A novel matrix approach for the stability and stabilization analysis of colored Petri nets

Abstract

In this study, the stability and stabilization problem of a colored Petri net based on the semitensor product of matrices is investigated. First, the marking evolution equation of the colored Petri net in a Boolean algebra framework is established, and the necessary and sufficient condition for the stability of the equilibrium point of the colored Petri net is given. Then, the concept of the pre-k steps reachability set is defined and is used to study the problem of marking feedback stabilization. Some properties of the pre-k steps reachability set are developed. The condition of the stabilization of the colored Petri net is given. The algorithm of the optimal marking feedback controller is designed. The proposed method in this paper could judge the stability and stabilization of the colored Petri net by matrix approach. The obtained results are simple and easy to implement by computer. An example is provided to illustrate the effectiveness of the proposed method.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Murata T. Petri nets: properties, analysis and applications. Proc IEEE, 1989, 77: 541–580

    Article  Google Scholar 

  2. 2

    Jensen K. Coloured Petri Nets: Basic Concepts, Analysis Methods and Practical Use, Volume 1, Basic Concepts. Berlin: Springer, 1992

    Google Scholar 

  3. 3

    Jensen K. Coloured Petri Nets: Basic Concepts, Analysis Methods and Practical Use, Volume 2, Analysis Methods. Berlin: Springer, 1995

    Google Scholar 

  4. 4

    Jensen K. Coloured Petri Nets: Basic Concepts, Analysis Methods and Practical Use, Volume 3, Practical Use. Berlin: Springer, 1996

    Google Scholar 

  5. 5

    Yamalidou K, Moody J, Lemmon M, et al. Feedback control of petri nets based on place invariants. Automatica, 1996, 32: 15–28

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Mahulea C, Júlvez J, Vázquez C R, et al. Continuous Petri nets: observability and diagnosis. Lect Notes Control Inf Sci, 2013, 433: 387–406

    MathSciNet  MATH  Google Scholar 

  7. 7

    Mahulea C, Recalde L, Silva M. Optimal observability for continuous Petri nets. IFAC Proc Volumes, 2005, 38: 37–42

    Article  Google Scholar 

  8. 8

    Vázquez C R, Ramírez A, Recalde L, et al. On controllability of timed continuous Petri nets. Hybrid Syst Comput Control, 2008, 4981: 528–541

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Giua A, Seatzu C. A systems theory view of Petri nets. IFAC Proc Volumes, 2004, 37: 17–19

    Article  MATH  Google Scholar 

  10. 10

    Zhao J T, Chen Z Q, Liu Z X. Modeling and analysis of colored Petri net based on the semi-tensor product of matrices. Sci China Inf Sci, 2018, 61: 010205

    MathSciNet  Article  Google Scholar 

  11. 11

    Cheng D Z. Semi-tensor product of matrices and its application to Morgens problem. Sci China Ser F-Inf Sci, 2001, 44: 195–212

    MathSciNet  Google Scholar 

  12. 12

    Cheng D Z, Qi H S. A linear representation of dynamics of Boolean networks. IEEE Trans Autom Control, 2010, 55: 2251–2258

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Cheng D Z, Qi H S. Analysis and control of Boolean networks: a semi-tensor product approach. Acta Autom Sin, 2011, 37: 529–540

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Xu X R, Hong Y G. Matrix expression and reachability analysis of finite automata. Control Theory Technol, 2012, 10: 210–215

    MathSciNet  Article  Google Scholar 

  15. 15

    Yan Y Y, Chen Z Q, Liu Z X. Semi-tensor product approach to controllability and stabilizability of finite automata. J Syst Eng Electron, 2015, 26: 134–141

    Article  Google Scholar 

  16. 16

    Cheng D Z, Qi H S, He F H, et al. Semi-tensor product approach to networked evolutionary games. Control Theory Technol, 2014, 12: 198–214

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Cheng D Z. On finite potential games. Automatica, 2014, 50: 1793–1801

    MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Cheng D Z, He F H, Qi H S, et al. Modeling, analysis and control of networked evolutionary games. IEEE Trans Autom Control, 2015, 60: 2402–2415

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Han X G, Chen Z Q, Liu Z X, et al. Calculation of siphons and minimal siphons in Petri nets based on semi-tensor product of matrices. IEEE Trans Syst Man Cybern Syst, 2017, 47: 531–536

    Article  Google Scholar 

  20. 20

    Han X G, Chen Z Q, Zhang K Z, et al. Modeling and reachability analysis of a class of Petri nets via semi-tensor product of matrices. In: Proceedings of the 34th Chinese Control Conference (CCC), 2015. 6586–6591

  21. 21

    Ross K A, Kenneth H. Discrete Mathematics-5th Editon. Englewood Cliffs: Prentice Hall. 1992

    Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 61573199).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zengqiang Chen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhao, J., Chen, Z. & Liu, Z. A novel matrix approach for the stability and stabilization analysis of colored Petri nets. Sci. China Inf. Sci. 62, 192202 (2019). https://doi.org/10.1007/s11432-018-9562-y

Download citation

Keywords

  • discrete event system
  • colored Petri net
  • semi-tensor product of matrices
  • stability
  • equilibrium point
  • stabilization