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Verifying average dwell time of hybrid systems

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Published:04 January 2009Publication History
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Abstract

Average dwell time (ADT) properties characterize the rate at which a hybrid system performs mode switches. In this article, we present a set of techniques for verifying ADT properties. The stability of a hybrid system A can be verified by combining these techniques with standard methods for checking stability of the individual modes of A.

We introduce a new type of simulation relation for hybrid automata—switching simulation—for establishing that a given automaton A switches more rapidly than another automaton B. We show that the question of whether a given hybrid automaton has ADT τa can be answered either by checking an invariant or by solving an optimization problem. For classes of hybrid automata for which invariants can be checked automatically, the invariant-based method yields an automatic method for verifying ADT; for automata that are outside this class, the invariant has to be checked using inductive techniques. The optimization-based method is automatic and is applicable to a restricted class of initialized hybrid automata. A solution of the optimization problem either gives a counterexample execution that violates the ADT property, or it confirms that the automaton indeed satisfies the property. The optimization and the invariant-based methods can be used in combination to find the unknown ADT of a given hybrid automaton.

References

  1. Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T. A., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., and Yovine, S. 1995. The algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138, 1, 3--34. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Alur R. and Henzinger, T. A., Eds. 1996. Verification of an audio protocol with bus collision using UPPAAL. In Proceedings of the 8th International Conference on Computer-Aided Verification (CAV'96). Springer-Verlag, Berlin, 411--414. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Alur, R., Henzinger, C. C. T. A., and Ho., P. H. 1993. Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In R. L. Grossman, A. Nerode, A. P. Ravn, and H. Rischel, Eds. Hybrid Systems. Lecture Notes in Computer Science, vol. 736. Springer-Verlag, 209--229. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Alur, R. and Pappas, G. J., Eds. 2004. Hybrid systems: computation and control. In Proceedings of the 7th International Workshop (HSCC'04). Springer, Berlin.Google ScholarGoogle Scholar
  5. Archer, M. 2001. TAME: PVS Strategies for special purpose theorem proving. Annals Math. AI 29, 1/4. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Bayen, A. M., Cruck, E., and Tomlin, C. 2002. Guaranteed overapproximations of unsafe sets for continuous and hybrid systems: solving the hamilton-jacobi equation using viability techniques. In Proceedings of the 5th International Workshop (HSCC'02), Springer, Berlin. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Bemporad, A., Bicchi, A., and Buttazzo, G. C., Eds. 2007. Proceedings of the 10th International Workshop, (HSCC'07). Springer, Berlin.Google ScholarGoogle Scholar
  8. Bemporad, A. and Morari, M. 1999. Verification of hybrid systems via mathematical programming. In Proceedings of the 2nd International Workshop on Hybrid Systems: Computation and Control (HSCC'99). Springer, Berlin, 31--45. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Branicky, M. 1995. Studies in hybrid systems: modeling, analysis, and control. Ph.D. Thesis, MIT, Cambridge, MA. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Branicky, M. 1998. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Auto. Control 43, 475--482.Google ScholarGoogle ScholarCross RefCross Ref
  11. Branicky, M., Borkar, V., and Mitter, S. 1998. A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Auto. Control 43, 1, 31--45.Google ScholarGoogle ScholarCross RefCross Ref
