Definitions of incremental stability for hybrid systems

Conference Paper (2015)
Author(s)

Romain Postoyan (Lorraine University)

J. J. Biemond (Katholieke Universiteit Leuven)

W. P M H Heemels (Eindhoven University of Technology)

N. van de Wouw (Eindhoven University of Technology, TU Delft - Team Bart De Schutter)

Research Group
Team Bart De Schutter
DOI related publication
https://doi.org/10.1109/CDC.2015.7403088
More Info
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Publication Year
2015
Language
English
Research Group
Team Bart De Schutter
Pages (from-to)
5544-5549
ISBN (electronic)
978-1-4799-7886-1

Abstract

The analysis of incremental stability properties typically involves measuring the distance between any pair of solutions of a given dynamical system, corresponding to different initial conditions, at the same time instant. This approach is not directly applicable for hybrid systems in general. Indeed, hybrid systems generate solutions that are defined with respect to hybrid times, which consist of both the continuous time elapsed and the discrete time, that is the number of jumps the solution has experienced. Two solutions of a hybrid system do not a priori have the same time domain, and we may therefore not be able to compare them at the same hybrid time instant. To overcome this issue, we invoke graphical closeness concepts. We present definitions for incremental stability depending on whether incremental asymptotic stability properties hold with respect to the hybrid time, the continuous time, or the discrete time, respectively. Examples are provided throughout the paper to illustrate these definitions, and the relations between these three incremental stability notions are investigated. The definitions are shown to be consistent with those available in the literature for continuous-time and discrete-time systems.

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