doi: 10.3934/dcdss.2022089
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Pinning detectability of Boolean control networks with injection mode

1. 

School of Mathematics, Shandong University, Jinan 250100, Shandong, China

2. 

School of Management, Shandong University, Jinan 250100, Shandong, China

* Corresponding author: Jun-E Feng

Received  November 2021 Revised  March 2022 Early access April 2022

This technical note presents analytical investigations on detectability of Boolean network with pinning control and injection mode (BNPCIM). Detectability represents the property to uniquely determine the current state of the system according to known input-output sequences. Using Cheng product of matrices, BNPCIM can be converted into a special algebraic form of BCNs with mix-valued logical control. Based on different research requirements, three types of detectability for BNPCIM are proposed: weak detectability, detectability and strong detectability. Under free and networked input conditions, a sequence of matrices are constructed to reflect output and state information by explicit forms. Then by using the established matrices, several necessary and sufficient conditions for three types of detectability are derived. Moreover, to avoid unnecessary calculations, the maximum steps to achieve different detectability are gained. Finally, two illustrative examples are given to demonstrate the effectiveness of the obtained results.

Citation: Tiantian Mu, Jun-E Feng, Biao Wang. Pinning detectability of Boolean control networks with injection mode. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022089
References:
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E. BonzonM. Lagasquie-SchiexJ. Lang and B. Zanuttini, Compact preference representation and Boolean games, Autonomous Agents and Multi-Agent Systems, 18 (2009), 1-35. 

[2]

H. Chen and J. Liang, Local synchronization of interconnected Boolean networks with stochastic disturbances, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 452-463.  doi: 10.1109/TNNLS.2019.2904978.

[3]

D. Cheng, Semi-tensor product of matrices and its application to Morgen's problem, Sci. China Ser. F, 44 (2001), 195-212. 

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D. Cheng, H. Qi and Z. Li, Analysis and Control of Boolean Networks: A Semi-tensor Product Approach, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-097-7.

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D. Cheng, H. Qi and Y. Zhao, An Introduction to Semi-tensor Product of Matrices and Its Application, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/8323.

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T. ChowdhuryS. Chakraborty and A. Nandan, GPU accelerated drug application on signaling pathways containing multiple faults using Boolean networks, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 19 (2020), 1-1.  doi: 10.1109/TCBB.2020.3014172.

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E. Fornasini and M. Valcher, Observability and reconstructibility of probabilistic Boolean networks, IEEE Control Syst. Lett., 4 (2020), 319-324.  doi: 10.1109/LCSYS.2019.2925870.

[8]

Y. GuoP. WangW. Gui and C. Yang, Set stability and set stabilization of Boolean control networks based on invariant subsets, Automatica J. IFAC, 61 (2015), 106-112.  doi: 10.1016/j.automatica.2015.08.006.

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T. IdekerT. Galiski and L. Hood, A new approach to decoding life: Systems biology, Annual Review of Genomics and Human Genetics, 2 (2001), 343-372.  doi: 10.1146/annurev.genom.2.1.343.

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

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K. Kobayashi and K. Hiraishi, Structural control of probabilistic Boolean networks and its application to design of real-time pricing systems, IFAC Proceedings, 47 (2014), 2442-2447.  doi: 10.3182/20140824-6-ZA-1003.02609.

[12]

F. LiJ. Lu and L. Shen, State feedback controller design for the synchronization of Boolean networks with time delays, Phys. A, 490 (2018), 1267-1276.  doi: 10.1016/j.physa.2017.08.041.

[13]

F. Li and T. Yang, Pinning controllability for a Boolean network with arbitrary disturbance inputs, IEEE Transactions on Cybernetics, 51 (2021), 3338-3347.  doi: 10.1109/TCYB.2019.2930734.

[14]

R. LiM. Yang and T. Chu, Observability conditions of Boolean control networks, Internat. J. Robust Nonlinear Control, 24 (2014), 2711-2723.  doi: 10.1002/rnc.3019.

[15]

X. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica J. IFAC, 124 (2020), 109336, 6 pp. doi: 10.1016/j.automatica.2020.109336.

