A Computationally Efficient Method for Optimal Input-Flow Control of Timed-Event Graphs Ensuring a Given Production Rate

  • J. R. S. Dias
  • C. A. Maia
  • V. F. LucenaJr.


We focus in this paper on input control of manufacturing systems modelled by timed-event graphs and max-plus algebra. We propose a method for controlling the inflow of material into a production system, and the parameters of a feedback controller are computed, delaying the material input and determining its input rate. The proposed method aims to minimize system inventory to meet demand specifications, allowing the controller to obtain the best control, which delays the occurrence of input events, and delaying the input of raw material in the system as much as possible. The proposed method guarantees that the controller is optimal and proceeds according to just-in-time production planning strategy. It can also be synthesized through efficient computation (polynomial time). In addition, we show that the proposed method outperforms feedback synthesis using model reference techniques.


Discrete-event system Control system Max-plus algebra Manufacturing Process 



The authors are grateful to FAPEAM, CNPq, FAPEMIG and CAPES for the support provided for the development of this work. The first author has a FAPEAM scholarship on the program to support the formation of human resources Post-Graduates of Amazonas State.


  1. Amari, S., Demongodin, I., Loiseau, J. J., & Martinez, C. (2012). Max-plus control design for temporal constraints meeting in timed event graphs. IEEE Transactions on Automatic Control, 57, 462–467.MathSciNetCrossRefGoogle Scholar
  2. Baccelli, F., Cohen, G., Olsder, G. J., & Quadrat, J. P. (1992). Synchronization and linearity. New York: Wiley.zbMATHGoogle Scholar
  3. Campos, J., Seatzu, C., & Xie, X. (2014). Formal methods in manufacturing. Boca Raton: CRC Press.Google Scholar
  4. Cassandras, C. G., & Lafortune, S. (1999). Introduction to discrete event systems. Dordrecht: Kluwer.CrossRefzbMATHGoogle Scholar
  5. Cohen, G., Gaubert, S., & Quadrat, J.-P. (1999). Max-plus algebra and system theory: Where we are and where to go now. Annual Reviews in Control, 23, 207–219.CrossRefGoogle Scholar
  6. Commault, C. (1998). Feedback stabilization of some event graph models. IEEE Transactions on Automatic Control, 43(10), 1419–1423.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cottenceau, B., Hardouin, L., Boimond, J.-L., & Ferrier, J.-L. (1999). Synthesis of greatest linear feedback for timed-event graphs in dioid. IEEE Transactions on Automatic Control, 44(6), 1258–1262.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cottenceau, B., Hardouin, L., Boimond, J.-L., & Ferrier, J.-L. (2001). Model reference control for timed event graphs in dioids. Automatica, 37(9), 1451–1458.CrossRefzbMATHGoogle Scholar
  9. Cottenceau, B., Lahaye, S., & Hardouin, L. (2014). Modeling of time-varying (max,+) systems by means of weighted timed event graphs. 12th IFAC International Workshop on Discrete Event Systems, WODES 2014.Google Scholar
  10. Cottenceau, B., Lhommeau, M., Hardouin, L., & Boimond, J.-L. (2003). on timed event graph stabilization by output feedback in dioid. Kybernetika, 39(2), 165–176.Google Scholar
  11. David-Henriet, X., Hardouin, L., Raisch, J., & Cottenceau, B. (2013). Optimal control for timed event graphs under partial synchronization. In Conference on decision and control, CDC, 2013 (pp. 7609–7614).Google Scholar
  12. David-Henriet, X., Hardouin, L., Raisch, J., & Cottenceau, B. (2014). Modeling and control for max-plus systems with partial synchronization. In 12th IFAC international workshop on discrete event systems, WODES 2014 (Vol. 12, pp. 105–110).Google Scholar
  13. Dias, J. R. S., Maia, C. A., & Lucena, V. F. (2012). Método para controle de fluxo de entrada em sistema max-plus lineares garantindo uma dada taxa de produção. In Anais do XIX Congresso Brasileiro de Automática, CBA, 2012 (pp. 1157–1164).Google Scholar
  14. Dias, J. R. S., Maia, C. A., & Lucena, V. F. (2013). Control of input stream of manufacturing systems modeled by timed event graphs and max-pus algebra. In 6th IFAC conference on management and control of production and logistics (Vol. 6, pp. 478–485).Google Scholar
  15. Heidergott, B., Olsder, G. J., & van der Woude, J. (2006). Max plus at work. Princeton Sries in Applied Mathematics.Google Scholar
  16. Katz, R. D. (2007). Max-plus (a, b)-invariant spaces and control of timed discrete-event systems. IEEE Transactions on Automatic Control, 52(2), 229–241.CrossRefGoogle Scholar
  17. Kim, C., & Lee, T.E. (2012). Feedback control design for cluster tools with wafer residency time constraints. In IEEE international conference on systems, man, and cybernetics, October 2012.Google Scholar
  18. Maia, C.A. (2003). Identificação e controle de sistemas a eventos discretos na álgebra (max,+). PhD thesis, UNICAMP.Google Scholar
  19. Maia, C. A., Andrade, C. R., & Hardouin, L. (2011). On the control of max-plus linear system subject to state restriction. Automatica, 47(5), 988–992.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Maia, C. A., Hardouin, L., Mendes, R. S., & Cottenceau, B. (2003). Optimal closed-loop control of timed event graphs in dioids. IEEE Transactions on Automatic Control, 48(12), 2284–2287.CrossRefGoogle Scholar
  21. Maia, C.A., Hardouin, L., Santos-Mendes, R., & Cottenceau, B. (2005). On the model reference control for max-plus linear systems. In Proceedings of the 44th IEEE conference on decision and control, and the European control conference 2005, December 12–15 2005.Google Scholar
  22. Maia, C. A., Hardouin, L., Santos-Mendes, R., & Loiseau, J. J. (2011). A super-eigenvector approach to control constrained max-plus linear systems. In 50th IEEE conference on decision and control and European control conference, December 12-15 2011.Google Scholar
  23. Maia, C.A., Mendes, R.S., Luders, R., & Hardouin, L. (2005). Estratégias de controle por modelo de referência de sistemas a eventos discretos max-plus lineares. Revista Controle e Automação.Google Scholar
  24. Murata, T. (1989). Petri nets: Properties, analysis and applications. Proceedings of the IEEE, 77(4), 541–580.CrossRefGoogle Scholar
  25. Max Plus. Second order theory of min-linear systems and its application to discrete event systems. In Proceedings of the 30th IEEE conference on decision and control, Brighton, England, December 1991.Google Scholar
  26. Seleim, A., & ElMaraghy, H. (2014). Max-plus modeling of manufacturing flow lines. Proceedings of the 47th CIRP Conference on Manufacturing Systems, 17, 71–75.Google Scholar

Copyright information

© Brazilian Society for Automatics--SBA 2015

Authors and Affiliations

  1. 1.PPGEE - UFMGBelo HorizonteBrazil
  2. 2.PPGEE - UFAMManausBrazil

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