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April  2021, 14(4): 1591-1605. doi: 10.3934/dcdss.2020413

State feedback for set stabilization of Markovian jump Boolean control networks

Sanmei Zhu 1, , Jun-e Feng 1,, and  Jianli Zhao 2,

1. 

School of Mathematics, Shandong University, Jinan, China
2. 

STP Center, Liaocheng University, Liaocheng, China

* Corresponding author: Jun-e Feng

Received  January 2020 Revised  April 2020 Published  April 2021 Early access  July 2020

In this paper, the set stabilization problem of Markovian jump Boolean control networks (MJBCNs) is investigated via semi-tensor product of matrices. First, the conception of set stabilization is proposed for MJBCNs. Then based on the algebraic expression of MJBCN, a necessary and sufficient condition for set stabilization is provided by a linear programming problem, which is simple to solve. Moreover, by solving this linear programming problem, an algorithm for designing a state feedback controller is developed. Finally, two examples are presented to illustrate the feasibility of the obtained results.

Keywords: Set stabilization, Markovian jump Boolean control networks, semi-tensor product, state feedback, control invariant subset.
Mathematics Subject Classification: Primary: 93D15, 93E03.
Citation: Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1591-1605. doi: 10.3934/dcdss.2020413
References:
[1]

H. Bei, L. Wang, Y. Ma, J. Sun and L. Zhang, A linear optimal feedback control for producing $1, 3$-propanediol via microbial fermentation, Discrete and Continous Dynamical Systems series S, 13 (2020), 1623-1635.  doi: 10.3934/dcdss.2020095.

[2]

D. Cheng, H. Qi and Z. Li, Analysis and Control of Boolean networks: A Semi-Tensor Product Approach, Springer, 2011. doi: 10.1007/978-0-85729-097-7.

[3]

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.

[4]

A. N. Churilov, Orbital stability of periodic solutions of an impulsive system with a linear continuous-time part, AIMS Mathematics, 5 (2019), 96-110.  doi: 10.3934/math.2020007.

[5]

O. L. Costa, M. D. Fragoso and R. P. Marques, Discrete-time Markov Jump Linear Systems, Springer, 2005. doi: 10.1007/b138575.

[6]

E. Fornasini and M. E. Valcher, Optimal control of Boolean control networks, IEEE Transactions on Automatic Control, 59 (2014), 1258-1270.  doi: 10.1109/TAC.2013.2294821.

[7]

D. G. Green, T. G. Leishman and S. Sadedin, The emergence of social consensus in Boolean networks, in 2007 IEEE Symposium on Artificial Life, (2007), 402–408.

[8]

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

[9]

Y. Guo, R. Zhou, Y. Wu, W. Gui and C. Yang, Stability and set stability in distribution of probabilistic Boolean networks, IEEE Transactions on Automatic Control, 64 (2019), 736-742.  doi: 10.1109/TAC.2018.2833170.

[10]

X. Hu, C. Huang, J. Lu and J. Cao, Stabilization of boolean control networks with stochastic impulses, Journal of the Franklin Institute, 356 (2019), 7164-7182.  doi: 10.1016/j.jfranklin.2019.06.039.

[11]

S. 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.

[12]

R. K. Layek, A. Datta and E. R. Dougherty, From biological pathways to regulatory networks, Molecular Biosystems, 7 (2011), 843-851.  doi: 10.1039/c0mb00263a.

[13]

F. Li and L. Xie, Set stabilization of probabilistic Boolean networks using pinning control, IEEE Transactions on Neural Networks and Learning Systems, 30 (2019), 2555-2561.  doi: 10.1109/TNNLS.2018.2881279.

[14]

X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Applied Mathematics and Computation, 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[15]

X. Li, Daniel W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.

[16]

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.

[17]

Y. Li, H. Li and W. Sun, Event-triggered control for robust set stabilization of logical control networks, Automatica, 95 (2018), 556-560.  doi: 10.1016/j.automatica.2018.06.030.

[18]

P. Li, J. Lam and Z. Shu, ${H}_{\infty}$ positive filtering for positive linear discrete-time systems: An augmentation approach, IEEE Transactions on Automatic Control, 55 (2010), 2337-2342.  doi: 10.1109/TAC.2010.2053471.

[19]

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

[20]

F. Li and Z. Yu, Feedback control and output feedback control for the stabilisation of switched Boolean networks, International Journal of Control, 89 (2016), 337-342.  doi: 10.1080/00207179.2015.1076938.

[21]

R. Liu, J. Lu, W. X. Zheng and J. Kurths, Output feedback control for set stabilization of Boolean control networks, IEEE Transactions on Neural Networks and Learning Systems, 31 (2020), 2129-2139.  doi: 10.1109/TNNLS.2019.2928028.

