Contraction Mapping Theorem 27
2.4 Nonlinear Differential Equations 33
2.5 Solution Estimates 46
3 Second-Order Systems 53
3.1 Preliminaries 53
3.2 Linearization Method 57
3.3 Periodic Solutions 67
3.4 Two Analytical Approximation Methods 79
4 Approximate Analysis Methods 88
4.1 Describing Functions 88
4.2 Periodic Solutions: Rigorous Arguments 109
4.3 Singular Perturbations 127
5 Lyapunov Stability 135
5.1 Stability Definitions 135
5.2 Some Preliminaries 147
5.3 Lyapunov's Linearization Method 157
5.4 Stability of Linear Systems 193
5.5 Lyapunov's Linearization Method 209
5.6 The Lur'e Problem 219
5.7 Converse Theorems 235
5.8 Applications of Converse Theorems 246
5.9 Discrete-Time Systems 264
6 Input-Output Stability 270
6.1 L [Subscript p] Spaces and their Extensions 271
6.2 Definitions of Input-Output Stability 277
6.3 Relationships Between I/O and Lyapunov Stability 284
6.4 Open-Loop Stability of Linear Systems 292
6.5 Linear Time-Invariant Feedback Systems 309
6.6 Time-Varying and/or Nonlinear Systems 337
6.7 Discrete-Time Systems 365
7 Differential Geometric Methods 376
7.1 Basics of Differential Geometry 377
7.2 Distributions, Frobenius Theorem 392
7.3 Reachability and Observability 399
7.4 Feedback Linearization: Single-Input Case 427
7.5 Feedback Linearization: Multi-Input Case 438
7.6 Input-Output Linearization 456
7.7 Stabilization of Linearizable Systems 464
A. Prevalence of Differential Equations With Unique Solutions 469
B. Proof of the Kalman-Yacubovitch Lemma 474
C. Proof of the Frobenius Theorem 476
References 486
Index 493
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