Über die Autorin bzw. den Autor

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.

Von der hinteren Coverseite

"This superb book unifies some of the key developments in hybrid dynamical systems from the last decade and, through elegant and clear technical content, introduces the necessary tools for understanding the stability of these systems. It will be a great resource for graduate students and researchers in the field."--Magnus Egerstedt, Georgia Institute of Technology

"With broad applications for science and engineering, this first-rate book develops solid foundations for a comprehensive theory of hybrid dynamical systems. Diverse literature is brought together for the first time, making a huge body of knowledge conveniently accessible."--Dennis S. Bernstein, University of Michigan

Aus dem Klappentext

"This superb book unifies some of the key developments in hybrid dynamical systems from the last decade and, through elegant and clear technical content, introduces the necessary tools for understanding the stability of these systems. It will be a great resource for graduate students and researchers in the field."--Magnus Egerstedt, Georgia Institute of Technology

"With broad applications for science and engineering, this first-rate book develops solid foundations for a comprehensive theory of hybrid dynamical systems. Diverse literature is brought together for the first time, making a huge body of knowledge conveniently accessible."--Dennis S. Bernstein, University of Michigan

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Hybrid Dynamical Systems

PRINCETON UNIVERSITY PRESS

Contents

Preface...........................................................................ix1 Introduction....................................................................11.1 The modeling framework........................................................11.2 Examples in science and engineering...........................................21.3 Control system examples.......................................................71.4 Connections to other modeling frameworks......................................151.5 Notes.........................................................................222 The solution concept............................................................252.1 Data of a hybrid system.......................................................252.2 Hybrid time domains and hybrid arcs...........................................262.3 Solutions and their basic properties..........................................292.4 Generators for classes of switching signals...................................352.5 Notes.........................................................................413 Uniform asymptotic stability, an initial treatment..............................433.1 Uniform global pre-asymptotic stability.......................................433.2 Lyapunov functions............................................................503.3 Relaxed Lyapunov conditions...................................................603.4 Stability from containment....................................................643.5 Equivalent characterizations..................................................683.6 Notes.........................................................................714 Perturbations and generalized solutions.........................................734.1 Differential and difference equations.........................................734.2 Systems with state perturbations..............................................764.3 Generalized solutions.........................................................794.4 Measurement noise in feedback control.........................................844.5 Krasovskii solutions are Hermes solutions.....................................884.6 Notes.........................................................................945 Preliminaries from set-valued analysis..........................................975.1 Set convergence...............................................................975.2 Set-valued mappings...........................................................1015.3 Graphical convergence of hybrid arcs..........................................1075.4 Differential inclusions.......................................................1115.5 Notes.........................................................................1156 Well-posed hybrid systems and their properties..................................1176.1 Nominally well-posed hybrid systems...........................................1176.2 Basic assumptions on the data.................................................1206.3 Consequences of nominal well-posedness........................................1256.4 Well-posed hybrid systems.....................................................1326.5 Consequences of well-posedness................................................1346.6 Notes.........................................................................1377 Asymptotic stability, an in-depth treatment.....................................1397.1 Pre-asymptotic stability for nominally well-posed systems.....................1417.2 Robustness concepts...........................................................1487.3 Well-posed systems............................................................1517.4 Robustness corollaries........................................................1537.5 Smooth Lyapunov functions.....................................................1567.6 Proof of robustness implies smooth Lyapunov functions.........................1617.7 Notes.........................................................................1678 Invariance principles...........................................................1698.1 Invariance and ω-limits.................................................1698.2 Invariance principles involving Lyapunov-like functions.......................1708.3 Stability analysis using invariance principles................................1768.4 Meagre-limsup invariance principles...........................................1788.5 Invariance principles for switching systems...................................1818.6 Notes.........................................................................1849 Conical approximation and asymptotic stability..................................1859.1 Homogeneous hybrid systems....................................................1859.2 Homogeneity and perturbations.................................................1899.3 Conical approximation and stability...........................................1929.4 Notes.........................................................................196Appendix: List of Symbols.........................................................199Bibliography......................................................................201Index.............................................................................211

Chapter One

The model of a hybrid system used in this book is informally presented in this section. The focus is on the data structure and on modeling. Several examples are given, including models of hybrid control systems. The model of a hybrid system is then related to other modeling frameworks, such as hybrid automata, impulsive differential equations, and switching systems. A formal presentation of the model, together with a rigorous definition of the solution, is postponed until Chapter 2.

1.1 THE MODELING FRAMEWORK

The model of a hybrid system used in this book can be represented in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)

A reader less familiar with set-valued mappings and differential or difference inclusions may choose to keep in mind a less general representation involving equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)

This representation suggests that the state of the hybrid system, represented by x, can change according to a differential inclusion [?? [member of] F(x) or differential equation [??] = f(x) while in the set ITLITL, and it can change according to a difference inclusion x⁺ [member of] G(x) or difference equation x⁺ = g(x) while in the set D. The notation [??] represents the velocity of the state x, while x⁺ represents the value of the state after an instantaneous change.

A rigorous statement of what constitutes a model of a hybrid system and what is a solution to the model is postponed until Chapter 2. This chapter focuses on modeling of various hybrid systems in the form (1.1) or (1.2).

To shorten the terminology, the behavior of a dynamical system that can be described by a differential equation or inclusion is referred to as flow. The behavior of a dynamical system that can be described by a difference equation or inclusion is referred to as jumps. This leads to the following names for the four objects involved in...