##
**A note on the observability of temporal Boolean control network.**
*(English)*
Zbl 1271.93031

Summary: Temporal Boolean network is a generalization of the Boolean network model that takes into account the time series nature of the data and tries to incorporate into the model the possible existence of delayed regulatory interactions among genes. This paper investigates the observability problem of temporal Boolean control networks. Using the semi-tensor product of matrices, the temporal Boolean networks can be converted into discrete time linear dynamic systems with time delays. Then, necessary and sufficient conditions on the observability via two kinds of inputs are obtained. An example is given to illustrate the effectiveness of the obtained results.

### MSC:

93B07 | Observability |

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

93C55 | Discrete-time control/observation systems |

PDFBibTeX
XMLCite

\textit{W. Shi} et al., Abstr. Appl. Anal. 2013, Article ID 631639, 9 p. (2013; Zbl 1271.93031)

Full Text:
DOI

### References:

[1] | Kauffman, S. A., Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22, 3, 437-467 (1969) |

[2] | Kauffman, S., The Origins of Order: Self-Organization and Selection in Evolution (1993), New York, NY, USA: Oxford University Press, New York, NY, USA |

[3] | Kauffman, S., At Home in the Universe (1995), New York, NY, USA: Oxford University Press, New York, NY, USA |

[4] | Cheng, D., Semi-tensor product of matrices and its applications: a survey, Proceedings of the 4th International Congress of Chinese Mathematicians |

[5] | Huang, S.; Ingber, D. E., Shape-dependent control of cell growth, differentiation, and apoptosis: Switching between attractors in cell regulatory networks, Experimental Cell Research, 261, 1, 91-103 (2000) |

[6] | Cheng, D., Input-state approach to Boolean networks, IEEE Transactions on Neural Networks, 20, 3, 512-521 (2009) |

[7] | Cheng, D.; Li, Z.; Qi, H., Realization of Boolean control networks, Automatica, 46, 1, 62-69 (2010) · Zbl 1214.93031 |

[8] | Cheng, D.; Qi, H., Controllability and observability of Boolean control networks, Automatica, 45, 7, 1659-1667 (2009) · Zbl 1184.93014 |

[9] | Cheng, D.; Qi, H., A linear representation of dynamics of Boolean networks, IEEE Transactions on Automatic Control, 55, 10, 2251-2258 (2010) · Zbl 1368.37025 |

[10] | Laschov, D.; Margaliot, M., Controllability of Boolean control networks via the Perron-Frobenius theory, Automatica, 48, 6, 1218-1223 (2012) · Zbl 1244.93026 |

[11] | Laschov, D.; Margaliot, M., A maximum principle for single-input Boolean control networks, IEEE Transactions on Automatic Control, 56, 4, 913-917 (2011) · Zbl 1368.93344 |

[12] | Laschov, D.; Margaliot, M., A pontryagin maximum principle for multi-input boolean control networks, Recent Advances in Dynamics and Control of Neural Networks (2011), Cambridge Scientific Publishers · Zbl 1368.93344 |

[13] | Guan, Z.-H.; Qian, T.-H.; Yu, X., Controllability and observability of linear time-varying impulsive systems, IEEE Transactions on Circuits and Systems I, 49, 8, 1198-1208 (2002) · Zbl 1368.93034 |

[14] | Liu, Y.; Zhao, S., Controllability for a class of linear time-varying impulsive systems with time delay in control input, IEEE Transactions on Automatic Control, 56, 2, 395-399 (2011) · Zbl 1368.93036 |

[15] | Liu, Y.; Zhao, S., Controllability analysis of linear time-varying systems with multiple time delays and impulsive effects, Nonlinear Analysis: Real World Applications, 13, 2, 558-568 (2012) · Zbl 1238.93016 |

[16] | Xie, G.; Wang, L., Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Transactions on Automatic Control, 49, 6, 960-966 (2004) · Zbl 1365.93049 |

[17] | Zhao, S.; Sun, J., Controllability and observability for time-varying switched impulsive controlled systems, International Journal of Robust and Nonlinear Control, 20, 12, 1313-1325 (2010) · Zbl 1206.93019 |

[18] | Zhao, S.; Sun, J., A geometric approach for reachability and observability of linear switched impulsive systems, Nonlinear Analysis, 72, 11, 4221-4229 (2010) · Zbl 1189.93021 |

[19] | Li, F.; Sun, J.; Wu, Q., Observability of boolean control networks with state time delays, IEEE Transactions on Neural Networks, 22, 6, 948-954 (2011) |

[20] | Lu, J.; Ho, D. W. C.; Kurths, J., Consensus over directed static networks with arbitrary finite communication delays, Physical Review E, 80, 6 (2009) |

[21] | Lu, J.; Ho, D. W. C.; Cao, J., Synchronization in an array of nonlinearly coupled chaotic neural networks with delay coupling, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 18, 10, 3101-3111 (2008) · Zbl 1165.34414 |

[22] | Lyu, S., Combining boolean method with delay times for determining behaviors of biological networks, Proceedings of the 31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society |

[23] | Cotta, C., On the evolutionary inference of temporal boolean networks, Computational Methods in Neural Modeling. Computational Methods in Neural Modeling, Lecture Notes in Computer Science, 2686, 494-501 (2003) |

[24] | Fogelberg, C.; Palade, V., Machine learning and genetic regulatory networks: a review and a roadmap, Foundations of Computational, Intelligence. Foundations of Computational, Intelligence, Studies in Computational Intelligence, 201, 3-34 (2009) |

[25] | Silvescu, A.; Honavar, V., Temporal Boolean network models of genetic networks and their inference from gene expression time series, Complex Systems, 13, 1, 61-78 (2001) · Zbl 1167.92335 |

[26] | Cheng, D.; Qi, H.; Li, Z., Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach (2011), New York, NY, USA: Springer, New York, NY, USA · Zbl 1209.93001 |

[27] | Liu, Y.; Lu, J.; Wu, B., Some necessary and sufficient conditions for the output controllability of temporal Boolean control networks · Zbl 1282.93055 |

[28] | Liu, Y.; Chen, H.; Wu, B., Controllability of Boolean control networks with impulsive effects and forbidden states, Mathematical Methods in the Applied Sciences (2013) · Zbl 1282.93054 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.