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X. LiJ. LuJ. QiuX. ChenX. Li and F. Alsaadi, Set stability for switched Boolean networks with open-loop and closed-loop switching signals, Science China (Information Sciences), 61 (2018), 232-240. 

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X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control, 63 (2018), 306-311.  doi: 10.1109/TAC.2016.2639819.

[18]

X. Li, X. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981.

[19]

Y. LiJ. Li and J. Feng, Set controllability of Boolean control networks with impulsive effects, Neurocomputing, 418 (2020), 263-269.  doi: 10.1016/j.neucom.2020.08.042.

[20]

Y. LiJ. Li and J. Feng, Output tracking of Boolean control networks with impulsive effects, IEEE Access, 8 (2020), 157793-157799.  doi: 10.1109/ACCESS.2020.3020124.

[21]

S. Liang, H. Li and S. Wang, Structural controllability of Boolean control networks with an unknown function structure, Sci. China Inf. Sci., 63 (2020), 219203, 3 pp. doi: 10.1007/s11432-018-9770-4.

[22]

Y. Liu, J. Cao, L. Wang and Z.-G. Wu, On pinning reachability of probabilistic Boolean control networks, Sci. China Inf. Sci., 63 (2020), 169201, 3 pp. doi: 10.1007/s11432-018-9575-4.

[23]

Y. Liu, J. Zhong, D. Ho and W. Gui, Minimal observability of Boolean networks, Science China (Information Sciences), in press, (2021).

[24]

Z. LiuD. Cheng and J. Liu, Pinning control of Boolean networks via injection mode, IEEE Trans. Control Netw. Syst., 8 (2020), 749-756.  doi: 10.1109/TCNS.2020.3037455.

[25]

Z. LiuJ. ZhongY. Liu and W. Gui, Weak stabilization of Boolean networks under state-flipped control, IEEE Transactions on Neural Networks and Learning Systems, (2021), 1-8.  doi: 10.1109/TNNLS.2021.3106918.

[26]

J. LuB. Li and J. Zhong, A novel synthesis method for reliable feedback shift registers via Boolean networks, Sci. China Inf. Sci., 64 (2021), 152207.  doi: 10.1007/s11432-020-2981-4.

[27]

J. LuR. LiuJ. Lou and Y. Liu, Pinning stabilization of Boolean control networks via a minimum number of controllers, IEEE Transactions On Cybernetics, 51 (2021), 373-381.  doi: 10.1109/TCYB.2019.2944659.

[28]

J. LuJ. YangJ. Lou and J. Qiu, Event-triggered sampled feedback synchronization in an array of output-coupled Boolean control networks, IEEE Transactions on Cybernetics, 51 (2019), 1-6.  doi: 10.1109/TCYB.2019.2939761.

[29]

J. LuJ. YangJ. Lou and J. Qiu, Event-Triggered sampled feedback synchronization in an array of output-coupled Boolean control networks., IEEE Transactions on Cybernetics, 51 (2021), 2278-2283.  doi: 10.1109/TCYB.2019.2939761.

[30]

Y. RongZ. ChenC. EvansH. Chen and G. Wang, Topology and dynamics of Boolean networks with strong inhibition, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1565-1575.  doi: 10.3934/dcdss.2011.4.1565.

[31]

P. SunL. Zhang and K. Zhang, Reconstructibility of Boolean control betworks with time delays in states, Kybernetika, 54 (2018), 1091-1104.  doi: 10.14736/kyb-2018-5-1091.

[32]

M. Toyoda and Y. Wu, Mayer-Type optimal control of probabilistic Boolean control network with uncertain selection probabilities, IEEE Transactions on Cybernetics, 51 (2021), 3079-3092. 

[33]

B. Wang and J. Feng, On detectability of probabilistic Boolean networks, Information Sciences, 483 (2019), 383-395.  doi: 10.1016/j.ins.2019.01.055.

[34]

B. Wang, J. Feng, H. Li and Y. Yu, On detectability of Boolean control networks, Nonlinear Anal. Hybrid Syst., 36 (2020), 100859, 18 pp. doi: 10.1016/j.nahs.2020.100859.

