Skip to main content
SpringerLink
Log in

Morgan’s problem of Boolean control networks

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

Abstract

This paper investigates the Morgan’s problem of Boolean control networks. Based on the matrix expression of logical functions, two key steps are proposed to solve the problem. First, the Boolean control network is converted into an outputdecomposed form by constructing a set of consistent output-friendly subspaces, and a necessary and sufficient condition for the existence of the consistent output-friendly subspaces is obtained. Secondly, a type of state feedback controllers are designed to solve the Morgan’s problem if it is solvable. By solving a set of matrix equations, a necessary and sufficient condition for converting an output-decomposed form to an input-output decomposed form is given, and by verifying the output controllability matrix, the solvability of Morgan’s problem is obtained.

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

    Article  MathSciNet  Google Scholar 

  2. D. Cheng, H. Qi, Z. Li. Analysis and Control of Boolean Networks–A Semi-tensor Product Approach. London: Springer, 2011.

    Book  MATH  Google Scholar 

  3. D. Cheng, H. Qi, Z. Li, et al. Stability and stabilization of Boolean networks. International Journal of Robust and Nonlinear Control, 2011, 21(2): 134–156.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Cheng, H. Qi. Controllability and observability of Boolean control networks. Automatica, 2009, 45(7): 1659–1667.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Laschov, M. Margaliot, G. Even. Observability of Boolean networks: A graph-theoretic approach. Automatica, 2013, 49(8): 2351–2362.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Cheng, Y. Zhao. Identification of Boolean control networks. Automatica, 2011, 47(4): 702–710.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Fornasini, M. E. Valcher. Optimal control of Boolean control networks. IEEE Transactions on Automatic Control, 2014, 59(5): 1258–1270.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Li, Y. Wang. Output tracking of switched Boolean networks under open-loop/closed-loop switching signals. Nonlinear Analysis: Hybrid Systems, 2016, 22: 137–146.

    MathSciNet  MATH  Google Scholar 

  9. M. Meng, J. Lam, J. Feng, et al. l1-gain analysis and model reduction problem for Boolean control networks. Information Sciences, 2016, 348: 68–83.

    Article  MathSciNet  Google Scholar 

  10. B. Morgan. The synthesis of linear multivariable systems by statevariable feedback. IEEE Transactions on Automatic Control, 1964, 9(4): 405–411.

    Article  MathSciNet  Google Scholar 

  11. W. M. Wonham. Linear Multivariable Control. Berlin: Springer, 1974.

    Book  MATH  Google Scholar 

  12. A. Glumineau, C. H. Moog. Nonlinear Morgan’s problem: Case of (p + 1) inputs and p outputs. IEEE transactions on Automatic Control, 1992, 37(7): 1067–1072.

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Nijmeijer. Feedback decomposition of nonlinear control systems. IEEE Transactions on Automatic Control, 1983, 28(8): 861–862.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Yang, R. Li, T. Chu. Controller design for disturbance decoupling of Boolean control networks. Automatica, 2013, 49(1): 273–277.

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. Zou, J. Zhu. System decomposition with respect to inputs for Boolean control networks. Automatica, 2014, 50(4): 1304–1309.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. E. Valcher. Input/output decoupling of Boolean control networks. IET Control Theory and Applications, 2017, 11(13): 2081–2088.

    Article  MathSciNet  Google Scholar 

  18. W. Respondek. On decomposition of nonlinear control systems. Systems & Control Letters, 1982, 1(5): 301–308.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Cheng, H. Qi, Y. Zhao. An Introduction to Semi-tensor Product of Matrices and Its Applications. Singapore: World Scientific, 2012.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan Shandong, 250061, China

    Shihua Fu & Daizhan Cheng

  2. School of Management Science and Engineering, Shandong Normal University, Jinan Shandong, 250014, China

    Yuanhua Wang

  3. Department of Computer Science and Information Systems, Bradley University, Peoria, IL, 61625, U.S.A.

    Jiangbo Liu

Authors
  1. Shihua Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuanhua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daizhan Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiangbo Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiangbo Liu.

Additional information

This paper is dedicated to Professor T. J. Tarn on the occasion of his 80th birthday.

This work was supported by the National Natural Science Foundation of China (No. 61333001).

Shihua FU received her M.Sc. degree from the Department of Mathematics, Liaocheng University, Liaocheng, China, in 2014. Since 2014 she has been pursuing her Ph.D. degree at the School of Control Science and Engineering, Shandong University. Her research interests include game theory, logical dynamic systems.

Yuanhua WANG received her B.Sc. degree and M.Sc. degree from the School of Control Science and Engineering, Shandong University, Jinan, China, in 2004 and 2007, respectively. Since 2013 she is pursuing her Ph.D. at the School of Control Science and Engineering, Shandong University. Currently, she is working in the School of Management Science and Engineering, Shandong Normal University. Her research interests include game theory, analysis and control of logical dynamic systems.

Daizhan CHENG (SM’01-F’06) received the B.Sc. degree from Department of Mechanics, Tsinghua University, in 1970, received the M.Sc. degree from Graduate School of Chinese Academy of Sciences in 1981, the Ph.D. degree from Washington University, St. Louis, in 1985. Since 1990, he is a Professor with Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He is the author/coauthor of over 200 journal papers, 9 books and 100 conference papers. He was Associate Editor of the International Journal of Mathematical Systems, Estimation and Control (1990–1993); Automatica (1999–2002); the Asian Journal of Control (2001–2004); Subject Editor of the International Journal of Robust and Nonlinear Control (2005–2008). He is currently Editor-in-Chief of the J. Control Theory and Applications and Deputy Editor-in-Chief of Control and Decision. He was the Chairman of IEEE CSS Beijing Chapter (2006–2008), Chairman of Technical Committee on Control Theory, Chinese Association of Automation, Program Committee Chair of annual Chinese Control Conference (2003–2010), IEEE Fellow (2005–) and IFAC Fellow (2008–). Prof. Cheng’s research interests include nonlinear system control, hamiltonian system, numerical method in system analysis and control, complex systems.

Jiangbo LIU received his M.Sc. and Ph.D. degrees from Washington University in St. Louis, in 1981 and 1985, respectively. Currently, he is a professor in the Computer Science and Information Systems Department, Bradley University. His research interests include computer networks, distributed computing, mobile computing, and linear and nonlinear control systems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, S., Wang, Y., Cheng, D. et al. Morgan’s problem of Boolean control networks. Control Theory Technol. 15, 316–326 (2017). https://doi.org/10.1007/s11768-017-7068-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11768-017-7068-z

Keywords

Search

Navigation