Cluster synchronization in finite/fixed time for semi-Markovian switching T-S fuzzy complex dynamical networks with discontinuous dynamic nodes

Zhengqi Zhang; Huaiqin Wu; Zhengqi Zhang; Huaiqin Wu

doi:10.3934/math.2022666

AIMS Mathematics

2022, Volume 7, Issue 7: 11942-11971. doi: 10.3934/math.2022666

Previous Article Next Article

Research article Special Issues

Cluster synchronization in finite/fixed time for semi-Markovian switching T-S fuzzy complex dynamical networks with discontinuous dynamic nodes

Zhengqi Zhang ,
Huaiqin Wu ^,

Department of Mathematics, Yanshan University, Qinhuangdao 066001, China

Received: 17 January 2022 Revised: 11 March 2022 Accepted: 30 March 2022 Published: 20 April 2022

In this paper, cluster synchronization in finite/fixed time for semi-Markovian switching complex dynamical networks (CDNs) with discontinuous dynamic nodes is studied. Firstly, the global fixed-time convergence principle of nonlinear systems with semi-Markovian switching is developed. Secondly, the novel state-feedback controllers, which include discontinuous factors and integral terms, are designed to achieve the global stochastic finite/fixed-time cluster synchronization. In the framework of Filippov stochastic differential inclusion, the Lyapunov-Krasovskii functional approach, Takagi-Sugeno(T-S) fuzzy theory, stochastic analysis theory, and inequality analysis techniques are applied, and the global stochastic finite/fixed time synchronization conditions are proposed in the form of linear matrix inequalities (LMIs). Moreover, the upper bound of the stochastic settling time is explicitly proposed. In addition, the correlations among the obtained results are interpreted analytically. Finally, two numerical examples are given to illustrate the correctness of the theoretical results.
- complex dynamical networks,
- finite/fixed time cluster synchronization,
- semi-Markovian switching,
- T-S fuzzy,
- discontinuous system
Citation: Zhengqi Zhang, Huaiqin Wu. Cluster synchronization in finite/fixed time for semi-Markovian switching T-S fuzzy complex dynamical networks with discontinuous dynamic nodes[J]. AIMS Mathematics, 2022, 7(7): 11942-11971. doi: 10.3934/math.2022666

Related Papers:

Abstract

In this paper, cluster synchronization in finite/fixed time for semi-Markovian switching complex dynamical networks (CDNs) with discontinuous dynamic nodes is studied. Firstly, the global fixed-time convergence principle of nonlinear systems with semi-Markovian switching is developed. Secondly, the novel state-feedback controllers, which include discontinuous factors and integral terms, are designed to achieve the global stochastic finite/fixed-time cluster synchronization. In the framework of Filippov stochastic differential inclusion, the Lyapunov-Krasovskii functional approach, Takagi-Sugeno(T-S) fuzzy theory, stochastic analysis theory, and inequality analysis techniques are applied, and the global stochastic finite/fixed time synchronization conditions are proposed in the form of linear matrix inequalities (LMIs). Moreover, the upper bound of the stochastic settling time is explicitly proposed. In addition, the correlations among the obtained results are interpreted analytically. Finally, two numerical examples are given to illustrate the correctness of the theoretical results.

