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State feedback for set stabilization of Markovian jump Boolean control networks
1. | School of Mathematics, Shandong University, Jinan, China |
2. | STP Center, Liaocheng University, Liaocheng, China |
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.
References:
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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. |
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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. |
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