[22]

X. Liu and J. Zhu, On potential equations of finite games, Automatica, 68 (2016), 245-253.  doi: 10.1016/j.automatica.2016.01.074.

[23]

Y. Liu, H. Chen, J. Lu and B. Wu, Controllability of probabilistic Boolean control networks based on transition probability matrices, Automatica, 52 (2015), 340-345.  doi: 10.1016/j.automatica.2014.12.018.

[24]

H. Lyu, W. Wang, X. Liu and Z. Wang, Modeling of multivariable fuzzy systems by semitensor product, IEEE Transactions on Fuzzy Systems, 28 (2020), 228-235.  doi: 10.1109/TFUZZ.2019.2902820.

[25]

H. Lyu, W. Wang and X. Liu, Universal approximation of multi-variable fuzzy systems by semi-tensor product, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1109/TFUZZ.2019.2946512.

[26]

J. Lu, H. Li, Y. Liu and F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory and Applications, 11 (2017), 2040-2047.  doi: 10.1049/iet-cta.2016.1659.

[27]

M. Meng, L. Liu and G. Feng, Stability and $l_1$ gain analysis of boolean networks with markovian jump parameters, IEEE Transactions on Automatic Control, 62 (2017), 4222-4228.  doi: 10.1109/TAC.2017.2679903.

[28]

M. Meng, J. Lam, J. Feng and K. C. Cheung, Stability and stabilization of Boolean networks with stochastic delays, IEEE Transactions on Automatic Control, 64 (2019), 790-796.  doi: 10.1109/TAC.2018.2835366.

[29]

M. Meng, G. Xiao, C. Zhai and G. Li, Controllability of Markovian jump Boolean control networks, Automatica, 106 (2019), 70-76.  doi: 10.1016/j.automatica.2019.04.028.

[30]

N. Megiddo, Linear programming in linear time when the dimension is fixed, Journal of the ACM, 31 (1984), 114-127.  doi: 10.1145/2422.322418.

[31]

K. Ratnavelu, M. Kalpana and P. Balasubramaniam, Asymptotic stability of Markovian switching genetic regulatory networks with leakage and mode-dependent time delays, Journal of the Franklin Institute, 353 (2016), 1615-1638.  doi: 10.1016/j.jfranklin.2016.01.015.

[32]

R. Rakkiyappan, A. Chandrasekar, F. A. Rihan and S. Lakshmanan, Exponentialstate estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays, Mathematical Biosciences, 251 (2014), 30-53.  doi: 10.1016/j.mbs.2014.02.008.

[33]

I. Shmulevich, E. R. Dougherty and W. Zhang, From boolean to probabilistic Boolean networks as models of genetic regulatory networks, Proceedings of the IEEE, 90 (2002), 1778-1792.  doi: 10.1109/JPROC.2002.804686.

[34]

X. Tan, J. Cao, X. Li and A. Alsaedi, Leader-following mean squareconsensus of stochastic multi-agent systems with input delay via event-triggered control, IET Control Theory Applications, 12 (2017), 299-309.  doi: 10.1049/iet-cta.2017.0462.

[35]

H. Tian and Y. Hou, State feedback design for set stabilization of probabilistic Boolean control networks, Journal of the Franklin Institute, 356 (2019), 4358-4377.  doi: 10.1016/j.jfranklin.2018.12.027.

[36]

H. Tian, H. Zhang, Z. Wang and Y. Hou, Stabilization of k-valued logical control networks by open-loop control via the reverse-transfer method, Automatica, 83 (2017), 387-390.  doi: 10.1016/j.automatica.2016.12.040.

[37]

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

[38]

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

[39]

M. Xu, Y. Liu, J. Lou, Z. Wu and J. Zhong, Set stabilization of probabilistic Boolean control networks: A sampled-data control approach, IEEE Transactions on Cybernetics, 2019, 1–8. doi: 10.1109/TCYB.2019.2940654.

[40]

D. Yang, X. 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.

[41]

X. Yang, X. Li, Q. 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.

[42]

Y. Yu, J. Feng, J. Pan and D. Cheng, Block decoupling of Boolean control networks, IEEE Transactions on Automatic Control, 64 (2019), 3129-3140.  doi: 10.1109/TAC.2018.2880411.

[43]

Y. Yu, M. Meng, and J. Feng, Observability of Boolean networks via matrix equations, Automatica, 111 (2020), 108621, 5 pp. doi: 10.1016/j.automatica.2019.108621.

[44]

Q. Zhang, J. Feng, J. Pan and J. Xia, Set controllability for switched Boolean control networks, Neurocomputing, 359 (2019), 476-482.  doi: 10.1016/j.neucom.2019.05.087.

[45]

R. Zhou, Y. Guo and W. Gui, Set reachability and observability of probabilistic Boolean networks, Automatica, 106 (2019), 230-241.  doi: 10.1016/j.automatica.2019.05.021.

