Skip to main content

Part of the book series: Advances in Industrial Control ((AIC))

  • 689 Accesses

Abstract

This chapter presents a mathematical treatment of Petri nets, including their formal definitions, structural and behavioral properties such as invariants, siphons, traps, reachability graphs, and state equations that are necessary to understand the subjects presented in this book. A number of important subclasses of Petri nets are introduced such as state machines and marked graphs. They are essential for the development of manufacturing-oriented Petri net models and the deadlock control strategies. The basics of automata are also covered in this chapter to facilitate the reader to understand well the deadlock prevention policy based on theory of regions. The concepts of a plant model, supervisor, and controlled system are defined.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abdallah, I.B., ElMaraghy, H.A., ElMekkawy, T. (1997) A logic programming approach for ñding minimal siphons in S3PR nets applied to manufacturing systems. In Proc. IEEE Int. Conf. on Systems, Man, and Cybernetics, pp.1710–1715.

    Google Scholar 

  2. Barkaoui, K., Lemaire, B. (1989) An effective characterization of minimal deadlocks and traps in Petri nets based on graph theory. In Proc. 10th Int. Conf. on Applications and Theory of Petri Nets, pp.1–21.

    Google Scholar 

  3. Barkaoui, K., Minoux, M. (1992) A polynomial time graph algorithm to decide liveness of some basic classes of Petri nets. In Proc. 13th Int. Conf. on Applications and Theory of Petri Nets, Lecture Notes in Computer Science, vol.616, pp.62–75.

    Google Scholar 

  4. Barkaoui, K., Pradat-Peyre, J.F. (1996) On liveness and controlled siphons in Petri nets. In Proc. 17th Int. Conf. on Applications and Theory of Petri Nets Lecture Notes in Computer Science, vol.1091, pp.57–72.

    Google Scholar 

  5. Boer, E.R., Murata, T. (1994) Generating basis siphons and traps of Petri nets using the sign incidence matrix. IEEE Transactions on Circuits and Systems I–Fundamental Theory and Applications, vol.41, no.4, pp.266–271.

    Article  MathSciNet  Google Scholar 

  6. Bogdan, S., Lewis, F.L., Kovacic, Z., Mireles, J. (2006) Manufacturing Systems Control Design. London: Springer.

    MATH  Google Scholar 

  7. Cassandras, C.G., Lafortune, S. (1999) Introduction to Discrete Event Systems. Boston, MA: Kluwer.

    MATH  Google Scholar 

  8. Cassandras, C.G., Lafortune, S. (2008) Introduction to Discrete Event Systems. Springer.

    Google Scholar 

  9. Chao, D.Y. (2006) Computation of elementary siphons in Petri nets for deadlock control. Computer Journal, vol.49, no.4, pp.470–479.

    Article  Google Scholar 

  10. Chao, D.Y. (2006) Searching strict minimal siphons for SNC-based resource allocation systems, Journal of Information Science and Engineering, vol.23, no.3, pp.853–867.

    Google Scholar 

  11. Cheung, K.S. (2004) New characterization for live and reversible augmented Petri nets. Information Processing Letters, vol.92, no.5, pp.239–243.

    Article  MathSciNet  Google Scholar 

  12. Chu, F., Xie, X.L. (1997) Deadlock analysis of Petri nets using siphons and mathematical programming. IEEE Transactions on Robotics and Automation, vol.13, no.6, pp.793–804.

    Article  Google Scholar 

  13. Colom, J.M., Campos, J., Silva, M. (1990) On liveness analysis through linear algebraic techniques. In Proc. of Annual General Meeting of ESPRIT Basic Research Action 3148 Design Methods Based on Nets DEMON.

    Google Scholar 

  14. Cordone, R., Ferrarini, L., Piroddi, L. (2003) Some results on the computation of minimal siphons in Petri nets. In Proc. 42nd IEEE Conf. on Decision and Control, pp.3754–3759.

