Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Stability Theory for Hybrid Dynamical Systems

  • Andrew R. TeelEmail author
Living reference work entry

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DOI: https://doi.org/10.1007/978-1-4471-5102-9_99-2

Abstract

This entry provides a short introduction to modeling of hybrid dynamical systems and then focuses on stability theory for these systems. It provides definitions of asymptotic stability, basin of attraction, and uniform asymptotic stability for a compact set. It points out mild assumptions under which different characterizations of asymptotic stability are equivalent, as well as when an asymptotically stable compact set exists. It also summarizes necessary and sufficient conditions for asymptotic stability in terms of Lyapunov functions.

Keywords

Asymptotic stability Basin of attraction Hybrid system Lyapunov function 
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References

  1. Bainov DD, Simeonov PS (1989) Systems with impulse effect: stability, theory, and applications. Ellis Horwood Limited, ChichesterzbMATHGoogle Scholar
  2. Branicky MS (1998) Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Control 43:1679–1684MathSciNetzbMATHGoogle Scholar
  3. DeCarlo RA, Branicky MS, Pettersson S, Lennartson B (2000) Perspectives and results on the stability and stabilizability of hybrid systems. Proc IEEE 88(7): 1069–1082CrossRefGoogle Scholar
  4. Goebel R, Sanfelice RG, Teel AR (2009) Hybrid dynamical systems. IEEE Control Syst Mag 29(2):28–93MathSciNetCrossRefGoogle Scholar
  5. Goebel R, Sanfelice RG, Teel AR (2012) Hybrid dynamical systems. Princeton University Press, PrincetonCrossRefGoogle Scholar
  6. Haddad W, Chellaboina V, Nersesov SG (2006) Impulsive and hybrid dynamical systems. Princeton University Press, PrincetonCrossRefGoogle Scholar
  7. Hespanha JP (2004) Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle. IEEE Trans Autom Control 49(4):470–482MathSciNetCrossRefGoogle Scholar
  8. Lakshmikantham V, Bainov DD, Simeonov PS (1989) Theory of impulsive differential equations. World Scientific, Singapore/TeaneckCrossRefGoogle Scholar
  9. Liberzon D (2003) Switching in systems and control. Birkhauser, BostonCrossRefGoogle Scholar
  10. Liberzon D, Morse AS (1999) Basic problems in stability and design of switched systems. IEEE Control Syst Mag 19(5):59–70CrossRefGoogle Scholar
  11. Lygeros J, Johansson KH, Simić SN, Zhang J, Sastry SS (2003) Dynamical properties of hybrid automata. IEEE Trans Autom Control 48(1):2–17MathSciNetCrossRefGoogle Scholar
  12. Matveev A, Savkin AV (2000) Qualitative theory of hybrid dynamical systems. Birkhauser, BostonCrossRefGoogle Scholar
  13. Michel AN, Hou L, Liu D (2008) Stability of dynamical systems: continuous, discontinuous, and discrete systems. Birkhauser, BostonzbMATHGoogle Scholar
  14. van der Schaft A, Schumacher H (2000) An introduction to hybrid dynamical systems. Springer, London/New YorkCrossRefGoogle Scholar
  15. Yang T (2001) Impulsive control theory. Springer, Berlin/ New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2020

Authors and Affiliations

  1. 1.Electrical and Computer Engineering DepartmentUniversity of CaliforniaSanta BarbaraUSA

Section editors and affiliations

  • Francoise Lamnabhi-Lagarrigue
    • 1
  1. 1.Laboratoire des Signaux et SystèmesCNRSGif-sur-YvetteFrance