[35]

S. Wang and H. Li, New results on the disturbance decoupling of Boolean control networks, IEEE Control Syst. Lett., 5 (2021), 1157-1162.  doi: 10.1109/LCSYS.2020.3017645.

[36]

Y. YangY. LiuJ. Lou and Z. Wang, Observability of switched Boolean control networks using algebraic forms, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1519-1533.  doi: 10.3934/dcdss.2020373.

[37]

J. YangW. Qian and Z. Li, Redefined reconstructibility and state estimation for Boolean networks, IEEE Trans. Control Netw. Syst., 7 (2020), 1882-1890.  doi: 10.1109/TCNS.2020.3007820.

[38]

Y. Yao and J. Sun, Optimal control of multi-task Boolean control networks via temporal logic, Systems Control Lett., 156 (2021), 105007, 9 pp. doi: 10.1016/j.sysconle.2021.105007.

[39]

J. ZhangJ. LuJ. CaoW. HuangJ. Guo and Y. Wei, Traffic congestion pricing via network congestion game approach, Discrete Contin. Dyn. Syst. Ser. S, 14 (2020), 1553-1567.  doi: 10.3934/dcdss.2020378.

[40]

X. ZhangM. MengY. Wang and D. Cheng, Criteria for observability and reconstructibility of Boolean control networks via set controllability, IEEE Transactions on Circuits and Systems-Ⅱ: Express Briefs, 68 (2021), 1263-1267.  doi: 10.1109/TCSII.2020.3021190.

[41]

X. ZhangY. Wang and D. Cheng, Output tracking of Boolean control networks, IEEE Trans. Automat. Control, 65 (2020), 2730-2735.  doi: 10.1109/TAC.2019.2944903.

[42]

Z. ZhangT. Leifeld and P. Zhang, Reconstructibility analysis and observer design for Boolean control networks, IEEE Trans. Control Netw. Syst., 7 (2020), 516-528.  doi: 10.1109/TCNS.2019.2926746.

[43]

Z. Zhang, C. Xia and Z. Chen, On the stabilization of nondeterministic finite automata via static output feedback, Appl. Math. Comput., 365 (2020), 124687, 11 pp. doi: 10.1016/j.amc.2019.124687.

[44]

S. ZhuJ. Feng and J. Zhao, State feedback for set stabilization of markovian jump Boolean control networks, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1591-1605.  doi: 10.3934/dcdss.2020413.

[45]

Y. ZouJ. Zhu and Y. Liu, State-feedback controller design for disturbance decoupling of Boolean control networks, IET Control Theory Appl., 11 (2017), 3233-3239.  doi: 10.1049/iet-cta.2017.0714.

show all references

References:
[1]

E. BonzonM. Lagasquie-SchiexJ. Lang and B. Zanuttini, Compact preference representation and Boolean games, Autonomous Agents and Multi-Agent Systems, 18 (2009), 1-35. 

[2]

H. Chen and J. Liang, Local synchronization of interconnected Boolean networks with stochastic disturbances, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 452-463.  doi: 10.1109/TNNLS.2019.2904978.

[3]

D. Cheng, Semi-tensor product of matrices and its application to Morgen's problem, Sci. China Ser. F, 44 (2001), 195-212. 

[4]

D. Cheng, H. Qi and Z. Li, Analysis and Control of Boolean Networks: A Semi-tensor Product Approach, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-097-7.

[5]

D. Cheng, H. Qi and Y. Zhao, An Introduction to Semi-tensor Product of Matrices and Its Application, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/8323.

[6]

T. ChowdhuryS. Chakraborty and A. Nandan, GPU accelerated drug application on signaling pathways containing multiple faults using Boolean networks, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 19 (2020), 1-1.  doi: 10.1109/TCBB.2020.3014172.

[7]

E. Fornasini and M. Valcher, Observability and reconstructibility of probabilistic Boolean networks, IEEE Control Syst. Lett., 4 (2020), 319-324.  doi: 10.1109/LCSYS.2019.2925870.

[8]

Y. GuoP. WangW. Gui and C. Yang, Set stability and set stabilization of Boolean control networks based on invariant subsets, Automatica J. IFAC, 61 (2015), 106-112.  doi: 10.1016/j.automatica.2015.08.006.

