Skip to main content

Bisimulation of Dynamical Systems

  • Conference paper
Hybrid Systems: Computation and Control (HSCC 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2993))

Included in the following conference series:

Abstract

A general notion of bisimulation is studied for dynamical systems. An algebraic characterization of bisimulation together with an algorithm for computing the maximal bisimulation relation is derived using geometric control theory. Bisimulation of dynamical systems is shown to be a concept which unifies the system-theoretic concepts of state space equivalence and state space reduction, and which allows to study equivalence of systems with non-minimal state space dimension. The notion of bisimulation is especially powerful for ‘non-deterministic’ dynamical systems, and leads in this case to a notion of equivalence which is finer than equality of external behavior. Furthermore, by merging bisimulation of dynamical systems with bisimulation of concurrent processes a notion of structural bisimulation is developed for hybrid systems with continuous input and output variables.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Alur, R., Henzinger, T.A., Lafferriere, G., Pappas, G.J.: Discrete abstractions of hybrid systems. Proceedings of the IEEE 88, 971–984 (2000)

    Article  Google Scholar 

  2. Basile, G., Marro, G.: Controlled and conditioned invariants in linear system theory. Prentice Hall, Englewood Cliffs (1992)

    MATH  Google Scholar 

  3. Haghverdi, E., Tabuada, P., Pappas, G.J.: Unifying bisimulation relations for discrete and continuous systems, Category Theory and Computer Science, Electronic Notes in Theoretical Computer Science (August 2002)

    Google Scholar 

  4. Henzinger, T.A.: Hybrid automata with finite bisimulations. In: Fülöp, Z., Gecseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 324–335. Springer, Heidelberg (1995)

    Google Scholar 

  5. Hermanns, H. (ed.): Interactive Markov Chains. LNCS, vol. 2428. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  6. Lafferriere, G., Pappas, G.J., Sastry, S.: Hybrid systems with finite bisimulations. In: Antsaklis, P., Kohn, W., Lemmon, M., Nerode, A., Sastry, S. (eds.) Hybrid Systems V. LNCS, Springer, Heidelberg (1998)

    Google Scholar 

  7. Lafferriere, G., Pappas, G.J., Sastry, S.: O-minimal hybrid systems. Math. Contr. Signals, Syst. 13, 1–21 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Milner, R.: Communication and Concurrency. Prentice Hall International Series in Computer Science (1989)

    Google Scholar 

  9. Milner, R.: Communication and Mobile Systems: the π- Calculus. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  10. Pappas, G.J., Lafferriere, G., Sastry, S.: Hierarchically consistent control systems. IEEE Transactions on Automatic Control 45(6), 1144–1160 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Pappas, G.J.: Bisimilar linear systems. Automatica 39, 2035–2047 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Pappas, G.J., Simic, S.: Consistent abstractions of affine control systems. IEEE Transactions on Automatic Control 47, 745–756 (2002)

    Article  MathSciNet  Google Scholar 

  13. Park, D.: Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) GI-TCS 1981. LNCS, vol. 104, Springer, Heidelberg (1981)

    Chapter  Google Scholar 

  14. Pola, G., van der Schaft, A.J., Di Benedetto, M.D.: Bisimulation theory for switching linear systems (in preparation)

    Google Scholar 

  15. van der Schaft, A.J.: Equivalence of dynamical systems by bisimulation., Technical Report Department of Applied Mathematics, University of Twente (October 2003) (submitted for publication)

    Google Scholar 

  16. van der Schaft, A.J., Schumacher, J.M.: An Introduction to Hybrid Dynamical Systems. Springer Lecture Notes in Control and Information Sciences, vol. 251. Springer, London (2000); Second revised edition to appear in Communications and Control Engineering Series, Springer, London (2004)

    MATH  Google Scholar 

  17. Tabuada, P., Pappas, G.J.: Bisimilar control affine systems. Systems and Control Letters (to appear)

    Google Scholar 

  18. Tabuada, P., Pappas, G.J., Lima, P.: Composing abstractions of hybrid systems. In: Tomlin, C., Greenstreet, M.R. (eds.) Hybrid Systems: Computation and Control. LNCS, pp. 436–450. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  19. Wonham, W.M.: Linear multivariable control: a geometric approach, 3rd edn. Springer, New York (1985)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

van der Schaft, A. (2004). Bisimulation of Dynamical Systems. In: Alur, R., Pappas, G.J. (eds) Hybrid Systems: Computation and Control. HSCC 2004. Lecture Notes in Computer Science, vol 2993. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24743-2_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-24743-2_37

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21259-1

  • Online ISBN: 978-3-540-24743-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics