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April  2021, 14(4): 1519-1533. doi: 10.3934/dcdss.2020373

Observability of switched Boolean control networks using algebraic forms

1. 

College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China

2. 

School of Information Engineering, Huzhou University, Huzhou, China

3. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, China

* Corresponding author: Zhen Wang

Received  October 2019 Revised  February 2020 Published  April 2021 Early access  May 2020

This paper addresses the observability of a switched Boolean control network (SBCN) using semi-tensor product (STP) of matrices. First, the observability of the SBCN is determined under desirable switching signals and arbitrary switching signals by encoding the switching signal as a boolean input. Then an algorithm is designed for determining the observability. Furthermore, feedback control laws are obtained to guarantee the observability of SBCNs. Examples and corresponding state trajectory graphs are given to illustrate the effectiveness of the given results.

Citation: Yvjing Yang, Yang Liu, Jungang Lou, Zhen Wang. Observability of switched Boolean control networks using algebraic forms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1519-1533. doi: 10.3934/dcdss.2020373
References:
[1]

D. Cheng, H. Qi and Z. Li, Analysis and Control of Boolean Networks: A Semi-tensor, Springer Science & Business Media, 2010.

[2]

D. Cheng and H. Qi, Controllability and observability of Boolean control networks, Automatica, 45 (2009), 1659-1667.  doi: 10.1016/j.automatica.2009.03.006.

[3]

H. ChenJ. LiangT. Huang and J. Cao, Synchronization of arbitrarily switched Boolean networks, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 612-619.  doi: 10.1109/TNNLS.2015.2497708.

[4]

S. ChenY. WuM. Macauley and X. Sun, Monostability and bistability of Boolean networks using semi-tensor products, IEEE Transactions on Control of Network Systems, 6 (2019), 1379-1390.  doi: 10.1109/TCNS.2018.2889015.

[5]

E. H. Davidson and H. Bolouri, A genomic regulatory network for development, Science, 295 (2002), 1669-1678.  doi: 10.1126/science.1069883.

[6]

E. Fornasini and M. E. Valcher, Observability, reconstructibility and state observers of Boolean control networks, IEEE Transactions on Automatic Control, 58 (2013), 1390-1401.  doi: 10.1109/TAC.2012.2231592.

[7]

E. Fornasini and M. E. Valcher, Fault detection analysis of Boolean control networks, IEEE Transactions on Automatic Control, 60 (2015), 2734-2739.  doi: 10.1109/TAC.2015.2396646.

[8]

S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22 (1969), 437-467.  doi: 10.1016/0022-5193(69)90015-0.

[9]

D. Laschov and M. Margaliot, Controllability of Boolean control networks via the Perron–Frobenius theory, Automatica, 48 (2012), 1218-1223.  doi: 10.1016/j.automatica.2012.03.022.

[10]

H. Li and Y. Wang, On reachability and controllability of switched Boolean control networks, Automatica, 48 (2012), 2917-2922.  doi: 10.1016/j.automatica.2012.08.029.

[11]

W. LiL. Cui and M. K. Ng, On computation of the steady-state probability distribution of probabilistic Boolean networks with gene perturbation, Journal of Computational and Applied Mathematics, 236 (2012), 4067-4081.  doi: 10.1016/j.cam.2012.02.022.

[12]

R. LiM. Yang and T. Chu, State feedback stabilization for Boolean control networks, IEEE Transactions on Automatic Control, 58 (2013), 1853-1857.  doi: 10.1109/TAC.2013.2238092.

[13]

H. LiY. Wang and Z. Liu, Simultaneous stabilization for a set of Boolean control networks, Systems & Control Letters, 62 (2013), 1168-1174.  doi: 10.1016/j.sysconle.2013.09.008.

[14]

H. LiY. Wang and Z. Liu, Stability analysis for switched Boolean networks under arbitrary switching signals, IEEE Transactions on Automatic Control, 59 (2014), 1978-1982.  doi: 10.1109/TAC.2014.2298731.

[15]

Y. LiuB. LiH. Chen and J. Cao, Function perturbations on singular Boolean networks, Automatica, 84 (2017), 36-42.  doi: 10.1016/j.automatica.2017.06.035.

