doi: 10.3934/dcdss.2022082
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Sampled-data based extended dissipative synchronization of stochastic complex dynamical networks

1. 

Graduate School of Advanced Science and Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, 739-8527, Japan

2. 

School of Cyber Science and Engineering, Southeast University, Nanjing 211189, China

3. 

Department of Electrical Engineering, Tafresh University, Tafresh 39518-79611, Iran

4. 

MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

5. 

Key Laboratory of Control and Optimization of Complex Systems, College of Hunan Province, Hunan Normal University, Changsha 410081, China

* Corresponding author: Quanxin Zhu

** Co-corresponding author: Yang Cao

Received  January 2022 Revised  February 2022 Early access March 2022

Fund Project: This work was supported by Grant-in-Aid for Research Activity Start-up (20K23328), funded by Japan Society for the Promotion of Science (JSPS), the National Natural Science Foundation of China (62173139, 62103103), the Science and Technology Innovation Program of Hunan Province (2021RC4030), Hunan Provincial Science and Technology Project Foundation (2019RS1033) and the Natural Science Foundation of Jiangsu Province of China (BK20210223)

In this paper, extended dissipative (ED) synchronization is considered for stochastic complex dynamical networks (SCDNs) with variable coupling delay via sampled-data control (SDC). First, a suitable Lyapunov–Krasovskii functional (LKF) is constructed, then a new synchronization criterion is obtained through stochastic integral inequality (SII) and linear matrix inequality (LMI) techniques. Moreover, the ED synchronization criteria are established, which consolidates passivity, dissipativity, $ H_\infty $, and $ L_2-L_\infty $ performances in a unified structure. SDC gain matrices are also designed for each performance in ED criteria. Finally, the feasibility and usefulness of the derived theoretical results are shown through numerical simulations.

Citation: Ramasamy Saravanakumar, Yang Cao, Ali Kazemy, Quanxin Zhu. Sampled-data based extended dissipative synchronization of stochastic complex dynamical networks. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022082
References:
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S. Aadhithiyan, R. Raja, Q. Zhu, J. Alzabut, M. Niezabitowski and C. P. Lim, Modified projective synchronization of distributive fractional order complex dynamic networks with model uncertainty via adaptive control, Chaos Solitons Fractals, 147 (2021), Paper No. 110853, 16 pp. doi: 10.1016/j.chaos.2021.110853.

[2]

S. AadhithiyanR. RajaQ. ZhuJ. AlzabutM. Niezabitowski and C. P. Lim, Exponential synchronization of nonlinear multi-weighted complex dynamic networks with hybrid time varying delays, Neural Processing Letters, 53 (2021), 1035-1063.  doi: 10.1007/s11063-021-10428-7.

[3]

Z. ChenK. Shi and S. Zhong, New synchronization criteria for complex delayed dynamical networks with sampled-data feedback control, ISA Transactions, 63 (2016), 154-169.  doi: 10.1016/j.isatra.2016.03.018.

[4]

K. Ding and Q. Zhu, A note on sampled-data synchronization of memristor networks subject to actuator failures and two different activations, IEEE Transactions on Circuits and Systems II: Express Briefs, 68 (2021), 2097-2101.  doi: 10.1109/TCSII.2020.3045172.

[5]

N. GunasekaranR. SaravanakumarY. H. Joo and H. S. Kim, Finite-time synchronization of sampled-data T–S fuzzy complex dynamical networks subject to average dwell-time approach, Fuzzy Sets and Systems, 374 (2019), 40-59.  doi: 10.1016/j.fss.2019.01.007.

[6]

H. HouQ. Zhang and M. Zheng, Cluster synchronization in nonlinear complex networks under sliding mode control, Nonlinear Dynam., 83 (2016), 739-749.  doi: 10.1007/s11071-015-2363-z.

[7]

W. JiangL. LiZ. Tu and Y. Feng, Semiglobal finite-time synchronization of complex networks with stochastic disturbance via intermittent control, Internat. J. Robust Nonlinear Control, 29 (2019), 2351-2363.  doi: 10.1002/rnc.4496.

[8]

A. Kazemy and M. Farrokhi, Synchronization of chaotic Lur'e systems with state and transmission line time delay: A linear matrix inequality approach, Transactions of the Institute of Measurement and Control, 39 (2017), 1703-1709.  doi: 10.1177/0142331216644497.

[9]

A. Kazemy and K. Shojaei, Adaptive synchronization of complex dynamical networks in presence of coupling connections with dynamical behavior, J. Comput. Nonlinear Dynam., 14 (2019), 061003, 8 pp. doi: 10.1115/1.4043146.

[10]

A. Kazemy and K. Shojaei, Synchronization of complex dynamical networks with dynamical behavior links, Asian J. Control, 22 (2020), 474-485.  doi: 10.1002/asjc.1910.

[11]

F. Kong and Q. Zhu, New fixed-time synchronization control of discontinuous inertial neural networks via indefinite lyapunov-krasovskii functional method, Internat. J. Robust Nonlinear Control, 31 (2021), 471-495.  doi: 10.1002/rnc.5297.

[12]

F. KongQ. ZhuR. Sakthivel and A. Mohammadzadeh, Fixed-time synchronization analysis for discontinuous fuzzy inertial neural networks with parameter uncertainties, Neurocomputing, 422 (2021), 295-313.  doi: 10.1016/j.neucom.2020.09.014.

[13]

T. H. LeeM.-J. ParkJ. H. ParkO.-M. Kwon and S.-M. Lee, Extended dissipative analysis for neural networks with time-varying delays, IEEE Transactions on Neural Networks and Learning Systems, 25 (2014), 1936-1941.  doi: 10.1109/TNNLS.2013.2296514.

[14]

S. H. LeeM.-J. ParkO. M. Kwon and R. Sakthivel, Advanced sampled-data synchronization control for complex dynamical networks with coupling time-varying delays, Information Sciences, 420 (2017), 454-465.  doi: 10.1016/j.ins.2017.08.071.

[15]

H. Li, Cluster synchronization stability for stochastic complex dynamical networks with probabilistic interval time-varying delays, J. Phys. A, 44 (2011), 105101, 24 pp. doi: 10.1088/1751-8113/44/10/105101.

[16]

H. LiW. K. Wong and Y. Tang, Global synchronization stability for stochastic complex dynamical networks with probabilistic interval time-varying delays, J. Optim. Theory Appl., 152 (2012), 496-516.  doi: 10.1007/s10957-011-9917-0.

[17]

J. LiY. Ma and L. Fu, Fault-tolerant passive synchronization for complex dynamical networks with Markovian jump based on sampled-data control, Neurocomputing, 350 (2019), 20-32.  doi: 10.1016/j.neucom.2019.03.059.

[18]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913. 

[19]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024-4034. 

[20]

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.

[21]

Y. LiuB.-Z. GuoJ. H. Park and S.-M. Lee, Non-fragile exponential synchronization of delayed complex dynamical networks with memory sampled-data control, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 118-128.  doi: 10.1109/TNNLS.2016.2614709.

[22]

Y.-A. LiuJ. XiaB. MengX. Song and H. Shen, Extended dissipative synchronization for semi-Markov jump complex dynamic networks via memory sampled-data control scheme, J. Franklin Inst., 357 (2020), 10900-10920.  doi: 10.1016/j.jfranklin.2020.08.023.

[23]

P. Lu and Y. Yang, Global asymptotic stability of a class of complex networks via decentralised static output feedback control, IET Control Theory Appl., 4 (2010), 2463-2470.  doi: 10.1049/iet-cta.2009.0416.

[24]

R. ManivannanR. SamiduraiJ. CaoA. Alsaedi and F. E. Alsaadi, Design of extended dissipativity state estimation for generalized neural networks with mixed time-varying delay signals, Inform. Sci., 424 (2018), 175-203.  doi: 10.1016/j.ins.2017.10.007.

[25]

T. Matsumoto, A chaotic attractor from Chua's circuit, IEEE Trans. Circuits and Systems, 31 (1984), 1055-1058.  doi: 10.1109/TCS.1984.1085459.

[26]

N. OzcanM. S. AliJ. YogambigaiQ. Zhu and S. Arik, Robust synchronization of uncertain Markovian jump complex dynamical networks with time-varying delays and reaction–diffusion terms via sampled-data control, J. Franklin Inst., 355 (2018), 1192-1216.  doi: 10.1016/j.jfranklin.2017.12.016.

[27]

R. Sasirekha and R. Rakkiyappan, Extended dissipativity state estimation for switched discrete-time complex dynamical networks with multiple communication channels: A sojourn probability dependent approach, Neurocomputing, 267 (2017), 55-68.  doi: 10.1016/j.neucom.2017.04.063.

[28]

P. SelvarajR. Sakthivel and O. M. Kwon, Finite-time synchronization of stochastic coupled neural networks subject to Markovian switching and input saturation, Neural Networks, 105 (2018), 154-165.  doi: 10.1016/j.neunet.2018.05.004.

[29]

K. Sivaranjani and R. Rakkiyappan, Pinning sampled-data synchronization of complex dynamical networks with Markovian jumping and mixed delays using multiple integral approach, Complexity, 21 (2016), 622-632.  doi: 10.1002/cplx.21777.

[30]

M. Syed AliM. UshaO. M. KwonN. Gunasekaran and G. K. Thakur, $H_\infty$ passive non-fragile synchronisation of Markovian jump stochastic complex dynamical networks with time-varying delays, Internat. J. Systems Sci., 52 (2021), 1270-1283.  doi: 10.1080/00207721.2020.1856445.

[31]

M. Syed Ali and J. Yogambigai, Extended dissipative synchronization of complex dynamical networks with additive time-varying delay and discrete-time information, J. Comput. Appl. Math., 348 (2019), 328-341.  doi: 10.1016/j.cam.2018.06.003.

[32]

J. WangL. SuH. ShenZ.-G. Wu and J. H. Park, Mixed $H_\infty$/passive sampled-data synchronization control of complex dynamical networks with distributed coupling delay, J. Franklin Inst., 354 (2017), 1302-1320.  doi: 10.1016/j.jfranklin.2016.11.035.

[33]

J. WangX.-M. ZhangY. LinX. Ge and Q.-L. Han, Event-triggered dissipative control for networked stochastic systems under non-uniform sampling, Information Sciences, 447 (2018), 216-228. 

[34]

X. WangX. LiuK. SheS. Zhong and L. Shi, Delay-dependent impulsive distributed synchronization of stochastic complex dynamical networks with time-varying delays, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 (2019), 1496-1504.  doi: 10.1109/TSMC.2018.2812895.

[35]

X. WangX. LiuK. SheS. Zhong and Q. Zhong, Extended dissipative memory sampled-data synchronization control of complex networks with communication delays, Neurocomputing, 347 (2019), 1-12.  doi: 10.1016/j.neucom.2018.10.073.

[36]

X. WangJ. H. ParkH. YangX. Zhang and S. Zhong, Delay-dependent fuzzy sampled-data synchronization of T–S fuzzy complex networks with multiple couplings, IEEE Transactions on Fuzzy Systems, 28 (2020), 178-189.  doi: 10.1109/TFUZZ.2019.2901353.

[37]

J. XiaoY. LiS. Zhong and F. Xu, Extended dissipative state estimation for memristive neural networks with time-varying delay, ISA Transactions, 64 (2016), 113-128. 

[38]

M. XingF. Deng and X. Zhao, Synchronization of stochastic complex dynamical networks under self-triggered control, Internat. J. Robust Nonlinear Control, 27 (2017), 2861-2878.  doi: 10.1002/rnc.3716.

[39]

H. YangL. ShuS. Zhong and X. Wang, Extended dissipative exponential synchronization of complex dynamical systems with coupling delay and sampled-data control, J. Franklin Inst., 353 (2016), 1829-1847.  doi: 10.1016/j.jfranklin.2016.03.003.

[40]

X. YangJ. Cao and J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control, IEEE Trans. Circuits Syst. I. Regul. Pap., 59 (2012), 371-384.  doi: 10.1109/TCSI.2011.2163969.

[41]

D. ZengR. ZhangS. ZhongJ. Wang and K. Shi, Sampled-data synchronization control for Markovian delayed complex dynamical networks via a novel convex optimization method, Neurocomputing, 266 (2017), 606-618.  doi: 10.1016/j.neucom.2017.05.070.

[42]

B. ZhangW. X. Zheng and S. Xu, Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans. Circuits Syst. I. Regul. Pap., 60 (2013), 1250-1263.  doi: 10.1109/TCSI.2013.2246213.

