Approximate Reachability for Linear Systems

  • Ashish Tiwari
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2623)


We describe new techniques to construct, and subsequently refine, over-approximations of the reachability sets for linear dynamical systems. Our approach extracts information from real eigenvectors and more generally, from certain vectors in the primary decomposition, to generate suitable invariants of the system and can be used in conjunction with other reachability computation methods. We also describe experimental results from using this technique inside the qualitative abstraction tool [18], where it helps to generate refined abstractions of hybrid systems with linear continuous dynamics. We illustrate this on a collision-avoidance example from automobile cruise control problem, which was handled completely automatically by our tool.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Alur, C. Courcoubetis, N. Halbwachs, T. A. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138(3):3–34, 1995.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    R. Alur and D. Dill. A theory of timed automata. Theoretical Computer Science, 126:183–235, 1994.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Rajeev Alur, Tom Henzinger, Gerardo Lafferriere, and George J. Pappas. Discrete abstractions of hybrid systems. Proceedings of the IEEE, 88(2):971–984, July 2000.Google Scholar
  4. [4]
    H. Anai and V. Weispfenning. Reach set computations using real quantifier elimination. In M. D. Di Benedetto and A. L. Sangiovanni-Vincentelli, editors, HSCC, volume 2034 of Lecture Notes in Computer Science, pages 63–76. Springer, 2001.Google Scholar
  5. [5]
    A. Chutinan and B. H. Krogh. Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In Vaandrager and van Schuppen [19], pages 76–90.Google Scholar
  6. [6]
    T. Dang and O. Maler. Reachability analysis via face lifting. In T. A. Henzinger and S. Sastry, editors, HSCC, volume 1386 of LNCS, pages 96–109. Springer, 1998.Google Scholar
  7. [7]
    D. Godbole and J. Lygeros. Longitudinal control of the lead car of a platoon. IEEE Transactions on Vehicular Technology, 43(4):1125–35, 1994.CrossRefMathSciNetGoogle Scholar
  8. [8]
    T. A. Henzinger, P. W. Kopke, A. Puri, and P. Varaiya. What’s decidable about hybrid automata? Journal of Computer and System Sciences, 57:94–124, 1998. A preliminary version appeared in the Proc. of the 27th Annual ACM Symposium on Theory of Computing (STOC 1995), pp. 373-382.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    K. Hoffman and R. Kunze. Linear Algebra. Prentice-Hall, second edition, 1971.Google Scholar
  10. [10]
    M. Jirstrand. Algebraic methods for modeling and design in control. Licentiate thesis LIU-TEK-LIC-1996:05 Linköping Studies in Science and Technology. Thesis No 540, Department of Electrical Engineering, Li, 1996.Google Scholar
  11. [11]
    A. B. Kurzhanski and P. Varaiya. Ellipsoidal techniques for reachability analysis. In Lynch and Krogh [14], pages 202–214.Google Scholar
  12. [12]
    G. Lafferriere, G. J. Pappas, and S. Yovine. A new class of decidable hybrid systems. In Vaandrager and van Schuppen [19], pages 137–151.Google Scholar
  13. [13]
    G. Lafferriere, G. J. Pappas, and S. Yovine. Symbolic reachability computations for families of linear vector fields. J. Symbolic Computation, 32(3):231–253, 2001.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    N. A. Lynch and B. H. Krogh, editors. Hybrid Systems: Computation and Control, Third International Workshop, HSCC 2000, Proceedings, volume 1790 of LNCS. Springer, 2000.MATHGoogle Scholar
  15. [15]
    I. Mitchell and C. Tomlin. Level set methods for computation in hybrid systems. In Lynch and Krogh [14].Google Scholar
  16. [16]
    A. Puri and P. Varaiya. Decidability of hybrid systems with rectangular differential inclusions. In D. L. Dill, editor, Computer Aided Verification, CAV, volume 818 of LNCS, pages 95–104. Springer Verlag, 1994.Google Scholar
  17. [17]
    A. Puri and P. Varaiya. Driving safely in smart cars. In Proceedings of the 1995 American Control Conference, 1995 Google Scholar
  18. [18]
    A. Tiwari and G. Khanna. Series of abstractions for hybrid automata. In C. Tomlin and M. R. Greenstreet, editors, HSCC, volume 2289 of Lecture Notes in Computer Science, pages 465–478. Springer, 2002.Google Scholar
  19. [19]
    F. W. Vaandrager and J. H. van Schuppen, editors. Hybrid Systems: Computation and Control, Second International Workshop, HSCC’99, Proceedings, volume 1569 of Lecture Notes in Computer Science. Springer, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ashish Tiwari
    • 1
  1. 1.SRI InternationalMenlo ParkUSA

Personalised recommendations