[16]

Y. LiuB. LiJ. Lu and J. Cao, Pinning control for the disturbance decoupling problem of Boolean networks, IEEE Transactions on Automatic Control, 62 (2017), 6595-6601.  doi: 10.1109/TAC.2017.2715181.

[17]

F. Li and Y. Tang, Set stabilization for switched Boolean control networks, Automatica, 78 (2017), 223-230.  doi: 10.1016/j.automatica.2016.12.007.

[18]

Y. Li, J. Zhong, J. Lu and Z. Wang, On robust synchronization of drive-response Boolean control networks with disturbances, Mathematical Problems in Engineering, 2018 (2018), Art. ID 1737685, 9 pp. doi: 10.1155/2018/1737685.

[19]

Y. LiJ. LouZ. Wang and F. E. Alsaadi, Synchronization of nonlinearly coupled dynamical networks under hybrid pinning impulsive controllers, Journal of the Franklin Institute, 355 (2018), 6520-6530.  doi: 10.1016/j.jfranklin.2018.06.021.

[20]

H. Li and Y. Zhen, Algebraic formulation and topological structure of Boolean networks with state-dependent delay, Journal of Computational and Applied Mathematics, 350 (2019), 87-97.  doi: 10.1016/j.cam.2018.10.003.

[21]

H. LiuY. Liu and J. Cao, Observability of SBCNs under arbitrary switching signals, 2018 37th Chinese Control Conference (CCC), 2019 (2019), 6299-6302.  doi: 10.23919/ChiCC.2018.8483499.

[22]

B. Li, J. Lou, Y. Liu and Z. Wang, Robust invariant set analysis of boolean networks, Complexity, 2019 (2019), Article ID 2731395, 8pp. doi: 10.1155/2019/2731395.

[23]

M. MengL. Liu and G. Feng, Stability and $L_1$ gain analysis of Boolean networks with markovian jump parameter, IEEE Transactions on Automatic Control, 62 (2017), 4222-4228.  doi: 10.1109/TAC.2017.2679903.

[24]

J. J. TysonA. Csikasz-Nagy and B. Novak, The dynamics of cell cycle regulation, Bioessays, 24 (2002), 1095-1109.  doi: 10.1002/bies.10191.

[25]

L. TongY. LiuY. LiJ. Lu and Z. Wang, Robust control invariance of probabilistic Boolean control networks via event-triggered control, IEEE Access, 6 (2018), 37767-37774.  doi: 10.1109/ACCESS.2018.2828128.

[26]

Y. Wu, J. Xu and X. Sun, Observability of Boolean multiplex control networks, Scientific Reports, 7 (2017), 46495. doi: 10.1038/srep46495.

[27]

S. Wang, J. Feng, Y. Yu and J. Zhao, Further results on dynamic-algebraic Boolean control networks, Science China Information Sciences, 62 (2019), 12208, 14pp. doi: 10.1007/s11432-018-9447-4.

[28]

Y. WuX. Sun and X. Zhao, Optimal control of Boolean control networks with average cost: A policy iteration approach, Automatica, 100 (2019), 378-387.  doi: 10.1016/j.automatica.2018.11.036.

[29]

X. YangX. LiQ. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences and Engineering, 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.

[30]

D. YangX. LiJ. Shen and Z. Zhou, State-dependent switching control of delayed switched systems with stable and unstable modes, Mathematical Methods in the Applied Sciences, 41 (2018), 6968-6983.  doi: 10.1002/mma.5209.

[31]

Y. YuB. Wang and J. Feng, Input observability of Boolean control networks, Neurocomputing, 333 (2019), 22-28.  doi: 10.1016/j.neucom.2018.12.014.

[32]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[33]

L. ZhangJ. Feng and J. Yao, Controllability and observability of switched Boolean control networks, IET Control Theory & Applications, 6 (2012), 2477-2484.  doi: 10.1049/iet-cta.2012.0362.

[34]

K. ZhangL. Zhang and L. Xie, Finite automata approach to observability of switched Boolean control networks, Nonlinear Analysis: Hybrid Systems, 19 (2016), 186-197.  doi: 10.1016/j.nahs.2015.10.002.