References

[1]	A. Bergman, M. Siegal, Evolutionary capacitance as a general feature of complex gene networks, Nature, 424 (2003), 549–552. https://doi.org/10.1038/nature01765 doi: 10.1038/nature01765
[2]	B. Huberman, L. Aadmic, Internet: Growth dynamics of the World-Wide Web, Nature, 401 (1999), 131.
[3]	Y. Zhang, H. Wu, J. Cao, Global Mittag-Leffler consensus for fractional singularly perturbed multiagent systems with discontinuous inherent dynamics via event-triggered control strategy, J. Frankl. Inst., 358 (2021), 2086–2114. https://doi.org/10.1016/j.jfranklin.2020.12.033 doi: 10.1016/j.jfranklin.2020.12.033
[4]	X. Peng, H. Wu, J. Cao, Global nonfragile synchronization in finite time for fractional-order discontinuous neural networks with nonlinear growth activations, IEEE Trans. Neural Netw. Learn. Syst., 30 (2019), 2123–2137. https://doi.org/10.1109/TNNLS.2018.2876726 doi: 10.1109/TNNLS.2018.2876726
[5]	M. Rosenblum, A. Pikovsky, J. Kurths, From phase to lag synchronization in coupled chaotic oscillators, Phys. Rev. Lett., 78 (1997), 4193–4196. https://doi.org/10.1103/PhysRevLett.78.4193 doi: 10.1103/PhysRevLett.78.4193
[6]	R. Li, H. Wu, J. Cao, Impulsive exponential synchronization of fractional-order complex dynamical networks with derivative couplings via feedback control based on discrete time state observations, Acta Mathematica Scientia., 42B (2022), 737–754. https://doi.org/10.1007/s10473-022-0219-4 doi: 10.1007/s10473-022-0219-4
[7]	Z. Sun, G. Si, F. Min, Adaptive modified function projective synchronization and parameter identification of uncertain hyperchaotic chaotic systems with identical or non-identical structures, Nonlinear Dyn., 68 (2018), 471–486. https://doi.org/10.1093/occmed/kqy104 doi: 10.1093/occmed/kqy104
[8]	X. Peng, H. Wu, Non-fragile robust finite-time stabilization and $H_{\infty}$ performance analysis for fractional-order delayed neural networks with discontinuous activations under the asynchronous switching, Neural. Comput. Appl., 32 (2020), 4045–4071. https://doi.org/10.1007/s00521-020-04887-7 doi: 10.1007/s00521-020-04887-7
[9]	W. Zhao, H. Wu, Fixed-time synchronization of semi-Markovian jumping neural networks with time-varying delays, Adv. Differ. Equ., 37 (2018), 256–268. https://doi.org/10.1186/s13662-018-1557-3 doi: 10.1186/s13662-018-1557-3
[10]	W. Zhang, C. Li, H. Li, X. Yang, Cluster stochastic synchronization of complex dynamical networks via fixed-time control scheme, Neural Networks, 124 (2020), 12–19. https://doi.org/10.1016/j.neunet.2019.12.019 doi: 10.1016/j.neunet.2019.12.019
[11]	H. Lu, Y. Hu, C. Guo, W. Zhou, Cluster synchronization for a class of complex dynamical network system with randomly occurring coupling delays via an improved event-triggered pinning control approach, J Franklin Inst., 357 (2020), 2167–2184. https://doi.org/10.1016/j.jfranklin.2019.11.076 doi: 10.1016/j.jfranklin.2019.11.076
[12]	M. Shen, D. Ye, S. Fei, Robust $H_{\infty}$ static output control of discrete Markov jump linear systems with norm bounded uncertainties, IET Control. Theory Appl., 8 (2014), 1449–1455. https://doi.org/10.1049/iet-cta.2013.1123 doi: 10.1049/iet-cta.2013.1123
[13]	L. Wu, X. Su, P. Shi, Output feedback control of Markovian jump repeated scalar nonlinear systems, IEEE Trans. Automat. Contr., 59 (2014), 199–204. https://doi.org/10.1109/TAC.2013.2267353 doi: 10.1109/TAC.2013.2267353
[14]	C. Morais, M. Braga, R. Oliveira, P. Peres, $H_{\infty}$ and $H_{2}$ control design for polytopic continuous-time Markov jump linear systems with uncertain transition rates, Int. J. Robust Nonlinear Control, 26 (2016), 599–612. https://doi.org/10.1002/rnc.3329 doi: 10.1002/rnc.3329
[15]	L. Zhang, $H_{\infty}$ estimation for piecewise homogeneous Markov jump linear systems, Automatica, 45 (2009), 2570–2576.
[16]	X. Wang, H. Wu, J. Cao, Global leader-following consensus in finite time for fractional-order multiagent systems with discontinuous inherent dynamics subject to nonlinear growth, Nonlinear Anal.: Hybrid Syst., 37 (2020), 100888. https://doi.org/10.1016/j.nahs.2020.100888 doi: 10.1016/j.nahs.2020.100888
[17]	Q. Yang, H. Wu, J. Cao, Global cluster synchronization in finite time for complex dynamical networks with hybrid couplings via aperiodically intermittent control, Optim. Control Appl. Methods, 41 (2020), 1097–1117. https://doi.org/10.1002/oca.2589 doi: 10.1002/oca.2589
