Cover image for Nonlinear systems stability analysis : Lyapunov-based approach
Nonlinear systems stability analysis : Lyapunov-based approach
Publication Information:
Boca Raton : CRC Press, Taylor & Franic Group, 2013
Physical Description:
xi, 307 pages ; 24 cm.


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30000010327949 QA402.35 N55 2013 Open Access Book Book

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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

Prefacep. ix
Acknowledgmentsp. xi
Chapter 1 Basic Conceptsp. 1
1.1 Mathematical Model for Nonlinear Systemsp. 1
1.1.1 Existence and Uniqueness of Solutionsp. 4
1.2 Qualitative Behavior of Second-Order Linear Time-Invariant Systemsp. 5
Chapter 2 Stability Analysis of Autonomous Systemsp. 11
2.1 System Preliminariesp. 11
2.2 Lyapunov's Second Method for Autonomous Systemsp. 12
2.2.1 Lyapunov Function Generation for Linear Systemsp. 15
2.3 Lyapunov Function Generation for Nonlinear Autonomous Systemsp. 16
2.3.1 Aizerman's Methodp. 19
2.3.2 Lure's Methodp. 21
2.3.3 Krasovskii's Methodp. 25
2.3.4 Szego's Methodp. 27
2.3.5 Ingwerson's Methodp. 34
2.3.6 Variable Gradient Method of Schultz and Gibsonp. 39
2.3.7 Reiss-Geiss's Methodp. 45
2.3.8 Infante-Clark's Methodp. 46
2.3.9 Energy Metric of Wall and Moep. 51
2.3.10 Zubov's Methodp. 53
2.3.11 Leighton's Methodp. 56
2.4 Relaxed Lyapunov Stability Conditionsp. 58
2.4.1 LaSalle Invariance Principlep. 59
2.4.2 Average Decrement of the V(x) Functionp. 61
2.4.3 Vector Lyapunov Functionp. 62
2.4.4 Higher-Order Derivatives a Lyapunov Function Candidatep. 67
2.4.5 Stability Analysis of Nonlinear Homogeneous Systemsp. 82 Homogeneityp. 82 Application of Higher-Order Derivatives of Lyapunov Functionsp. 84 Polynomial ¿-Homogeneous Systems of Order k = 0p. 88 The ¿-Homogeneous Polar Coordinatep. 91 Numerical Examplesp. 93
2.5 New Stability Theoremsp. 96
2.5.1 Fathabadi-Nikravesh's Methodp. 96 Low-Order Systemsp. 96 Linear Systemsp. 101 Higher-Order Systemsp. 102
2.6 Lyapunov Stability Analysis of a Transformed Nonlinear Systemp. 106
Endnotesp. 116
Chapter 3 Stability Analysis of Nonautonomous Systemsp. 119
3.1 Preliminariesp. 119
3.2 Relaxed Lyapunov Stability Conditionsp. 122
3.2.1 Average Decrement of Functionp. 122
3.2.2 Vector Lyapunov Functionp. 124
3.2.3 Higher-Order Derivatives of a Lyapunov Function Candidatep. 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 Analysisp. 143
3.4.1 Unified Stability Theory for Nonlinear Time-Varying Systemsp. 149
Chapter 4 Stability Analysis of Time-Delayed Systemsp. 155
4.1 Preliminariesp. 155
4.2 Stability Analysis of Linear Time-Delayed Systemsp. 159
4.2.1 Stability Analysis of Linear Time-Varying Time-Delayed Systemsp. 160
4.3 Delay-Dependent Stability Analysis of Nonlinear Time-Delayed Systemsp. 166
4.3.1 Vali-Nikravesh Method of Generating the Lyapunov-Krasovskii Functional for Delay-Dependent System Stability Analysisp. 167
Chapter 5 An Introduction to Stability Analysis of Linguistic Fuzzy Dynamic Systemsp. 187
5.1 TSK Fuzzy Model System's Stability Analysisp. 187
5.2 Linguistic Fuzzy Stability Analysis Using a Fuzzy Petri Netp. 190
5.2.1 Review of a Petri Net and Fuzzy Petri Netp. 190
5.2.2 Appropriate Models for Linguistic Stability Analysisp. 192 The Infinite Place Modelp. 192 The BIBO Stability in the Infinite Place Modelp. 193 The Variation Modelp. 193
5.2.3 The Necessary and Sufficient Condition for Stability Analysis of a First-Order Linear System Using Variation Modelsp. 194
5.2.4 Stability Criterionp. 196
5.3 Linguistic Model Stability Analysisp. 199
5.3.1 Definitions in Linguistic Calculusp. 199
5.3.2 A Necessary and Sufficient Condition for Stability Analysis of a Class of Applied Mechanical Systemsp. 201
5.3.3 A Necessary and Sufficient Condition for Stability Analysis of a Class of Linguistic Fuzzy Modelsp. 204
5.4 Stability Analysis of Fuzzy Relational Dynamic Systemsp. 208
5.4.1 Model Representation and Configurationp. 209
5.4.2 Stability in an FRDS: An Analytical Glancep. 211
5.5 Asymptotic Stability in a Sum-Prod FRDSp. 216
5.6 Asymptotic Convergence to the Equilibrium Statep. 231
Referencesp. 239
Appendix Al

p. 245

Appendix A2

p. 257

Appendix A3

p. 265

Appendix A4

p. 269

Appendix A5

p. 287

Indexp. 299