Skip to main content

Advertisement

Springer Nature Link
Log in

A compositional modelling and analysis framework for stochastic hybrid systems

  • Published:
Formal Methods in System Design Aims and scope Submit manuscript

Abstract

The theory of hybrid systems is well-established as a model for real-world systems consisting of continuous behaviour and discrete control. In practice, the behaviour of such systems is also subject to uncertainties, such as measurement errors, or is controlled by randomised algorithms. These aspects can be modelled and analysed using stochastic hybrid systems. In this paper, we present HModest, an extension to the Modest modelling language—which is originally designed for stochastic timed systems without complex continuous aspects—that adds differential equations and inclusions as an expressive way to describe the continuous system evolution. Modest is a high-level language inspired by classical process algebras, thus compositional modelling is an integral feature. We define the syntax and semantics of HModest and show that it is a conservative extension of Modest that retains the compositional modelling approach. To allow the analysis of HModest models, we report on the implementation of a connection to recently developed tools for the safety verification of stochastic hybrid systems, and illustrate the language and the tool support with a set of small, but instructive case studies.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. Our implementation also supports fixed-size arrays and user-defined data structures, which are technical extensions but not conceptually relevant for this paper.

  2. Since we omitted the details of the expression syntax, we assume type correctness in assignments, guards, weights etc. instead of providing the (standard) type checking rules in detail.

  3. The semantics of par { ::Tank()::Controller() } itself contains additional locations because the two process calls are not syntactically equal to the behaviours of the processes called. The semantics shown above can be obtained as the semantics of the entire model by replacing the do construct in the Controller process by a (tail-)recursive process call and the call Tank() in the parallel composition by a direct call to TankOff().

  4. http://www.mono-poject.org/.

  5. Computations were performed on an AMD Athlon II X4 620 system with 4 GB RAM.

  6. Computations were performed on an Intel Core i7 860 system with 8 GB RAM.

  7. Computations were performed on an Intel Core i7 860 system with 8 GB RAM.

References

  1. Abate A, Katoen J, Lygeros J, Prandini M (2010) Approximate model checking of stochastic hybrid systems. Eur J Control 16(6):624–641

    Article  MathSciNet  MATH  Google Scholar 

  2. Abate A, Prandini M, Lygeros J, Sastry S (2008) Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems. Automatica 44(11):2724–2734

    Article  MathSciNet  MATH  Google Scholar 

  3. Altman E, Gaitsgory V (1997) Asymptotic optimization of a nonlinear hybrid system governed by a Markov decision process. SIAM J Control Optim 35(6):2070–2085

    Article  MathSciNet  MATH  Google Scholar 

  4. Alur R, Courcoubetis C, Halbwachs N, Henzinger TA, Ho PH, Nicollin X, Olivero A, Sifakis J, Yovine S (1995) The algorithmic analysis of hybrid systems. Theor Comput Sci 138:3–34

    Article  MATH  Google Scholar 

  5. Alur R, Dang T, Esposito JM, Hur Y, Ivancic F, Kumar V, Lee I, Mishra P, Pappas GJ, Sokolsky O (2003) Hierarchical modeling and analysis of embedded systems. Proc IEEE 91(1):11–28

    Article  Google Scholar 

  6. Alur R, Dang T, Ivancic F (2006) Predicate abstraction for reachability analysis of hybrid systems. ACM Trans Embed Comput Syst 5(1):152–199

    Article  Google Scholar 

  7. Alur R, Dill DL (1994) A theory of timed automata. Theor Comput Sci 126(2):183–235

    Article  MathSciNet  MATH  Google Scholar 

  8. Baró Graf H, Hermanns H, Kulshrestha J, Peter J, Vahldiek A, Vasudevan A (2011) A verified wireless safety critical hard real-time design. In: IEEE int symp on a world of wireless, mobile and multimedia networks (WoWMoM). IEEE Press, New York

    Google Scholar 

  9. van Beek DA, Man KL, Reniers MA, Rooda JE, Schiffelers RRH (2006) Syntax and consistent equation semantics of hybrid Chi. J Log Algebr Program 68(1–2):129–210

    Article  MathSciNet  MATH  Google Scholar 

  10. Behrmann G, David A, Larsen KG (2004) A tutorial on uppaal. In: Formal methods for the design of real-time systems (SFM-RT). LNCS, vol 3185. Springer, Berlin, pp 200–236

    Chapter  Google Scholar 

  11. Berendsen J, Jansen DN, Katoen JP (2006) Probably on time and within budget: on reachability in priced probabilistic timed automata. In: Quantitative evaluation of systems (QEST). IEEE Comput Soc, Los Alamitos, pp 311–322

