Preface ......................................................... i
Acknowledgments ................................................ xi
Chapter 1 Basic Concepts ....................................... 1
1.1 Mathematical Model for Nonlinear Systems ................... 1
1.1.1 Existence and Uniqueness of Solutions ............... 4
1.2 Qualitative Behavior of Second-Order Linear Time-
Invariant Systems .......................................... 5
Chapter 2 Stability Analysis of Autonomous Systems ............ 11
2.1 System Preliminaries ...................................... 11
2.2 Lyapunov's Second Method for Autonomous Systems ........... 12
2.2.1 Lyapunov Function Generation for Linear Systems .... 15
2.3 Lyapunov Function Generation for Nonlinear Autonomous
Systems ................................................... 16
2.3.1 Aizerman's Method .................................. 19
2.3.2 Lure's Method ...................................... 21
2.3.3 Krasovskii's Method ................................ 25
2.3.4 Szego's Method ..................................... 27
2.3.5 Ingwerson's Method ................................. 34
2.3.6 Variable Gradient Method of Schultz and Gibson ..... 39
2.3.7 Reiss-Geiss's Method ............................... 45
2.3.8 Infante-Clark's Method ............................. 46
2.3.9 Energy Metric of Wall and Мое ...................... 51
2.3.10 Zubov's Method ..................................... 53
2.3.11 Leighton's Method .................................. 56
2.4 Relaxed Lyapunov Stability Conditions ..................... 58
2.4.1 LaSalle Invariance Principle ....................... 59
2.4.2 Average Decrement of the V(x) Function ............. 61
2.4.3 Vector Lyapunov Function ........................... 62
2.4.4 Higher-Order Derivatives of a Lyapunov Function
Candidate .......................................... 67
2.4.5 Stability Analysis of Nonlinear Homogeneous
Systems ............................................ 82
2.4.5.1 Homogeneity ............................... 82
2.4.5.2 Application of Higher-Order Derivatives
of Lyapunov Functions ..................... 84
2.4.5.3 Polynomial Δ-Homogeneous Systems of
Order k = 0 ............................... 88
2.4.5.4 The Δ-Homogeneous Polar Coordinate ........ 91
2.4.5.5 Numerical Examples ........................ 93
2.5 New Stability Theorems .................................... 96
2.5.1 Fathabadi-Nikravesh's Method ....................... 96
2.5.1.1 Low-Order Systems ......................... 96
2.5.1.2 Linear Systems ........................... 101
2.5.1.3 Higher-Order Systems ..................... 102
2.6 Lyapunov Stability Analysis of a Transformed
Nonlinear System .................................... 106
Endnotes ................................................. 116
Chapter 3 Stability Analysis of Nonautonomous Systems ........ 119
3.1 Preliminaries ............................................ 119
3.2 Relaxed Lyapunov Stability Conditions .................... 122
3.2.1 Average Decrement of Function ..................... 122
3.2.2 Vector Lyapunov Function .......................... 124
3.2.3 Higher-Order Derivatives of a Lyapunov Function
Candidate ......................................... 126
3.3 New Stability Theorems (Fathabadi-Nikravesh Time-
Varying Method) .......................................... 138
3.4 Application of Partial Stability Theory in Nonlinear
Nonautonomous System Stability Analysis .................. 143
3.4.1 Unified Stability Theory for Nonlinear Time-
Varying Systems .......................................... 149
Chapter 4 Stability Analysis of Time-Delayed Systems ......... 155
4.1 Preliminaries ............................................ 155
4.2 Stability Analysis of Linear Time-Delayed Systems ........ 159
4.2.1 Stability Analysis of Linear Time-Varying Time-
Delayed Systems ................................... 160
4.3 Delay-Dependent Stability Analysis of Nonlinear Time-
Delayed Systems .......................................... 166
4.3.1 Vali-Nikravesh Method of Generating the
Lyapunov-Krasovskii Functional for Delay-
Dependent System Stability Analysis ............... 167
Chapter 5 An Introduction to Stability Analysis of
Linguistic Fuzzy Dynamic Systems .............................. 187
5.1 TSK Fuzzy Model System's Stability Analysis .............. 187
5.2 Linguistic Fuzzy Stability Analysis Using a Fuzzy Petri
Net ...................................................... 190
5.2.1 Review of a Petri Net and Fuzzy Petri Net ......... 190
5.2.2 Appropriate Models for Linguistic Stability
Analysis .......................................... 192
5.2.2.1 The Infinite Place Model ................. 192
5.2.2.2 The BIBO Stability in the Infinite
Place Model .............................. 193
5.2.2.3 The Variation Model ...................... 193
5.2.3 The Necessary and Sufficient Condition for
Stability Analysis of a First-Order Linear
System Using Variation Models ..................... 194
5.2.4 Stability Criterion ............................... 196
5.3 Linguistic Model Stability Analysis ...................... 199
5.3.1 Definitions in Linguistic Calculus ................ 199
5.3.2 A Necessary and Sufficient Condition for
Stability Analysis of a Class of Applied
Mechanical Systems ................................ 201
5.3.3 A Necessary and Sufficient Condition for
Stability Analysis of a Class of Linguistic
Fuzzy Models ...................................... 204
5.4 Stability Analysis of Fuzzy Relational Dynamic Systems ... 208
5.4.1 Model Representation and Configuration ............ 209
5.4.2 Stability in an FRDS: An Analytical Glance ........ 211
5.5 Asymptotic Stability in a Sum-Prod FRDS .................. 216
5.6 Asymptotic Convergence to the Equilibrium State .......... 231
References ............................................... 239
Appendix A1 ................................................... 245
Appendix A2 ................................................... 257
Appendix A3 ................................................... 265
Appendix A4 ................................................... 269
Appendix A5 ................................................... 287
Index ......................................................... 299
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