Approximated timed reachability graphs for the robust control of discrete event systems

  • Dimitri LefebvreEmail author


This paper is about control sequences design for Discrete Event Systems (DES) modeled with Time Petri nets (TPN) including a set of temporal specifications. Petri nets are known as efficient mathematical and graphical models that are widely used to describe distributed DES including choices, synchronizations and parallelisms. The domains of application include but are not restricted to manufacturing systems, computer science and transportation networks. Incorporating the time in the model is important to consider many control problems such as scheduling and planning. This paper solves some control issues in timed context and uncertain environments that include unexpected events modeled with uncontrollable transitions. To deal with such uncertainties, we propose first to build an Approximated Timed Reachability Graph that includes the time specifications and model all feasible timed trajectories at a given accuracy under earliest firing policy. Then, this graph is used to search optimal paths by using an approach based on Markov Decision Processes that encode the environment uncertainties. Such optimal paths lead to near-optimal solutions for the TPN. Several simulations illustrate the benefit of the proposed method from the performance and computational points of view.


Discrete event system Time petri net Reachability graph Control design Markov decision process 



The Project MRT MADNESS 2016-2019 has been funded with the support from the European Union with the European Regional Development Fund (ERDF) and from the Regional Council of Normandie.


  1. Alur R, Henzinger T (1999) Reactive modules. Formal Methods in System Design 15(1):7–48CrossRefGoogle Scholar
  2. Baker KR, Trietsch D (2009) Principles of Sequencing and Scheduling. WileyGoogle Scholar
  3. Basile F, Cabasino MP, Seatzu C (2015) State estimation and fault diagnosis of labeled time petri net systems with unobservable transitions. IEEE Trans Autom Control 60(4):997–1009MathSciNetCrossRefzbMATHGoogle Scholar
  4. Beccuti M, Franceschinis G, Haddad S (2007a) Markov decision petri net and Markov decision well-formed net formalisms. LNCS 4546:43–62MathSciNetzbMATHGoogle Scholar
  5. Beccuti M., Codetta-Raiteri D, Franceschinis G, Haddad S (2007b) A framework to design and solve Markov Decision Well-formed Net models. In Proc. of the 4th IEEE Int. Conf. on Quantitative Evaluation of Systems (QEST’07); 165–166, Edinburgh, Scotland, UKGoogle Scholar
  6. Beccuti M, Franceschinis G, Haddad S (2011) A framework to design and solve Markov decision petri nets. IJPE 7(5):417–442Google Scholar
  7. Bellman RE (1957) Dynamic programming. Princeton University Press, PrincetonzbMATHGoogle Scholar
  8. Berthomieu B, Diaz M (1991) Modeling and verification of time dependent systems using time petri nets. IEEE T Software Eng 17(3):259–273MathSciNetCrossRefGoogle Scholar
  9. Berthomieu B, Menasche M (1983) An enumerative approach for analyzing time petri nets. In IFIP Congress, pages 41–46Google Scholar
  10. Berthomieu B, Vernadat F (2003) State class constructions for branching analysis of time petri nets. In TACAS 2003, volume 2619 of LNCS, pages 442–457. SpringerGoogle Scholar
  11. Cassandras C (1993) Discrete event systems: modeling and performances analysis. Aksen Ass. Inc. Pub.Google Scholar
  12. Chen Y, Li Z, Khalgui M, Mosbahi O (2011) Design of a Maximally Permissive Liveness-Enforcing Petri net Supervisor for flexible manufacturing systems. IEEE Trans Autom Sci Eng 8(2):374–393CrossRefGoogle Scholar
  13. Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms. MIT Press and McGraw-HillGoogle Scholar
  14. Daoui C, Abbad M, Tkiouat M (2010) Exact decomposition approaches for Markov decision processes: a survey. Advances in Operations Research 2010:1–19CrossRefzbMATHGoogle Scholar
  15. David R, Alla H (1992) Petri nets and grafcet – tools for modelling discrete events systems. Prentice Hall, LondonzbMATHGoogle Scholar