    Google Scholar 

  15. Cordone, R., Ferrarini, L., Piroddi, L. (2005) Enumeration algorithms for minimal siphons in Petri nets based on place constraints. IEEE Transactions on Systems, Man and Cybernetics, Part A, vol.35, no.6, pp.844–854.

    Article  Google Scholar 

  16. Desel, J., Esparza, J. (1995) Free Choice Petri Nets. London: Cambridge University Press.

    MATH  Google Scholar 

  17. Desel, J., Reisig, W. (1998) Place/transition Petri nets. In Lectures on Petri Nets I: Basic Models, Lecture Notes in Computer Science, vol.1491, W. Reisig and G. Rozenberg (Eds.), pp.122–174.

    Google Scholar 

  18. Desel, J. (1998) Basic linear algebraic techniques for place/transition nets. In Lectures on Petri Nets I: Basic Models, Lecture Notes in Computer Science, vol.1491, W. Reisig and G. Rozenberg (Eds.), pp.257–308.

    Google Scholar 

  19. Desrocher, A.A., AI-Jaar, R.Y. (1995) Applications of Petri Nets in Manufacturing Systems: Modeling, Control, and Performance Analysis, Piscataway, NJ: IEEE Press.

    Google Scholar 

  20. Ding, Z.J., Jiang, C.J., Zhou, M.C. (2008) Deadlock checking for one-place unbounded Petri nets based on modiêd reachability trees. IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol.38, no.3, pp.881–882.

    Article  Google Scholar 

  21. Esparza, J., Silva, M. (1992) A polynomial-time algorithm to decide liveness of bounded free choice nets. Theoretical Computer Sciences, vol.102, no.1, pp.185–205.

    Article  MATH  MathSciNet  Google Scholar 

  22. Ezpeleta, J., Couvreur, J.M., Silva, M. (1993) A new technique for ñding a generating family of siphons, traps, and st-components: Application to colored Petri nets. In Advances in Petri Nets, Lecture Notes in Computer Science, vol.674, G. Rozenberg (Ed.), pp.126–147.

    Google Scholar 

  23. Ezpeleta, J, Colom, J.M., Martinez, J. (1995) A Petri net based deadlock prevention policy for êxible manufacturing systems. IEEE Transactions on Robotics and Automation, vol.11, no.2, pp.173–184.

    Article  Google Scholar 

  24. Giua, A., Seatzu, C. (2007) A systems theory view of Petri nets. In Advances in Control Theory and Applications, Lecture Notes in Control and Information Science, vol.353, C. Bonivento et al. (Eds.), pp.99–127.

    Google Scholar 

  25. Holloway, L.E., Krogh, B.H., Giua, A. (1997) A survey of Petri net methods for controlled discrete event systems. Discrete Event Dynamic Systems: Theory and Applications, vol.7, no.2, pp.151–190.

    Article  MATH  Google Scholar 

  26. Hopcroft, J.E., Motwani, R., Ullman, J.D. (2000) Introduction to Automata Theory, Languages, and Computation, 2nd ed., New York: Addison-Wesley.

    Google Scholar 

  27. Hruz, B., Zhou, M.C (2007) Modeling and Control of Discrete-Event Dynamic Systems: With Petri Nets and Other Tools. London: Springer.

    MATH  Google Scholar 

  28. Jeng, M.D, Peng, M.Y. (1999) Augmented reachability trees for 1-place-unbounded generalized Petri nets. IEEE Transactions on Systems, Man, and Cybernetics, Part A, vol.29, no.2, pp.173–183.

    Article  Google Scholar 

  29. Jeng, M.D., Peng, M.Y., Huang, Y.S. (1999) An algorithm for calculating minimal siphons and traps in Petri nets. International Journal of Intelligent Control and Systems, vol.3, no.3, pp.263–275.

    Google Scholar 

  30. Kumar, R. Garg, V. (1995) Modeling and Control of Logical Discrete Event Systems. Boston, MA: Kluwer.

    MATH  Google Scholar 

  31. Lautenbach, K. (1987) Linear algebraic calculation of deadlocks and traps. In Concurrency and Nets, K. Voss, H. J. Genrich and G. Rozenberg (Eds.), pp.315–336.

