Advertisement

An Introduction to Hybrid Automata, Numerical Simulation and Reachability Analysis

Chapter

Abstract

Hybrid automata combine finite state models with continuous variables that are governed by differential equations. Hybrid automata are used to model systems in a wide range of domains such as automotive control, robotics, electronic circuits, systems biology, and health care. Numerical simulation approximates the evolution of the variables with a sequence of points in discretized time. This highly scalable technique is widely used in engineering and design, but it is difficult to simulate all representative behaviors of a system. To ensure that no critical behaviors are missed, reachability analysis aims at accurately and quickly computing a cover of the states of the system that are reachable from a given set of initial states. Reachability can be used to formally show safety and bounded liveness properties. This chapter outlines the major concepts and discusses advantages and shortcomings of the different techniques.

Keywords

Convex Hull Hybrid System Reachable State Location Extension Outer Approximation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Althoff. Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets. In Hybrid systems: computation and control (HSCC'19), pages 173–182. ACM, 2013.Google Scholar
  2. 2.
    M. Althoff and B. H. Krogh. Avoiding geometric intersection operations in reachability analysis of hybrid systems. In Hybrid Systems: Computation and Control (HSCC'lli), pages 4 & -54. ACM, 2012.Google Scholar
  3. 3.
    M. Althoff, , H. Krogh, and O. Stursberg. Analyzing reachability of linear dynamic systems with parametric uncertainties. In A. Rauh and E. Auer, editors, Modeling, Design, and Simulation of Systems with Uncertainties. Springer, 2011.Google Scholar
  4. 4.
    R. AIur. Formal verification of hybrid systems. In S. Chakraborty, A. Jerraya, S. , Baruah, and S. Fischmeister, editors, EMSOFT, pages 273–278. ACM, 2011.Google Scholar
  5. 5.
    R. Alur, C. Courcoubetis, N. Halbwachs, T. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138:3–34, 1995.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    R. Alur, C. Courcoubetis, T. A. Henzinger, and P.-H. Ho. Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems. In Hybrid Systems, LNCS 736, pages 209–229. Springer, 1993.Google Scholar
  7. 7.
    E. Asarin, T. Dang, and A. Girard. Hybridization methods for the analysis of nonlinear systems. Acta Inf., 43(7):451–476, 2007.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    E. Asarin, T. Dang, O. Maler, and O. Bournez. Approximate reachability analysis of piecewise-linear dynamical systems. In Hybrid Systems: Computation and Control (HSCC’00), volume 1790 of LNCS, pages 20–31. Springer, 2000.Google Scholar
  9. 9.
    E. Asarin, T. Dang, O. Maler, and R. Testylier. Using redundant constraints for refinement. In Automated Technology for Verification and Analysis, pages 37–51. Springer, 2010.Google Scholar
  10. 10.
    R. Bagnara, P. M. Hill, and E. Zaffanella. The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Science of Computer Programming, 72(1–2):3–21, 2008.MathSciNetCrossRefGoogle Scholar
  11. 11.
    O. Bouissou, A. Chapoutot, S. Mimram, and B. Strazzulla. Set-based simulation for design and verification of simulink models. In Embedded Real Time Software and Systems (ERTS’tf), 2014.Google Scholar
  12. 12.
    O. Bouissou, S. Mimram, and A. Chapoutot. Hyson: Set-based simulation of hybrid systems. In RSP, pages 79–85. IEEE, October 2012.Google Scholar