[18]	Q. Gan, F. Xiao, Y. Qin, Fixed-time cluster synchronization of discontinuous directed community networks via periodically or aperiodically switching control, IEEE Access, 7 (2019), 83306–83318. https://doi.org/10.1109/ACCESS.2019.2924661 doi: 10.1109/ACCESS.2019.2924661
[19]	W. Zhang, C. Li, H. Li, Cluster stochastic synchronization of complex dynamical networks via fixed-time control scheme, Neural Networks, 124 (2020), 12–19. https://doi.org/10.1016/j.neunet.2019.12.019 doi: 10.1016/j.neunet.2019.12.019
[20]	Q. He, Y. Ma, Quantized adaptive pinning control for fixed/preassigned-time cluster synchronization of multi-weighted complex networks with stochastic disturbances, Neural Networks, Nonlinear Anal.: Hybrid Syst., 44 (2022), 101157124.
[21]	O. Costa, M. Fragoso, M. Todorov, Continuous-time Markovian jump linear systems, Springer-Verlag, Society for Industrial and Applied Mathematics, 44 (2006), 801–815. https://doi.org/10.1137/S0363012903436259 doi: 10.1137/S0363012903436259
[22]	Z. Shu, J. Lam, J. Xiong, Static output-feedback stabilization of discrete-time Markovian jump linear systems: a system augmentation approach, Automatica, 46 (2010), 687–694. https://doi.org/10.1016/j.automatica.2010.02.001 doi: 10.1016/j.automatica.2010.02.001
[23]	A. Fioravanti, A. Goncalves, J. Geromel, Discrete-time $H_{\infty}$ output feedback for Markov jump systems with uncertain transition probabilities, nt. J. Robust Nonlinear Control, 23 (2013), 894–902. https://doi.org/10.1016/j.sbi.2013.07.006 doi: 10.1016/j.sbi.2013.07.006
[24]	M. Shen, D. Ye, S. Fei, Robust $H_{\infty}$ static output control of discrete Markov jump linear systems with norm bounded uncertainties, IET Control. Theory Appl., 8 (2014), 1449–1455. https://doi.org/10.1049/iet-cta.2013.1123 doi: 10.1049/iet-cta.2013.1123
[25]	Y. Zhang, H. Wu, J. Cao, Group consensus in finite time for fractional multiagent systems with discontinuous inherent dynamics subject to Hölder growth, IEEE Transactions on Cybernetics, doi.org/10.1109/TCYB.2020.3023704
[26]	C. Morais, M. Braga, R. Oliveira, P. Peres, $H_{\infty}$ and $H_{2}$ control design for polytopic continuoustime Markov jump linear systems with uncertain transition rates, Int. J. Robust Nonlinear Control, 26 (2016), 599–612. https://doi.org/10.1002/rnc.3329 doi: 10.1002/rnc.3329
[27]	L. Zhang, $H_{\infty}$ estimation for piecewise homogeneous Markov jump linear systems, Automatica, 45 (2009), 2570–2576.
[28]	M. Shen, S. Yan, G. Zhang, J. Park, Finite-time $H_{\infty}$ static output control of Markov jump systems with an auxiliary approach, Appl. Math. Comput., 273 (2016), 553–561. https://doi.org/10.1016/j.amc.2015.10.038 doi: 10.1016/j.amc.2015.10.038
[29]	E.K. Boukas, Static output feedback control for stochastic hybrid systems: LMI approach, Automatica, 42 (2006), 183–188. https://doi.org/10.1016/j.automatica.2005.08.012 doi: 10.1016/j.automatica.2005.08.012
[30]	J. Zhang, Y. Xia, E.K. Boukas, New approach to $H_{\infty}$ control for Markovian jump singular systems, IET Control. Theory Appl., 4 (2010), 2273–2284. https://doi.org/10.1049/iet-cta.2009.0231 doi: 10.1049/iet-cta.2009.0231
[31]	Y. Zhang, C. Liu, X. Mu, Robust finite-time $H_{\infty}$ control of singular stochastic systems via static output feedback, Appl. Math. Comput., 218 (2012), 5629–5640. https://doi.org/10.1016/j.amc.2011.11.057 doi: 10.1016/j.amc.2011.11.057
[32]	Z. Wu, H. Su, J. Chu, Output feedback stabilization for discrete singular systems with random abrupt changes, Int. J. Robust Nonlinear Control, 20 (2010), 1945–1957. https://doi.org/10.1002/rnc.1560 doi: 10.1002/rnc.1560
[33]	R. Sakthivel, M. Joby, K. Mathiyalagan, S. Santra, Mixed $H_{\infty}$ and passive control for singular Markovian jump systems with time delays, J Franklin Inst., 352 (2015), 4446–4466. https://doi.org/10.1016/j.jfranklin.2015.06.008 doi: 10.1016/j.jfranklin.2015.06.008
[34]	F. Li, L. Wu, P. Shi, C. Lim, State estimation and sliding mode control for semi-Markovian jump systems with mismatched uncertainties, Automatica, 51 (2015), 385–393. https://doi.org/10.1016/j.automatica.2014.10.065 doi: 10.1016/j.automatica.2014.10.065