    Google Scholar 

  32. Lautenbach, K., Ridder, H. (1993) Liveness in bounded Petri nets which are covered by T- invariants. In Proc. 13th Int. Conf. on Applications and Theory of Petri Nets, Lecture Notes in Computer Science, vol.815, R. Valette (Ed.), pp.358–375.

    Google Scholar 

  33. Li, Z.W., Zhou, M.C. (2004) Elementary siphons of Petri nets and their application to deadlock prevention in êxible manufacturing systems. IEEE Transactions on Systems, Man, and Cybernetics, Part A, vol.34, no.1, pp.38–51.

    Article  Google Scholar 

  34. Li, Z.W., Wei, N. (2007) Deadlock control of êxible manufacturing systems via invariantcontrolled elementary siphons of Petri nets. International Journal of Advanced Manufacturing Technology, vol.33, no.1–2, pp.24–35.

    Google Scholar 

  35. Li, Z.W., Zhou, M.C. (2008) On siphon computation for deadlock control in a class of Petri nets. IEEE Transactions on Systems, Man, and Cybernetics, A., vol.38, no.3, pp.667–679.

    Google Scholar 

  36. Minoux, M., Barkaoui, K. (1990) Deadlocks and traps in Petri nets as horn-satisffiability solutions and some related polynomially solvable problems. Discrete Mathematics, vol.29, no.2– 3, pp.195–210.

    MATH  MathSciNet  Google Scholar 

  37. Murata, T. (1989) Petri nets: Properties, analysis, and applications. Proceedings of the IEEE, vol.77, no.4, pp.541–580.

    Article  Google Scholar 

  38. Peterson, J.L. (1977) Petri nets. Computing Surveys, vol.9, no.3, pp.223–252.

    Article  MATH  MathSciNet  Google Scholar 

  39. Peterson, J.L. (1981) Petri Net Theory and the Modeling of Systems. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  40. Piroddi, L., Cordone, R., Fumagalli, I. (2008) Selective siphon control for deadlock prevention in Petri nets, IEEE Transactions on Systems, Man, and Cybernetics, Part A, vol. 38, no. 6, pp.1337–1348.

    Article  Google Scholar 

  41. Ramadge, P., Wonham, W.M. (1989) The control of discrete event systems. Proceedings of the IEEE, vol.77, no.1, pp.81–89.

    Article  Google Scholar 

  42. Recalde, L., Teruel, E., Silva, M., (1998) On linear algebraic techniques for liveness analysis of P/T systems. Journal of Circuits, Systems, and Computers, vol.8, no.1, pp.223–265.

    Article  MathSciNet  Google Scholar 

  43. Reisig, W. (1985) Petri Nets: An Introduction. New York: Springer.

    MATH  Google Scholar 

  44. Reutenauer, C. (1990) The Mathematics of Petri Nets. Translated by I. Varig, Englewood Cliffs, NJ: Prentice-Hall.

    MATH  Google Scholar 

  45. Ru, Y., Wu, W.M., Hadjicostis, C.N. (2006) Comments on “A modiêd reachability tree approach to analysis of unbounded Petri nets”. IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol.36, no.5, p.1210.

    Article  Google Scholar 

  46. Silva, M., Teruel, E. (1996) A systems theory perspective of discrete event dynamic systems: The Petri net paradigm. In P. Borne, J. C. Gentina, E. Craye, and S. El Khattabi, (Eds.), Symposium on Discrete Events and Manufacturing Systems, IMACS Multiconference, Lille, France, pp.1–12.

    Google Scholar 

  47. Starke, P. H. (2003) INA: Integrated Net Analyzer. http://www2.informatik.hu-berlin.de/∼star ke/ina.html.

    Google Scholar 

  48. Tanimoto, S., Yamauchi, M., Watanabe, T. (1996) Finding minimal siphons in general Petri nets. IEICE Transactions on Fundamentals, vol.E79-A, no.11, pp.1817–1824.