[43]

L. ZouZ. WangH. Gao and X. Liu, Event-triggered state estimation for complex networks with mixed time delays via sampled data information: The continuous-time case, IEEE Transactions on Cybernetics, 45 (2015), 2804-2815.  doi: 10.1109/TCYB.2014.2386781.

show all references

References:
[1]

S. Aadhithiyan, R. Raja, Q. Zhu, J. Alzabut, M. Niezabitowski and C. P. Lim, Modified projective synchronization of distributive fractional order complex dynamic networks with model uncertainty via adaptive control, Chaos Solitons Fractals, 147 (2021), Paper No. 110853, 16 pp. doi: 10.1016/j.chaos.2021.110853.

[2]

S. AadhithiyanR. RajaQ. ZhuJ. AlzabutM. Niezabitowski and C. P. Lim, Exponential synchronization of nonlinear multi-weighted complex dynamic networks with hybrid time varying delays, Neural Processing Letters, 53 (2021), 1035-1063.  doi: 10.1007/s11063-021-10428-7.

[3]

Z. ChenK. Shi and S. Zhong, New synchronization criteria for complex delayed dynamical networks with sampled-data feedback control, ISA Transactions, 63 (2016), 154-169.  doi: 10.1016/j.isatra.2016.03.018.

[4]

K. Ding and Q. Zhu, A note on sampled-data synchronization of memristor networks subject to actuator failures and two different activations, IEEE Transactions on Circuits and Systems II: Express Briefs, 68 (2021), 2097-2101.  doi: 10.1109/TCSII.2020.3045172.

[5]

N. GunasekaranR. SaravanakumarY. H. Joo and H. S. Kim, Finite-time synchronization of sampled-data T–S fuzzy complex dynamical networks subject to average dwell-time approach, Fuzzy Sets and Systems, 374 (2019), 40-59.  doi: 10.1016/j.fss.2019.01.007.

[6]

H. HouQ. Zhang and M. Zheng, Cluster synchronization in nonlinear complex networks under sliding mode control, Nonlinear Dynam., 83 (2016), 739-749.  doi: 10.1007/s11071-015-2363-z.

[7]

W. JiangL. LiZ. Tu and Y. Feng, Semiglobal finite-time synchronization of complex networks with stochastic disturbance via intermittent control, Internat. J. Robust Nonlinear Control, 29 (2019), 2351-2363.  doi: 10.1002/rnc.4496.

[8]

A. Kazemy and M. Farrokhi, Synchronization of chaotic Lur'e systems with state and transmission line time delay: A linear matrix inequality approach, Transactions of the Institute of Measurement and Control, 39 (2017), 1703-1709.  doi: 10.1177/0142331216644497.

[9]

A. Kazemy and K. Shojaei, Adaptive synchronization of complex dynamical networks in presence of coupling connections with dynamical behavior, J. Comput. Nonlinear Dynam., 14 (2019), 061003, 8 pp. doi: 10.1115/1.4043146.

[10]

A. Kazemy and K. Shojaei, Synchronization of complex dynamical networks with dynamical behavior links, Asian J. Control, 22 (2020), 474-485.  doi: 10.1002/asjc.1910.

[11]

F. Kong and Q. Zhu, New fixed-time synchronization control of discontinuous inertial neural networks via indefinite lyapunov-krasovskii functional method, Internat. J. Robust Nonlinear Control, 31 (2021), 471-495.  doi: 10.1002/rnc.5297.

[12]

F. KongQ. ZhuR. Sakthivel and A. Mohammadzadeh, Fixed-time synchronization analysis for discontinuous fuzzy inertial neural networks with parameter uncertainties, Neurocomputing, 422 (2021), 295-313.  doi: 10.1016/j.neucom.2020.09.014.

[13]

T. H. LeeM.-J. ParkJ. H. ParkO.-M. Kwon and S.-M. Lee, Extended dissipative analysis for neural networks with time-varying delays, IEEE Transactions on Neural Networks and Learning Systems, 25 (2014), 1936-1941.  doi: 10.1109/TNNLS.2013.2296514.

[14]

S. H. LeeM.-J. ParkO. M. Kwon and R. Sakthivel, Advanced sampled-data synchronization control for complex dynamical networks with coupling time-varying delays, Information Sciences, 420 (2017), 454-465.  doi: 10.1016/j.ins.2017.08.071.

[15]

H. Li, Cluster synchronization stability for stochastic complex dynamical networks with probabilistic interval time-varying delays, J. Phys. A, 44 (2011), 105101, 24 pp. doi: 10.1088/1751-8113/44/10/105101.

