# Hybrid Dynamical Systems: Modeling, Stability, and Robustness

Rafal Goebel
Ricardo G. Sanfelice
Andrew R. Teel
Pages: 232
https://www.jstor.org/stable/j.ctt7s02z

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xiv)
Rafal Goebel, Ricardo G. Sanfelice and Andrew R. Teel
4. Chapter One Introduction
(pp. 1-24)

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.

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

$\left\{ {\begin{array}{*{20}c} {x \in C} & {\dot x \in F(x)} \\ {x \in D} & {x^ + G(x).} \\ \end{array} } \right.(1.1)$

5. Chapter Two The solution concept
(pp. 25-42)

A rigorous development of the concept of a solution to a hybrid system is the topic of this chapter. The data of a hybrid system is defined, and a generalized concept of time is introduced. Solutions to a hybrid system are defined and basic properties of solutions, like their existence and uniqueness, are addressed. The concept of a solution is further illustrated by hybrid models of switching systems under broad families of switching signals.

From now on, a hybrid system is identified with the model describing it, in the form (1.1) or (1.2). Data of a hybrid system is formally...

6. Chapter Three Uniform asymptotic stability, an initial treatment
(pp. 43-72)

This chapter focuses on uniform asymptotic stability of a closed set. Studying this property provides another opportunity to illustrate the concept of a solution to a hybrid system. Asymptotic stability is a fundamental property of dynamical systems, one that is usually desired in natural and engineered systems. It provides qualitative information about solutions, especially a characterization of the solutions’ long-term trends. Asymptotic stability of a closed set, rather than of an equilibrium point, is significant since the solutions of a hybrid system often do not settle down to an equilibrium point. In a sample-and-hold control system, for example, the controlled...

7. Chapter Four Perturbations and generalized solutions
(pp. 73-96)

This chapter discusses the effect of state perturbations on solutions to a hybrid system. It is shown that state perturbations, of arbitrarily small size, can dramatically change the behavior of solutions. While such a phenomenon is also present in continuous-time and discrete-time dynamical systems, it is magnified in the hybrid setting, due to the flows and the jumps being constrained to the flow and the jump sets, respectively.

Perturbations affecting the whole state of a hybrid system are usually considered. The resulting behaviors are quite representative of what may occur if perturbations come from state measurement error in a hybrid...

8. Chapter Five Preliminaries from set-valued analysis
(pp. 97-116)

Further developments in the theory of hybrid systems, for example, making rigorous the concept of convergence of sequences of hybrid arcs or considering perturbations of the flow and the jump sets, can be conveniently carried out with the help of some concepts from set-valued analysis. This chapter includes the necessary background. Section 5.1 presents the concept of convergence of sets. Section 5.2 deals with set-valued mappings and their continuity properties. Section 5.3 specializes some of the concepts, such as graphical convergence, to hybrid arcs and provides further details in such a setting. Finally, Section 5.4 discusses differential inclusions. The presentation...

9. Chapter Six Well-posed hybrid systems and their properties
(pp. 117-138)

In a classical setting, for example in differential equations or in optimization, a well-posed problem is often defined as one in which a solution exists, is unique, and depends continuously on parameters. For hybrid dynamical systems, insisting on uniqueness of solutions and on their continuous dependence on initial conditions (and possibly, on perturbations) is very restrictive and, as it turns out, not necessary to develop a reasonable stability theory. In fact, stability theory results such as converse smooth Lyapunov results and invariance principles are possible for a quite general class of hybrid systems. In what follows,nominally well-posed hybrid systems...

10. Chapter Seven Asymptotic stability, an in-depth treatment
(pp. 139-168)

Chapter 3 defined global uniform pre-asymptotic stability for a closed set in a hybrid system and gave numerous sufficient conditions for it. Those sufficient conditions did not require the system to be well-posed, as in Definition 6.29, or even nominally well-posed, as in Definition 6.2. This chapter defines local pre-asymptotic stability for a compact (closed and bounded) set and studies its properties for systems that are nominally well-posed or well-posed. For nominally well-posed hybrid systems, pre-asymptotic stability turns out to be equivalent to uniform pre-asymptotic stability. For well-posed systems, pre-asymptotic stability turns out to be equivalent to uniform, robust pre-asymptotic...

11. Chapter Eight Invariance principles
(pp. 169-184)

Invariance principles characterize the sets to which precompact solutions to a dynamical system must converge. They rely on invariance properties of ω-limit sets of solutions, as defined in Definition 6.17, and additionally on Lyapunov-like functions, which do not increase along solutions, or output functions. Invariance principles which rely on Lyapunov-like functions are presented in Section 8.2. Applications of these invariance principles to analysis of asymptotic stability are described in Section 8.3. Section 8.4 states an invariance principle involving not a Lyapunov-like function, but an output function having a certain property not along all solutions, but only along the solution whose...

12. Chapter Nine Conical approximation and asymptotic stability
(pp. 185-198)

The goal of this chapter is to present a technique of approximating a hybrid system with a conical hybrid system: a system with conical flow and jump sets and with constant or linear flow and jump maps. The main result, Theorem 9.11, deduces pre-asymptotic stability for the original system from pre-asymptotic stability for the conical approximation. This result generalizes, to a hybrid system, the result that asymptotic stability for the linearization of a differential equation implies asymptotic stability for the differential equation. In many cases, the analysis of the conical approximation is simpler than of the original hybrid system; this...

13. Appendix: List of Symbols
(pp. 199-200)
14. Bibliography
(pp. 201-210)
15. Index
(pp. 211-212)