  12. Chatterjee, D. and Liberzon, D. 2006. Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions. SIAM J. Control Optimiz. 45, 1, 174--206. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Cormen, T. H., Leiserson, C. E., and Rivest, R. L. 1990. Introduction to Algorithms. MIT Press/McGraw-Hill, Cambridge, MA. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Cruz, R. L. 1991. A calculus for network delay, part i: Network elements in isolation. IEEE Trans. Inform. Theory 37, 1, 114--131.Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Floyd, R. 1967. Assigning meanings to programs. In Proceedings of the Symposium on Applied Mathematics. Mathematical Aspects of Computer Science. American Mathematical Society, 19--32.Google ScholarGoogle ScholarCross RefCross Ref
  16. Frehse, G. 2005. Phaver: algorithmic verification of hybrid systems past hytech. In Proceedings of the 8th International Workshop on Hybrid Systems: Computation and Control (HSCC'05). Springer, Berlin.Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. GNU. GLPK—GNU linear programming kit. http://www.gnu.org/directory/libs/glpk.html.Google ScholarGoogle Scholar
  18. Heitmeyer, C. and Lynch, N. 1994. The generalized railroad crossing: A case study in formal verification of real-time system. In Proceedings of the 15th IEEE Real-Time Systems Symposium, (San Juan, Puerto Rico). IEEE, Los Alamitos, CA.Google ScholarGoogle Scholar
  19. Henzinger, T. A., Ho, P.-H., and Wong-Toi, H. 1997. Hytech: A model checker for hybrid systems. In Proceedings of the 9th International Conference on Computer Aided Verification (CAV'97). Springer, Berlin, 460--483. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Henzinger, T. A. and Kopke, P. W. 1996. State equivalences for rectangular hybrid automata. In Proceedings of the International Conference on Concurrency Theory (CONCUR'96). Springer, Berlin, 530--545. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Henzinger, T. A., Kopke, P. W., Puri, A., and Varaiya, P. 1995. What' decidable about hybrid automata? In Proceedings of the 27th Annual ACM Symposium on Theory of Computing (SOTC'95). ACM, New York, 373--382. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Henzinger, T. A. and Majumdar, R. 2000. Symbolic model checking for rectangular hybrid systems. In Proceedings of the 6th International Workshop on Tools and Algorithms for the Construction and Analysis of Systems (TACAS'00). Springer, Berlin, 142--156. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Hespanha, J., Liberzon, D., and Morse, A. 2003. Hysteresis-based switching algorithms for supervisory control of uncertain systems. Automatica 39, 263--272.Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Hespanha, J. and Morse, A. 1999. Stability of switched systems with average dwell-time. In Proceedings of 38th IEEE Conference on Decision and Control. IEEE, Los Alamitos, CA, 2655--2660.Google ScholarGoogle Scholar
  25. Hespanha, J. P. and Tiwari, A., Eds. 2006. Proceedings of the 9th International Workshop on Hybrid Systems: Computation and Control (HSCC'06). Springer, Berlin.Google ScholarGoogle Scholar
  26. Khalil, H. K. 2002. Nonlinear Systems 3rd Ed. Prentice Hall, Upper Saddle River, NJ.Google ScholarGoogle Scholar
  27. Kurzhanski, A. B. and Varaiya, P. 2000. Ellipsoidal techniques for reachability analysis. In Proceedings of the 3rd International Workshop on Hybrid Systems: Computation and Control (HSCC'00). Springer, Berlin, 202--214. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Lafferriere, G., Pappas, G. J., and Yovine, S. 1999. A new class of decidable hybrid systems. In Proceedings of the 2nd International Workshop on Hybrid Systems: Computation and Control (HSCC'99). Springer, Berlin, 137--151. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Liberzon, D. 2003. Switching in Systems and Control. Systems and Control: Foundations and Applications. Birkhauser, Boston, MA.Google ScholarGoogle Scholar
  30. Livadas, C., Lygeros, J., and Lynch, N. A. 1999. High-level modeling and analysis of TCAS. In Proceedings of the 20th IEEE Real-Time Systems Symposium (RTSS'99). IEEE, Los Alamitos, CA, 115--125. Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. Lynch, N. 1996. A three-level analysis of a simple acceleration maneuver, with uncertainties. In Proceedings of the 3rd AMAST Workshop on Real-Time Systems. World Scientific Publishing Company, Mountain View, CA, 1--22.Google ScholarGoogle Scholar
