Skip to main content
Log in

Dynamic output feedback stabilization of deterministic finite automata via the semi-tensor product of matrices approach

  • Research Article
  • Published:
Control Theory and Technology Aims and scope Submit manuscript

Abstract

This paper deals with the dynamic output feedback stabilization problem of deterministic finite automata (DFA). The static form of this problem is defined and solved in previous studies via a set of equivalent conditions. In this paper, the dynamic output feedback (DOF) stabilization of DFAs is defined in which the controller is supposed to be another DFA. The DFA controller will be designed to stabilize the equilibrium point of the main DFA through a set of proposed equivalent conditions. It has been proven that the design problem of DOF stabilization is more feasible than the static output feedback (SOF) stabilization. Three simulation examples are provided to illustrate the results of this paper in more details. The first example considers an instance DFA and develops SOF and DOF controllers for it. The example explains the concepts of the DOF controller and how it will be implemented in the closed-loop DFA. In the second example, a special DFA is provided in which the DOF stabilization is feasible, whereas the SOF stabilization is not. The final example compares the feasibility performance of the SOF and DOF stabilizations through applying them to one hundred random-generated DFAs. The results reveal the superiority of the DOF stabilization.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. Kobetski, A., & Fabian, M. (2009). Time-optimal coordination of flexible manufacturing systems using deterministic finite automata and mixed integer linear programming. Discrete Event Dynamic Systems, 19(3), 287–315.

    Article  MathSciNet  Google Scholar 

  2. Passino, K., Michel, A., & Antsaklis, P. (1994). Lyapunov stability of a class of discrete event systems. IEEE Transaction on Automatic Control, 39(2), 269–279.

    Article  MathSciNet  Google Scholar 

  3. Daniel, R., & Markus, L. (2014). Automata with modulo counters and nondeterministic counter bounds. Kybernetika, 50(1), 66–94.

    MathSciNet  MATH  Google Scholar 

  4. Tiwari, S. P., & Srivastava, A. K. (2005). On a decomposition of fuzzy automata. Fuzzy Sets Systems, 151(3), 503–511.

    Article  MathSciNet  Google Scholar 

  5. Lu, J., Li, H., Liu, Y., & Li, F. (2017). Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems. IET Control Theory and Applications, 11(13), 2040–2047.

    Article  MathSciNet  Google Scholar 

  6. Kauffman, S. A. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22(3), 437–467.

    Article  MathSciNet  Google Scholar 

  7. Cheng, D., He, F., Qi, H., & Xu, T. (2015). Modeling, analysis and control of networked evolutionary games. IEEE Transaction on Automatic Control, 60(9), 2402–2415.

    Article  MathSciNet  Google Scholar 

  8. Li, H., & Wang, Y. (2012). Boolean derivative calculation with application to fault detection of combinational circuits via the semi-tensor product method. Automatica, 48(4), 688–693.

    Article  MathSciNet  Google Scholar 

  9. Ozveren, F., & Willsky, A. (1991). Output stabilizability of discrete-event dynamic systems. IEEE Transaction on Automatic Control, 36(8), 925–935. https://doi.org/10.1109/9.133186.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bof, N., Fornasini, E., & Valcher, M. (2015). Output feedback stabilization of Boolean control networks. Automatica, 57, 21–28.

    Article  MathSciNet  Google Scholar 

  11. Li, H., & Wang, Y. (2013). Output feedback stabilization control design for Boolean control networks. Automatica, 49(12), 3641–3645.

    Article  MathSciNet  Google Scholar 

  12. Yan, Y., Chen, Z., & Liu, Z. (2014). Solving type-2 fuzzy relation equations via semi-tensor product of matrices. Control Theory and Technology, 12(2), 173–186.

    Article  MathSciNet  Google Scholar 

  13. Cheng, D. (2011). Disturbance decoupling of Boolean control networks. IEEE Transaction on Automatic Control, 56(1), 2–10.

    Article  MathSciNet  Google Scholar 

  14. Zhang, Z., Xia, C., & Chen, Z. (2020). On the stabilization of nondeterministic finite automata via static output feedback. Applied Mathematics and Computation, 365, 124687. https://doi.org/10.1016/j.amc.2019.124687.

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhang, Z., Chen, Z., Han, X., & Liu, Z. (2019). On the static output feedback stabilisation of discrete event dynamic systems based upon the approach of semi-tensor product of matrices. International Journal of Systems Science, 50(8), 1595–1608.

    Article  MathSciNet  Google Scholar 

  16. Syrmos, V., Abdallah, C., & Dorato, P. (1997). Static output feedback—A survey. Automatica, 33(2), 125–137.

    Article  MathSciNet  Google Scholar 

  17. Kong, X. S., Wang, S. L., Li, H. T., & Alsaadi, F. E. (2020). New developments in control design techniques of logical control networks. Frontiers of Information Technology and Electronic Engineering, 21, 220–233.

    Article  Google Scholar 

  18. Meng, M., Liu, L., & Feng, G. (2017). Stability and \(l_1\) gain analysis of Boolean networks with Markovian jump parameters. IEEE Transactions on Automatic Control, 62(8), 4222–4228.

    Article  MathSciNet  Google Scholar 

  19. Possieri, C., & Teel, A. R. (2017). Asymptotic stability in probability for stochastic Boolean networks. Automatica, 83, 1–9.

    Article  MathSciNet  Google Scholar 

  20. Li, H., & Ding, X. (2019). A control Lyapunov function approach to feedback stabilization of logical control networks. SIAM Journal on Control and Optimization, 57(2), 810–831.

    Article  MathSciNet  Google Scholar 

  21. Lu, J., Liu, R., Lou, J., & Liu, Y. (2019). Pinning stabilization of Boolean control networks via a minimum number of controllers. IEEE Transactions on Cybernetics,. https://doi.org/10.1109/TCYB.2019.2944659.

    Article  Google Scholar 

  22. Wang, X. F., & Chen, G. (2002). Pinning control of scale-free dynamical networks. Physica A: Statistical Mechanics and its Applications, 310(3–4), 521–531.

    Article  MathSciNet  Google Scholar 

  23. Li, F. (2015). Pinning control design for the stabilization of Boolean networks. IEEE Transactions on Neural Networks and Learning Systems, 27(7), 1585–1590.

    Article  MathSciNet  Google Scholar 

  24. Zhang, Z., Chen, Z., Han, X., & Liu, Z. (2018). On the static output feedback stabilization of deterministic finite automata based upon the approach of semi-tensor product of matrices. Kybernetika, 54(1), 41–60.

    MathSciNet  MATH  Google Scholar 

  25. Cheng, D., & Qi, H. (2010). A linear representation of dynamics of Boolean networks. IEEE Transaction on Automatic Control, 55(10), 2251–2258.

    Article  MathSciNet  Google Scholar 

  26. Xu, X., & Hong, Y. (2012). Matrix expression and reachability analysis of finite automata. Journal of Control Theory and Applications, 10(2), 210–215.

    Article  MathSciNet  Google Scholar 

  27. Han, X., Chen, Z., Liu, Z., & Zhang, Q. (2018). The detection and stabilisation of limit cycle for deterministic finite automata. International Journal of Control, 91(4), 874–886.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mohsen Raji.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abolpour, R., Raji, M. & Moradi, P. Dynamic output feedback stabilization of deterministic finite automata via the semi-tensor product of matrices approach. Control Theory Technol. 19, 170–182 (2021). https://doi.org/10.1007/s11768-020-00026-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11768-020-00026-8

Keywords

Navigation