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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010327949 | QA402.35 N55 2013 | Open Access Book | Book | Searching... |
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Summary
Summary
The equations used to describe dynamic properties of physical systems are often nonlinear, and it is rarely possible to find their solutions. Although numerical solutions are impractical and graphical techniques are not useful for many types of systems, there are different theorems and methods that are useful regarding qualitative properties of nonlinear systems and their solutions--system stability being the most crucial property. Without stability, a system will not have value.
Nonlinear Systems Stability Analysis: Lyapunov-Based Approach introduces advanced tools for stability analysis of nonlinear systems. It presents the most recent progress in stability analysis and provides a complete review of the dynamic systems stability analysis methods using Lyapunov approaches. The author discusses standard stability techniques, highlighting their shortcomings, and also describes recent developments in stability analysis that can improve applicability of the standard methods. The text covers mostly new topics such as stability of homogonous nonlinear systems and higher order Lyapunov functions derivatives for stability analysis. It also addresses special classes of nonlinear systems including time-delayed and fuzzy systems.
Presenting new methods, this book provides a nearly complete set of methods for constructing Lyapunov functions in both autonomous and nonautonomous systems, touching on new topics that open up novel research possibilities. Gathering a body of research into one volume, this text offers information to help engineers design stable systems using practice-oriented methods and can be used for graduate courses in a range of engineering disciplines.
Author Notes
Seyyed Kamaleddin Yadavar Nikravesh, Ph.D., is a professor in the electrical engineering department at Amirkabir University of Technology. His research interests include dynamic and biomedical modeling, system stability, and system optimization. He has published five different books on electrical circuit analysis, optimal control systems, industrial control system analysis, industrial control system synthesis and design, and system stability analysis: Lyapounov-based approach. He has also published more than 180 journal and conference papers, mostly in systems modeling, and system stability analysis and synthesis, which form the main structure of his present book.
Table of Contents
| Preface | p. ix |
| Acknowledgments | p. xi |
| Chapter 1 Basic Concepts | p. 1 |
| 1.1 Mathematical Model for Nonlinear Systems | p. 1 |
| 1.1.1 Existence and Uniqueness of Solutions | p. 4 |
| 1.2 Qualitative Behavior of Second-Order Linear Time-Invariant Systems | p. 5 |
| Chapter 2 Stability Analysis of Autonomous Systems | p. 11 |
| 2.1 System Preliminaries | p. 11 |
| 2.2 Lyapunov's Second Method for Autonomous Systems | p. 12 |
| 2.2.1 Lyapunov Function Generation for Linear Systems | p. 15 |
| 2.3 Lyapunov Function Generation for Nonlinear Autonomous Systems | p. 16 |
| 2.3.1 Aizerman's Method | p. 19 |
| 2.3.2 Lure's Method | p. 21 |
| 2.3.3 Krasovskii's Method | p. 25 |
| 2.3.4 Szego's Method | p. 27 |
| 2.3.5 Ingwerson's Method | p. 34 |
| 2.3.6 Variable Gradient Method of Schultz and Gibson | p. 39 |
| 2.3.7 Reiss-Geiss's Method | p. 45 |
| 2.3.8 Infante-Clark's Method | p. 46 |
| 2.3.9 Energy Metric of Wall and Moe | p. 51 |