  32. Lynch, N., Segala, R., and Vaandrager, F. 2003. Hybrid I/O automata. Inform. Compu. 185, 1, 105--157. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. Lynch, N. and Vaandrager, F. 1996. Forward and backward simulations—part II: Timingbased systems. Inform. Comput. 128, 1, 1--25. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. Maler, O. and Pnueli, A., Eds. 2003. Proceedings of the 6th International Workshop on Hybrid Systems: Computation and Control (HSCC'03). Springer, Berlin.Google ScholarGoogle Scholar
  35. Mitchell, I. and Tomlin, C. 2000. Level set methods for computation in hybrid systems. In Proceedings of the 3rd International Workshop on Hybrid Systems: Computation and Control (HSCC'00). Springer, Berlin, 310--323. Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. Mitra, S. 2007. A verification framework for hybrid systems. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Mitra, S. and Archer, M. 2005. PVS strategies for proving abstraction properties of automata. Electronic Notes Theore. Comput. Sci. 125, 2, 45--65.Google ScholarGoogle ScholarDigital LibraryDigital Library
  38. Mitra, S. and Lynch, N. A. 2007. Trace-based semantics for probabilistic timed i/o automata. In Proceedings of the 10th International Workshop (HSCC'07), (Pisa, Italy), April 3--5. Springer, Berlin, 718--722.Google ScholarGoogle Scholar
  39. Mitra, S., Wang, Y., Lynch, N., and Feron, E. 2003. Safety verification of model helicopter controller using hybrid Input/Output automata. In Proceedings of the 6th International Workshop on Hybrid Systems: Computation and Control (HSCC'03). Springer, Berlin, 343--358.Google ScholarGoogle Scholar
  40. Morari, M. and Thiele, L., Eds. 2005. Proceedings of the 8th International Workshop on Hybrid Systems: Computation and Control (HSCC'05). Springer, Berlin.Google ScholarGoogle Scholar
  41. Morse, A. S. 1996. Supervisory control of families of linear set-point controllers, part 1: Exact matching. IEEE Trans. Auto. Control 41, 1413--1431.Google ScholarGoogle ScholarCross RefCross Ref
  42. Owre, S., Rajan, S., Rushby, J., Shankar, N., and Srivas, M. 1996. PVS: Combining specification, proof checking, and model checking. In Proceedings of Computer-Aided Verification (CAV'96). Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. Prajna, S. and Jadbabaie, A. 2004. Safety verification of hybrid systems using barrier certificates. In Proceedings of the 7th International Workshop on Hybrid Systems: Computation and Control (HSCC'04). Springer, Berlin.Google ScholarGoogle Scholar
  44. Tomlin, C. and Greenstreet, M. R., Eds. 2002. Proceedings of the 5th International Workshop on Hybrid Systems: Computation and Control (HSCC'02). Springer, Berlin. Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Umeno, S. and Lynch, N. A. 2007. Safety verification of an aircraft landing protocol: A refinement approach. In Proceedings of the 10th International Workshop, (HSCC'07). Springer, Berlin, 557--572.Google ScholarGoogle Scholar
  46. van der Schaft, A. and Schumacher, H. 2000. An Introduction to Hybrid Dynamical Systems. Springer, Berlin.Google ScholarGoogle Scholar
  47. Vu, L., Chatterjee, D., and Liberzon, D. 2007. Input-to-state stability of switched systems and switching adaptive control. Automatica 43, 4, 639--646. Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. Weinberg, H. B. and Lynch, N. 1996. Correctness of vehicle control systems—a case study. In Proceeding of the 17th IEEE Real-Time Systems Symposium. IEEE, Los Alamitos, CA, 62--72. Google ScholarGoogle ScholarDigital LibraryDigital Library
  49. Weinberg, H. B., Lynch, N., and Delisle, N. 1995. Verification of automated vehicle protection systems. In Proceedings of the 3rd International Workshop on Hybrid Systems III: Verification and Control Workshop on Verification and Control of Hybrid Systems. Springer, Berlin, 101--113. Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. Williams, H. 1990. Model Building in Mathematical Programming 3rd Ed. John Wiley, New York.Google ScholarGoogle Scholar
  51. Zhai, G., Hu, B., Yasuda, K., and Michel, A. 2000. Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach. In Proceedings of the American Control Conference (AACC).Google ScholarGoogle Scholar

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