Skip to main content

Ellipsoidal Techniques for Reachability Analysis

  • Conference paper
  • First Online:
Hybrid Systems: Computation and Control (HSCC 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1790))

Included in the following conference series:


This report describes the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and hard bounds on the controls through tight external and internal ellipsoidal approximations. These approximating tubes touch the reach tubes from outside and inside respectively at every point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The proposed approximation scheme induces a very small computational burden compared with other methods of reach set calculation.

In particular such approximations may be expressed through ordinary differential equations with coefficients given in explicit analytical form. This yields exact parametric representation of reach tubes through families of external and internal ellipsoidal tubes. The proposed techniques, combined with calculation of external and internal approximations for intersections of ellipsoids, provide an approach to reachability problems for hybrid systems.

Research supported by National Science Foundation Grant ECS 9725148

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others


  1. Botchkarev O., Tripakis S., Verification of Hybrid Systems with Linear Differential Inclusions using Ellipsoidal Approximations. Proceedings of this Conference, Pittsburg, 2000.

    Google Scholar 

  2. Boyd S., El Ghaoui L., Feron E., Balakrishnan V., Linear Matrix Inequalities in System and Control Theory, SIAM, Studies in Applied Mathematics, 1994.

    Google Scholar 

  3. Chernousko F.L., State Estimation for Dynamic Systems, CRC Press, 1994.

    Google Scholar 

  4. Henzinger T.A., Kopke P.W., Puri A. and Varaiya P., What’s Decidable about Hybrid Automata ? Proc. 27-th STOC, pp.373–382, 1995.

    Google Scholar 

  5. Kurzhanski A.B., Filippova T.F., On the Theory of Trajectory Tubes: a Mathematical Formalism for Uncertain Dynamics, Viability and Control, in: Advances in Nonlinear Dynamics and Control, ser. PSCT 17, pp.122–188, Birkhäuser, Boston, 1993.

    MathSciNet  Google Scholar 

  6. Kurzhanski A. B., Vályi I. Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, ser. SCFA, 1997.

    Google Scholar 

  7. Kurzhanski A. B., Varaiya P., Ellipsoidal techniques for reachability analysis. To appear.

    Google Scholar 

  8. Kurzhanski A. B., Varaiya P., Ellipsoidal techniques for reachability analysis. Internal approximations. To appear.

    Google Scholar 

  9. Lee E.B., Marcus L., Foundations of Optimal Control Theory, Wiley, NY,, 1967.

    MATH  Google Scholar 

  10. Leitmann G., Optimality and reachability via feedback controls. In:Dynamic Systems and Mycrophysics, Blaquiere A., Leitmann G. eds.,1982.

    Google Scholar 

  11. Lempio F., Veliov V., Discrete Approximations of Di_erential Inclusions, Bayreuther Mathematische Schriften, Heft 54, pp. 149–232, 1998.

    MATH  MathSciNet  Google Scholar 

  12. Pappas G.L., Sastry S., Straightening out differential inclusions. System and Control Letters, 35(2), pp.79–85, Sept.1998.

    Article  MATH  MathSciNet  Google Scholar 

  13. Puri A., Borkar V. and Varaiya P., -Approximations of Differential Inclusions, in: R. Alur, T.A. Henzinger, and E.D. Sonntag eds., Hybrid Systems, pp. 109–123, LNCS 1201, Springer, 1996.

    Google Scholar 

  14. Puri A., Varaiya P., Decidability of hybrid systems with rectangular inclusions. In D. Dill ed., Proc. CAV-94, LNCS 1066, Springer, 1996.

    Google Scholar 

  15. Rockafellar, R. T., Convex Analysis, 2-nd ed., Princeton University Press, 1999.

    Google Scholar 

  16. Varaiya P. Reach Set Computation Using Optimal Control, in Proc.of KIT Workshop on Verification of Hybrid Systems, Verimag, Grenoble, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kurzhanski, A.B., Varaiya, P. (2000). Ellipsoidal Techniques for Reachability Analysis. In: Lynch, N., Krogh, B.H. (eds) Hybrid Systems: Computation and Control. HSCC 2000. Lecture Notes in Computer Science, vol 1790. Springer, Berlin, Heidelberg.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67259-3

  • Online ISBN: 978-3-540-46430-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics