American Institute of Mathematical Sciences

Advanced
doi: 10.3934/era.2020104

Finite/fixed-time synchronization for complex networks via quantized adaptive control

Yu-Jing Shi , and  Yan Ma

Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China

* Corresponding author: Yu-Jing Shi

Received  May 2020 Revised  July 2020 Published  September 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant no. 61673141) and Natural Science Foundation of Heilongjiang Province (Grant no. A2018007)

In this paper, a unified theoretical method is presented to implement the finite/fixed-time synchronization control for complex networks with uncertain inner coupling. The quantized controller and the quantized adaptive controller are designed to reduce the control cost and save the channel resources, respectively. By means of the linear matrix inequalities technique, two sufficient conditions are proposed to guarantee that the synchronization error system of the complex networks is finite/fixed-time stable in virtue of the Lyapunov stability theory. Moreover, two types of setting time, which are dependent and independent on the initial values, are given respectively. Finally, the effectiveness of the control strategy is verified by a simulation example.

Keywords: Complex network, finite/fixed-time, synchronization, uncertain internal coupling, quantized adaptive control.
Mathematics Subject Classification: 94C30.
Citation: Yu-Jing Shi, Yan Ma. Finite/fixed-time synchronization for complex networks via quantized adaptive control. Electronic Research Archive, doi: 10.3934/era.2020104
References:
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H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar

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[26]

C. Xu, X. Yang, J. Lu, J. Feng, F.E. Alsaadi and T. Hayat, Finite-time synchronization of networks via quantized intermittent pinning control, IEEE Trans. Cybern., 48 (2018), 3021-3027.  doi: 10.1109/TCYB.2017.2749248.  Google Scholar

[27]

X. Yang, J. Cao, C. Xu and J. Feng, Finite-time stabilization of switched dynamical networks with quantized couplings via quantized controller, Sci. China Technol. Sci., 61 (2018), 299-308.  doi: 10.1007/s11431-016-9054-y.  Google Scholar

[28]

W. Yin, W. Yang and H. Liu, A neural network scheme for recovering scattering obstacles with limited phaseless far-field data, J. Comput. Phys., 417 (2020), 18pp. doi: 10.1016/j.jcp.2020.109594.  Google Scholar

[29]

D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 21pp. doi: 10.1088/1361-6420/aaccda.  Google Scholar

[30]

W. Zhang, H. Li, C. Li, Z. Li and X. Yang, Fixed-time synchronization criteria for complex networks via quantized pinning control, ISA Trans., 91 (2019), 151-156.  doi: 10.1016/j.isatra.2019.01.032.  Google Scholar

[31]

W. Zhang, X. Yang and C. Li, Fixed-time stochastic synchronization of complex networks via continuous control, IEEE Trans. Cybern., 49 (2019), 3099-3104.  doi: 10.1109/TCYB.2018.2839109.  Google Scholar

[32]

C. Zhou, W. Zhang, X. Yang, C. Xu and J. Feng, Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations, Neural Process. Lett., 46 (2017), 271-291.  doi: 10.1007/s11063-017-9590-x.  Google Scholar

show all references

References:
[1]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.  Google Scholar

[2]

L. O. Chua, M. Itoh, L. Kocarev and K. Eckert, Chaos synchronization in Chua's circuit, J. Circuits Systems Comput., 3 (1993), 93-108.  doi: 10.1142/S0218126693000071.  Google Scholar

[3]

Y. Deng, J. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.  Google Scholar

[4]

Y. Deng, H. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar

[5]

Z. Ding and Z. Li, Distributed adaptive consensus control of nonlinear output-feedback systems on directed graphs, Automatica J. IFAC, 72 (2016), 46-52.  doi: 10.1016/j.automatica.2016.05.014.  Google Scholar

[6]

N. Elia and S. K. Mitter, Stabilization of linear systems with limited information, IEEE Trans. Automat. Contr., 46 (2001), 1384-1400.  doi: 10.1109/9.948466.  Google Scholar

[7]

J. Feng, F. Yu and Y. Zhao, Exponential synchronization of nonlinearly coupled complex networks with hybrid time-varying delays via impulsive control, Nonlinear Dynam., 85 (2016), 621-632.  doi: 10.1007/s11071-016-2711-7.  Google Scholar

[8]

Q. Gan, F. Xiao, Y. Qin and J. Yang, Fixed-time cluster synchronization of discontinuous directed community networks via periodically or aperiodically switching control, IEEE Access, 7 (2019), 83306-83318.  doi: 10.1109/ACCESS.2019.2924661.  Google Scholar

[9]

X. Ge, F. Yang and Q.-L. Han, Distributed networked control systems: A brief overview, Inf. Sci., 380 (2017), 117-131.  doi: 10.1016/j.ins.2015.07.047.  Google Scholar

[10]

H. Hou, Q. 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.  Google Scholar

[11]

C.-C. Hwang, J.-Y. Hsieh and R.-S. Lin, A linear continuous feedback control of Chua's circuit, Chaos Solitons Fract., 8 (1997), 1507-1516.  doi: 10.1016/S0960-0779(96)00150-6.  Google Scholar

[12]

A. Khan and M. Shahzad, Synchronization of circular restricted three body problem with Lorenz hyper chaotic system using a robust adaptive sliding mode controller, Complexity, 18 (2013), 58-64.  doi: 10.1002/cplx.21459.  Google Scholar

[13]

Q. Li, J. Guo, C. Sun, Y. Wu and Z. Ding, Finite-time synchronization for a class of dynamical complex networks with nonidentical nodes and uncertain disturbance, J. Syst. Sci. Complex., 32 (2019), 818-834.  doi: 10.1007/s11424-018-8141-5.  Google Scholar

[14]

J. Li, H. Liu and S. Ma, Determining a random Schrödinger operator: Both potential and source are random, preprint, arXiv: 1906.01240. Google Scholar

[15]

J. Li, H. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[16]

Q. Li, B. Shen, J. Liang and H. Shu, Event-triggered synchronization control for complex networks with uncertain inner coupling, Int. J. Gen. Syst., 44 (2015), 212-225.  doi: 10.1080/03081079.2014.973725.  Google Scholar

[17]

X. Liu and T. Chen, Finite-time and fixed-time cluster synchronization with or without pinning control, IEEE Trans. Cybern., 48 (2018), 240-252.  doi: 10.1109/TCYB.2016.2630703.  Google Scholar

[18]

X. Liu and T. Chen, Cluster synchronization in directed networks via intermittent pinning control, IEEE Trans. Neural Netw., 22 (2011), 1009-1020.  doi: 10.1109/TNN.2011.2139224.  Google Scholar

[19]

X. Liu, D. W. C. Ho, Q. Song and J. Cao, Finite/fixed-time robust stabilization of switched discontinuous systems with disturbances, Nonlinear Dynam., 90 (2017), 2057-2068.  doi: 10.1007/s11071-017-3782-9.  Google Scholar

[20]

X. Liu, D. W. C. Ho, Q. Song and W. Xu, Finite/fixed-time pinning synchronization of complex networks with stochastic disturbances, IEEE Trans. Cybern., 49 (2019), 2398-2403.  doi: 10.1109/TCYB.2018.2821119.  Google Scholar

[21]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 10pp. doi: 10.1088/0266-5611/31/10/105005.  Google Scholar

[22]

H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[23]

A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Automat. Control, 57 (2012), 2106-2110.  doi: 10.1109/TAC.2011.2179869.  Google Scholar

[24]

S. Qiu, Y. Huang and S. Ren, Finite-time synchronization of multi-weighted complex dynamical networks with and without coupling delay, Neurocomputing, 275 (2018), 1250-1260.  doi: 10.1016/j.neucom.2017.09.073.  Google Scholar

[25]

J. Wang, T. Ru, J. Xia, Y. Wei and Z. Wang, Finite-time synchronization for complex dynamic networks with semi-Markov switching topologies: An $H_{\infty}$ event-triggered control scheme, Appl. Math. Comput., 356 (2019), 235-251.  doi: 10.1016/j.amc.2019.03.037.  Google Scholar

[26]

C. Xu, X. Yang, J. Lu, J. Feng, F.E. Alsaadi and T. Hayat, Finite-time synchronization of networks via quantized intermittent pinning control, IEEE Trans. Cybern., 48 (2018), 3021-3027.  doi: 10.1109/TCYB.2017.2749248.  Google Scholar

[27]

X. Yang, J. Cao, C. Xu and J. Feng, Finite-time stabilization of switched dynamical networks with quantized couplings via quantized controller, Sci. China Technol. Sci., 61 (2018), 299-308.  doi: 10.1007/s11431-016-9054-y.  Google Scholar

[28]

W. Yin, W. Yang and H. Liu, A neural network scheme for recovering scattering obstacles with limited phaseless far-field data, J. Comput. Phys., 417 (2020), 18pp. doi: 10.1016/j.jcp.2020.109594.  Google Scholar

[29]

D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 21pp. doi: 10.1088/1361-6420/aaccda.  Google Scholar

[30]

W. Zhang, H. Li, C. Li, Z. Li and X. Yang, Fixed-time synchronization criteria for complex networks via quantized pinning control, ISA Trans., 91 (2019), 151-156.  doi: 10.1016/j.isatra.2019.01.032.  Google Scholar

[31]

W. Zhang, X. Yang and C. Li, Fixed-time stochastic synchronization of complex networks via continuous control, IEEE Trans. Cybern., 49 (2019), 3099-3104.  doi: 10.1109/TCYB.2018.2839109.  Google Scholar

[32]

C. Zhou, W. Zhang, X. Yang, C. Xu and J. Feng, Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations, Neural Process. Lett., 46 (2017), 271-291.  doi: 10.1007/s11063-017-9590-x.  Google Scholar

Figure 1.  Chua's circuit
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Figure 2.  The trajectories of synchronization error without control
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Figure 3.  The trajectories of finite-time synchronization errors $ e_{i}(t) \;(i = 1, \cdots, 5) $ with adaptive control
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Figure 4.  The trajectories of fixed-time synchronization errors $ e_{i}(t)\; (i = 1, \cdots, 5) $ with adaptive control
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Figure 5.  The trajectory of adaptive parameter $ \xi_{i}(t)\; (i = 1, \cdots, 5) $ for finite-time synchronization
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Figure 6.  The trajectory of adaptive parameter $ \xi_{i}(t)\; (i = 1, \cdots, 5) $ for fixed-time synchronization
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