    Google Scholar 

  12. Bernadsky M, Sharykin R, Alur R (2004) Structured modeling of concurrent stochastic hybrid systems. In: Formal modelling and analysis of timed systems, and formal techniques in real-time and fault-tolerant systems (FORMATS/FTRTFT). LNCS, vol 3253. Springer, Berlin, pp 309–324

    Chapter  Google Scholar 

  13. Berrang P, Bogdoll J, Hahn EM, Hartmanns A, Hermanns H (2012) Dependability results for power grids with decentralized stabilization strategies. Reports of SFB/TR 14 AVACS 83, SFB/TR 14 AVACS, ISSN: 1860-9821. www.avacs.org

  14. Blom H, Lygeros J (2006) Stochastic hybrid systems: theory and safety critical applications. Lecture notes in control and information sciences, vol 337. Springer, Berlin

    Book  Google Scholar 

  15. Bogdoll J, David A, Hartmanns A, Hermanns H (2012) mctau: bridging the gap between Modest and UPPAAL. In: Model checking software—19th international workshop, SPIN 2012, Oxford, UK, July 23–24. LNCS, vol 7385. Springer, Berlin. ISBN 978-3-642-31758-3

    Chapter  Google Scholar 

  16. Bogdoll J, Fioriti LMF, Hartmanns A, Hermanns H (2011) Partial order methods for statistical model checking and simulation. In: Formal techniques for distributed systems (FMOODS/FORTE). LNCS, vol 6722. Springer, Berlin, pp 59–74

    Chapter  Google Scholar 

  17. Bohnenkamp HC, D’Argenio PR, Hermanns H, Katoen JP (2006) MoDeST: a compositional modeling formalism for hard and softly timed systems. IEEE Trans Softw Eng 32(10):812–830

    Article  Google Scholar 

  18. Bohnenkamp HC, Gorter J, Guidi J, Katoen JP (2005) Are you still there?—A lightweight algorithm to monitor node presence in self-configuring networks. In: Dependable systems and networks (DSN). IEEE Comput Soc, Los Alamitos, pp 704–709

    Google Scholar 

  19. Brinksma E, Krilavicius T, Usenko YS (2005) A process-algebraic approach to hybrid systems. In: 16th IFAC world congress. IFAC, Laxenburg

    Google Scholar 

  20. Bujorianu ML (2004) Extended stochastic hybrid systems and their reachability problem. In: Hybrid systems: computation and control (HSCC). LNCS, vol 2993. Springer, Berlin, pp 234–249

    Chapter  Google Scholar 

  21. Bujorianu ML, Lygeros J, Bujorianu MC (2005) Bisimulation for general stochastic hybrid systems. In: Hybrid systems: computation and control (HSCC). LNCS, vol 3414. Springer, Berlin, pp 198–214

    Chapter  Google Scholar 

  22. Clarke E, Fehnker A, Han Z, Krogh B, Stursberg O, Theobald M (2003) Verification of hybrid systems based on counterexample-guided abstraction refinement. In: Tools and algorithms for the construction and analysis of systems (TACAS). LNCS, vol 2619. Springer, Berlin, pp 192–207

    Chapter  Google Scholar 

  23. Cuijpers PJL, Reniers MA (2005) Hybrid process algebra. J Log Algebr Program 62(2):191–245

    Article  MathSciNet  MATH  Google Scholar 

  24. Dang T, Maler O (1998) Reachability analysis via face lifting. In: Hybrid systems: computation and control (HSCC). LNCS, vol 1386. Springer, Berlin, pp 96–109

    Chapter  Google Scholar 

  25. D’Argenio PR, Wolovick N, Terraf PS, Celayes P (2009) Nondeterministic labeled Markov processes: bisimulations and logical characterization. In: Quantitative evaluation of systems (QEST). IEEE Comput Soc, Los Alamitos, pp 11–20

    Google Scholar 

  26. Davis MHA (1993) Markov models and optimization. Chapman & Hall, London

    Book  MATH  Google Scholar 

  27. Desharnais J, Edalat A, Panangaden P (2002) Bisimulation for labelled Markov processes. Inf Comput 179(2):163–193

    Article  MathSciNet  MATH  Google Scholar 

  28. Edwards S, Lavagno L, Lee EA, Sangiovanni-Vincentelli A (1997) Design of embedded systems: formal models, validation, and synthesis. Proc IEEE 85(3):366–390