  16. Dijkstra EW (1971) A short introduction to the art of programmingGoogle Scholar
  17. Eboli MG, Cozman FG (2010) Markov decision processes from colored petri nets, SBIA 2010. Lecture Notes in Computer Science, vol 6404. SpringerGoogle Scholar
  18. Gardey G, Roux OH, Roux OF (2003) Using zone graph method for computing the state space of a time petri net. In FORMATS 2003, volume 2791 of LNCS, pages 246–259. SpringerGoogle Scholar
  19. Haddad S, Moreaux P (2009) Stochastic Petri Nets (Chapter 7), In Petri Nets: Fundamental Models and Applications. WileyGoogle Scholar
  20. Heidari P, Boucheneb H (2012) Maximally permissive controller synthesis for time petri nets. Int J ControlGoogle Scholar
  21. Heidari P, Boucheneb H (2013) Controller synthesis of time petri nets using stopwatch, Journal of Engineering, Hindawi Publishing Corporation, Article ID 970487Google Scholar
  22. Jeng MD, Chen SC (1998) Heuristic search approach using approximate solutions to petri net state equations for scheduling flexible manufacturing systems. Int J FMS 10(2):139–162Google Scholar
  23. Klai K, Aber N, Petrucci L (2013) A new approach to abstract reachability state space of time petri nets. 20th International Symposium on Temporal Representation and ReasoningGoogle Scholar
  24. Lee DY, DiCesare F (1994) Scheduling flexible manufacturing systems using petri nets and heuristic search. IEEE Trans Robot Autom 10(2):123–133CrossRefGoogle Scholar
  25. Lefebvre D (2016a) Approaching minimal time control sequences for timed petri nets. IEEE Trans Autom Sci Eng 13(2):1215–1221CrossRefGoogle Scholar
  26. Lefebvre D (2016b) Deadlock-free scheduling for Timed Petri Net models combined with MPC and backtracking. Proc. IEEE WODES 2016, Invited session “Control, Observation, Estimation and Diagnosis with Timed PN”, pp. 466-471, Xi’an, ChinaGoogle Scholar
  27. Lefebvre D, Leclercq E (2015) Control design for trajectory tracking with untimed petri nets. IEEE Trans Autom Control 60(7):1921–1926MathSciNetCrossRefzbMATHGoogle Scholar
  28. Lei H, Xing K, Han L, Xiong F, Ge Z (2014) Deadlock-free scheduling for flexible manufacturing systems using petri nets and heuristic search. Comput Ind Eng 72:297–305CrossRefGoogle Scholar
  29. Leung Y-T (2004) Handbook of Scheduling: Algorithms, Models, and Performance Analysis. Chapman & Hall/CRC Computer & Information Science SeriesGoogle Scholar
  30. Lime D, Roux OH (2006) Model checking of time petri nets using the state class timed automaton. DEDS 16(2):179–205MathSciNetzbMATHGoogle Scholar
  31. Lopez P, Roubellat F (2008) Production scheduling. ISTE/Wiley, LondonCrossRefGoogle Scholar
  32. Mejia G, Nino K (2017) A new hybrid filtered beam search algorithm for deadlock-free scheduling of flexible manufacturing systems using petri nets. Comput Ind Eng 108:165–176CrossRefGoogle Scholar
  33. Merlin P, Faber DJ (1976) Recoverability of communication protocols. IEEE Trans Commun 24(9):1036–1043CrossRefzbMATHGoogle Scholar
  34. Pan L, Ding Z, Zhou MC (2014) A configurable state class method for temporal analysis of time petri nets. IEEE Trans Syst Man Cybern Syst 44(4):482–493CrossRefGoogle Scholar
  35. Puterman M (1994) Markov decision processes: discrete stochastic dynamic programming. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  36. Ramchandani C (1973) Analysis of asynchronous concurrent systems by timed petri nets. Ph. D, MIT, USAGoogle Scholar
  37. Reyes-Moro A, Hu H, Kelleher G (2002) Hybrid heuristic search for the scheduling of flexible manufacturing systems using petri nets. IEEE Trans Robot Autom 18(2):240–245CrossRefGoogle Scholar
  38. Tarek A, Lopez-Benitez N (2004) Optimal legal firing sequence of petri nets using linear programming. Optim Eng 5:25–43MathSciNetCrossRefzbMATHGoogle Scholar
  39. Tuncel G, Bayhan GM (2007) Applications of petri nets in production scheduling: a review. Int J Adv Manuf Technol 34:762–773CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.UNIHAVRE, GREAHNormandie UniversityLe HavreFrance

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