    Google Scholar 

  49. Thistle, J.G. (1996) Supervisory control of discrete event systems. Mathematical and Computer and Modeling, vol.23, no.11–12, pp.25–53.

    Article  MATH  MathSciNet  Google Scholar 

  50. Tricas, F., Garacía-Vallés, F., Colom, J.M., Ezpeleta, J. (1998) A partial approach to the problem of deadlocks in processes with resources. Research Report, GISI-RR-97-05, Departamento de Informática e Ingeniería de Sistemas, Universidad de Zaragoza, Spain.

    Google Scholar 

  51. Tricas, F., García-Vallés, F., Colom, J.M., Ezpeleta, J. (1998) A structural approach to the problem of deadlock prevention in processes with shared resources. In Proc. 4th Workshop on Discrete Event Systems, pp.273–278.

    Google Scholar 

  52. Tricas, F., Ezpeleta, J. (1999) A Petri net solution to the problem of deadlocks in systems of processes with resources. In Proc. IEEE Int. Conf. on Emerging Technologies and Factory Automation, pp.1047–1056.

    Google Scholar 

  53. Tricas, F., Ezpeleta, J. (2003) Some results on siphon computation for deadlock prevention in resource allocation systems modeled with Petri nets. In Proc. IEEE Int. Conf. on Emerging Technologies and Factory Automation, pp.322–329.

    Google Scholar 

  54. Tricas, F., Ezpeleta, J. (2006) Computing minimal siphons in Petri net models of resource allocation systems: A parallel solution. IEEE Transactions on Systems, Man, and Cybernetics, Part A, vol.36, no.3, pp.532–539.

    Article  Google Scholar 

  55. Tsuji, K., Murata, T. (1993) On reachability conditions for unrestricted Petri nets. In Proc. IEEE Int. Symp. on Circuits and Systems, pp.2713–2716.

    Google Scholar 

  56. Wang, F.Y., Gao, Y.Q., Zhou, M.C. (2004) A modiêd reachability tree approach to analysis of unbounded Petri nets. IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol.34, no.1, pp.303–308.

    Article  Google Scholar 

  57. Watanabe, T., Yamauchi, M., Tanimoto, S. (1998) Extracting siphons containing speciêd set of places in Petri nets. In Proc. IEEE Int. Conf. on Systems, Man, and Cybernetics, pp.142– 147.

    Google Scholar 

  58. Xing, K.Y., Hu, B.S., Chen, H.X. (1996) Deadlock avoidance policy for Petri-net modelling of êxible manufacturing systems with shared resources. IEEE Transactions on Automatic Control, vol.41, no.2, pp.289–295.

    Article  MATH  MathSciNet  Google Scholar 

  59. Xing, K.Y., Hu, B.S. (2005) Optimal liveness Petri net controllers with minimal structures for automated manufacturing systems. In Proc. IEEE Int. Conf. on Systems, Man and Cybernetics, pp.282–287.

    Google Scholar 

  60. Yamauchi, M., Watanabe, T. (1999) Algorithms for extracting minimal siphons containing speciêd places in a general Petri net. IEICE Transactions on Fundamentals, vol.E82-A, no.11, pp.2566–2575.

    Google Scholar 

  61. Zhou, M.C., DiCesare, F. (1991) Parallel and sequential exclusions for Petri net modeling for manufacturing systems. IEEE Transactions on Robotics and Automation, vol.7, no.4, pp.515– 527.

    Article  Google Scholar 

  62. Zhou, M.C., DiCesare, F. (1993) Petri Net Synthesis for Discrete Event Control of Manufacturing Systems. Boston, MA: Kluwer.

    MATH  Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer London

About this chapter

Cite this chapter

(2009). Petri Nets. In: Deadlock Resolution in Automated Manufacturing Systems. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-84882-244-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-84882-244-3_2

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84882-243-6

  • Online ISBN: 978-1-84882-244-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics