Hybrid dynamical systems : modeling, stability, and robustness

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 Systems unifies 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 Notes

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.

Preface	p. ix
1 Introduction	p. 1
1.1 The modeling framework	p. 1
1.2 Examples in science and engineering	p. 2
1.3 Control system examples	p. 7
1.4 Connections to other modeling frameworks	p. 15
1.5 Notes	p. 22
2 The solution concept	p. 25
2.1 Data of a hybrid system	p. 25
2.2 Hybrid time domains and hybrid arcs	p. 26
2.3 Solutions and their basic properties	p. 29
2.4 Generators for classes of switching signals	p. 35
2.5 Notes	p. 41
3 Uniform asymptotic stability, an initial treatment	p. 43
3.1 Uniform global pre-asymptotic stability	p. 43
3.2 Lyapunov functions	p. 50
3.3 Relaxed Lyapunov conditions	p. 60
3.4 Stability from containment	p. 64
3.5 Equivalent characterizations	p. 68
3.6 Notes	p. 71
4 Perturbations and generalized solutions	p. 73
4.1 Differential and difference equations	p. 73
4.2 Systems with state perturbations	p. 76
4.3 Generalized solutions	p. 79
4.4 Measurement noise in feedback control	p. 84
4.5 Krasovskii solutions are Hermes solutions	p. 88
4.6 Notes	p. 94
5 Preliminaries from set-valued analysis	p. 97
5.1 Set convergence	p. 97
5.2 Set-valued mappings	p. 101
5.3 Graphical convergence of hybrid arcs	p. 107
5.4 Differential inclusions	p. 111
5.5 Notes	p. 115
6 Well-posed hybrid systems and their properties	p. 117
6.1 Nominally well-posed hybrid systems	p. 117
6.2 Basic assumptions on the data	p. 120
6.3 Consequences of nominal well-posedness	p. 125
6.4 Well-posed hybrid systems	p. 132
6.5 Consequences of well-posedness	p. 134
6.6 Notes	p. 137
7 Asymptotic stability, an in-depth treatment	p. 139
7.1 Pre-asymptotic stability for nominally well-posed systems	p. 141
7.2 Robustness concepts	p. 148
7.3 Well-posed systems	p. 151
7.4 Robustness corollaries	p. 153
7.5 Smooth Lyapunov functions	p. 156
7.6 Proof of robustness implies smooth Lyapunov functions	p. 161
7.7 Notes	p. 167
8 Invariance principles	p. 169
8.1 Invariance and ¿-limits	p. 169
8.2 Invariance principles involving Lyapunov-like functions	p. 170
8.3 Stability analysis using invariance principles	p. 176
8.4 Meagre-limsup invariance principles	p. 178
8.5 Invariance principles for switching systems	p. 181
8.6 Notes	p. 184
9 Conical approximation and asymptotic stability	p. 185
9.1 Homogeneous hybrid systems	p. 185
9.2 Homogeneity and perturbations	p. 189
9.3 Conical approximation and stability	p. 192
9.4 Notes	p. 196
Appendix: List of Symbols	p. 199
Bibliography	p. 201
Index	p. 211

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