| 2.3.10 Zubov's Method | p. 53 |
| 2.3.11 Leighton's Method | p. 56 |
| 2.4 Relaxed Lyapunov Stability Conditions | p. 58 |
| 2.4.1 LaSalle Invariance Principle | p. 59 |
| 2.4.2 Average Decrement of the V(x) Function | p. 61 |
| 2.4.3 Vector Lyapunov Function | p. 62 |
| 2.4.4 Higher-Order Derivatives a Lyapunov Function Candidate | p. 67 |
| 2.4.5 Stability Analysis of Nonlinear Homogeneous Systems | p. 82 |
| 2.4.5.1 Homogeneity | p. 82 |
| 2.4.5.2 Application of Higher-Order Derivatives of Lyapunov Functions | p. 84 |
| 2.4.5.3 Polynomial ¿-Homogeneous Systems of Order k = 0 | p. 88 |
| 2.4.5.4 The ¿-Homogeneous Polar Coordinate | p. 91 |
| 2.4.5.5 Numerical Examples | p. 93 |
| 2.5 New Stability Theorems | p. 96 |
| 2.5.1 Fathabadi-Nikravesh's Method | p. 96 |
| 2.5.1.1 Low-Order Systems | p. 96 |
| 2.5.1.2 Linear Systems | p. 101 |
| 2.5.1.3 Higher-Order Systems | p. 102 |
| 2.6 Lyapunov Stability Analysis of a Transformed Nonlinear System | p. 106 |
| Endnotes | p. 116 |
| Chapter 3 Stability Analysis of Nonautonomous Systems | p. 119 |
| 3.1 Preliminaries | p. 119 |
| 3.2 Relaxed Lyapunov Stability Conditions | p. 122 |
| 3.2.1 Average Decrement of Function | p. 122 |
| 3.2.2 Vector Lyapunov Function | p. 124 |
| 3.2.3 Higher-Order Derivatives of a Lyapunov Function Candidate | p. 126 |
| 3.3 New Stability Theorems (Fathabadi-Nikravesh Time-Varying Method) | p. 138 |
| 3.4 Application of Partial Stability Theory in Nonlinear Nonautonomous System Stability Analysis | p. 143 |
| 3.4.1 Unified Stability Theory for Nonlinear Time-Varying Systems | p. 149 |
| Chapter 4 Stability Analysis of Time-Delayed Systems | p. 155 |
| 4.1 Preliminaries | p. 155 |
| 4.2 Stability Analysis of Linear Time-Delayed Systems | p. 159 |
| 4.2.1 Stability Analysis of Linear Time-Varying Time-Delayed Systems | p. 160 |
| 4.3 Delay-Dependent Stability Analysis of Nonlinear Time-Delayed Systems | p. 166 |
| 4.3.1 Vali-Nikravesh Method of Generating the Lyapunov-Krasovskii Functional for Delay-Dependent System Stability Analysis | p. 167 |
| Chapter 5 An Introduction to Stability Analysis of Linguistic Fuzzy Dynamic Systems | p. 187 |
| 5.1 TSK Fuzzy Model System's Stability Analysis | p. 187 |
| 5.2 Linguistic Fuzzy Stability Analysis Using a Fuzzy Petri Net | p. 190 |
| 5.2.1 Review of a Petri Net and Fuzzy Petri Net | p. 190 |
| 5.2.2 Appropriate Models for Linguistic Stability Analysis | p. 192 |
| 5.2.2.1 The Infinite Place Model | p. 192 |
| 5.2.2.2 The BIBO Stability in the Infinite Place Model | p. 193 |
| 5.2.2.3 The Variation Model | p. 193 |
| 5.2.3 The Necessary and Sufficient Condition for Stability Analysis of a First-Order Linear System Using Variation Models | p. 194 |
| 5.2.4 Stability Criterion | p. 196 |
| 5.3 Linguistic Model Stability Analysis | p. 199 |
| 5.3.1 Definitions in Linguistic Calculus | p. 199 |
| 5.3.2 A Necessary and Sufficient Condition for Stability Analysis of a Class of Applied Mechanical Systems | p. 201 |
| 5.3.3 A Necessary and Sufficient Condition for Stability Analysis of a Class of Linguistic Fuzzy Models | p. 204 |
| 5.4 Stability Analysis of Fuzzy Relational Dynamic Systems | p. 208 |
| 5.4.1 Model Representation and Configuration | p. 209 |
| 5.4.2 Stability in an FRDS: An Analytical Glance | p. 211 |
| 5.5 Asymptotic Stability in a Sum-Prod FRDS | p. 216 |
| 5.6 Asymptotic Convergence to the Equilibrium State | p. 231 |
| References | p. 239 |
| Appendix Al p. 245 | |
| Appendix A2 p. 257 | |
| Appendix A3 p. 265 | |
| Appendix A4 p. 269 | |
| Appendix A5 p. 287 | |
| Index | p. 299 |