  13. 13.
    D. Bruck, H. Elmqvist, S. E. Mattsson, and H. Olsson. Dymola for multiengineering modeling and simulation. In Proceedings of Modelica, 2002.Google Scholar
  14. 14.
    R. P. Canale and S. C. Chapra. Numerical methods for engineers. Mc Graw Hill, New York, 1998.Google Scholar
  15. 15.
    C. G. Cassandras and J. Lygeros. Stochastic hybrid systems. CRC Press, 2006.Google Scholar
  16. 16.
    C. Chase, J. Serrano, and P. J. Ramadge. Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous systems. Automatic Control, IEEE Transactions on, 38(1):70–83, 1993.MATHMathSciNetGoogle Scholar
  17. 17.
    X. Chen, E. Abraham, and S. Sankaranarayanan. Taylor model flowpipe construction for non-linear hybrid systems. In RTSS, pages 183–192. IEEE Computer Society, 2012.Google Scholar
  18. 18.
    A. Chutinan and B. H. Krogh. Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In F. W. Vaandrager and J. H. van Schuppen, editors, HSCC, volume 1569 of LNCS, pages 76–90. Springer, 1999.Google Scholar
  19. 19.
    T. Dang and R. Testylier. Reachability analysis for polynomial dynamical systems using the bernstein expansion. Reliable Computing, 17(2):128–152, 2012.MathSciNetGoogle Scholar
  20. 20.
    A. Donze. Breach, a toolbox for verification and parameter synthesis of hybrid systems. In Computer Aided Verification, pages 167–170. Springer, 2010.Google Scholar
  21. 21.
    J. Eker, J. W. Janneck, E. A. Lee, J. Liu, X. Liu, J. Ludvig, S. Neuendorffer, S. Sachs, and Y. Xiong. Taming heterogeneity-the ptolemy approach. Proceedings of the IEEE, 91(1):127–144, 2003.Google Scholar
  22. 22.
    G. E. Fainekos, A. Girard, and G. J. Pappas. Temporal logic verification using simulation. In Formal Modeling and Analysis of Timed Systems, pages 171–186. Springer, 2006.Google Scholar
  23. 23.
    G. Frehse. PHAVer: algorithmic verification of hybrid systems past HyTech. STTT, 10(3):263–279, 2008.MathSciNetCrossRefGoogle Scholar
  24. 24.
    G. Frehse, C. L. Guernic, A. Donze, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang, and O. Maler. SpaceEx: Scalable verification of hybrid systems. In G. Gopalakrishnan and S. Qadeer, editors, CAV, LNCS. Springer, 2011.Google Scholar
  25. 25.
    G. Frehse, R. Kateja, and C. Le Guernic. Flowpipe approximation and clustering in space-time. In Hybrid systems: computation and control (HSCC’13), pages 203212. ACM, 2013.Google Scholar
  26. 26.
    A. Girard. Reachability of uncertain linear systems using zonotopes. In M. Morari and L. Thiele, editors, HSCC, volume 3414 of LNCS, pages 291–305. Springer, 2005.Google Scholar
  27. 27.
    A. Girard, C. L. Guernic, and O. Maler. Efficient computation of reachable sets of linear time-invariant systems with inputs. In J. P. Hespanha and A. Tiwari, editors, HSCC, volume 3927 of LNCS, pages 257–271. Springer, 2006.Google Scholar
  28. 28.
    A. Girard and G. Zheng. Verification of safety and liveness properties of metric transition systems. ACM Transactions on Embedded Computing Systems (TECS), 11(S2):54, 2012.CrossRefGoogle Scholar
  29. 29.
    M. R. Greenstreet. Verifying safety properties of differential equations. In Computer Aided Verification, pages 277–287. Springer, 1996.Google Scholar
  30. 30.
    C. L. Guernic and A. Girard. Reachability analysis of hybrid systems using support functions. In A. Bouajjani and O. Maler, editors, CAV, volume 5643 of LNCS, pages 540–554. Springer, 2009.Google Scholar
  31. 31.
    N. Halbwachs, Y.-E. Proy, and P. Raymond. Verification of linear hybrid systems by means of convex approximations. In International Static Analysis Symposium, SAS’94, Namur (Belgium), September 1994.Google Scholar