[46]

S. Zhu, J. Lu, Y. Liu, T. Huang and J. Kurths, Output tracking of probabilistic Boolean networks by output feedback control, Information Sciences, 483 (2019), 96-105.  doi: 10.1016/j.ins.2018.12.087.

[47]

R. Zhou and Y. Guo, Set stabilization in distribution of probabilistic Boolean control networks, in 2018 13th World Congress on Intelligent Control and Automation (WCICA), (2018), 274–279.

show all references

References:
[1]

H. Bei, L. Wang, Y. Ma, J. Sun and L. Zhang, A linear optimal feedback control for producing $1, 3$-propanediol via microbial fermentation, Discrete and Continous Dynamical Systems series S, 13 (2020), 1623-1635.  doi: 10.3934/dcdss.2020095.

[2]

D. Cheng, H. Qi and Z. Li, Analysis and Control of Boolean networks: A Semi-Tensor Product Approach, Springer, 2011. doi: 10.1007/978-0-85729-097-7.

[3]

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.

[4]

A. N. Churilov, Orbital stability of periodic solutions of an impulsive system with a linear continuous-time part, AIMS Mathematics, 5 (2019), 96-110.  doi: 10.3934/math.2020007.

[5]

O. L. Costa, M. D. Fragoso and R. P. Marques, Discrete-time Markov Jump Linear Systems, Springer, 2005. doi: 10.1007/b138575.

[6]

E. Fornasini and M. E. Valcher, Optimal control of Boolean control networks, IEEE Transactions on Automatic Control, 59 (2014), 1258-1270.  doi: 10.1109/TAC.2013.2294821.

[7]

D. G. Green, T. G. Leishman and S. Sadedin, The emergence of social consensus in Boolean networks, in 2007 IEEE Symposium on Artificial Life, (2007), 402–408.

[8]

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

[9]

Y. Guo, R. Zhou, Y. Wu, W. Gui and C. Yang, Stability and set stability in distribution of probabilistic Boolean networks, IEEE Transactions on Automatic Control, 64 (2019), 736-742.  doi: 10.1109/TAC.2018.2833170.

[10]

X. Hu, C. Huang, J. Lu and J. Cao, Stabilization of boolean control networks with stochastic impulses, Journal of the Franklin Institute, 356 (2019), 7164-7182.  doi: 10.1016/j.jfranklin.2019.06.039.

[11]

S. 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.

[12]

R. K. Layek, A. Datta and E. R. Dougherty, From biological pathways to regulatory networks, Molecular Biosystems, 7 (2011), 843-851.  doi: 10.1039/c0mb00263a.

[13]

F. Li and L. Xie, Set stabilization of probabilistic Boolean networks using pinning control, IEEE Transactions on Neural Networks and Learning Systems, 30 (2019), 2555-2561.  doi: 10.1109/TNNLS.2018.2881279.

[14]

X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Applied Mathematics and Computation, 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[15]

X. Li, Daniel W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.

[16]

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.

[17]

Y. Li, H. Li and W. Sun, Event-triggered control for robust set stabilization of logical control networks, Automatica, 95 (2018), 556-560.  doi: 10.1016/j.automatica.2018.06.030.

[18]

P. Li, J. Lam and Z. Shu, ${H}_{\infty}$ positive filtering for positive linear discrete-time systems: An augmentation approach, IEEE Transactions on Automatic Control, 55 (2010), 2337-2342.  doi: 10.1109/TAC.2010.2053471.

[19]

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

[20]

F. Li and Z. Yu, Feedback control and output feedback control for the stabilisation of switched Boolean networks, International Journal of Control, 89 (2016), 337-342.  doi: 10.1080/00207179.2015.1076938.

[21]

R. Liu, J. Lu, W. X. Zheng and J. Kurths, Output feedback control for set stabilization of Boolean control networks, IEEE Transactions on Neural Networks and Learning Systems, 31 (2020), 2129-2139.  doi: 10.1109/TNNLS.2019.2928028.

[22]

X. Liu and J. Zhu, On potential equations of finite games, Automatica, 68 (2016), 245-253.  doi: 10.1016/j.automatica.2016.01.074.

[23]

Y. Liu, H. Chen, J. Lu and B. Wu, Controllability of probabilistic Boolean control networks based on transition probability matrices, Automatica, 52 (2015), 340-345.  doi: 10.1016/j.automatica.2014.12.018.

[24]

H. Lyu, W. Wang, X. Liu and Z. Wang, Modeling of multivariable fuzzy systems by semitensor product, IEEE Transactions on Fuzzy Systems, 28 (2020), 228-235.  doi: 10.1109/TFUZZ.2019.2902820.

[25]

H. Lyu, W. Wang and X. Liu, Universal approximation of multi-variable fuzzy systems by semi-tensor product, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1109/TFUZZ.2019.2946512.

