Hybrid Dynamical Systems

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  • Format: Hardcover
  • Copyright: 2012-02-27
  • Publisher: Princeton Univ Pr

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?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

Prefacep. ix
Introductionp. 1
The modeling frameworkp. 1
Examples in science and engineeringp. 2
Control system examplesp. 7
Connections to other modeling frameworksp. 15
Notesp. 22
The solution conceptp. 25
Data of a hybrid systemp. 25
Hybrid time domains and hybrid arcsp. 26
Solutions and their basic propertiesp. 29
Generators for classes of switching signalsp. 35
Notesp. 41
Uniform asymptotic stability, an initial treatmentp. 43
Uniform global pre-asymptotic stabilityp. 43
Lyapunov functionsp. 50
Relaxed Lyapunov conditionsp. 60
Stability from containmentp. 64
Equivalent characterizationsp. 68
Notesp. 71
Perturbations and generalized solutionsp. 73
Differential and difference equationsp. 73
Systems with state perturbationsp. 76
Generalized solutionsp. 79
Measurement noise in feedback controlp. 84
Krasovskii solutions are Hermes solutionsp. 88
Notesp. 94
Preliminaries from set-valued analysisp. 97
Set convergencep. 97
Set-valued mappingsp. 101
Graphical convergence of hybrid arcsp. 107
Differential inclusionsp. 111
Notesp. 115
Well-posed hybrid systems and their propertiesp. 117
Nominally well-posed hybrid systemsp. 117
Basic assumptions on the datap. 120
Consequences of nominal well-posednessp. 125
Well-posed hybrid systemsp. 132
Consequences of well-posednessp. 134
Notesp. 137
Asymptotic stability, an in-depth treatmentp. 139
Pre-asymptotic stability for nominally well-posed systemsp. 141
Robustness conceptsp. 148
Well-posed systemsp. 151
Robustness corollariesp. 153
Smooth Lyapunov functionsp. 156
Proof of robustness implies smooth Lyapunov functionsp. 161
Notesp. 167
Invariance principlesp. 169
Invariance and -limitsp. 169
Invariance principles involving Lyapunov-like functionsp. 170
Stability analysis using invariance principlesp. 176
Meagre-limsup invariance principlesp. 178
Invariance principles for switching systemsp. 181
Notesp. 184
Conical approximation and asymptotic stabilityp. 185
Homogeneous hybrid systemsp. 185
Homogeneity and perturbationsp. 189
Conical approximation and stabilityp. 192
Notesp. 196
Appendix: List of Symbolsp. 199
Bibliographyp. 201
Indexp. 211
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