[35]	Y. Wei, J. Park, J. Qiu, Sliding mode control for semi-markovian jump systems via output feedback, Automatica, 81 (2017), 133–141. https://doi.org/10.1016/j.automatica.2017.03.032 doi: 10.1016/j.automatica.2017.03.032
[36]	H. Shen, J. Park, Z. Wu, Z. Zhang, Finite-time $H_{\infty}$ synchronization for complex networks with semi-Markov jump topology, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 40–51.
[37]	C. Zheng, S. Liu, H. Meng, Event-triggered synchronization for semi-Markov jump complex dynamic networks with time-varying delay, Neurocomputing, 458 (2021), 390–402. https://doi.org/10.1016/j.neucom.2021.06.022 doi: 10.1016/j.neucom.2021.06.022
[38]	M. Bucolo, S. Fazzino, M. L. Rosa, L. Fortuna, Small-world networks of fuzzy chaotic oscillators, Chaos Solitons Fractals, 17 (2003), 557–565. https://doi.org/10.1016/S0960-0779(02)00398-3 doi: 10.1016/S0960-0779(02)00398-3
[39]	L. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst., 3 (1973), 28–44. https://doi.org/10.1109/TSMC.1973.5408575 doi: 10.1109/TSMC.1973.5408575
[40]	T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, it IEEE Trans. Syst., 15 (1985), 116–132. https://doi.org/10.1109/TSMC.1985.6313399
[41]	H. Gao, T. Chen, Stabilization of nonlinear systems under variable sampling: A fuzzy control approach, IEEE Trans. Fuzzy Syst., 15 (2007), 972–983. https://doi.org/10.1109/TFUZZ.2006.890660 doi: 10.1109/TFUZZ.2006.890660
[42]	H. Dong, Z. Wang. Z, J. Lam, Fuzzy model based robust fault detection with stochastic mixed time delays and successive packet dropouts, IEEE Trans. Fuzzy Syst., 42 (2012), 365. https://doi.org/10.1109/TSMCB.2011.2163797 doi: 10.1109/TSMCB.2011.2163797
[43]	H. Li, H. Liu, H. Gao, P. Shi, Reliable fuzzy control for active suspension systems with actuator delay and fault, IEEE Trans. Fuzzy Syst., 20 (2012), 342–357. https://doi.org/10.1109/TFUZZ.2011.2174244 doi: 10.1109/TFUZZ.2011.2174244
[44]	X. Zhu, B. Chen, D. Yue, Y. Wang, An improved input delay approach to stabilization of fuzzy systems under variable sampling, IEEE Trans. Fuzzy Syst., 20 (2012), 342–357. https://doi.org/10.1109/TFUZZ.2011.2174244 doi: 10.1109/TFUZZ.2011.2174244
[45]	J. Qiu, G. Feng, H. Gao, Static-output-feedback $H_{\infty}$ control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions, IEEE Trans. Fuzzy Syst., 21 (2013), 245–261. https://doi.org/10.1109/TFUZZ.2012.2210555 doi: 10.1109/TFUZZ.2012.2210555
[46]	H. Shen, L. Su, J. H. Park, Reliable mixed $H_{\infty}$ passive control for T-S fuzzy delayed systems based on a semi-Markov jump model approach, IEEE Trans. Fuzzy Syst., 314 (2017), 79–98. https://doi.org/10.1016/j.fss.2016.09.007 doi: 10.1016/j.fss.2016.09.007
[47]	Y. Tang, J. Fang, M. Xia, X. Gu, Synchronization of Takagi Sugeno fuzzy stochastic discrete-time complex networks with mixed time-varying delays, Appl. Math. Model., 34(2010), 843–855.
[48]	J. Liu, D. Yue, Asymptotic and robust stability of T-S fuzzy genetic regulatory networks with time-varying delays, Int. J. Robust Nonlinear Control, 22 (2012), 827–840. https://doi.org/10.1002/rnc.1729 doi: 10.1002/rnc.1729
[49]	J. Tranthi, T. Botmart, W. Weera, New results on robust exponential stability of Takagi-Sugeno fuzzy for neutral differential systems with mixed time-varying delays, Math. Comput. Simul., doi.org/10.1016/j.matcom.2021.09.018
[50]	C. Chen, L. Li, H. Peng, A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks, Neural networks, 123 (2020), 412–419. https://doi.org/10.1016/j.neunet.2019.12.028 doi: 10.1016/j.neunet.2019.12.028
[51]	F. Kong, Q. Zhu, R. Sakthivel, Finite-time and fixed-time synchronization control of fuzzy CohenGrossberg neural networks, Fuzzy Sets. Syst., 394 (2020), 87–109. https://doi.org/10.1016/j.fss.2019.12.002 doi: 10.1016/j.fss.2019.12.002
[52]	S. Yang, C. Li, T. Huang, Fixed-time consensus of complex dynamical networks with nonlinear coupling and state-dependent uncertainties, Fuzzy Sets. Syst., 394 (2019), 81–97. https://doi.org/10.1016/j.fss.2018.05.005 doi: 10.1016/j.fss.2018.05.005
[53]	Z. Wang, H. Wu, Global synchronization in fixed time for semi-Markovian switching complex dynamical networks with hybrid couplings and time-varying delays, Nonlinear Dyn., 95 (2019), 2031–2062.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)