Constructions of Strict Lyapunov Functions (eBook)

Preface 7
Contents 10
Part I Background 16
Chapter 1 Background on Nonlinear Systems 17
1.1 Preliminaries 17
1.2 Families of Nonlinear Systems 18
1.3 Notions of Stability 22
1.4 Stabilization 27
1.5 Examples 29
1.6 Comments 34
Chapter 2 Review of Lyapunov Functions 38
2.1 Strict Lyapunov Function 38
2.2 Non-strict Lyapunov Function 44
2.3 Discrete Time Lyapunov Function 48
2.4 Illustrations 49
2.5 Basin of Attraction Revisited 57
2.6 L2 Gains 58
2.7 Lyapunov Functions with Bounded Gradients 60
2.8 Comments 66
Part II Time-Invariant Case 71
Chapter 3 Matrosov Conditions: Simple Case 72
3.1 Motivation 72
3.2 Continuous Time Theorem 75
3.3 Proof of Continuous Time Theorem 77
3.4 Discrete Time Theorem 80
3.5 Proof of Discrete Time Theorem 82
3.6 Illustrations 86
3.7 Comments 91
Chapter 4 Jurdjevic-Quinn Conditions 93
4.1 Motivation 93
4.2 Control Affine Case 95
4.3 General Case 103
4.5 Hamiltonian Systems 108
4.6 Robustness 112
4.7 Illustrations 117
4.8 Comments 124
Chapter 5 Systems Satisfying the Conditions of LaSalle 126
5.1 Background and Motivation 126
5.2 First Method: Iterated Lie Derivatives 127
5.3 Discussion and Extensions of First Method 133
5.4 Second Method: Matrosov Conditions 135
5.5 Application: Lotka-Volterra Model 138
5.6 Comments 145
Part III Time-Varying Case 147
Chapter 6 Stricti.cation: Basic Results 148
6.1 Background 148
6.2 Motivating Example 150
6.3 Time-Varying Stricti.cation Theorem 152
6.4 Remarks on Rate of Convergence 157
6.6 Equivalent Characterizations of Non-strict ISS 162
6.7 Input-to-Output Stability 165
6.8 Illustrations 167
6.9 Comments 178
Chapter 7 Backstepping for Time-Varying Systems 182
7.1 Motivation: PVTOL 182
7.2 Classical Backstepping 184
7.3 Backstepping for Nonautonomous Systems 193
7.4 Linear Time-Varying Systems 199
7.5 Nonlinear Time-Varying Systems 217
7.6 Bounded Backstepping 220
7.7 Two-Dimensional Example 226
7.8 PVTOL Revisited 228
7.9 Comments 234
Chapter 8 Matrosov Conditions: General Case 238
8.1 Motivation 238
8.2 Preliminaries and Matrosov Assumptions 240
8.3 One Auxiliary Function 242
8.4 Several Auxiliary Functions 245
8.5 Persistency of Excitation 247
8.6 Applications 249
8.7 Sign Constrained Controller 252
8.8 Comments 259
Chapter 9 Adaptively Controlled Systems 260
9.1 Overview of Adaptive Control 260
9.2 Motivating Example 261
9.3 Assumptions and Main Construction 263
9.4 Robustness 267
9.5 Rössler System Revisited 271
9.6 Lorenz System 272
9.7 Extension: More General Feedbacks 275
9.8 Comments 278
Part IV Systems with Multiple Time Scales 280
Chapter 10 Rapidly Time-Varying Systems 281
10.1 Motivation 281
10.2 Overview of Methods 282
10.3 Assumptions and Lemmas 283
10.4 Main Lyapunov Function Construction 285
10.5 Alternative Stricti.cation Result 290
10.6 Illustrations 292
10.7 Further Illustrations: Stricti.cation Approach 298
Chapter 11 Slowly Time-Varying Systems 302
11.1 Motivation 302
11.2 Overview of Methods 303
11.3 Assumptions and Lemmas 304
11.4 Main Lyapunov Construction 304
11.5 More General Decay Rates 306
11.6 Illustrations 307
11.7 Input-to-State Stability 315
11.8 Comments 317
Chapter 12 Hybrid Time-Varying Systems 321
12.1 Motivation 321
12.2 Preliminaries 324
12.3 Stricti.cation for Time-Varying Systems 331
12.4 Matrosov Constructions for Time-Varying Systems 335
12.5 Illustrations 342
12.6 Comments 344
Part V Appendices 346
Appendix A Some Lemmas 347
A.1 Useful Families of Functions 347
A.2 Some Useful Inequalities 356
A.3 A Lower Bound for the Lotka-Volterra Model 358
A.4 ISS and iISS for the Lotka-Volterra Model 360
A.5 Useful Integral 363
A.6 Continuous Time Matrosov Result with PE 366
Appendix B Converse Theory 369
B.1 Converse Lyapunov Function Theorem 369
B.2 Time-Varying Converse ISS Result 372
References 374
Index 383