[16]

H. LiW. K. Wong and Y. Tang, Global synchronization stability for stochastic complex dynamical networks with probabilistic interval time-varying delays, J. Optim. Theory Appl., 152 (2012), 496-516.  doi: 10.1007/s10957-011-9917-0.

[17]

J. LiY. Ma and L. Fu, Fault-tolerant passive synchronization for complex dynamical networks with Markovian jump based on sampled-data control, Neurocomputing, 350 (2019), 20-32.  doi: 10.1016/j.neucom.2019.03.059.

[18]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913. 

[19]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024-4034. 

[20]

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.

[21]

Y. LiuB.-Z. GuoJ. H. Park and S.-M. Lee, Non-fragile exponential synchronization of delayed complex dynamical networks with memory sampled-data control, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 118-128.  doi: 10.1109/TNNLS.2016.2614709.

[22]

Y.-A. LiuJ. XiaB. MengX. Song and H. Shen, Extended dissipative synchronization for semi-Markov jump complex dynamic networks via memory sampled-data control scheme, J. Franklin Inst., 357 (2020), 10900-10920.  doi: 10.1016/j.jfranklin.2020.08.023.

[23]

P. Lu and Y. Yang, Global asymptotic stability of a class of complex networks via decentralised static output feedback control, IET Control Theory Appl., 4 (2010), 2463-2470.  doi: 10.1049/iet-cta.2009.0416.

[24]

R. ManivannanR. SamiduraiJ. CaoA. Alsaedi and F. E. Alsaadi, Design of extended dissipativity state estimation for generalized neural networks with mixed time-varying delay signals, Inform. Sci., 424 (2018), 175-203.  doi: 10.1016/j.ins.2017.10.007.

[25]

T. Matsumoto, A chaotic attractor from Chua's circuit, IEEE Trans. Circuits and Systems, 31 (1984), 1055-1058.  doi: 10.1109/TCS.1984.1085459.

[26]

N. OzcanM. S. AliJ. YogambigaiQ. Zhu and S. Arik, Robust synchronization of uncertain Markovian jump complex dynamical networks with time-varying delays and reaction–diffusion terms via sampled-data control, J. Franklin Inst., 355 (2018), 1192-1216.  doi: 10.1016/j.jfranklin.2017.12.016.

[27]

R. Sasirekha and R. Rakkiyappan, Extended dissipativity state estimation for switched discrete-time complex dynamical networks with multiple communication channels: A sojourn probability dependent approach, Neurocomputing, 267 (2017), 55-68.  doi: 10.1016/j.neucom.2017.04.063.

[28]

P. SelvarajR. Sakthivel and O. M. Kwon, Finite-time synchronization of stochastic coupled neural networks subject to Markovian switching and input saturation, Neural Networks, 105 (2018), 154-165.  doi: 10.1016/j.neunet.2018.05.004.

[29]

K. Sivaranjani and R. Rakkiyappan, Pinning sampled-data synchronization of complex dynamical networks with Markovian jumping and mixed delays using multiple integral approach, Complexity, 21 (2016), 622-632.  doi: 10.1002/cplx.21777.

[30]

M. Syed AliM. UshaO. M. KwonN. Gunasekaran and G. K. Thakur, $H_\infty$ passive non-fragile synchronisation of Markovian jump stochastic complex dynamical networks with time-varying delays, Internat. J. Systems Sci., 52 (2021), 1270-1283.  doi: 10.1080/00207721.2020.1856445.

[31]

M. Syed Ali and J. Yogambigai, Extended dissipative synchronization of complex dynamical networks with additive time-varying delay and discrete-time information, J. Comput. Appl. Math., 348 (2019), 328-341.  doi: 10.1016/j.cam.2018.06.003.

[32]

J. WangL. SuH. ShenZ.-G. Wu and J. H. Park, Mixed $H_\infty$/passive sampled-data synchronization control of complex dynamical networks with distributed coupling delay, J. Franklin Inst., 354 (2017), 1302-1320.  doi: 10.1016/j.jfranklin.2016.11.035.

[33]

J. WangX.-M. ZhangY. LinX. Ge and Q.-L. Han, Event-triggered dissipative control for networked stochastic systems under non-uniform sampling, Information Sciences, 447 (2018), 216-228. 

[34]

X. WangX. LiuK. SheS. Zhong and L. Shi, Delay-dependent impulsive distributed synchronization of stochastic complex dynamical networks with time-varying delays, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 (2019), 1496-1504.  doi: 10.1109/TSMC.2018.2812895.

