doi: 10.3934/dcdss.2022068
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Synchronization for singularity-perturbed complex networks via event-triggered impulsive control

Key Laboratory of Smart Manufacturing in Energy Chemical Processes, Ministry of Education, East China University of Science and Technology, 200237 Shanghai, China

* Corresponding author: Wangli He

Received  November 2021 Revised  January 2022 Early access March 2022

Fund Project: This work is supported by National Natural Science Foundation of China (61922030), Shanghai International Science Technology Cooperation Program (21550712400) and Fundamental Research Funds for the Central Universities (222202217006)

This paper studies synchronization of singularity-perturbed complex networks (SPCNs) with a small singular perturbation parameter (SPP) via event-triggered impulsive control (ETIC). A novel dynamic event-triggered mechanism is proposed where an auxiliary impulse parameter is introduced to regulate the triggering threshold dynamically for saving the network resource. Based on SPP-dependent Lyapunov function, some sufficient conditions involving the impulsive gain, triggering parameters and singular perturbation parameter (SPP) are obtained to synchronize the SPCNs, and the upper bound of SPP is also determined. Moreover, it proves that the Zeno behavior can be excluded. Finally, two simulations are provided to demonstrate the validity of the obtained results.

Citation: Kun Liang, Wangli He, Yang Yuan, Liyu Shi. Synchronization for singularity-perturbed complex networks via event-triggered impulsive control. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022068
References:
[1]

M. S. Ali and J. Yogambigai, Finite-time robust stochastic synchronization of uncertain Markovian complex dynamical networks with mixed time-varying delays and reaction–diffusion terms via impulsive control, Journal of The Franklin Institute, 354 (2017), 2415-2436.  doi: 10.1016/j.jfranklin.2016.12.014.

[2]

C. CaiZ. WangJ. XuX. Liu and F. E. Alsaadi, An integrated approach to global synchronization and state estimation for nonlinear singularly perturbed complex networks, IEEE Transactions on Cybernetics, 45 (2015), 1597-1609.  doi: 10.1109/TCYB.2014.2356560.

[3]

L. Chen and K. Aihara, A model of periodic oscillation for genetic regulatory systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49 (2002), 1429-1436.  doi: 10.1109/TCSI.2002.803354.

[4]

W.-H. ChenG. Yuan and W. X. Zheng, Robust stability of singularly perturbed impulsive systems under nonlinear perturbation, IEEE Transactions on Automatic Control, 58 (2012), 168-174.  doi: 10.1109/TAC.2012.2203029.

[5]

L. DingP. YuZ.-W. LiuZ.-H. Guan and G. Feng, Consensus of second-order multi-agent systems via impulsive control using sampled hetero-information, Automatica J. IFAC, 49 (2013), 2881-2886.  doi: 10.1016/j.automatica.2013.06.014.

[6]

Y. GaoG. Lu and Z. Wang, Passivity analysis of uncertain singularly perturbed systems, IEEE Transactions on Circuits and Systems II: Express Briefs, 57 (2010), 486-490. 

[7]

E. G. Gilbert and G. A. Harasty, A class of fixed-time fuel-optimal impulsive control problems and an efficient algorithm for their solution, IEEE Transactions on Automatic Control, AC-16 (1971), 1-11.  doi: 10.1109/tac.1971.1099656.

[8]

J. GongD. NingX. Wu and G. He, Bounded leader-following consensus of heterogeneous directed delayed multi-agent systems via asynchronous impulsive control, IEEE Transactions on Circuits and Systems II: Express Briefs, 68 (2021), 2680-2684.  doi: 10.1109/TCSII.2021.3054374.

[9]

W. He and Z. Mo, Secure event-triggered consensus control of linear multiagent systems subject to sequential scaling attacks, IEEE Transactions on Cybernetics, (2021), http://dx.doi.org/10.1109/TCYB.2021.3070356.

[10]

W. HeZ. MoQ.-L. Han and F. Qian, Secure impulsive synchronization in lipschitz-type multi-agent systems subject to deception attacks, IEEE/CAA Journal of Automatica Sinica, 7 (2020), 1326-1334. 

[11]

W. HeF. QianQ.-L. Han and G. Chen, Almost sure stability of nonlinear systems under random and impulsive sequential attacks, IEEE Transactions on Automatic Control, 65 (2020), 3879-3886.  doi: 10.1109/TAC.2020.2972220.

[12]

W. HeF. QianJ. LamG. ChenQ.-L. Han and J. Kurths, Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design, Automatica J. IFAC, 62 (2015), 249-262.  doi: 10.1016/j.automatica.2015.09.028.

[13]

W. HeB. XuQ.-L. Han and F. Qian, Adaptive consensus control of linear multiagent systems with dynamic event-triggered strategies, IEEE Transactions on Cybernetics, 50 (2020), 2996-3008.  doi: 10.1109/TCYB.2019.2920093.

[14]

B. Jiang, J. Lu, X. Li and J. Qiu, Event-triggered impulsive stabilization of systems with external disturbances, IEEE Transactions on Automatic Control, (2021), 1–1. doi: 10.1109/TAC.2021.3108123.

[15] P. V. KokotovicH. K. Khalil and J. O'reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. 
[16]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Transactions on Automatic Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[17]

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.

[18]

X. LiuJ.-W. XiaoD. Chen and Y.-W. Wang, Dynamic consensus of nonlinear time-delay multi-agent systems with input saturation: an impulsive control algorithm, Nonlinear Dynamics, 97 (2019), 1699-1710. 

[19]

X. Liu and K. Zhang, Input-to-state stability of time-delay systems with delay-dependent impulses, IEEE Transactions on Automatic Control, 65 (2019), 1676-1682.  doi: 10.1109/TAC.2019.2930239.

[20]

Y. LiuR. TangC. ZhouZ. Xiang and X. Yang, Event-triggered leader-following consensus of multiple mechanical systems with switched dynamics, International Journal of Systems Science, 51 (2020), 3563-3572.  doi: 10.1080/00207721.2020.1818146.

[21]

X. LvJ. CaoX. LiM. Abdel-Aty and U. A. Al-Juboori, Synchronization analysis for complex dynamical networks with coupling delay via event-triggered delayed impulsive control, IEEE transactions on Cybernetics, 51 (2020), 5269-5278.  doi: 10.1109/TCYB.2020.2974315.

[22]

J. B. RejebI.-C. Morărescu and J. Daafouz, Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems, Automatica J. IFAC, 91 (2018), 89-97.  doi: 10.1016/j.automatica.2018.01.019.

[23]

J. B. RejebI.-C. MorărescuA. Girard and J. Daafouz, Stability analysis of a general class of singularly perturbed linear hybrid systems, Automatica J. IFAC, 90 (2018), 98-108.  doi: 10.1016/j.automatica.2017.12.019.

[24]

H. ShenF. LiZ.-G. WuJ. H. Park and V. Sreeram, Fuzzy-model-based nonfragile control for nonlinear singularly perturbed systems with semi-Markov jump parameters, IEEE Transactions on Fuzzy systems, 26 (2018), 3428-3439.  doi: 10.1109/TFUZZ.2018.2832614.

[25]

X. TanJ. Cao and X. Li, Consensus of leader-following multiagent systems: A distributed event-triggered impulsive control strategy, IEEE Transactions on Cybernetics, 49 (2018), 792-801.  doi: 10.1109/TCYB.2017.2786474.

[26]

X. WanZ. WangM. Wu and X. Liu, ${H}_{\infty }$ state estimation for discrete-time nonlinear singularly perturbed complex networks under the round-robin protocol, IEEE Transactions on Neural Networks and Learning Systems, 30 (2018), 415-426.  doi: 10.1109/TNNLS.2018.2839020.

[27]

Y. WangP. Shi and H. Yan, Reliable control of fuzzy singularly perturbed systems and its application to electronic circuits, IEEE Transactions on Circuits and Systems I: Regular Papers, 65 (2018), 3519-3528.  doi: 10.1109/TCSI.2018.2834481.