[35]

K. Zhang and L. Zhang, Observability of Boolean control networks: A unified approach based on finite automata, IEEE Transactions on Automatic Control, 61 (2016), 2733-2738.  doi: 10.1109/TAC.2015.2501365.

[36]

J. ZhongJ. LuT. Huang and D. W. C. Ho, Controllability and synchronization analysis of Identical-Hierarchy mixed-valued logical control networks, IEEE Transactions on Cybernetics, 47 (2017), 3482-3493.  doi: 10.1109/TCYB.2016.2560240.

[37]

Q. Zhu, Y. Liu, J. Lu and J. Cao, Observability of Boolean control networks, Science China Information Sciences, 61 (2018), 092201, 12pp. doi: 10.1007/s11432-017-9135-4.

[38]

J. Zhang and J. Sun, Exponential synchronization of complex networks with continuous dynamics and Boolean mechanism, Neurocomputing, 307 (2018), 146-152.  doi: 10.1016/j.neucom.2018.03.061.

[39]

S. Zhu, J. Lou, Y. Liu, Y. Li and Z. Wang, Event-Triggered control for the stabilization of probabilistic Boolean control networks, Complexity, 2018 (2018), Article ID 9259348, 7pp. doi: 10.1155/2018/9259348.

[40]

Q. ZhuY. LiuJ. Lu and J. Cao, On the optimal control of Boolean control networks, SIAM J. on Control and Optimization, 56 (2018), 1321-1341.  doi: 10.1137/16M1070281.

[41]

Q. ZhuY. LiuJ. Lu and J. Cao, Further results on the controllability of Boolean control networks, IEEE Transactions on Automatic Control, 64 (2019), 440-442.  doi: 10.1109/TAC.2018.2830642.

[42]

J. ZhongY. LiuK. Ia. KouL. Sun and J. Cao, On the ensemble controllability of Boolean control networks using STP method, Applied Mathematics and Computation, 358 (2019), 51-62.  doi: 10.1016/j.amc.2019.03.059.

[43]

J. ZhongB. LiY. Liu and W. Gui, Output feedback stabilizer design of Boolean networks based on network structure, Frontiers of Information Technology & Electronic Engineering, 21 (2020), 247-259.  doi: 10.1631/FITEE.1900229.

show all references

References:
[1]

D. Cheng, H. Qi and Z. Li, Analysis and Control of Boolean Networks: A Semi-tensor, Springer Science & Business Media, 2010.

[2]

D. Cheng and H. Qi, Controllability and observability of Boolean control networks, Automatica, 45 (2009), 1659-1667.  doi: 10.1016/j.automatica.2009.03.006.

[3]

H. ChenJ. LiangT. Huang and J. Cao, Synchronization of arbitrarily switched Boolean networks, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 612-619.  doi: 10.1109/TNNLS.2015.2497708.

[4]

S. ChenY. WuM. Macauley and X. Sun, Monostability and bistability of Boolean networks using semi-tensor products, IEEE Transactions on Control of Network Systems, 6 (2019), 1379-1390.  doi: 10.1109/TCNS.2018.2889015.

[5]

E. H. Davidson and H. Bolouri, A genomic regulatory network for development, Science, 295 (2002), 1669-1678.  doi: 10.1126/science.1069883.

[6]

E. Fornasini and M. E. Valcher, Observability, reconstructibility and state observers of Boolean control networks, IEEE Transactions on Automatic Control, 58 (2013), 1390-1401.  doi: 10.1109/TAC.2012.2231592.

[7]

E. Fornasini and M. E. Valcher, Fault detection analysis of Boolean control networks, IEEE Transactions on Automatic Control, 60 (2015), 2734-2739.  doi: 10.1109/TAC.2015.2396646.

[8]

S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22 (1969), 437-467.  doi: 10.1016/0022-5193(69)90015-0.

[9]

D. Laschov and M. Margaliot, Controllability of Boolean control networks via the Perron–Frobenius theory, Automatica, 48 (2012), 1218-1223.  doi: 10.1016/j.automatica.2012.03.022.

[10]

H. Li and Y. Wang, On reachability and controllability of switched Boolean control networks, Automatica, 48 (2012), 2917-2922.  doi: 10.1016/j.automatica.2012.08.029.