[35]

X. WangX. LiuK. SheS. Zhong and Q. Zhong, Extended dissipative memory sampled-data synchronization control of complex networks with communication delays, Neurocomputing, 347 (2019), 1-12.  doi: 10.1016/j.neucom.2018.10.073.

[36]

X. WangJ. H. ParkH. YangX. Zhang and S. Zhong, Delay-dependent fuzzy sampled-data synchronization of T–S fuzzy complex networks with multiple couplings, IEEE Transactions on Fuzzy Systems, 28 (2020), 178-189.  doi: 10.1109/TFUZZ.2019.2901353.

[37]

J. XiaoY. LiS. Zhong and F. Xu, Extended dissipative state estimation for memristive neural networks with time-varying delay, ISA Transactions, 64 (2016), 113-128. 

[38]

M. XingF. Deng and X. Zhao, Synchronization of stochastic complex dynamical networks under self-triggered control, Internat. J. Robust Nonlinear Control, 27 (2017), 2861-2878.  doi: 10.1002/rnc.3716.

[39]

H. YangL. ShuS. Zhong and X. Wang, Extended dissipative exponential synchronization of complex dynamical systems with coupling delay and sampled-data control, J. Franklin Inst., 353 (2016), 1829-1847.  doi: 10.1016/j.jfranklin.2016.03.003.

[40]

X. YangJ. Cao and J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control, IEEE Trans. Circuits Syst. I. Regul. Pap., 59 (2012), 371-384.  doi: 10.1109/TCSI.2011.2163969.

[41]

D. ZengR. ZhangS. ZhongJ. Wang and K. Shi, Sampled-data synchronization control for Markovian delayed complex dynamical networks via a novel convex optimization method, Neurocomputing, 266 (2017), 606-618.  doi: 10.1016/j.neucom.2017.05.070.

[42]

B. ZhangW. X. Zheng and S. Xu, Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans. Circuits Syst. I. Regul. Pap., 60 (2013), 1250-1263.  doi: 10.1109/TCSI.2013.2246213.

[43]

L. ZouZ. WangH. Gao and X. Liu, Event-triggered state estimation for complex networks with mixed time delays via sampled data information: The continuous-time case, IEEE Transactions on Cybernetics, 45 (2015), 2804-2815.  doi: 10.1109/TCYB.2014.2386781.

