Science China Information Sciences

, 61:092201 | Cite as

Observability of Boolean control networks

Research Paper

Abstract

We show some new results on the observability of Boolean control networks (BCNs). First, to study the observability, we combine two BCNs with the same transition matrix into a new BCN. Then, we propose the concept of a reachable set that results in a given set of initial states, and we derive four additional necessary and sufficient conditions for the observability of BCNs. In addition, we present an algorithm and construct an observability graph to determine the observability of BCNs. Finally, we illustrate the obtained results using three numerical examples.

Keywords

Boolean control network observability semi-tensor product 

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11671361, 61573096, 61573102, 61573110), China Postdoctoral Science Foundation (Grant Nos. 2016T90406, 2015M580378), National Training Programs of Innovation and Entrepreneurship for Undergraduates (Grant No. 201610345020), and Natural Science Foundation of Jiangsu Province of China (Grant No. BK20170019).

References

  1. 1.
    Kauffman S A. Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol, 1969, 22: 437–467CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cheng D Z, Qi H S, Li Z Q. Analysis and Control of Boolean Networks: A Semi-tensor Product Approach. Berlin: Springer, 2011.CrossRefMATHGoogle Scholar
  3. 3.
    Chen H W, Liang J L, Wang Z D. Pinning controllability of autonomous Boolean control networks. Sci China Inf Sci, 2016, 59: 070107.CrossRefGoogle Scholar
  4. 4.
    Liu Y, Chen H W, Wu B. Controllability of Boolean control networks with impulsive effects and forbidden states. Math Meth Appl Sci, 2014, 37: 1–9CrossRefMATHGoogle Scholar
  5. 5.
    Lu J Q, Zhong J, Ho D W C, et al. On controllability of delayed Boolean control networks. SIAM J Control Opt, 2016, 54: 475–494CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Lu J Q, Zhong J, Huang C, et al. On pinning controllability of Boolean control networks. IEEE Trans Autom Control, 2016, 61: 1658–1663CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cheng D Z, Li Z Q, Qi H S. Realization of Boolean control networks. Automatica, 2010, 46: 62–69CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Fornasini E, Valcher M E. Observability, reconstructibility and state observers of Boolean control networks. IEEE Trans Autom Control, 2013, 58: 1390–1401CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Laschov D, Margaliot M. A maximum principle for single-input Boolean control networks. IEEE Trans Autom Control, 2011, 56: 913–917CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Laschov D, Margaliot M. Minimum-time control of Boolean networks. SIAM J Control Opt, 2012, 51: 2869–2892CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Zou Y L, Zhu J D. System decomposition with respect to inputs for Boolean control networks. Automatica, 2014, 50: 1304–1309CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Zou Y L, Zhu J D. Kalman decomposition for Boolean control networks. Automatica, 2015, 54: 65–71CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Li F F. Pinning control design for the stabilization of Boolean networks. IEEE Trans Neural Netw Learn Syst, 2016, 27: 1585–1590CrossRefMathSciNetGoogle Scholar
  14. 14.
    Li R, Yang M, Chu T G. State feedback stabilization for Boolean control networks. IEEE Trans Autom Control, 2013, 58: 1853–1857CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Li R, Yang M, Chu T G. State feedback stabilization for probabilistic Boolean networks. Automatica, 2014, 50: 1272–1278CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Liu Y, Sun L J, Lu J Q, et al. Feedback controller design for the synchronization of Boolean control networks. IEEE Trans Neural Netw Learn Syst, 2016, 27: 1991–1996CrossRefMathSciNetGoogle Scholar
  17. 17.
    Zhong J, Lu J Q, Liu Y, et al. Synchronization in an array of output-coupled Boolean networks with time delay. IEEE Trans Neural Netw Learn Syst, 2014, 25: 2288–2294CrossRefGoogle Scholar
  18. 18.
    Liu Y, Li B W, Chen H W, et al. Function perturbations on singular Boolean networks. Automatica, 2017, 84: 36–42CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Li H T, Xie L H, Wang Y Z. On robust control invariance of Boolean control networks. Automatica, 2016, 68: 392–396CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Fornasini E, Valcher M E. On the periodic trajectories of Boolean control networks. Automatica, 2013, 49: 1506–1509CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Fornasini E, Valcher M E. Fault detection analysis of Boolean control networks. IEEE Trans Autom Control, 2015, 60: 2734–2739CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Liu Y, Chen H W, Lu J Q, et al. Controllability of probabilistic Boolean control networks based on transition probability matrices. Automatica, 2015, 52: 340–345CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Li H T, Wang Y Z. Controllability analysis and control design for switched Boolean networks with state and input constraints. SIAM J Control Opt, 2015, 53: 2955–2979CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Li H T, Wang Y Z, Xie L H. Output tracking control of Boolean control networks via state feedback: constant reference signal case. Automatica, 2015, 59: 54–59CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Zhao Y, Cheng D Z. On controllability and stabilizability of probabilistic Boolean control networks. Sci China Inf Sci, 2014, 57: 012202.MATHMathSciNetGoogle Scholar
  26. 26.
    Cheng D Z. Semi-tensor product of matrices and its application to morgen’s problem. Sci China Ser F-Inf Sci, 2001, 44: 195–212MATHMathSciNetGoogle Scholar
  27. 27.
    Cheng D Z, Xu T T, Qi H S. Evolutionarily stable strategy of networked evolutionary games. IEEE Trans Neural Netw Learn Syst, 2014, 25: 1335–1345CrossRefGoogle Scholar
  28. 28.
    Guo P L, Wang Y Z, Li H T. Stable degree analysis for strategy profiles of evolutionary networked games. Sci China Inf Sci, 2016, 59: 052204.CrossRefGoogle Scholar
  29. 29.
    Lu J Q, Li H T, Liu Y, et al. A survey on semi-tensor product method with its applications in logical networks and other finite-valued systems. IET Control Theory Appl, 2017, 13: 2040–2047CrossRefGoogle Scholar
  30. 30.
    Cheng D Z, Qi H S, Li Z Q. Controllability and observability of Boolean control networks. Automatica, 2009, 45: 1659–1667CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Cheng D Z, Zhao Y. Identification of Boolean control networks. Automatica, 2011, 47: 702–710CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Zhao Y, Qi H S, Cheng D Z. Input-state incidence matrix of Boolean control networks and its applications. Syst Control Lett, 2010, 59: 767–774CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Zhang K, Zhang L. Observability of Boolean control networks: a unified approach based on finite automata. IEEE Trans Autom Control, 2016, 61: 2733–2738CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Cheng D Z, Qi H S, Liu T, et al. A note on observability of Boolean control networks. Syst Control Lett, 2016, 87: 76–82CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Laschov D, Margaliot M E. Controllability of Boolean control networks via perron-frobenius theory. Automatica, 2012, 48: 1218–1223CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Qunxi Zhu
    • 1
    • 2
  • Yang Liu
    • 1
    • 3
  • Jianquan Lu
    • 3
  • Jinde Cao
    • 3
    • 4
  1. 1.College of Mathematics, Physics and Information EngineeringZhejiang Normal UniversityJinhuaChina
  2. 2.College of Mathematical SciencesFudan UniversityShanghaiChina
  3. 3.School of MathematicsSoutheast UniversityNanjingChina
  4. 4.School of Mathematical SciencesShandong Normal UniversityJi’nanChina

Personalised recommendations