[11]

W. LiL. Cui and M. K. Ng, On computation of the steady-state probability distribution of probabilistic Boolean networks with gene perturbation, Journal of Computational and Applied Mathematics, 236 (2012), 4067-4081.  doi: 10.1016/j.cam.2012.02.022.

[12]

R. LiM. Yang and T. Chu, State feedback stabilization for Boolean control networks, IEEE Transactions on Automatic Control, 58 (2013), 1853-1857.  doi: 10.1109/TAC.2013.2238092.

[13]

H. LiY. Wang and Z. Liu, Simultaneous stabilization for a set of Boolean control networks, Systems & Control Letters, 62 (2013), 1168-1174.  doi: 10.1016/j.sysconle.2013.09.008.

[14]

H. LiY. Wang and Z. Liu, Stability analysis for switched Boolean networks under arbitrary switching signals, IEEE Transactions on Automatic Control, 59 (2014), 1978-1982.  doi: 10.1109/TAC.2014.2298731.

[15]

Y. LiuB. LiH. Chen and J. Cao, Function perturbations on singular Boolean networks, Automatica, 84 (2017), 36-42.  doi: 10.1016/j.automatica.2017.06.035.

[16]

Y. LiuB. LiJ. Lu and J. Cao, Pinning control for the disturbance decoupling problem of Boolean networks, IEEE Transactions on Automatic Control, 62 (2017), 6595-6601.  doi: 10.1109/TAC.2017.2715181.

[17]

F. Li and Y. Tang, Set stabilization for switched Boolean control networks, Automatica, 78 (2017), 223-230.  doi: 10.1016/j.automatica.2016.12.007.

[18]

Y. Li, J. Zhong, J. Lu and Z. Wang, On robust synchronization of drive-response Boolean control networks with disturbances, Mathematical Problems in Engineering, 2018 (2018), Art. ID 1737685, 9 pp. doi: 10.1155/2018/1737685.

[19]

Y. LiJ. LouZ. Wang and F. E. Alsaadi, Synchronization of nonlinearly coupled dynamical networks under hybrid pinning impulsive controllers, Journal of the Franklin Institute, 355 (2018), 6520-6530.  doi: 10.1016/j.jfranklin.2018.06.021.

[20]

H. Li and Y. Zhen, Algebraic formulation and topological structure of Boolean networks with state-dependent delay, Journal of Computational and Applied Mathematics, 350 (2019), 87-97.  doi: 10.1016/j.cam.2018.10.003.

[21]

H. LiuY. Liu and J. Cao, Observability of SBCNs under arbitrary switching signals, 2018 37th Chinese Control Conference (CCC), 2019 (2019), 6299-6302.  doi: 10.23919/ChiCC.2018.8483499.

[22]

B. Li, J. Lou, Y. Liu and Z. Wang, Robust invariant set analysis of boolean networks, Complexity, 2019 (2019), Article ID 2731395, 8pp. doi: 10.1155/2019/2731395.

[23]

M. MengL. Liu and G. Feng, Stability and $L_1$ gain analysis of Boolean networks with markovian jump parameter, IEEE Transactions on Automatic Control, 62 (2017), 4222-4228.  doi: 10.1109/TAC.2017.2679903.

[24]

J. J. TysonA. Csikasz-Nagy and B. Novak, The dynamics of cell cycle regulation, Bioessays, 24 (2002), 1095-1109.  doi: 10.1002/bies.10191.

[25]

L. TongY. LiuY. LiJ. Lu and Z. Wang, Robust control invariance of probabilistic Boolean control networks via event-triggered control, IEEE Access, 6 (2018), 37767-37774.  doi: 10.1109/ACCESS.2018.2828128.

[26]

Y. Wu, J. Xu and X. Sun, Observability of Boolean multiplex control networks, Scientific Reports, 7 (2017), 46495. doi: 10.1038/srep46495.

[27]

S. Wang, J. Feng, Y. Yu and J. Zhao, Further results on dynamic-algebraic Boolean control networks, Science China Information Sciences, 62 (2019), 12208, 14pp. doi: 10.1007/s11432-018-9447-4.

[28]

Y. WuX. Sun and X. Zhao, Optimal control of Boolean control networks with average cost: A policy iteration approach, Automatica, 100 (2019), 378-387.  doi: 10.1016/j.automatica.2018.11.036.

