Malisoff / Mazenc Constructions of Strict Lyapunov Functions

1. Auflage 2009
ISBN: 978-1-84882-535-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 386 Seiten

Reihe: Communications and Control Engineering

Converse Lyapunov function theory guarantees the existence of strict Lyapunov functions in many situations, but the functions it provides are often abstract and nonexplicit, and therefore may not lend themselves to engineering applications. Often, even when a system is known to be stable, one still needs explicit Lyapunov functions; however, once an appropriate strict Lyapunov function has been constructed, many robustness and stabilization problems can be solved through standard feedback designs or robustness arguments. Non-strict Lyapunov functions are often readily constructed. This book contains a broad repertoire of Lyapunov constructions for nonlinear systems, focusing on methods for transforming non-strict Lyapunov functions into strict ones. Their explicitness and simplicity make them suitable for feedback design, and for quantifying the effects of uncertainty. Readers will benefit from the authors' mathematical rigor and unifying, design-oriented approach, as well as the numerous worked examples.

Autoren/Hrsg.

Malisoff, Michael

Mazenc, Frédéric

Inhaltsverzeichnis

1;Preface;7
2;Contents;10
3;Part I Background;16
3.1;Chapter 1 Background on Nonlinear Systems;17
3.1.1;1.1 Preliminaries;17
3.1.2;1.2 Families of Nonlinear Systems;18
3.1.3;1.3 Notions of Stability;22
3.1.4;1.4 Stabilization;27
3.1.5;1.5 Examples;29
3.1.6;1.6 Comments;34
3.2;Chapter 2 Review of Lyapunov Functions;38
3.2.1;2.1 Strict Lyapunov Function;38
3.2.2;2.2 Non-strict Lyapunov Function;44
3.2.3;2.3 Discrete Time Lyapunov Function;48
3.2.4;2.4 Illustrations;49
3.2.5;2.5 Basin of Attraction Revisited;57
3.2.6;2.6 L2 Gains;58
3.2.7;2.7 Lyapunov Functions with Bounded Gradients;60
3.2.8;2.8 Comments;66
4;Part II Time-Invariant Case;71
4.1;Chapter 3 Matrosov Conditions: Simple Case;72
4.1.1;3.1 Motivation;72
4.1.2;3.2 Continuous Time Theorem;75
4.1.3;3.3 Proof of Continuous Time Theorem;77
4.1.4;3.4 Discrete Time Theorem;80
4.1.5;3.5 Proof of Discrete Time Theorem;82
4.1.6;3.6 Illustrations;86
4.1.7;3.7 Comments;91
4.2;Chapter 4 Jurdjevic-Quinn Conditions;93
4.2.1;4.1 Motivation;93
4.2.2;4.2 Control Affine Case;95
4.2.3;4.3 General Case;103
4.2.4;4.5 Hamiltonian Systems;108
4.2.5;4.6 Robustness;112
4.2.6;4.7 Illustrations;117
4.2.7;4.8 Comments;124
4.3;Chapter 5 Systems Satisfying the Conditions of LaSalle;126
4.3.1;5.1 Background and Motivation;126
4.3.2;5.2 First Method: Iterated Lie Derivatives;127
4.3.3;5.3 Discussion and Extensions of First Method;133
4.3.4;5.4 Second Method: Matrosov Conditions;135
4.3.5;5.5 Application: Lotka-Volterra Model;138
4.3.6;5.6 Comments;145
5;Part III Time-Varying Case;147
5.1;Chapter 6 Stricti.cation: Basic Results;148
5.1.1;6.1 Background;148
5.1.2;6.2 Motivating Example;150
5.1.3;6.3 Time-Varying Stricti.cation Theorem;152
5.1.4;6.4 Remarks on Rate of Convergence;157
5.1.5;6.6 Equivalent Characterizations of Non-strict ISS;162
5.1.6;6.7 Input-to-Output Stability;165
5.1.7;6.8 Illustrations;167
5.1.8;6.9 Comments;178
5.2;Chapter 7 Backstepping for Time-Varying Systems;182
5.2.1;7.1 Motivation: PVTOL;182
5.2.2;7.2 Classical Backstepping;184
5.2.3;7.3 Backstepping for Nonautonomous Systems;193
5.2.4;7.4 Linear Time-Varying Systems;199
5.2.5;7.5 Nonlinear Time-Varying Systems;217
5.2.6;7.6 Bounded Backstepping;220
5.2.7;7.7 Two-Dimensional Example;226
5.2.8;7.8 PVTOL Revisited;228
5.2.9;7.9 Comments;234
5.3;Chapter 8 Matrosov Conditions: General Case;238
5.3.1;8.1 Motivation;238
5.3.2;8.2 Preliminaries and Matrosov Assumptions;240
5.3.3;8.3 One Auxiliary Function;242
5.3.4;8.4 Several Auxiliary Functions;245
5.3.5;8.5 Persistency of Excitation;247
5.3.6;8.6 Applications;249
5.3.7;8.7 Sign Constrained Controller;252
5.3.8;8.8 Comments;259
5.4;Chapter 9 Adaptively Controlled Systems;260
5.4.1;9.1 Overview of Adaptive Control;260
5.4.2;9.2 Motivating Example;261
5.4.3;9.3 Assumptions and Main Construction;263
5.4.4;9.4 Robustness;267
5.4.5;9.5 Rössler System Revisited;271
5.4.6;9.6 Lorenz System;272
5.4.7;9.7 Extension: More General Feedbacks;275
5.4.8;9.8 Comments;278
6;Part IV Systems with Multiple Time Scales;280
6.1;Chapter 10 Rapidly Time-Varying Systems;281
6.1.1;10.1 Motivation;281
6.1.2;10.2 Overview of Methods;282
6.1.3;10.3 Assumptions and Lemmas;283
6.1.4;10.4 Main Lyapunov Function Construction;285
6.1.5;10.5 Alternative Stricti.cation Result;290
6.1.6;10.6 Illustrations;292
6.1.7;10.7 Further Illustrations: Stricti.cation Approach;298
6.2;Chapter 11 Slowly Time-Varying Systems;302
6.2.1;11.1 Motivation;302
6.2.2;11.2 Overview of Methods;303
6.2.3;11.3 Assumptions and Lemmas;304
6.2.4;11.4 Main Lyapunov Construction;304
6.2.5;11.5 More General Decay Rates;306
6.2.6;11.6 Illustrations;307
6.2.7;11.7 Input-to-State Stability;315
6.2.8;11.8 Comments;317
6.3;Chapter 12 Hybrid Time-Varying Systems;321
6.3.1;12.1 Motivation;321
6.3.2;12.2 Preliminaries;324
6.3.3;12.3 Stricti.cation for Time-Varying Systems;331
6.3.4;12.4 Matrosov Constructions for Time-Varying Systems;335
6.3.5;12.5 Illustrations;342
6.3.6;12.6 Comments;344
7;Part V Appendices;346
7.1;Appendix A Some Lemmas;347
7.1.1;A.1 Useful Families of Functions;347
7.1.2;A.2 Some Useful Inequalities;356
7.1.3;A.3 A Lower Bound for the Lotka-Volterra Model;358
7.1.4;A.4 ISS and iISS for the Lotka-Volterra Model;360
7.1.5;A.5 Useful Integral;363
7.1.6;A.6 Continuous Time Matrosov Result with PE;366
7.2;Appendix B Converse Theory;369
7.2.1;B.1 Converse Lyapunov Function Theorem;369
7.2.2;B.2 Time-Varying Converse ISS Result;372
8;References;374
9;Index;383