  32. 32.
    T. Henzinger, P.-H. Ho, and H. Wong-Toi. HyTech: A model checker for hybrid systems. Software Tools for Technology Transfer, pages 110–122, 1997.Google Scholar
  33. 33.
    T. A. Henzinger, P.-H. Ho, and H. Wong-Toi. HyTech: A model checker for hybrid systems. In O. Grumberg, editor, CAV, volume 1254 of LNCS, pages 460–463. Springer, 1997.Google Scholar
  34. 34.
    T. A. Henzinger, P.-H. Ho, and H. Wong-Toi. Algorithmic analysis of nonlinear hybrid systems. IEEE Transactions on Automatic Control, 43:540–554, 1998.MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    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.MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    P.-H. Ho. Automatic Analysis of Hybrid Systems. PhD thesis, Cornell University, Aug. 1995. Technical Report CSD-TR95–1536.Google Scholar
  37. 37.
    A. A. Julius, G. E. Fainekos, M. Anand, I. Lee, and G. J. Pappas. Robust test generation and coverage for hybrid systems. In Hybrid Systems: Computation and Control, pages 329–342. Springer, 2007.Google Scholar
  38. 38.
    W. Kuhn. Rigorously computed orbits of dynamical systems without the wrapping effect. Computing, 61(1):47–67, 1998.MathSciNetCrossRefGoogle Scholar
  39. 39.
    A. B. Kurzhanski and P. Varaiya. Dynamics and Control of Trajectory Tubes. Springer, 2014.Google Scholar
  40. 40.
    A. A. Kurzhanskiy and P. Varaiya. Ellipsoidal toolbox (et). In Decision and Control, 2006 45th IEEE Conference on, pages 1498–1503. IEEE, 2006.Google Scholar
  41. 41.
    C. Le Guernic. Reachability analysis of hybrid systems with linear continuous dynamics. PhD thesis, Universite Grenoble 1 – Joseph Fourier, 2009.Google Scholar
  42. 42.
    A. V. Lotov, V. A. Bushenkov, and G. K. Kamenev. Interactive Decision Maps, volume 89 of Applied Optimization. Kluwer, 2004.Google Scholar
  43. 43.
    O. Maler. Algorithmic verification of continuous and hybrid systems. In Int. Workshop on Verification of Infinite-Stale System (Infinity), 2013.Google Scholar
  44. 44.
    MapleSoft. Maplesim 7: Advanced system-level modeling. http://www.maplesoft.com/products/maplesim, 2015.
  45. 45.
    MathWorks. Mathworks simulink: Simulation et model-based design, Mar. 2014. www.mathworks.fr/products/simulink.
  46. 46.
    S. E. Mattsson, H. Elmqvist, and M. Otter. Physical system modeling with mod- elica. Control Engineering Practice, 6(4):501–510, 1998.CrossRefGoogle Scholar
  47. 47.
    P. Prabhakar and M. Viswanathan. A dynamic algorithm for approximate flow computations. In E. Frazzoli and R. Grosu, editors, HSCC, pages 133–142. ACM, 2011.Google Scholar
  48. 48.
    W. H. Press. Numerical recipes 3rd edition: The art of scientific computing. Cambridge University Press, 2007.Google Scholar
  49. 49.
    S. Sankaranarayanan, T. Dang, and F. Ivancic. Symbolic model checking of hybrid systems using template polyhedra. In Tools and Algorithms for the Construction and Analysis of Systems, pages 188–202. Springer, 2008.Google Scholar
  50. 50.
    P. Tabuada. Verification and Control of Hybrid Systems: A Symbolic Approach. Springer, 2009.Google Scholar
  51. 51.
    F. Zhang, M. Yeddanapudi, and P. Mosterman. Zero-crossing location and detection algorithms for hybrid system simulation. In IFAC World Congress, pages 7967–7972, 2008.Google Scholar

Copyright information

© Springer Fachmedien Wiesbaden 2015

Authors and Affiliations

  1. 1.Verimag, Université Joseph Fourier – Grenoble 1GiéresFrance

Personalised recommendations