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Reachability Analysis of Multi-affine Systems

  • Marius Kloetzer
  • Calin Belta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3927)

Abstract

We present a technique for reachability analysis of continuous multi-affine systems based on rectangular partitions. The method is iterative. At each step, finer partitions and larger discrete quotients are produced. We exploit some interesting convexity properties of multi-affine functions on rectangles to show that the construction of the discrete quotient at each step requires only the evaluation of the vector field at the set of all vertices of all rectangles in the partition and finding the roots of a finite set of scalar affine functions. The methodology promises to be easily extendable to rectangular hybrid automata with multi-affine vector fields and is expected to find important applications in analysis of biological networks and robot control.

Keywords

Hybrid System Hybrid Automaton Reachability Analysis Initial Partition Reachability Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Milner, R.: Communication and Concurrency. Prentice Hall, Englewood Cliffs (1989)MATHGoogle Scholar
  2. 2.
    Pappas, G.J.: Bisimilar linear systems. Automatica 39(12), 2035–2047 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Haghverdi, E., Tabuada, P., Pappas, G.: Bisimulation relations for dynamical and control systems. In: Blute,, Selinger, e.P. (eds.) Electronic Notes in Theoretical Computer Science, vol. 69, Elsevier, Amsterdam (2003)Google Scholar
  4. 4.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What is decidable about hybrid automata? J. Comput. Syst. Sci. 57, 94–124 (1998)CrossRefMATHGoogle Scholar
  5. 5.
    Alur, R., Dill, D.L.: A theory of timed automata. Theoret. Comput. Sci. 126, 183–235 (1994)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.H.: Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems. LNCS, vol. 736, pp. 209–229. Springer, New York (1993)Google Scholar
  7. 7.
    Nicolin, X., Olivero, A., Sifakis, J., Yovine, S.: An approach to the description and analysis of hybrid automata. LNCS, vol. 736, pp. 149–178. Springer, New York (1993)Google Scholar
  8. 8.
    Puri, A., Varaiya, P.: Decidability of hybrid systems with rectangular differential inclusions. Computer Aided Verification, 95–104 (1994)Google Scholar
  9. 9.
    Lafferriere, G., Pappas, G.J., Sastry, S.: O-minimal hybrid systems. Math. Control, Signals, Syst 13(1), 1–21 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lafferriere, G., Pappas, G.J., Yovine, S.: A new class of decidable hybrid systems. LNCS, vol. 1569, pp. 137–151. Springer, New York (1999)MATHGoogle Scholar
  11. 11.
    Lafferriere, G., Pappas, G.J., Yovine, S.: Reachability computation for linear hybrid systems. In: Proc. 14th IFAC World Congress, Beijing, P.R.C (July 1999)Google Scholar
  12. 12.
    Alur, R., Dang, T., Ivancic, F.: Reachability analysis of hybrid systems via predicate abstraction. In: Fifth International Workshop on Hybrid Systems: Computation and Control, Stanford (2002)Google Scholar
  13. 13.
    Tiwari, A., Khanna, G.: Series of abstractions for hybrid automata. In: Fifth International Workshop on Hybrid Systems: Computation and Control, Stanford (2002)Google Scholar
  14. 14.
    Ghosh, R., Tiwari, A., Tomlin, C.: Automated symbolic reachability analysis; with application to delta-notch signaling automata. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 233–248. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Habets, L., van Schuppen, J.: A control problem for affine dynamical systems on a full-dimensional polytope. Automatica 40, 21–35 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Belta, C., Habets, L.: Constructing decidable hybrid systems with velocity bounds. In: 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas (2004)Google Scholar
  17. 17.
    Belta, C., Isler, V., Pappas, G.J.: Discrete abstractions for robot planning and control in polygonal environments. IEEE Trans. on Robotics 21(5), 864–874 (2005)CrossRefGoogle Scholar
  18. 18.
    Belta, C., Habets, L.: Control of a class of nonlinear systems on rectangles. IEEE Transactions on Automatic Control (to appear, 2005)Google Scholar
  19. 19.
    Belta, C.: On controlling aircraft and underwater vehicles. In: IEEE International Conference on Robotics and Automation, New Orleans (2004)Google Scholar
  20. 20.
    Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926)CrossRefMATHGoogle Scholar
  21. 21.
    Lotka, A.: Elements of physical biology. Dover Publications, Inc., New York (1925)MATHGoogle Scholar
  22. 22.
    Kloetzer, M., Belta, C.: Reachability analysis of multi-affine systems. Boston University, Brookline, MA, Technical report CISE-2005-IR-0070 (October 2005) [Online]. Available: http://www.bu.edu/systems/research/publications/2005/2005-IR-0070.pdf
  23. 23.
    Tabuada, P., Pappas, G.: Model checking LTL over controlable linear systems is decidable. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, Springer, Heidelberg (2003)Google Scholar
  24. 24.
    Kloetzer, M., Belta, C.: Reachability analysis of multi-affine systems (ramas), http://iasi.bu.edu/~software/reach-ma.htm

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Marius Kloetzer
    • 1
  • Calin Belta
    • 1
  1. 1.Center for Information and Systems EngineeringBoston UniversityBrooklineUSA

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