Paper:

# Reachability and Controllability Analysis of Periodic Switched Boolean Control Networks

## Zhiqiang Li, Jinli Song, and Huimin Xiao

School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou Henan 450046, China

The reachability and controllability of switched Boolean (control) network are discussed in this paper. Based on semi-tensor product, using the vector form of Boolean logical variable, the switched Boolean (control) network is expressed as a discrete time system with state and control variables. For the switched Boolean network without control, the stabilization by suitable switching signal is discussed. Also, the controllability of the periodic switching signal is learned, and the conditions for stability and controllability of periodic switched Boolean networks avoiding states set *C* are obtained.

*J. Robot. Mechatron.*, Vol.26, No.5, pp. 573-579, 2014.

- [1] S. A. Kauffman, “Metabolic stability and epigenesis in randomly constructed genetic nets,” J. Theoretical Biology, Vol.22, pp. 437-467, 1969.
- [2] B. Elspas, “The theory of autonomous linear sequential networks,” IRE Trans. on Circuit Theory, Vol.6, No.1 pp. 45-60, 1959.
- [3] C. Farrow, J. Heidel, H. Maloney, and J. Rogers, “Scalar equations for synchronous Boolean networks with biological applications,” IEEE Trans. Neural Networks Vol.15, No.2, pp. 348-354, 2004.
- [4] J. Heidel, J. Maloney, C. Farrow, and J. A. Rogers, “Finding cycles in synchronous boolean networks with applications to biochemical systems,” I. J. Bifurcation and Chaos, Vol.13, No.3, pp. 535-552, 2003.
- [5] D. Cheng and H. Qi, “A linear representation of dynamics of Boolean networks,” IEEE Trans. Auto. Contr., Vol.55, No.10, pp. 2251-2258, 2010.
- [6] Y. Zhao, H. Qi, and D. Cheng, “Input-state incidence matrix of Boolean control networks and its applications,” Systems Control Lett., Vol.59, pp. 767-774, 2010.
- [7] D. Laschov and M. Margaliot, “Controllability of Boolean Control Networks via Perron-Frobenius Theory,” Automatica, Vol.48, No.6, pp. 1218-1223, 2012.
- [8] E. Fornasini and M. E. Valcher, “Observability, Reconstructibility and State Observers of Boolean Control Networks,” IEEE Trans. on Automatic Control, Vol.58, No.6, pp. 1390-1401, 2013.
- [9] D. Laschov, M. Margaliot, and G. Even, “Observability of Boolean networks: A graph-theoretic approach,” Automatica, Vol.49, pp. 2351-2362, 2013.
- [10] D. Laschov and M. Margaliot, “A maximum principle for singleinput Boolean control networks,” IEEE Trans. Aut. Contr., Vol.5, pp. 913-917, 2011.
- [11] D. Laschov and M. Margaliot, “Minimum-time control of Boolean networks,” SIAM J. Contr. Opt., Vol.51, pp. 2869-2892, 2013.
- [12] T. Akutsu, S. Miyano, and S. Kuhara, “Inferring qualitative relations in genetic networks and metabolic pathways,” Bioinformatics, Vol.16, No.8, pp. 727-734, 2000.
- [13] R. Albert and A. Barabási, “Dynamics of complex systems: scaling laws for the period of Boolean networks,” Phys. Rev. Lett., Vol.84, No.24, pp. 5660-5663, 2000.
- [14] J. Feng, J. Yao, and P. Cui, “Singular Boolean networks: Semitensor product approach,” SCIENCE CHINA Information Sciences, Vol.56, No.11, pp. 1-14, 2013.
- [15] Y. Zhao and D. Z. Cheng, “On controllability and stabilizability of probabilistic Boolean control networks,” Sci China Inf. Sci., Vol.57, No.1, pp. 1-14, 2014. DOI: 10.1007/s11432-013-4851-4
- [16] F. Li and J. Sun, “Controllability of probabilistic Boolean control networks,” Automatica, Vol.47, pp. 2765-2771, 2011.
- [17] M. Aldana, S. Coppersmith, and L. P. Kadanofff, “Boolean dynamics with random couplings,” in E. Kaplan, J. Marsden and K. Sreenivasan (Eds.), Perspectives and problems in nonlinear science, Springer, New York, 2003.
- [18] T. Akutsu, M. Hayashida, W. K. Ching, and M. K. Ng, “Control of Boolean networks: Hardness results and algorithms for tree structured networks,” J. Theoretical Biology, Vol.244, pp. 670-679, 2007.
- [19] Z. Li and D. Cheng, “Algebraic approach to dynamics of multivalued networks,” Int. J. of Bifurcation and Chaos, Vol.20, pp. 561-582, 2010.
- [20] A. Adamatzky, “On dynamically non-trivial three-valued logics: oscillatory and bifurcatory species,” Chaos Solitons and Fractals, Vol.18, pp. 917-936, 2003.
- [21] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Syst. Mag., Vol.19, No.5, pp. 59-70, Oct. 1999.
- [22] Z. Sun and D. Zheng, “On reachability and stabilization of switched linear systems,” IEEE Trans. Aut. Contr., Vol.46, No.2, pp. 291-295, 2001.
- [23] A. Irawan and K. Nonami, “Compliant Walking Control for Hydraulic Driven Hexapod Robot on Rough Terrain,” J. of Robotics and Mechatronics, Vol.23, No.1, 2011.
- [24] N. Motoi, K. Sasahara, and A. Kawamura, “Switching Control Method for Stable Landing by Legged Robot Based on Zero Moment Point,” J. of Robotics and Mechatronics, Vol.25, No.5, pp. 831-839, 2013.
- [25] Z. Sun and S. S. Ge, “Analysis and synthesis of switched linear control systems,” Automatica, Vol.41, No.2, pp. 181-195, 2005.
- [26] R. A. DeCarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, “Perspectives and results on the stability and stabilizability of hybrid systems,” Proc. IEEE, Vol.88, No.7, pp. 1069-1082, Jul. 2000.
- [27] L. Zhang, J. Feng, and M. Meng, “Controllability of higher order switched boolean control networks,” 2013 9th Asian Control Conf. (ASCC), pp. 1-6, June 23-26, 2013.
- [28] H. Li and Y.Wang, “On reachability and controllability of switched Boolean control networks,” Automatica, Vol.48 pp. 2917-2922, 2012.
- [29] D. Cheng and H. Qi, “Semi-tensor Product of Matrix – Theory and Applications,” Science Press, Beijing, 2007 (in Chinese).
- [30] Z. Q. Li and J. L. Song, “Controllability of Boolean control networks avoiding states set,” Sci. China Inf. Sci., Vol.57, No.3, pp. 1-13, 2014. DOI: 10.1007/s11432-013-4839-0