[28]

J. XuC.-C. Lim and P. Shi, Sliding mode control of singularly perturbed systems and its application in quad-rotors, International Journal of Control, 92 (2019), 1325-1334.  doi: 10.1080/00207179.2017.1393102.

[29]

C. Yang and Q. Zhang, Multiobjective control for T-S fuzzy singularly perturbed systems, IEEE Transactions on Fuzzy Systems, 17 (2009), 104-115. 

[30]

G.-H. Yang and J. Dong, Control synthesis of singularly perturbed fuzzy systems, IEEE Transactions on Fuzzy Systems, 16 (2008), 615-629. 

[31]

T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272. Springer-Verlag, Berlin, 2001.

[32]

W. YangY.-W. WangZ.-H. Guan and C. Wen, Controllability of impulsive singularly perturbed systems and its application to a class of multiplex networks, Nonlinear Analysis: Hybrid Systems, 31 (2019), 123-134.  doi: 10.1016/j.nahs.2018.08.011.

[33]

X. YangX. LiJ. Lu and Z. Cheng, Synchronization of time-delayed complex networks with switching topology via hybrid actuator fault and impulsive effects control, IEEE transactions on Cybernetics, 50 (2019), 4043-4052.  doi: 10.1109/TCYB.2019.2938217.

[34]

X. YangX. WanC. ZunshuiJ. CaoY. Liu and L. Rutkowski, Synchronization of switched discrete-time neural networks via quantized output control with actuator fault, IEEE Transactions on Neural Networks and Learning Systems, 32 (2020), 4191-4201.  doi: 10.1109/TNNLS.2020.3017171.

[35]

S. Zhai and X.-S. Yang, Bounded synchronisation of singularly perturbed complex network with an application to power systems, IET Control Theory Appl., 8 (2014), 61-66.  doi: 10.1049/iet-cta.2013.0453.

[36]

W. ZhuD. WangL. Liu and G. Feng, Event-based impulsive control of continuous-time dynamic systems and its application to synchronization of memristive neural networks, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 3599-3609.  doi: 10.1109/TNNLS.2017.2731865.

show all references

References:
[1]

M. S. Ali and J. Yogambigai, Finite-time robust stochastic synchronization of uncertain Markovian complex dynamical networks with mixed time-varying delays and reaction–diffusion terms via impulsive control, Journal of The Franklin Institute, 354 (2017), 2415-2436.  doi: 10.1016/j.jfranklin.2016.12.014.

[2]

C. CaiZ. WangJ. XuX. Liu and F. E. Alsaadi, An integrated approach to global synchronization and state estimation for nonlinear singularly perturbed complex networks, IEEE Transactions on Cybernetics, 45 (2015), 1597-1609.  doi: 10.1109/TCYB.2014.2356560.

[3]

L. Chen and K. Aihara, A model of periodic oscillation for genetic regulatory systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49 (2002), 1429-1436.  doi: 10.1109/TCSI.2002.803354.

[4]

W.-H. ChenG. Yuan and W. X. Zheng, Robust stability of singularly perturbed impulsive systems under nonlinear perturbation, IEEE Transactions on Automatic Control, 58 (2012), 168-174.  doi: 10.1109/TAC.2012.2203029.

[5]

L. DingP. YuZ.-W. LiuZ.-H. Guan and G. Feng, Consensus of second-order multi-agent systems via impulsive control using sampled hetero-information, Automatica J. IFAC, 49 (2013), 2881-2886.  doi: 10.1016/j.automatica.2013.06.014.

[6]

Y. GaoG. Lu and Z. Wang, Passivity analysis of uncertain singularly perturbed systems, IEEE Transactions on Circuits and Systems II: Express Briefs, 57 (2010), 486-490. 

[7]

E. G. Gilbert and G. A. Harasty, A class of fixed-time fuel-optimal impulsive control problems and an efficient algorithm for their solution, IEEE Transactions on Automatic Control, AC-16 (1971), 1-11.  doi: 10.1109/tac.1971.1099656.