[29]

X. YangX. LiQ. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences and Engineering, 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.

[30]

D. YangX. LiJ. Shen and Z. Zhou, State-dependent switching control of delayed switched systems with stable and unstable modes, Mathematical Methods in the Applied Sciences, 41 (2018), 6968-6983.  doi: 10.1002/mma.5209.

[31]

Y. YuB. Wang and J. Feng, Input observability of Boolean control networks, Neurocomputing, 333 (2019), 22-28.  doi: 10.1016/j.neucom.2018.12.014.

[32]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[33]

L. ZhangJ. Feng and J. Yao, Controllability and observability of switched Boolean control networks, IET Control Theory & Applications, 6 (2012), 2477-2484.  doi: 10.1049/iet-cta.2012.0362.

[34]

K. ZhangL. Zhang and L. Xie, Finite automata approach to observability of switched Boolean control networks, Nonlinear Analysis: Hybrid Systems, 19 (2016), 186-197.  doi: 10.1016/j.nahs.2015.10.002.

[35]

K. Zhang and L. Zhang, Observability of Boolean control networks: A unified approach based on finite automata, IEEE Transactions on Automatic Control, 61 (2016), 2733-2738.  doi: 10.1109/TAC.2015.2501365.

[36]

J. ZhongJ. LuT. Huang and D. W. C. Ho, Controllability and synchronization analysis of Identical-Hierarchy mixed-valued logical control networks, IEEE Transactions on Cybernetics, 47 (2017), 3482-3493.  doi: 10.1109/TCYB.2016.2560240.

[37]

Q. Zhu, Y. Liu, J. Lu and J. Cao, Observability of Boolean control networks, Science China Information Sciences, 61 (2018), 092201, 12pp. doi: 10.1007/s11432-017-9135-4.

[38]

J. Zhang and J. Sun, Exponential synchronization of complex networks with continuous dynamics and Boolean mechanism, Neurocomputing, 307 (2018), 146-152.  doi: 10.1016/j.neucom.2018.03.061.

[39]

S. Zhu, J. Lou, Y. Liu, Y. Li and Z. Wang, Event-Triggered control for the stabilization of probabilistic Boolean control networks, Complexity, 2018 (2018), Article ID 9259348, 7pp. doi: 10.1155/2018/9259348.

[40]

Q. ZhuY. LiuJ. Lu and J. Cao, On the optimal control of Boolean control networks, SIAM J. on Control and Optimization, 56 (2018), 1321-1341.  doi: 10.1137/16M1070281.

[41]

Q. ZhuY. LiuJ. Lu and J. Cao, Further results on the controllability of Boolean control networks, IEEE Transactions on Automatic Control, 64 (2019), 440-442.  doi: 10.1109/TAC.2018.2830642.

[42]

J. ZhongY. LiuK. Ia. KouL. Sun and J. Cao, On the ensemble controllability of Boolean control networks using STP method, Applied Mathematics and Computation, 358 (2019), 51-62.  doi: 10.1016/j.amc.2019.03.059.

[43]

J. ZhongB. LiY. Liu and W. Gui, Output feedback stabilizer design of Boolean networks based on network structure, Frontiers of Information Technology & Electronic Engineering, 21 (2020), 247-259.  doi: 10.1631/FITEE.1900229.

Figure 1.  The state trajectory graph of $ \delta_{64}^{2} $, $ \delta_{64}^{20} $, $ \delta_{64}^{38} $, $ \delta_{64}^{56} $, where $ \delta_{64}^0 $ is a virtual state, number $ i $ in each cycle denote state $ \delta_{64}^i $, number $ j $ beside each edge stand for wight $ \delta_8^j $, and $ c\in\Lambda_0 $
Figure 2.  The state trajectory graph of $ \delta_{64}^{6} $, $ \delta_{64}^{15} $, $ \delta_{64}^{29} $, $ \delta_{64}^{23} $, $ \delta_{64}^{11} $, where number $ s $ in each cycle denote state $ \delta_{64}^s $, weight $ (i,j) $ beside each edge stands for weight $ (\delta_2^i, j) $, $ \delta_2^i $ refers to input, and $ j $ means the $ jth $ subsystem
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