An alternative to automata for untimed models of DES is provided by Petri nets. These models were first developed by C. A. Petri in the early 1960s. As we will see, Petri nets are related to automata in the sense that they also explicitly represent the transition function of DES. Like an automaton, a Petri net is a device that manipulates events according to certain rules. One of its features is that it includes explicit conditions under which an event can be enabled; this allows the representation of very general DES whose operation depends on potentially complex control schemes. This representation is conveniently described graphically, at least for small systems, resulting in Petri net graphs; Petri net graphs are intuitive and capture a lot of structural information about the system. We will see that an automaton can always be represented as a Petri net; on the other hand, not all Petri nets can be represented as finite-stateautomata. Consequently, Petri nets can represent a larger class of languages than the class of regular languages, R. Another motivation for considering Petri net models of DES is the body of analysis techniques that have been developed for studying them. These techniques include reachability analysis, similarly to the case of automata, as well as linear-algebraic techniques. Such techniques cover not only untimed Petri net models but timed Petri net models as well; in this regard, we will see in the next chapter that there is a well-developed theory, called the “max-plus algebra,” for a certain class of timed Petri nets (cf. Sect. 5.4). Finally, we mention that control of Petri nets is an active research area and there are controller synthesis techniques that exploit the structural properties of Petri nets.


Coverability Tree Reachable State State Transition Diagram State Transition Function Reachability Tree 
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Selected References

∎ Introductory books or survey papers on Petri nets

  1. — David, R., and H. Alla, Petri Nets & Grafcet: Tools for Modelling Discrete Event Systems, Prentice-Hall, New York, 1992.MATHGoogle Scholar
  2. — Murata, T., “Petri Nets: Properties, Analysis and Applications,” Proceedings of the IEEE, Vol. 77, No. 4, pp. 541–580, 1989.CrossRefGoogle Scholar
  3. — Peterson, J.L., Petri Net Theory and the Modeling of Systems, Prentice Hall, Englewood Cliffs, 1981.Google Scholar

∎ Control of Petri nets

  1. — Holloway, L.E., B.H. Krogh, and A. Giua, “A Survey of Petri Net Methods for Controlled Discrete Event Systems,” Journal of Discrete Event Dynamic Sys tems: Theory and Applications, Vol. 7, No. 2, pp. 151–190, April 1997.MATHCrossRefGoogle Scholar
  2. — Iordache, M.V., and P.J. Antsaklis, Supervisory Control of Concurrent Systems - A Petri Net Structural Approach, Birkhäuser, Boston, 2006.MATHGoogle Scholar
  3. — Moody, J.O., and P.J. Antsaklis, Supervisory Control of Discrete Event Systems Using Petri Nets, Kluwer Academic Publishers, Boston, 1998.MATHGoogle Scholar
  4. — Stremersch, G., Supervision of Petri Nets, Kluwer Academic Publishers, Boston, 2001.MATHGoogle Scholar

∎ Miscellaneous

  1. — Desrochers, A.A., and R.Y. Al-Jaar, Applications of Petri Nets in Automated Manufacturing Systems: Modeling, Control, and Performance Analysis, IEEE Press, Piscataway, NJ, 1995.Google Scholar
  2. — Giua, A., and F. DiCesare, “Blocking and Controllability of Petri Nets in Su pervisory Control,” IEEE Transactions on Automatic Control, Vol. 39, No. 4, pp. 818–823, April 1994.MATHCrossRefMathSciNetGoogle Scholar
  3. — Sreenivas, R.S., and B.H. Krogh, “On Petri Net Models of Infinite State Supervisors,” IEEE Transactions on Automatic Control, Vol. 37, No. 2, pp. 274—277, February 1992.CrossRefMathSciNetGoogle Scholar
  4. — Zhou, M.C., and F. DiCesare, Petri Net Synthesis for Discrete Event Control of Manufacturing Systems, Kluwer Academic Publishers, Boston, 1993.MATHGoogle Scholar

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