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Hybrid Control Systems

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Book cover Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Notation

Definition of the Subject

Introduction

Well-posed Hybrid Dynamical Systems

Modeling Hybrid Control Systems

Stability Theory

Design Tools

Applications

Discussion and Final Remarks

Future Directions

Bibliography

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Abbreviations

Global asymptotic stability:

The typical closed‐loop objective of a hybrid controller. Often, the hybrid controller achieves global asymptotic stability of a compact set rather than of a point. This is the property that solutions starting near the set remain near the set for all time and all solutions tend toward the set asymptotically. This property is robust, in a practical sense, for well-posed hybrid dynamical systems .

(Well-posed) Hybrid dynamical system:

System that combines behaviors typical of continuous‐time and discrete‐time dynamical systems, that is, combines both flows and jumps. The system is said to be well-posed if the data used to describe the evolution (consisting of a flow map, flow set, jump map, and jump set) satisfy mild regularity conditions; see conditions (C1)–(C3) in Subsect. “Conditions for Well‐posedness”.

Hybrid controller:

Algorithm that takes, as inputs, measurements from a system to be controlled (called the plant) and combines behaviors of continuous‐time and discrete‐time controllers (i. e. flows and jumps) to produce, as outputs, signals that are to control the plant.

Hybrid closed‐loop system:

The hybrid system resulting from the interconnection of a plant and a controller, at least one of which is a hybrid dynamical system.

Invariance principle:

A tool for studying asymptotic properties of bounded solutions to (hybrid) dynamical systems, applicable when asymptotic stability is absent. It characterizes the sets to which such solutions must converge, by relying in part on invariance properties of such sets.

Lyapunov stability theory:

A tool for establishing global asymptotic stability of a compact set without solving for the solutions to the hybrid dynamical system. A Lyapunov function is one that takes its minimum, which is zero, on the compact set, that grows unbounded as its argument grows unbounded, and that decreases in the direction of the flow map on the flow set and via the jump map on the jump set.

Supervisor of hybrid controllers:

A hybrid controller that coordinates the actions of a family of hybrid controllers in order to achieve a certain stabilization objective. Patchy control Lyapunov functions provide a means of constructing supervisors.

Temporal regularization:

A modification to a hybrid controller to enforce a positive lower bound on the amount of time between jumps triggered by the hybrid control algorithm.

Zeno (and discrete) solutions:

A solution (to a hybrid dynamical system) that has an infinite number of jumps in a finite amount of time. It is discrete if, moreover, the solution never flows, i. e., never changes continuously.

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Author information

Authors and Affiliations

  1. Electrical and Computer Engineering Department, University of California, Santa Barbara, USA

    Andrew R. Teel

  2. Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, USA

    Ricardo G. Sanfelice

  3. Department of Mathematics and Statistics, Loyola University, Chicago, USA

    Rafal Goebel

Authors
  1. Andrew R. Teel
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  2. Ricardo G. Sanfelice
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  3. Rafal Goebel
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Teel, A.R., Sanfelice, R.G., Goebel, R. (2012). Hybrid Control Systems. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_43

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