[26]

J. Lu, H. Li, Y. Liu and F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory and Applications, 11 (2017), 2040-2047.  doi: 10.1049/iet-cta.2016.1659.

[27]

M. Meng, L. Liu and G. Feng, Stability and $l_1$ gain analysis of boolean networks with markovian jump parameters, IEEE Transactions on Automatic Control, 62 (2017), 4222-4228.  doi: 10.1109/TAC.2017.2679903.

[28]

M. Meng, J. Lam, J. Feng and K. C. Cheung, Stability and stabilization of Boolean networks with stochastic delays, IEEE Transactions on Automatic Control, 64 (2019), 790-796.  doi: 10.1109/TAC.2018.2835366.

[29]

M. Meng, G. Xiao, C. Zhai and G. Li, Controllability of Markovian jump Boolean control networks, Automatica, 106 (2019), 70-76.  doi: 10.1016/j.automatica.2019.04.028.

[30]

N. Megiddo, Linear programming in linear time when the dimension is fixed, Journal of the ACM, 31 (1984), 114-127.  doi: 10.1145/2422.322418.

[31]

K. Ratnavelu, M. Kalpana and P. Balasubramaniam, Asymptotic stability of Markovian switching genetic regulatory networks with leakage and mode-dependent time delays, Journal of the Franklin Institute, 353 (2016), 1615-1638.  doi: 10.1016/j.jfranklin.2016.01.015.

[32]

R. Rakkiyappan, A. Chandrasekar, F. A. Rihan and S. Lakshmanan, Exponentialstate estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays, Mathematical Biosciences, 251 (2014), 30-53.  doi: 10.1016/j.mbs.2014.02.008.

[33]

I. Shmulevich, E. R. Dougherty and W. Zhang, From boolean to probabilistic Boolean networks as models of genetic regulatory networks, Proceedings of the IEEE, 90 (2002), 1778-1792.  doi: 10.1109/JPROC.2002.804686.

[34]

X. Tan, J. Cao, X. Li and A. Alsaedi, Leader-following mean squareconsensus of stochastic multi-agent systems with input delay via event-triggered control, IET Control Theory Applications, 12 (2017), 299-309.  doi: 10.1049/iet-cta.2017.0462.

[35]

H. Tian and Y. Hou, State feedback design for set stabilization of probabilistic Boolean control networks, Journal of the Franklin Institute, 356 (2019), 4358-4377.  doi: 10.1016/j.jfranklin.2018.12.027.

[36]

H. Tian, H. Zhang, Z. Wang and Y. Hou, Stabilization of k-valued logical control networks by open-loop control via the reverse-transfer method, Automatica, 83 (2017), 387-390.  doi: 10.1016/j.automatica.2016.12.040.

[37]

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

[38]

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

[39]

M. Xu, Y. Liu, J. Lou, Z. Wu and J. Zhong, Set stabilization of probabilistic Boolean control networks: A sampled-data control approach, IEEE Transactions on Cybernetics, 2019, 1–8. doi: 10.1109/TCYB.2019.2940654.

[40]

D. Yang, X. 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.

[41]

X. Yang, X. Li, Q. 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.

[42]

Y. Yu, J. Feng, J. Pan and D. Cheng, Block decoupling of Boolean control networks, IEEE Transactions on Automatic Control, 64 (2019), 3129-3140.  doi: 10.1109/TAC.2018.2880411.

[43]

Y. Yu, M. Meng, and J. Feng, Observability of Boolean networks via matrix equations, Automatica, 111 (2020), 108621, 5 pp. doi: 10.1016/j.automatica.2019.108621.

[44]

Q. Zhang, J. Feng, J. Pan and J. Xia, Set controllability for switched Boolean control networks, Neurocomputing, 359 (2019), 476-482.  doi: 10.1016/j.neucom.2019.05.087.

[45]

R. Zhou, Y. Guo and W. Gui, Set reachability and observability of probabilistic Boolean networks, Automatica, 106 (2019), 230-241.  doi: 10.1016/j.automatica.2019.05.021.

[46]

S. Zhu, J. Lu, Y. Liu, T. Huang and J. Kurths, Output tracking of probabilistic Boolean networks by output feedback control, Information Sciences, 483 (2019), 96-105.  doi: 10.1016/j.ins.2018.12.087.

[47]

R. Zhou and Y. Guo, Set stabilization in distribution of probabilistic Boolean control networks, in 2018 13th World Congress on Intelligent Control and Automation (WCICA), (2018), 274–279.

Figure 1.  Trajectory of $ h(t) $, where $ x(t) = \delta_8^{h(t)} $
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Figure 2.  Trajectory of $ h(t) $, where $ x(t) = \delta_8^{h(t)} $
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