Incremental stability of hybrid dynamical systems

J.J.B. Biemond, Romain Postoyan, W.P.M.H. Heemels, Nathan van de Wouw

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)

Abstract

The analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. Defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump- and flow incremental asymptotic stability, for which Lyapunov conditions are provided. Various examples are provided throughout the paper.

Original languageEnglish
Pages (from-to)4094 - 4109
JournalIEEE Transactions on Automatic Control
Volume63
Issue number12
DOIs
Publication statusPublished - Dec 2018

Fingerprint

Asymptotic stability
Hybrid systems
Dynamical systems

Keywords

  • Hybrid systems
  • Incremental stability
  • Lyapunov stability

Cite this

Biemond, J.J.B. ; Postoyan, Romain ; Heemels, W.P.M.H. ; van de Wouw, Nathan. / Incremental stability of hybrid dynamical systems. In: IEEE Transactions on Automatic Control. 2018 ; Vol. 63, No. 12. pp. 4094 - 4109.
@article{63495add931d49c1869df911d389fac8,
title = "Incremental stability of hybrid dynamical systems",
abstract = "The analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. Defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump- and flow incremental asymptotic stability, for which Lyapunov conditions are provided. Various examples are provided throughout the paper.",
keywords = "Hybrid systems, Incremental stability, Lyapunov stability",
author = "J.J.B. Biemond and Romain Postoyan and W.P.M.H. Heemels and {van de Wouw}, Nathan",
year = "2018",
month = "12",
doi = "10.1109/TAC.2018.2830506",
language = "English",
volume = "63",
pages = "4094 -- 4109",
journal = "IEEE Transactions on Automatic Control",
issn = "0018-9286",
publisher = "Institute of Electrical and Electronics Engineers",
number = "12",

}

Incremental stability of hybrid dynamical systems. / Biemond, J.J.B.; Postoyan, Romain; Heemels, W.P.M.H.; van de Wouw, Nathan.

In: IEEE Transactions on Automatic Control, Vol. 63, No. 12, 12.2018, p. 4094 - 4109.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Incremental stability of hybrid dynamical systems

AU - Biemond, J.J.B.

AU - Postoyan, Romain

AU - Heemels, W.P.M.H.

AU - van de Wouw, Nathan

PY - 2018/12

Y1 - 2018/12

N2 - The analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. Defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump- and flow incremental asymptotic stability, for which Lyapunov conditions are provided. Various examples are provided throughout the paper.

AB - The analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. Defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump- and flow incremental asymptotic stability, for which Lyapunov conditions are provided. Various examples are provided throughout the paper.

KW - Hybrid systems

KW - Incremental stability

KW - Lyapunov stability

UR - http://www.scopus.com/inward/record.url?scp=85045980966&partnerID=8YFLogxK

U2 - 10.1109/TAC.2018.2830506

DO - 10.1109/TAC.2018.2830506

M3 - Article

AN - SCOPUS:85045980966

VL - 63

SP - 4094

EP - 4109

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 12

ER -