    Article  Google Scholar 

  29. Fränzle M, Hahn EM, Hermanns H, Wolovick N, Zhang L (2011) Measurability and safety verification for stochastic hybrid systems. In: Hybrid systems: computation and control (HSCC). ACM, New York, pp 43–52

    Google Scholar 

  30. Fränzle M, Herde C, Teige T, Ratschan S, Schubert T (2007) Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. J. Satisf. Boolean Model. Comput. 1(3–4):209–236

    Google Scholar 

  31. Frehse G (2008) Phaver: algorithmic verification of hybrid systems past HyTech. Int J Softw Tools Technol Transf 10(3):263–279

    Article  MathSciNet  Google Scholar 

  32. Frehse G, Guernic CL, Donzé A, Cotton S, Ray R, Lebeltel O, Ripado R, Girard A, Dang T, Maler O (2011) Spaceex: scalable verification of hybrid systems. In: Computer-aided verification (CAV). LNCS, vol 6806. Springer, Berlin, pp 379–395

    Chapter  Google Scholar 

  33. Giry M (1982) A categorical approach to probability theory. In: Categorical aspects of topology and analysis. Springer, Berlin, pp 68–85

    Chapter  Google Scholar 

  34. Groß C, Hermanns H, Pulungan R (2007) Does clock precision influence Zigbee’s energy consumptions? In: Principles of distributed systems (OPODIS). LNCS, vol 4878. Springer, Berlin, pp 174–188

    Chapter  Google Scholar 

  35. Grosu R, Stauner T (2002) Modular and visual specification of hybrid systems: an introduction to HyCharts. Form Methods Syst Des 21(1):5–38

    Article  MATH  Google Scholar 

  36. Hartmanns A (2010) Model-checking and simulation for stochastic timed systems. In: FMCO. LNCS, vol 6957. Springer, Berlin, pp 372–391

    Google Scholar 

  37. Hartmanns A, Hermanns H (2009) A Modest approach to checking probabilistic timed automata. In: Quantitative evaluation of systems (QEST). IEEE Comput Soc, Los Alamitos, pp 187–196

    Google Scholar 

  38. Henzinger TA (1996) The theory of hybrid automata. In: IEEE symp on logic in computer science (LICS), pp 278–292

    Google Scholar 

  39. Henzinger TA, Ho PH, Wong-Toi H (1997) HYTECH: a model checker for hybrid systems. Int J Softw Tools Technol Transf 1(1–2):110–122

    Article  MATH  Google Scholar 

  40. Herde C, Eggers A, Fränzle M, Teige T (2008) Analysis of hybrid systems using HySAT. In: International conference on systems (ICONS). IEEE Comput Soc, Los Alamitos, pp 196–201

    Chapter  Google Scholar 

  41. Hermanns H, Herzog U, Katoen JP (2002) Process algebra for performance evaluation. Theor Comput Sci 274(1–2):43–87

    Article  MathSciNet  MATH  Google Scholar 

  42. Hillston J (1994) A compositional approach to performance modelling. PhD thesis, Univ of Edinburgh

  43. Hu J, Lygeros J, Sastry S (2000) Towards a theory of stochastic hybrid systems. In: Hybrid systems: computation and control (HSCC). LNCS, vol 1790. Springer, Berlin, pp 160–173

    Chapter  Google Scholar 

  44. Kwiatkowska M, Norman G, Parker D (2011) PRISM 4.0: verification of probabilistic real-time systems. In: Computer aided verification (CAV’11). LNCS, vol 6806. Springer, Berlin, pp 585–591

    Chapter  Google Scholar 

  45. Kwiatkowska M, Norman G, Segala R, Sproston J (2000) Verifying quantitative properties of continuous probabilistic timed automata. In: Concurrency theory (CONCUR’00). LNCS, vol 1877. Springer, Berlin, pp 123–137

    Google Scholar 

  46. Kwiatkowska MZ, Norman G, Segala R, Sproston J (2002) Automatic verification of real-time systems with discrete probability distributions. Theor Comput Sci 282(1):101–150

    Article  MathSciNet  MATH  Google Scholar 

  47. Lee EA (2002) Embedded software. In: Zelkowitz M (ed) Advances in computers, vol 56. Academic Press, San Diego

    Google Scholar 

  48. Legay A, Delahaye B, Bensalem S (2010) Statistical model checking: an overview. In: Runtime verification (RV). LNCS, vol 6418. Springer, Berlin, pp 122–135