Figure 1.  Synchronization error (Example 1)
Figure 2.  Generated control signals (Example 1)
Figure 3.  Synchronization error (Example 2)
Figure 4.  Generated control signals (Example 2)
Figure 5.  Synchronization error of the first node for different cases (Example 2)
Figure 6.  Generated control signals of the first node for different cases (Example 2)
Table 1.  Matrix parameters for each cases in ED synchronization
Criteria $ \Sigma_1 $ $ \Sigma_2 $ $ \Sigma_3 $ $ \Sigma_4 $
Passivity 0 1 $ \gamma = 0.5 $ 0
Dissipativity -1 1 2 0
$ H_\infty $ performance -1 0 $ \gamma^2 $ 0
$ L_2 $-$ L_\infty $ performance 0 0 $ \gamma^2 $ 1
Criteria $ \Sigma_1 $ $ \Sigma_2 $ $ \Sigma_3 $ $ \Sigma_4 $
Passivity 0 1 $ \gamma = 0.5 $ 0
Dissipativity -1 1 2 0
$ H_\infty $ performance -1 0 $ \gamma^2 $ 0
$ L_2 $-$ L_\infty $ performance 0 0 $ \gamma^2 $ 1
Table 2.  SDC gain matrices for each performance in ED synchronization
Criteria SDC gain matrices
Passivity $ \begin{array}{l} K_1 = K_5 = \left[ \begin{array} {ccc} -0.8104 &-0.1674\\0.0499&-1.6699 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} 0.6884&-0.1835\\0.0605&-1.6412 \end{array} \right] , \\ \;\;\;\;\; K_3 = \left[ \begin{array} {ccc} 0.6969&-0.1761\\0.0581&-1.6111 \end{array} \right] \end{array}$
Dissipativity $\begin{array}{l} K_1 = K_5 = \left[ \begin{array} {ccc} -0.7750&-0.1747\\-0.0155&-1.6717 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} -0.7588&-0.1733\\-0.0169&-1.6445 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.7567&-0.1693\\-0.0167&-1.6214 \end{array} \right]\end{array} $
$ H_\infty $ performance $ \begin{array}{l}K_1 = K_5 = \left[ \begin{array} {ccc} -0.7310&-0.1789\\0.0532&-1.6689 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} 0.5810&-0.2004\\0.0642&-1.6405 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.5843&-0.1930\\0.0651&-1.6107 \end{array} \right] \end{array}$
$ L_2 $-$ L_\infty $ performance $ \begin{array}{l}K_1 = K_5 = \left[ \begin{array} {ccc} -0.7715&-0.1847\\0.0286&-1.6624 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} -0.6395&-0.2060\\0.0318&-1.6327 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.6434&-0.1989\\0.0315&-1.6036 \end{array} \right] \end{array}$
Criteria SDC gain matrices
Passivity $ \begin{array}{l} K_1 = K_5 = \left[ \begin{array} {ccc} -0.8104 &-0.1674\\0.0499&-1.6699 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} 0.6884&-0.1835\\0.0605&-1.6412 \end{array} \right] , \\ \;\;\;\;\; K_3 = \left[ \begin{array} {ccc} 0.6969&-0.1761\\0.0581&-1.6111 \end{array} \right] \end{array}$
Dissipativity $\begin{array}{l} K_1 = K_5 = \left[ \begin{array} {ccc} -0.7750&-0.1747\\-0.0155&-1.6717 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} -0.7588&-0.1733\\-0.0169&-1.6445 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.7567&-0.1693\\-0.0167&-1.6214 \end{array} \right]\end{array} $
$ H_\infty $ performance $ \begin{array}{l}K_1 = K_5 = \left[ \begin{array} {ccc} -0.7310&-0.1789\\0.0532&-1.6689 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} 0.5810&-0.2004\\0.0642&-1.6405 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.5843&-0.1930\\0.0651&-1.6107 \end{array} \right] \end{array}$
$ L_2 $-$ L_\infty $ performance $ \begin{array}{l}K_1 = K_5 = \left[ \begin{array} {ccc} -0.7715&-0.1847\\0.0286&-1.6624 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} -0.6395&-0.2060\\0.0318&-1.6327 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.6434&-0.1989\\0.0315&-1.6036 \end{array} \right] \end{array}$
Table 3.  SDC gain matrices for each performance in ED synchronization
Criteria SDC gain matrices
Passivity $ K_1 = \left[ \begin{array} {ccc} -1.4426& -0.0287 \\ -0.0390& -1.5145 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -1.2444& 0.0080 \\ 0.0350& -1.3375 \end{array} \right] $
Dissipativity $ K_1 = \left[ \begin{array} {ccc} -1.2143& -0.1133 \\ -0.1427& -1.4406 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -0.8945& -0.1379\\ -0.1690& -1.2041 \end{array} \right] $
$ H_\infty $ performance $ K_1 = \left[ \begin{array} {ccc} -1.5024& 0.0015 \\ 0.0212& -1.5711 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -1.2859& 0.0376\\ 0.1014& -1.3978 \end{array} \right] $
$ L_2 $-$ L_\infty $ performance $ K_1 = \left[ \begin{array} {ccc} -1.3752& -0.0691 \\ -0.0757& -1.4975 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -0.9550& -0.1274\\ -0.1197& -1.2459 \end{array} \right] $
Criteria SDC gain matrices
Passivity $ K_1 = \left[ \begin{array} {ccc} -1.4426& -0.0287 \\ -0.0390& -1.5145 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -1.2444& 0.0080 \\ 0.0350& -1.3375 \end{array} \right] $
Dissipativity $ K_1 = \left[ \begin{array} {ccc} -1.2143& -0.1133 \\ -0.1427& -1.4406 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -0.8945& -0.1379\\ -0.1690& -1.2041 \end{array} \right] $
$ H_\infty $ performance $ K_1 = \left[ \begin{array} {ccc} -1.5024& 0.0015 \\ 0.0212& -1.5711 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -1.2859& 0.0376\\ 0.1014& -1.3978 \end{array} \right] $
$ L_2 $-$ L_\infty $ performance $ K_1 = \left[ \begin{array} {ccc} -1.3752& -0.0691 \\ -0.0757& -1.4975 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -0.9550& -0.1274\\ -0.1197& -1.2459 \end{array} \right] $
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