State Space and Subspaces

  • Daizhan Cheng
  • Hongsheng Qi
  • Zhiqiang Li
Part of the Communications and Control Engineering book series (CCE)


This chapter presents a systematic description of state spaces and subspaces of Boolean (control) networks. After defining the state space and subspaces, the coordinate transformation of Boolean (control) networks is proposed. Using coordinate transformation, some useful kinds of subspaces, including regular subspaces and invariant subspaces, are investigated in detail. Moreover, the indistinct rolling gear structure of Boolean networks is revealed. This state-space description makes a state-space approach, similar to that of the modern control theory, applicable to the analysis of Boolean networks and the synthesis of Boolean control systems. This chapter is based on Cheng and Qi (IEEE Trans. Neural. Netw. 21(4):584–594, 2010).


State Space Coordinate Transformation Invariant Subspace Coordinate Frame Boolean Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aracena, J., Demongeot, J., Goles, E.: On limit cycles of monotone functions with symmetric connection graph. Theor. Comput. Sci. 322, 237–244 (2004) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Cheng, D.: Input-state approach to Boolean networks. IEEE Trans. Neural Netw. 20(3), 512–521 (2009) CrossRefGoogle Scholar
  3. 3.
    Cheng, D., Qi, H.: A linear representation of dynamics of Boolean networks. IEEE Trans. Automat. Contr. 55(10), 2251–2258 (2010) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cheng, D., Qi, H.: State-space analysis of Boolean networks. IEEE Trans. Neural Netw. 21(4), 584–594 (2010) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Farrow, C., Heidel, J., Maloney, J., Rogers, J.: Scalar equations for synchronous Boolean networks with biological applications. IEEE Trans. Neural Netw. 15(2), 348–354 (2004) CrossRefGoogle Scholar
  6. 6.
    Heidel, J., Maloney, J., Farrow, C., Rogers, J.: Finding cycles in synchronous Boolean networks with applications to biochemical systems. Int. J. Bifurc. Chaos 13(3), 535–552 (2003) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Kalman, R.E.: On the general theory of control systems. In: Automatic and Remote Control, Proc. First Internat. Congress, International Federation of Automatic Control, (IFAC), Moscow, 1960, vol. 1, pp. 481–492. Butterworth, Stoneham (1961) Google Scholar
  8. 8.
    Shih, M.H., Dong, J.L.: A combinatorial analogue of the Jacobian problem in automata networks. Adv. Appl. Math. 34, 30–46 (2005) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

There are no affiliations available

Personalised recommendations