[9]

T. IdekerT. Galiski and L. Hood, A new approach to decoding life: Systems biology, Annual Review of Genomics and Human Genetics, 2 (2001), 343-372.  doi: 10.1146/annurev.genom.2.1.343.

[10]

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

[11]

K. Kobayashi and K. Hiraishi, Structural control of probabilistic Boolean networks and its application to design of real-time pricing systems, IFAC Proceedings, 47 (2014), 2442-2447.  doi: 10.3182/20140824-6-ZA-1003.02609.

[12]

F. LiJ. Lu and L. Shen, State feedback controller design for the synchronization of Boolean networks with time delays, Phys. A, 490 (2018), 1267-1276.  doi: 10.1016/j.physa.2017.08.041.

[13]

F. Li and T. Yang, Pinning controllability for a Boolean network with arbitrary disturbance inputs, IEEE Transactions on Cybernetics, 51 (2021), 3338-3347.  doi: 10.1109/TCYB.2019.2930734.

[14]

R. LiM. Yang and T. Chu, Observability conditions of Boolean control networks, Internat. J. Robust Nonlinear Control, 24 (2014), 2711-2723.  doi: 10.1002/rnc.3019.

[15]

X. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica J. IFAC, 124 (2020), 109336, 6 pp. doi: 10.1016/j.automatica.2020.109336.

[16]

X. LiJ. LuJ. QiuX. ChenX. Li and F. Alsaadi, Set stability for switched Boolean networks with open-loop and closed-loop switching signals, Science China (Information Sciences), 61 (2018), 232-240. 

[17]

X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control, 63 (2018), 306-311.  doi: 10.1109/TAC.2016.2639819.

[18]

X. Li, X. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981.

[19]

Y. LiJ. Li and J. Feng, Set controllability of Boolean control networks with impulsive effects, Neurocomputing, 418 (2020), 263-269.  doi: 10.1016/j.neucom.2020.08.042.

[20]

Y. LiJ. Li and J. Feng, Output tracking of Boolean control networks with impulsive effects, IEEE Access, 8 (2020), 157793-157799.  doi: 10.1109/ACCESS.2020.3020124.

[21]

S. Liang, H. Li and S. Wang, Structural controllability of Boolean control networks with an unknown function structure, Sci. China Inf. Sci., 63 (2020), 219203, 3 pp. doi: 10.1007/s11432-018-9770-4.

[22]

Y. Liu, J. Cao, L. Wang and Z.-G. Wu, On pinning reachability of probabilistic Boolean control networks, Sci. China Inf. Sci., 63 (2020), 169201, 3 pp. doi: 10.1007/s11432-018-9575-4.

[23]

Y. Liu, J. Zhong, D. Ho and W. Gui, Minimal observability of Boolean networks, Science China (Information Sciences), in press, (2021).

[24]

Z. LiuD. Cheng and J. Liu, Pinning control of Boolean networks via injection mode, IEEE Trans. Control Netw. Syst., 8 (2020), 749-756.  doi: 10.1109/TCNS.2020.3037455.

[25]

Z. LiuJ. ZhongY. Liu and W. Gui, Weak stabilization of Boolean networks under state-flipped control, IEEE Transactions on Neural Networks and Learning Systems, (2021), 1-8.  doi: 10.1109/TNNLS.2021.3106918.

[26]

J. LuB. Li and J. Zhong, A novel synthesis method for reliable feedback shift registers via Boolean networks, Sci. China Inf. Sci., 64 (2021), 152207.  doi: 10.1007/s11432-020-2981-4.

[27]

J. LuR. LiuJ. Lou and Y. Liu, Pinning stabilization of Boolean control networks via a minimum number of controllers, IEEE Transactions On Cybernetics, 51 (2021), 373-381.  doi: 10.1109/TCYB.2019.2944659.

[28]

J. LuJ. YangJ. Lou and J. Qiu, Event-triggered sampled feedback synchronization in an array of output-coupled Boolean control networks, IEEE Transactions on Cybernetics, 51 (2019), 1-6.  doi: 10.1109/TCYB.2019.2939761.

