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Reachability of Uncertain Linear Systems Using Zonotopes

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Hybrid Systems: Computation and Control (HSCC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3414))

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Abstract

We present a method for the computation of reachable sets of uncertain linear systems. The main innovation of the method consists in the use of zonotopes for reachable set representation. Zonotopes are special polytopes with several interesting properties : they can be encoded efficiently, they are closed under linear transformations and Minkowski sum. The resulting method has been used to treat several examples and has shown great performances for high dimensional systems. An extension of the method for the verification of piecewise linear hybrid systems is proposed.

Research partially supported by the Région Rhône-Alpes (Projet CalCel).

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References

  1. Alur, R., Dang, T., Ivancic, F.: Reachability analysis of hybrid systems via predicate abstraction. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 35–48. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  2. Asarin, E., Bournez, O., Dang, T., Maler, O.: Approximate reachability analysis of piecewise linear dynamical systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 21–31. Springer, Heidelberg (2000)

    Google Scholar 

  3. Asarin, E., Dang, T., Maler, O.: d/dt: A verification tool for hybrid systems. In: The Proc. of CDC 2001 (2001)

    Google Scholar 

  4. Asarin, E., Schneider, G., Yovine, S.: Towards computing phase portraits of polygonal differential inclusions. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 49–61. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  5. Asarin, E., Dang, T., Girard, A.: Reachability of non-linear systems using conservative approximations. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 22–35. Springer, Heidelberg (2003)

    Google Scholar 

  6. Asarin, E., Dang, T.: Abstraction by projection and application to multi-affine systems. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 32–47. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  7. Chutinan, A., Krogh, B.H.: Verification of polyhedral invariant hybrid automata using polygonal flow pipe approximations. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 76–90. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  8. Chutinan, A., Krogh, B.H.: Computational techniques for hybrid system verification. IEEE Trans. on Automatic Control 48(1), 64–75 (2003)

    Article  MathSciNet  Google Scholar 

  9. Combastel, C.: A state bounding observer based on zonotopes. In: Proc. of European Control Conference (2003)

    Google Scholar 

  10. Dang, T.: Vérification et synthèse des systèmes hybrides, Thèse de Doctorat, Institut National Polytechnique de Grenoble (2000)

    Google Scholar 

  11. Guibas, L.J., Nguyen, A., Zhang, L.: Zonotopes as bounding. In: Proc. of Symposium on Discrete Algorithms, pp. 803–812

    Google Scholar 

  12. Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. of Computation 64, 1541–1555 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kühn, W.: Zonotope dynamics in numerical quality control. In: Hege, H.-C., Polthier, K. (eds.) Mathematical Visualization, pp. 125–134. Springer, Heidelberg (1998)

    Google Scholar 

  14. Kurzhanski, A., Varaiya, P.: Ellipsoidal tehcniques for reachability analysis. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790. Springer, Heidelberg (2000)

    Google Scholar 

  15. Lafferriere, G., Pappas, G., Yovine, S.: Reachability computation for linear systems. Proc. IFAC World Congress E, 7–12 (1999)

    Google Scholar 

  16. Mitchell, I., Tomlin, C.: Level set methods for computation in hybrid systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790. Springer, Heidelberg (2000)

    Google Scholar 

  17. Rubensson, M., Lennartson, B., Pettersson, S.: Convergence to limit cycles in hybrid systems: an example. Large Scale Systems: Theory and Applications, 704–709 (1998)

    Google Scholar 

  18. Stursberg, O., Krogh, B.H.: Efficient representation and computation of reachable sets for hybrid systems. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 482–497. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  19. Tomlin, C., Mitchell, I., Bayen, A., Oishi, M.: Computational techniques for the verification and control of hybrid systems. Proc. of the IEEE 91(7), 986–1001 (2003)

    Article  Google Scholar 

  20. Tiwari, A., Khanna, G.: Series of abstractions for hybrid automata. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 465–478. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  21. Yazarel, H., Pappas, G.J.: Geometric programming relaxations for linear system reachability. In: Proc. American Control Conference (2004)

    Google Scholar 

  22. Ziegler, G.M.: Lectures on polytopes. Graduate texts in Mathematics. Springer, Heidelberg (1995)

    MATH  Google Scholar 

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Girard, A. (2005). Reachability of Uncertain Linear Systems Using Zonotopes. In: Morari, M., Thiele, L. (eds) Hybrid Systems: Computation and Control. HSCC 2005. Lecture Notes in Computer Science, vol 3414. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31954-2_19

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  • DOI: https://doi.org/10.1007/978-3-540-31954-2_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25108-8

  • Online ISBN: 978-3-540-31954-2

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