Skip to main content
SpringerLink
Log in

Further results on dynamic-algebraic Boolean control networks

  • Research Paper
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

Restricted coordinate transformation, controllability, observability and topological structures of dynamic-algebraic Boolean control networks are investigated under an assumption. Specifically, given the input-state at some point, assume that the subsequent state is certain or does not exist. First, the system can be expressed in a new form after numbering the elements in admissible set. Then, restricted coordinate transformation is raised, which allows the dimension of new coordinate frame to be different from that of the original one. The system after restricted coordinate transformation is derived in the proposed form. Afterwards, three types of incidence matrices are constructed and the results of controllability, observability and topological structures are obtained. Finally, two practical examples are shown to demonstrate the theory in this paper.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kauffman S A. Metabolic stability and epigenesis in randomly constructed genetic nets. J Theory Biol, 1969, 22: 437–467

    MathSciNet  Google Scholar 

  2. Akutsu T, Miyano S, Kuhara S. Inferring qualitative relations in genetic networks and metabolic pathways. Bioinformatics, 2000, 16: 727–734

    Google Scholar 

  3. Albert R, Barabási A L. Dynamics of complex systems: scaling laws for the period of Boolean networks. Phys Rev Lett, 2000, 84: 5660–5663

    Google Scholar 

  4. Zhang S Q, Ching W K, Chen X, et al. Generating probabilistic Boolean networks from a prescribed stationary distribution. Inf Sci, 2010, 180: 2560–2570

    MathSciNet  MATH  Google Scholar 

  5. Zhao Q C. A remark on “scalar equations for synchronous Boolean networks with biological applications” by C. Farrow, J. Heidel, J. Maloney, and J. Rogers. IEEE Trans Neural Netw, 2005, 16: 1715–1716

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. 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  Google Scholar 

  9. Liu G J, Jiang C J. Observable liveness of Petri nets with controllable and observable transitions. Sci China Inf Sci, 2017, 60: 118102

    Google Scholar 

  10. Cheng D Z, Qi H S. Controllability and observability of Boolean control networks. Automatica, 2009, 45: 1659–1667

    MathSciNet  MATH  Google Scholar 

  11. Cheng D Z, Li Z Q, Qi H S. Realization of Boolean control networks. Automatica, 2010, 46: 62–69

    MathSciNet  MATH  Google Scholar 

  12. Cheng D Z, Zhao Y. Identification of Boolean control networks. Automatica, 2011, 47: 702–710

    MathSciNet  MATH  Google Scholar 

  13. Cheng D Z, Qi H S, Li Z Q, et al. Stability and stabilization of Boolean networks. Int J Robust Nonlinear Control, 2011, 21: 134–156

    MathSciNet  MATH  Google Scholar 

  14. Cheng D Z. Disturbance decoupling of Boolean control networks. IEEE Trans Autom Control, 2011, 56: 2–10

    MathSciNet  MATH  Google Scholar 

  15. Liu Y, Li B W, Lu J Q, et al. Pinning control for the disturbance decoupling problem of Boolean networks. IEEE Trans Autom Control, 2017, 62: 6595–6601

    MathSciNet  MATH  Google Scholar 

  16. Meng M, Lam J, Feng J E, et al. l1-gain analysis and model reduction problem for Boolean control networks. Inf Sci, 2016, 348: 68–83

    MATH  Google Scholar 

  17. Cheng D Z, Qi H S, Liu T, et al. A note on observability of Boolean control networks. Syst Control Lett, 2016, 87: 76–82

    MathSciNet  MATH  Google Scholar 

  18. Zhang K Z, Zhang L J. Observability of Boolean control networks: a unified approach based on the theories of finite automata. IEEE Trans Autom Control, 2014, 61: 6854–6861

    Google Scholar 

  19. Zhu Q X, Liu Y, Lu J Q, et al. Observability of Boolean control networks. Sci China Inf Sci, 2018, 61: 092201

    MathSciNet  Google Scholar 

  20. Cheng D Z. Input-state approach to Boolean networks. IEEE Trans Neural Netw, 2009, 20: 512–521

    Google Scholar 

  21. Zhao Y, Qi H S, Cheng D Z. Input-state incidence matrix of Boolean control networks and its applications. Syst Control Lett, 2010, 59: 767–774

    MathSciNet  MATH  Google Scholar 

  22. Liu G B, Xu S Y, Wei Y L, et al. New insight into reachable set estimation for uncertain singular time-delay systems. Appl Math Comput, 2018, 320: 769–780

