Hybrid Dynamical Systems

by Goebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R.

• ISBN13: 9780691153896
• ISBN10: 0691153892
• Format: Hardcover
• Copyright: 2012-02-27
• Publisher: Princeton Univ Pr

Summary

?Hybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. Filled with a wealth of examples to illustrate concepts, this book presents a complete theory of robust asymptotic stability for hybrid dynamical systems that is applicable to the design of hybrid control algorithms--algorithms that feature logic, timers, or combinations of digital and analog components. With the tools of modern mathematical analysis, Hybrid Dynamical Systemsunifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems. It presents hybrid system versions of the necessary and sufficient Lyapunov conditions for asymptotic stability, invariance principles, and approximation techniques, and examines the robustness of asymptotic stability, motivated by the goal of designing robust hybrid control algorithms. This self-contained and classroom-tested book requires standard background in mathematical analysis and differential equations or nonlinear systems. It will interest graduate students in engineering as well as students and researchers in control, computer science, and mathematics.

Author Biography

Rafal Goebel is an assistant professor in the Department of Mathematics and Statistics at Loyola University, Chicago. Ricardo G. Sanfelice is an assistant professor in the Department of Aerospace and Mechanical Engineering at the University of Arizona. Andrew R. Teel is a professor in the Electrical and Computer Engineering Department at the University of California, Santa Barbara.

Table of Contents

Preface	p. ix
Introduction	p. 1
The modeling framework	p. 1
Examples in science and engineering	p. 2
Control system examples	p. 7
Connections to other modeling frameworks	p. 15
Notes	p. 22
The solution concept	p. 25
Data of a hybrid system	p. 25
Hybrid time domains and hybrid arcs	p. 26
Solutions and their basic properties	p. 29
Generators for classes of switching signals	p. 35
Notes	p. 41
Uniform asymptotic stability, an initial treatment	p. 43
Uniform global pre-asymptotic stability	p. 43
Lyapunov functions	p. 50
Relaxed Lyapunov conditions	p. 60
Stability from containment	p. 64
Equivalent characterizations	p. 68
Notes	p. 71
Perturbations and generalized solutions	p. 73
Differential and difference equations	p. 73
Systems with state perturbations	p. 76
Generalized solutions	p. 79
Measurement noise in feedback control	p. 84
Krasovskii solutions are Hermes solutions	p. 88
Notes	p. 94
Preliminaries from set-valued analysis	p. 97
Set convergence	p. 97
Set-valued mappings	p. 101
Graphical convergence of hybrid arcs	p. 107
Differential inclusions	p. 111
Notes	p. 115
Well-posed hybrid systems and their properties	p. 117
Nominally well-posed hybrid systems	p. 117
Basic assumptions on the data	p. 120
Consequences of nominal well-posedness	p. 125
Well-posed hybrid systems	p. 132
Consequences of well-posedness	p. 134
Notes	p. 137
Asymptotic stability, an in-depth treatment	p. 139
Pre-asymptotic stability for nominally well-posed systems	p. 141
Robustness concepts	p. 148
Well-posed systems	p. 151
Robustness corollaries	p. 153
Smooth Lyapunov functions	p. 156
Proof of robustness implies smooth Lyapunov functions	p. 161
Notes	p. 167
Invariance principles	p. 169
Invariance and ¿-limits	p. 169
Invariance principles involving Lyapunov-like functions	p. 170
Stability analysis using invariance principles	p. 176
Meagre-limsup invariance principles	p. 178
Invariance principles for switching systems	p. 181
Notes	p. 184
Conical approximation and asymptotic stability	p. 185
Homogeneous hybrid systems	p. 185
Homogeneity and perturbations	p. 189
Conical approximation and stability	p. 192
Notes	p. 196
Appendix: List of Symbols	p. 199
Bibliography	p. 201
Index	p. 211
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