Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Hybrid Systems: Computation and Control

HSCC 2001: Hybrid Systems: Computation and Control pp 63–76Cite as

Reach Set Computations Using Real Quantifier Elimination

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2034))

Abstract

Reach set computations are of fundamental importance in control theory. We consider the reach set problem for open-loop systems described by parametric inhomogeneous linear differential systems and use real quantifier elimination methods to get exact and approximate solutions. The method employs a reduction of the forward and backward reach set and control parameter set problems to the transcendental implicitization problems for the components of special solutions of simpler non-parametric systems. For simple elementary functions we give an exact calculation of the cases where exact semialgebraic transcendental implicitization is possible. For the negative cases we provide approximate alternating using discrete point checking or safe estimations of reach sets and control parameter sets. Examples are computed using the redlog and qepcad packages.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. L. Alling. Real elliptic curves. North-Holland, 1981.

    Google Scholar 

  2. R. Alur, C. Coucoubetis, N. Halbwachs, T. Henzinger, P. Ho, X. Nicolin, A. Olivere, J. Safakis, and S. Yovine. The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138:3–34, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Anai. Algebraic approach to analysis of discrete-time polynomial systems. In Proc. of European Control Conference 1999, 1999.

    Google Scholar 

  4. H. Anai and S. Hara. Fixed-structure robust controller synthesis based on sign definite condition by a special quantifier elimination. In Proceedings of American Control Conference 2000, pages 1312–1316, 2000.

    Google Scholar 

  5. H. Anai and V. Weispfenning. Reach set computation using real quantifier elimination. Technical Report MIP-0012, FMI, Universität Passau, D-94030 Passau, Germany, Oct. 2000.

    Google Scholar 

  6. B. Anderson, N. Bose, and E. Jury. Output feedback stabilization and related problems — solution via decision methods. IEEE Trans. Auto. Control, 20(1):53–65, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. J. ACM, 43(6):1002–1045, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  8. T. Becker, V. Weispfenning, and H. Kredel. Gröbner Bases, a Computational Approach to Commutative Algebra, volume 141 of Graduate Texts in Mathematics. Springer, New York, corrected second printing edition, 1998.

    Google Scholar 

  9. J. Bochnak, M. Coste, and M.-F. Roy. Géometrie algébrique réelle. Springer, Berlin, Heidelberg, New York, 1987.

    MATH  Google Scholar 

  10. M. Braun. Differential Equations and Their Applications, volume 15 of Applied Mathematical Science. Springer, 3rd edition, 1983.

    Google Scholar 

  11. B. Caviness and J. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer, Berlin, Heidelberg, New York, 1998.

    Google Scholar 

  12. G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12(3):299–328, Sept. 1991.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Craig. Introduction to Robotics. Addison-Wesley, 1986.

    Google Scholar 

  14. A. Dolzmann and T. Sturm. REDLOG: Computer algebra meets computer logic. ACM SIGSAM Bulletin, 31(2):2–9, June 1997.

    Article  MathSciNet  Google Scholar 

  15. P. Dorato, W. Yang, and C. Abdallah. Application of quantifier elimination theory to robust multi-object feedback design. J. Symb. Comp. 11, pages 1–6, 1995.

    Google Scholar 

  16. L. v. d. Dries. Tame Topology and o-minimal structures. Cambridge University Press, 1998.

    Google Scholar 

  17. T. Henzinger, P. Kopke, A. Puri, and P. Varaiya. What’s decidable about hybrid automata? In Proceedings of the 27th Annual Symposium on Theory of Computing, STOC’95, pages 373–382. ACM Press, 1995.

    Google Scholar 

  18. T. Henzinger and S. Sastry, editors. Hybrid Systems: Computation and Control, volume 1386 of LNCS. Springer-Verlag, 1998.

    MATH  Google Scholar 

  19. M. Jirstrand. Nonlinear control system design by quantifier elimination. Journal of Symbolic Computation, 24(2):137–152, Aug. 1997. Special issue on applicationsof quantifier elimination.