    Chapter  Google Scholar 

  49. Lynch NA, Segala R, Vaandrager FW (2003) Hybrid i/o automata. Inf Comput 185(1):105–157

    Article  MathSciNet  MATH  Google Scholar 

  50. Mader A, Bohnenkamp HC, Usenko YS, Jansen DN, Hurink J, Hermanns H (2010) Synthesis and stochastic assessment of cost-optimal schedules. Int J Softw Tools Technol Transf 12(5):305–318

    Article  Google Scholar 

  51. Meseguer J, Sharykin R (2006) Specification and analysis of distributed object-based stochastic hybrid systems. In: Hybrid systems: computation and control (HSCC). LNCS, vol 3927. Springer, Berlin, pp 460–475

    Chapter  Google Scholar 

  52. Panangaden P (2008) Labelled Markov processes. World Scientific, Singapore

    Google Scholar 

  53. Penna GD, Intrigila B, Melatti I, Tronci E, Zilli MV (2006) Finite horizon analysis of Markov chains with the Murphy verifier. Int J Softw Tools Technol Transf 8(4–5):397–409

    Article  Google Scholar 

  54. Platzer A (2011) Stochastic differential dynamic logic for stochastic hybrid programs. In: Bjørner N, Sofronie-Stokkermans V (eds) CADE. LNCS, vol 6803. Springer, Berlin, pp 446–460

    Google Scholar 

  55. Preußig J, Kowalewski S, Wong-Toi H, Henzinger T (1998) An algorithm for the approximative analysis of rectangular automata. In: Formal techniques in fault tolerant and real time systems (FTRTFT). LNCS, vol 1486. Springer, Berlin, pp 228–240

    Chapter  Google Scholar 

  56. Ratschan S, She Z (2007) Safety verification of hybrid systems by constraint propagation-based abstraction refinement. ACM Trans Embed Comput Syst 6(1):8

    Article  Google Scholar 

  57. Segala R (1995) Modelling and verification of randomized distributed real-time systems. PhD thesis, MIT, Cambridge, MA, USA

  58. Segala R, Lynch NA (1995) Probabilistic simulations for probabilistic processes. Nord J Comput 2(2):250–273

    MathSciNet  MATH  Google Scholar 

  59. Sproston J (2000) Decidable model checking of probabilistic hybrid automata. In: Formal techniques in real-time and fault-tolerant systems (FTRTFT). LNCS, vol 1926. Springer, Berlin, pp 31–45

    Chapter  Google Scholar 

  60. Strubbe S, van der Schaft A (2006) Compositional modelling of stochastic hybrid systems. In: Cassandras CG, Lygeros J (eds) Stochastic hybrid systems. Control engineering series. Taylor & Francis, London, pp 47–77

    Chapter  Google Scholar 

  61. Wolovick N (2012) Continuous probability and nondeterminism in labeled transition systems. PhD thesis, FaMAF, UNC, Córdoba, Argentina

  62. Yue H, Bohnenkamp HC, Kampschulte M, Katoen JP (2011) Analysing and improving energy efficiency of distributed slotted aloha. In: Smart spaces and next generation wired/wireless networking (NEW2AN). LNCS, vol 6869. Springer, Berlin, pp 197–208

    Chapter  Google Scholar 

  63. Zhang L, She Z, Ratschan S, Hermanns H, Hahn E (2010) Safety verification for probabilistic hybrid systems. In: Computer aided verification. LNCS, vol 6174. Springer, Berlin, pp 196–211

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors thank Pedro D’Argenio for discussions on the language design and Nicolás Wolovick (both from University of Cordoba, Argentina) for his support in the development of the concrete semantics.

Author information

Authors and Affiliations

  1. Saarland University – Computer Science, Campus E1 3, 66123, Saarbrücken, Germany

    Ernst Moritz Hahn, Arnd Hartmanns & Holger Hermanns

  2. LS2: Software Modelling and Verification, RWTH Aachen University, 52056, Aachen, Germany

    Joost-Pieter Katoen

Authors
  1. Ernst Moritz Hahn
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Arnd Hartmanns
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Holger Hermanns
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Joost-Pieter Katoen
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Arnd Hartmanns.

Additional information

This work has been supported by the DFG as part of SFB/TR 14 AVACS, by the EU FP7 project MoVeS, by the DFG/NWO bilateral research project ROCKS and has received funding from the European Union Seventh Framework Programme under grant agreement number 295261 as part of the MEALS project.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hahn, E.M., Hartmanns, A., Hermanns, H. et al. A compositional modelling and analysis framework for stochastic hybrid systems. Form Methods Syst Des 43, 191–232 (2013). https://doi.org/10.1007/s10703-012-0167-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10703-012-0167-z

Keywords