A note on observability of Boolean control networks

The observability of Boolean control networks is investigated. The pairs of states are classified into three classes: (i)diagonal, (ii) h-distinguishable, and h-indistinguishable. For h-indistinguishable pairs, we construct a matrix W called the transferable matrix, which indicates the control-transferability among h-indistinguishable pairs. Modifying W yields a matrix U0, which is used as the initial matrix for an iterative algorithm. After finite iterations a stable Uk* is reached, which is called the observability matrix. It is proved that a Boolean control network is observable, if and only if, the last column for all rows of Uk* are one. Some numerical examples are presented.

[1]  Michael Margaliot,et al.  Controllability of Boolean control networks via the Perron-Frobenius theory , 2012, Autom..

[2]  Lijun Zhang,et al.  Controllability of time-variant Boolean control networks and its application to Boolean control networks with finite memories , 2012, Science China Information Sciences.

[3]  Daizhan Cheng,et al.  Realization of Boolean control networks , 2010, Autom..

[4]  M. Margaliot.  Controllability and observability of Boolean control networks , 2012 .

[5]  Michael Margaliot,et al.  Observability of Boolean networks: A graph-theoretic approach , 2013, Autom..

[6]  Lijun Zhang,et al.  Observability of Boolean Control Networks: A Unified Approach Based on Finite Automata , 2014, IEEE Transactions on Automatic Control.

[7]  Jun-ichi Imura,et al.  Polynomial-Time Algorithm for Controllability Test of a Class of Boolean Biological Networks , 2010, EURASIP J. Bioinform. Syst. Biol..

[8]  Fangfei Li,et al.  Controllability of probabilistic Boolean control networks , 2011, Autom..

[9]  M. Ng,et al.  Control of Boolean networks: hardness results and algorithms for tree structured networks. , 2007, Journal of theoretical biology.

[10]  Wenping Shi,et al.  A Note on the Observability of Temporal Boolean Control Network , 2013 .

[11]  Yuzhen Wang,et al.  Reachability/controllability of high order mix-valued logical networks , 2013, J. Syst. Sci. Complex..

[12]  Jitao Sun,et al.  Observability analysis of Boolean control networks with impulsive effects , 2011 .

[13]  A. Datta,et al.  External Control in Markovian Genetic Regulatory Networks , 2003, Proceedings of the 2003 American Control Conference, 2003..

[14]  Fangfei Li,et al.  Controllability of Boolean control networks with time delays in states , 2011, Autom..

[15]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[16]  Daizhan Cheng,et al.  State–Space Analysis of Boolean Networks , 2010, IEEE Transactions on Neural Networks.

[17]  Chen Hao,et al.  A new approach for global controllability of higher order Boolean control network , 2012, Proceedings of the 31st Chinese Control Conference.

[18]  Daizhan Cheng,et al.  Analysis and Control of Boolean Networks , 2011 .

[19]  Daizhan Cheng,et al.  Input-state incidence matrix of Boolean control networks and its applications , 2010, Syst. Control. Lett..

[20]  Ettore Fornasini,et al.  Observability, Reconstructibility and State Observers of Boolean Control Networks , 2013, IEEE Transactions on Automatic Control.

[21]  Xian-Da Zhang,et al.  Matrix Analysis and Applications , 2017 .

[22]  Kobayashi Koichi,et al.  Polynomial-time Algorithm for Controllability Test of Boolean Networks , 2009 .

[23]  Ettore Fornasini,et al.  On the periodic trajectories of Boolean control networks , 2013, Autom..

[24]  Lijun Zhang,et al.  Controllability and Observability of Boolean Control Networks With Time-Variant Delays in States , 2013, IEEE Transactions on Neural Networks and Learning Systems.

[25]  Daizhan Cheng,et al.  Modeling, Analysis and Control of Networked Evolutionary Games , 2015, IEEE Transactions on Automatic Control.

[26]  Aniruddha Datta,et al.  External control in Markovian genetic regulatory networks: the imperfect information case , 2004, Bioinform..

[27]  Daizhan Cheng,et al.  A Linear Representation of Dynamics of Boolean Networks , 2010, IEEE Transactions on Automatic Control.

[28]  D. Cheng,et al.  An Introduction to Semi-Tensor Product of Matrices and Its Applications , 2012 .

[29]  W. Feng,et al.  observability ? , 2013 .

[30]  Lequn Zhang,et al.  Controllability and observability of switched Boolean control networks , 2012 .

[31]  Jitao Sun,et al.  A new approach for global controllability of higher order Boolean control network , 2013, Neural Networks.

[32]  Yuzhen Wang,et al.  On reachability and controllability of switched Boolean control networks , 2012, Autom..

[33]  B. Stigler,et al.  Boolean Models Can Explain Bistability in the lac Operon , 2008, J. Comput. Biol..

[34]  Kuize Zhang,et al.  Observability of Boolean control networks: A unified approach based on the theories of finite automata and formal languages , 2014, CCC 2014.

[35]  T. Ideker,et al.  A new approach to decoding life: systems biology. , 2001, Annual review of genomics and human genetics.

[36]  Daizhan Cheng,et al.  On controllability and stabilizability of probabilistic Boolean control networks , 2012, Science China Information Sciences.