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Kovács. Computer algebra in robot-kinematics. In Proceeding of the workshop:“Computer Algebra in Science and Engineering”, Bielefeld, Aug’94, pages 303–316, Singapore, 1994. World Scientific.

    Google Scholar 

  21. A. Kurzhanski and I. Vĺyi. Ellipsoidal Calculus for Estimation and Control. Birkhäuser, 1997.

    Google Scholar 

  22. G. Lafferriere, G. Pappas, and S. Sastry. O-minimal hybrid systems. Mathematics of control, Signals and Systems, 13(1):1–21, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  23. G. Lafferriere, G. Pappas, and S. Yovine. A new class of decidable hybrid systems. In Hybrid Systems: Computation and Control, volume 1569 of LNCS, pages 137–151. Springer, 1999.

    Chapter  Google Scholar 

  24. G. Lafferriere, G. Pappas, and S. Yovine. Reachability computation for linear hybrid systems. In Proceedings of the 14th IFAC World Congress, volume E, pages 7–12, Beijin, P.R.China, 1999.

    Google Scholar 

  25. R. Loos and V. Weispfenning. Applying linear quantifier elimination. The Computer Journal, 36(5):450–462, 1993. Special issue on computational quantifier elimination.

    Article  MATH  MathSciNet  Google Scholar 

  26. N. A. Lynch and B. H. Krogh, editors. Proceedings Third International Workshop on Hybrid Systems: Computation and Control (HSCC 2000), Pittsburgh, PA, USA, volume 1790 of Lecture Notes in Computer Science. Springer-Verlag, Mar. 2000.

    Google Scholar 

  27. D. Manocha and J. Canny. Algorithm for implicitizing rational parametric surfaces. In R. Barnhill and W. Boehm, editors, Computer Aided Geometric Design, volume 9. North-Holland, 1992.

    Google Scholar 

  28. D. Nešić. Two algorithms arising in analysis of polynomial models. In Proceedings of 1998 American Control Conference, pages 1889–1893, 1998.

    Google Scholar 

  29. G. Pappas and S. Yovine. Decidable hybrid systems. Technical report, University of California at Berkeley, 1998.

    Google Scholar 

  30. F. Vaandrager and J. von Schuppen, editors. Hybrid Systems: Computation and Control, volume 1569 of LNCS. Springer-Verlag, 1999.

    MATH  Google Scholar 

  31. V. Weispfenning. Quanti_er elimination for real algebra—the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing, 8(2):85–101, Feb. 1997.

    Article  MATH  MathSciNet  Google Scholar 

  32. V. Weispfenning. Semilinear motion planning in REDLOG. Technical report, Universität Passau, D-94030 Passau, May 1999. available at the electronic proceedings of IMACS-ACA’99, http://math.unm.edu/aca.html.

Download references

Author information

Authors and Affiliations

  1. Computer System Laboratories, Fujitsu Laboratories Ltd, Kamikodanaka 4-1-1, Nakahara-ku Kawasaki, 211-8588, Japan

    Hirokazu Anai

  2. Fakultät für Mathematik und Informatik, Universität Passau, D-94030, Passau, Germany

    Volker Weispfenning

Authors
  1. Hirokazu Anai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Volker Weispfenning
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Ingegneria Elettrica, Universitá dell’Aquila, Poggio di Roio, 67040, L’Aquila, Italy

    Maria Domenica Di Benedetto

  2. University of California at Berkeley, Berkeley, CA, 94720

    Alberto Sangiovanni-Vincentelli

  3. PARADES, Via San Pantaleo, 66, 00186, Roma, Italy

    Alberto Sangiovanni-Vincentelli

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Anai, H., Weispfenning, V. (2001). Reach Set Computations Using Real Quantifier Elimination. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A. (eds) Hybrid Systems: Computation and Control. HSCC 2001. Lecture Notes in Computer Science, vol 2034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45351-2_9

Download citation

  • DOI: https://doi.org/10.1007/3-540-45351-2_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41866-5

  • Online ISBN: 978-3-540-45351-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation