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(s): Seyed Kamaleddin Yadavar Nikravesh
Publisher: CRC Press
Year: 2013
Language: English
Pages: xii+308
Nonlinear Systems Stability Analysis: Lyapunov-Based Approach......Page 4
Contents......Page 6
Preface......Page 10
Acknowledgments......Page 12
1.1 Mathematical Model for Nonlinear Systems......Page 14
1.1.1 Existence and Uniqueness of Solutions [k1]......Page 17
1.2 Qualitative Behavior of Second-Order Linear Time-Invariant Systems......Page 18
Problems......Page 21
2.1 System Preliminaries......Page 24
2.2 Lyapunov’s Second Method for Autonomous Systems......Page 25
2.2.1 Lyapunov Function Generation for Linear Systems......Page 28
2.3 Lyapunov Function Generation for Nonlinear Autonomous Systems [n1]³......Page 29
2.3.1 Aizerman’s Method [n1,l1]......Page 32
2.3.2 Lure’s Method......Page 34
2.3.3 Krasovskii’s Method [a11,n1,o1]......Page 38
2.3.4 Szego’s Method [n1,l1,r1]......Page 40
2.3.5 Ingwerson’s Method [n1,i1,b1,g2,l3,r2]......Page 47
2.3.6 Variable Gradient Method of Schultz and Gibson [b5, n1,l1,o1,g1,g2,l3,s3]......Page 52
2.3.7 Reiss–Geiss’s Method [n1,r4]......Page 58
2.3.8 Infante–Clark’s Method [n1,i2]......Page 59
2.3.9 Energy Metric of Wall and Moe [n1,w2,p2]......Page 64
2.3.10 Zubov’s Method [n1,b1,o1,g2,l3,r2,m1,z2]......Page 66
2.3.11 Leighton’s Method [n1,l4,a1]......Page 69
2.4 Relaxed Lyapunov Stability Conditions......Page 71
2.4.1 LaSalle Invariance Principle......Page 72
2.4.2 Average Decrement of the V(x) Function......Page 74
2.4.3 Vector Lyapunov Function......Page 75
2.4.4 Higher-Order Derivatives of a Lyapunov Function Candidate [m13]......Page 80
2.4.5.1 Homogeneity......Page 95
2.4.5.2 Application of Higher-Order Derivatives of Lyapunov Functions......Page 97
2.4.5.3 Polynomial Δ-Homogeneous Systems of Order k=0......Page 101
2.4.5.4 The Δ-Homogeneous Polar Coordinate......Page 104
2.4.5.5 Numerical Examples......Page 106
2.5.1.1 Low-Order Systems......Page 109
2.5.1.2 Linear Systems......Page 114
2.5.1.3 Higher-Order Systems......Page 115
2.6 Lyapunov Stability Analysis of a Transformed Nonlinear System......Page 119
Problems......Page 125
Endnotes......Page 129
3.1 Preliminaries......Page 132
3.2.1 Average Decrement of Function......Page 135
3.2.2 Vector Lyapunov Function......Page 137
3.2.3 Higher-Order Derivatives of a Lyapunov Function Candidate......Page 139
3.3 New Stability Theorems (Fathabadi–Nikravesh Time-Varying Method)......Page 151
3.4 Application of Partial Stability Theory in Nonlinear Nonautonomous System Stability Analysis [c1]......Page 156
3.4.1 Unified Stability Theory for Nonlinear Time-Varying Systems......Page 162
Problems......Page 166
4.1 Preliminaries [h4]......Page 168
4.2 Stability Analysis of Linear Time-Delayed Systems [p4]......Page 172
4.2.1 Stability Analysis of Linear Time-Varying Time-Delayed Systems......Page 173
4.3 Delay-Dependent Stability Analysis of Nonlinear Time-Delayed Systems [v2,v3]......Page 179
4.3.1 Vali–Nikravesh Method of Generating the Lyapunov–Krasovskii Functional for Delay-Dependent System Stability Analysis......Page 180
Problems......Page 197
5.1 TSK Fuzzy Model System’s Stability Analysis......Page 200
5.2.1 Review of a Petri Net and Fuzzy Petri Net......Page 203
5.2.2.1 The Infinite Place Model......Page 205
5.2.2.3 The Variation Model......Page 206
5.2.3 The Necessary and Sufficient Condition for Stability Analysis of a First-Order Linear System Using Variation Models......Page 207
5.2.4 Stability Criterion......Page 209
5.3.1 Definitions in Linguistic Calculus......Page 212
5.3.2 A Necessary and Sufficient Condition for Stability Analysis of a Class of Applied Mechanical Systems......Page 214
5.3.3 A Necessary and Sufficient Condition for Stability Analysis of a Class of Linguistic Fuzzy Models......Page 217
5.4 Stability Analysis of Fuzzy Relational Dynamic Systems......Page 221
5.4.1 Model Representation and Configuration......Page 222
5.4.2 Stability in an FRDS: An Analytical Glance......Page 224
5.5 Asymptotic Stability in a Sum-Prod FRDS......Page 229
5.6 Asymptotic Convergence to the Equilibrium State......Page 244
Problems......Page 250
References......Page 252
Appendix A1: Application of VLF in Nonlinear Power System Stabilization......Page 258
A1.1 Application to Multi-Machine Power Systems......Page 263
Appendix A2: Proof of Theorem 3.8......Page 270
Appendix A3: Stability Analysis of Nonlinear Systems via Δ-Homogeneous Approximation......Page 278
A4.1 Stabilizing Predictive Control of Input Constrained Nonlinear Time-Delayed Systems......Page 282
A4.1.2 Proposed Model Predictive Controller......Page 283
A4.1.3 Stability Analysis of a Closed-Loop System......Page 284
A4.1.4 Calculation of Terminal Region and Terminal Cost......Page 288
A4.2.1 Proposed Predictive Controller......Page 294
A4.2.2 Stability Analysis......Page 296
A5.1 Lemmas and Theorems......Page 300
Index......Page 312
