Abstract
We look at the problem of proving inevitability of continuous dynamical systems. An inevitability property says that a region of the state space will eventually be reached: this is a type of liveness property from the computer science viewpoint, and is related to attractivity of sets in dynamical systems. We consider a method of Maler and Batt to make an abstraction of a continuous dynamical system to a timed automaton, and show that a potentially infinite number of splits will be made if the splitting of the state space is made arbitrarily. To solve this problem, we define a method which creates a finite-sized timed automaton abstraction for a class of linear dynamical systems, and show that this timed abstraction proves inevitability.
This paper has been made under the framework of the EPSRC-funded project “DYVERSE: A New Kind of Control for Hybrid Systems” (EP/I001689/1). The second author is also grateful for the support of the RCUK (EP/E50048/1).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Batt, G., Belta, C., Weiss, R.: Model Checking Liveness Properties of Genetic Regulatory Networks. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 323–338. Springer, Heidelberg (2007)
Behrmann, G., David, A., Larsen, K.G.: A Tutorial on Uppaal. In: Bernardo, M., Corradini, F. (eds.) SFM-RT 2004. LNCS, vol. 3185, pp. 200–236. Springer, Heidelberg (2004)
Clarke, E.M., Fehnker, A., Han, Z., Krogh, B., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and Counterexample-Guided Refinement in Model Checking of Hybrid Systems. International Journal of Foundations of Computer Science 14(4), 583–604 (2003)
D’Innocenzo, A., Julius, A.A., Di Benedetto, M.D., Pappas, G.J.: Approximate timed abstractions of hybrid automata. In: 46th IEEE Conference on Decision and Control, pp. 4045–4050 (2007)
Duggirala, P.S., Mitra, S.: Lyapunov Abstractions for Inevitability of Hybrid Systems. In: Hybrid Systems: Computation and Control (HSCC), pp. 115–123 (2012)
Heinrich, R., Neel, B.G., Rapoport, T.A.: Mathematical Models of Protein Kinase Signal Transduction. Molecular Cell 9(5), 957–970 (2002)
Johansson, K.H., Egerstedt, M., Lygeros, J., Sastry, S.: On the regularization of Zeno hybrid automata. Systems & Control Letters 38(3), 141–150 (1999)
Lyapunov, A.M.: The general problem of the stability of motion. PhD thesis, Moscow University (1892); Reprinted in English in the International Journal of Control 55(3) (1992)
Maler, O., Batt, G.: Approximating Continuous Systems by Timed Automata. In: Fisher, J. (ed.) FMSB 2008. LNCS (LNBI), vol. 5054, pp. 77–89. Springer, Heidelberg (2008)
Mitchell, I., Tomlin, C.J.: Level Set Methods for Computation in Hybrid Systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 310–323. Springer, Heidelberg (2000)
Olivero, A., Sifakis, J., Yovine, S.: Using Abstractions for the Verification of Linear Hybrid Systems. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 81–94. Springer, Heidelberg (1994)
Pnueli, A.: The temporal logic of programs. In: Proceedings of the 18th International Symposium on the Foundations of Computer Science, pp. 46–57 (1977)
Ratschan, S., She, Z.: Safety verification of hybrid systems by constraint propagation-based abstraction refinement. ACM Transactions on Embedded Computing Systems 6(1), 573–589 (2007)
Sloth, C., Wisniewski, R.: Verification of continuous dynamical systems by timed automata. Formal Methods in System Design 39(1), 47–82 (2011)
Stursberg, O., Kowalewski, S., Engell, S.: On the Generation of Timed Discrete Approximations for Continuous Systems. Mathematical and Computer Modelling of Dynamical Systems: Methods, Tools and Applications in Engineering and Related Sciences 6(1), 51–70 (2000)
Tiwari, A.: Abstractions for hybrid systems. Formal Methods in System Design 32(1), 57–83 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Carter, R., Navarro-López, E.M. (2012). Dynamically-Driven Timed Automaton Abstractions for Proving Liveness of Continuous Systems. In: Jurdziński, M., Ničković, D. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2012. Lecture Notes in Computer Science, vol 7595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33365-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-33365-1_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33364-4
Online ISBN: 978-3-642-33365-1
eBook Packages: Computer ScienceComputer Science (R0)