    MathSciNet  MATH  Google Scholar 

  23. Liu L S, Li H D, Wu Y H, et al. Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary conditions. J Nonlinear Sci Appl, 2017, 10: 243–262

    MathSciNet  MATH  Google Scholar 

  24. Liu L S, Sun F L, Zhang X G, et al. Bifurcation analysis for a singular differential system with two parameters via to topological degree theory. Nonlinear Anal Model Control, 2017, 22: 31–50

    MathSciNet  MATH  Google Scholar 

  25. Zheng Z W, Kong Q K. Friedrichs extensions for singular Hamiltonian operators with intermediate deficiency indices. J Math Anal Appl, 2018, 461: 1672–1685

    MathSciNet  MATH  Google Scholar 

  26. Cheng D Z, Zhao Y, Xu X R. Mix-valued logic and its applications. J Shandong Univ (Natl Sci), 2011, 46: 32–44

    MathSciNet  MATH  Google Scholar 

  27. Feng J E, Yao J, Cui P. Singular Boolean networks: semi-tensor product approach. Sci China Inf Sci, 2013, 56: 112203

    MathSciNet  Google Scholar 

  28. Meng M, Feng J E. Optimal control problem of singular Boolean control networks. Int J Control Autom Syst, 2015, 13: 266–273

    Google Scholar 

  29. Liu Y, Li B W, Chen H W, et al. Function perturbations on singular Boolean networks. Automatica, 2017, 84: 36–42

    MathSciNet  MATH  Google Scholar 

  30. Qiao Y P, Qi H S, Cheng D Z. Partition-based solutions of static logical networks with applications. IEEE Trans Neural Netw Learn Syst, 2018, 29: 1252–1262

    Google Scholar 

  31. Guo Y X. Nontrivial periodic solutions of nonlinear functional differential systems with feedback control. Turkish J Math, 2010, 34: 35

    MathSciNet  MATH  Google Scholar 

  32. Ma C Q, Li T, Zhang J F. Consensus control for leader-following multi-agent systems with measurement noises. J Syst Sci Complex, 2010, 23: 35–49

    MathSciNet  MATH  Google Scholar 

  33. Qin H Y, Liu J W, Zuo X, et al. Approximate controllability and optimal controls of fractional evolution systems in abstract spaces. Adv Diff Equ, 2014, 2014: 322

    MathSciNet  MATH  Google Scholar 

  34. Sun W W, Peng L H. Observer-based robust adaptive control for uncertain stochastic Hamiltonian systems with state and input delays. Nonlinear Anal Model Control, 2014, 19: 626–645

    MathSciNet  MATH  Google Scholar 

  35. Khatri C G, Rao C R. Solutions to some functional equations and their applications to characterization of probability distributions. Indian J Stat Ser A, 1968, 30: 167–180

    MathSciNet  MATH  Google Scholar 

  36. Heidel J, Maloney J, Farrow C, et al. Finding cycles in synchronous Boolean networks with applications to biochemical systems. Int J Bifurcation Chaos, 2003, 13: 535–552

    MathSciNet  MATH  Google Scholar 

  37. Ashenhurst R L. The decomposition of switching functions. In: Proceedings of an International Symposium on the Theory of Switching, 1957. 74–116

    Google Scholar 

  38. Curtis H A. A New Approach to the Design of Switching Circuits. New York: Van Nostrand Reinhold, 1962

    Google Scholar 

  39. Sasao T, Butler J T. On Bi-Decompositions of Logic Functions. Technical Report, DTIC Document, 1997

    MATH  Google Scholar 

  40. Sasao T. Application of multiple-valued logic to a serial decomposition of plas. In: Proceedings of the 19th International Symposium on Multiple-Valued Logic, 1989. 264–271

    Google Scholar 

  41. Muroga S. Logic Design and Switching Theory. New York: Wiley, 1979

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 61773371).

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Sen Wang, Jun-E. Feng & Yongyuan Yu

  2. School of Mathematical Science, Liaocheng University, Liaocheng, 252000, China

    Jianli Zhao

Authors
  1. Sen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-E. Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongyuan Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianli Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun-E. Feng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Feng, JE., Yu, Y. et al. Further results on dynamic-algebraic Boolean control networks. Sci. China Inf. Sci. 62, 12208 (2019). https://doi.org/10.1007/s11432-018-9447-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11432-018-9447-4

Keywords

Search

Navigation