[29]

J. LuJ. YangJ. Lou and J. Qiu, Event-Triggered sampled feedback synchronization in an array of output-coupled Boolean control networks., IEEE Transactions on Cybernetics, 51 (2021), 2278-2283.  doi: 10.1109/TCYB.2019.2939761.

[30]

Y. RongZ. ChenC. EvansH. Chen and G. Wang, Topology and dynamics of Boolean networks with strong inhibition, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1565-1575.  doi: 10.3934/dcdss.2011.4.1565.

[31]

P. SunL. Zhang and K. Zhang, Reconstructibility of Boolean control betworks with time delays in states, Kybernetika, 54 (2018), 1091-1104.  doi: 10.14736/kyb-2018-5-1091.

[32]

M. Toyoda and Y. Wu, Mayer-Type optimal control of probabilistic Boolean control network with uncertain selection probabilities, IEEE Transactions on Cybernetics, 51 (2021), 3079-3092. 

[33]

B. Wang and J. Feng, On detectability of probabilistic Boolean networks, Information Sciences, 483 (2019), 383-395.  doi: 10.1016/j.ins.2019.01.055.

[34]

B. Wang, J. Feng, H. Li and Y. Yu, On detectability of Boolean control networks, Nonlinear Anal. Hybrid Syst., 36 (2020), 100859, 18 pp. doi: 10.1016/j.nahs.2020.100859.

[35]

S. Wang and H. Li, New results on the disturbance decoupling of Boolean control networks, IEEE Control Syst. Lett., 5 (2021), 1157-1162.  doi: 10.1109/LCSYS.2020.3017645.

[36]

Y. YangY. LiuJ. Lou and Z. Wang, Observability of switched Boolean control networks using algebraic forms, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1519-1533.  doi: 10.3934/dcdss.2020373.

[37]

J. YangW. Qian and Z. Li, Redefined reconstructibility and state estimation for Boolean networks, IEEE Trans. Control Netw. Syst., 7 (2020), 1882-1890.  doi: 10.1109/TCNS.2020.3007820.

[38]

Y. Yao and J. Sun, Optimal control of multi-task Boolean control networks via temporal logic, Systems Control Lett., 156 (2021), 105007, 9 pp. doi: 10.1016/j.sysconle.2021.105007.

[39]

J. ZhangJ. LuJ. CaoW. HuangJ. Guo and Y. Wei, Traffic congestion pricing via network congestion game approach, Discrete Contin. Dyn. Syst. Ser. S, 14 (2020), 1553-1567.  doi: 10.3934/dcdss.2020378.

[40]

X. ZhangM. MengY. Wang and D. Cheng, Criteria for observability and reconstructibility of Boolean control networks via set controllability, IEEE Transactions on Circuits and Systems-Ⅱ: Express Briefs, 68 (2021), 1263-1267.  doi: 10.1109/TCSII.2020.3021190.

[41]

X. ZhangY. Wang and D. Cheng, Output tracking of Boolean control networks, IEEE Trans. Automat. Control, 65 (2020), 2730-2735.  doi: 10.1109/TAC.2019.2944903.

[42]

Z. ZhangT. Leifeld and P. Zhang, Reconstructibility analysis and observer design for Boolean control networks, IEEE Trans. Control Netw. Syst., 7 (2020), 516-528.  doi: 10.1109/TCNS.2019.2926746.

[43]

Z. Zhang, C. Xia and Z. Chen, On the stabilization of nondeterministic finite automata via static output feedback, Appl. Math. Comput., 365 (2020), 124687, 11 pp. doi: 10.1016/j.amc.2019.124687.

[44]

S. ZhuJ. Feng and J. Zhao, State feedback for set stabilization of markovian jump Boolean control networks, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1591-1605.  doi: 10.3934/dcdss.2020413.

[45]

Y. ZouJ. Zhu and Y. Liu, State-feedback controller design for disturbance decoupling of Boolean control networks, IET Control Theory Appl., 11 (2017), 3233-3239.  doi: 10.1049/iet-cta.2017.0714.

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