[8]

J. GongD. NingX. Wu and G. He, Bounded leader-following consensus of heterogeneous directed delayed multi-agent systems via asynchronous impulsive control, IEEE Transactions on Circuits and Systems II: Express Briefs, 68 (2021), 2680-2684.  doi: 10.1109/TCSII.2021.3054374.

[9]

W. He and Z. Mo, Secure event-triggered consensus control of linear multiagent systems subject to sequential scaling attacks, IEEE Transactions on Cybernetics, (2021), http://dx.doi.org/10.1109/TCYB.2021.3070356.

[10]

W. HeZ. MoQ.-L. Han and F. Qian, Secure impulsive synchronization in lipschitz-type multi-agent systems subject to deception attacks, IEEE/CAA Journal of Automatica Sinica, 7 (2020), 1326-1334. 

[11]

W. HeF. QianQ.-L. Han and G. Chen, Almost sure stability of nonlinear systems under random and impulsive sequential attacks, IEEE Transactions on Automatic Control, 65 (2020), 3879-3886.  doi: 10.1109/TAC.2020.2972220.

[12]

W. HeF. QianJ. LamG. ChenQ.-L. Han and J. Kurths, Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design, Automatica J. IFAC, 62 (2015), 249-262.  doi: 10.1016/j.automatica.2015.09.028.

[13]

W. HeB. XuQ.-L. Han and F. Qian, Adaptive consensus control of linear multiagent systems with dynamic event-triggered strategies, IEEE Transactions on Cybernetics, 50 (2020), 2996-3008.  doi: 10.1109/TCYB.2019.2920093.

[14]

B. Jiang, J. Lu, X. Li and J. Qiu, Event-triggered impulsive stabilization of systems with external disturbances, IEEE Transactions on Automatic Control, (2021), 1–1. doi: 10.1109/TAC.2021.3108123.

[15] P. V. KokotovicH. K. Khalil and J. O'reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. 
[16]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Transactions on Automatic Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[17]

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.

[18]

X. LiuJ.-W. XiaoD. Chen and Y.-W. Wang, Dynamic consensus of nonlinear time-delay multi-agent systems with input saturation: an impulsive control algorithm, Nonlinear Dynamics, 97 (2019), 1699-1710. 

[19]

X. Liu and K. Zhang, Input-to-state stability of time-delay systems with delay-dependent impulses, IEEE Transactions on Automatic Control, 65 (2019), 1676-1682.  doi: 10.1109/TAC.2019.2930239.

[20]

Y. LiuR. TangC. ZhouZ. Xiang and X. Yang, Event-triggered leader-following consensus of multiple mechanical systems with switched dynamics, International Journal of Systems Science, 51 (2020), 3563-3572.  doi: 10.1080/00207721.2020.1818146.

[21]

X. LvJ. CaoX. LiM. Abdel-Aty and U. A. Al-Juboori, Synchronization analysis for complex dynamical networks with coupling delay via event-triggered delayed impulsive control, IEEE transactions on Cybernetics, 51 (2020), 5269-5278.  doi: 10.1109/TCYB.2020.2974315.

[22]

J. B. RejebI.-C. Morărescu and J. Daafouz, Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems, Automatica J. IFAC, 91 (2018), 89-97.  doi: 10.1016/j.automatica.2018.01.019.

[23]

J. B. RejebI.-C. MorărescuA. Girard and J. Daafouz, Stability analysis of a general class of singularly perturbed linear hybrid systems, Automatica J. IFAC, 90 (2018), 98-108.  doi: 10.1016/j.automatica.2017.12.019.

[24]

H. ShenF. LiZ.-G. WuJ. H. Park and V. Sreeram, Fuzzy-model-based nonfragile control for nonlinear singularly perturbed systems with semi-Markov jump parameters, IEEE Transactions on Fuzzy systems, 26 (2018), 3428-3439.  doi: 10.1109/TFUZZ.2018.2832614.

[25]

X. TanJ. Cao and X. Li, Consensus of leader-following multiagent systems: A distributed event-triggered impulsive control strategy, IEEE Transactions on Cybernetics, 49 (2018), 792-801.  doi: 10.1109/TCYB.2017.2786474.

[26]

X. WanZ. WangM. Wu and X. Liu, ${H}_{\infty }$ state estimation for discrete-time nonlinear singularly perturbed complex networks under the round-robin protocol, IEEE Transactions on Neural Networks and Learning Systems, 30 (2018), 415-426.  doi: 10.1109/TNNLS.2018.2839020.

[27]

Y. WangP. Shi and H. Yan, Reliable control of fuzzy singularly perturbed systems and its application to electronic circuits, IEEE Transactions on Circuits and Systems I: Regular Papers, 65 (2018), 3519-3528.  doi: 10.1109/TCSI.2018.2834481.

[28]

J. XuC.-C. Lim and P. Shi, Sliding mode control of singularly perturbed systems and its application in quad-rotors, International Journal of Control, 92 (2019), 1325-1334.  doi: 10.1080/00207179.2017.1393102.

[29]

C. Yang and Q. Zhang, Multiobjective control for T-S fuzzy singularly perturbed systems, IEEE Transactions on Fuzzy Systems, 17 (2009), 104-115. 

[30]

G.-H. Yang and J. Dong, Control synthesis of singularly perturbed fuzzy systems, IEEE Transactions on Fuzzy Systems, 16 (2008), 615-629. 

[31]

T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272. Springer-Verlag, Berlin, 2001.

[32]

W. YangY.-W. WangZ.-H. Guan and C. Wen, Controllability of impulsive singularly perturbed systems and its application to a class of multiplex networks, Nonlinear Analysis: Hybrid Systems, 31 (2019), 123-134.  doi: 10.1016/j.nahs.2018.08.011.

[33]

X. YangX. LiJ. Lu and Z. Cheng, Synchronization of time-delayed complex networks with switching topology via hybrid actuator fault and impulsive effects control, IEEE transactions on Cybernetics, 50 (2019), 4043-4052.  doi: 10.1109/TCYB.2019.2938217.

[34]

X. YangX. WanC. ZunshuiJ. CaoY. Liu and L. Rutkowski, Synchronization of switched discrete-time neural networks via quantized output control with actuator fault, IEEE Transactions on Neural Networks and Learning Systems, 32 (2020), 4191-4201.  doi: 10.1109/TNNLS.2020.3017171.

[35]

S. Zhai and X.-S. Yang, Bounded synchronisation of singularly perturbed complex network with an application to power systems, IET Control Theory Appl., 8 (2014), 61-66.  doi: 10.1049/iet-cta.2013.0453.

[36]

W. ZhuD. WangL. Liu and G. Feng, Event-based impulsive control of continuous-time dynamic systems and its application to synchronization of memristive neural networks, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 3599-3609.  doi: 10.1109/TNNLS.2017.2731865.

Figure 1.  Framework of the SPCN with ETIC strategy
Figure 2.  Lyapunov function
Figure 3.  Trajectories of $ \eta_{is} $ ($ i = 1, 2, 3, 4, 5, s = 1, 2, 3 $) of error system (5) without dynamic ETIC
Figure 4.  Trajectories of $ \eta_{is} $ ($ i = 1, 2, 3, 4, 5, s = 1, 2, 3 $) of the error system (5) with dynamic ETIC
Table 1.  The relationship between the upper bound of SPP $ \bar{\varepsilon} $ and the nonlinear function $ f\left( x_{i}\left( t\right) \right) $
$ h $ 0.65 0.70 0.75 0.85 0.90
$ \bar{\varepsilon} $ 1.5737 0.9119 0.2226 0.0305 infeasible
$ h $ 0.65 0.70 0.75 0.85 0.90
$ \bar{\varepsilon} $ 1.5737 0.9119 0.2226 0.0305 infeasible
Table 2.  The number of triggering times
$ \beta $ 0.06 0.05 0.04 0.03 0.02 0.01 0.00
number 17 19 21 26 30 39 50
$ \beta $ 0.06 0.05 0.04 0.03 0.02 0.01 0.00
number 17 19 21 26 30 39 50
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