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doi: 10.3934/dcdss.2020357

Fault-tolerant anti-synchronization control for chaotic switched neural networks with time delay and reaction diffusion

Jianping Zhou 1, , Yamin Liu 1, , Ju H. Park 2,, , Qingkai Kong 2, and  Zhen Wang 3,

1. 

School of Computer Science and Technology, Anhui University of Technology, Ma'anshan 243032, China
2. 

Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea
3. 

College of Mathematics and Systems Science, Shandong University of Science & Technology, Qingdao 266590, China

* Corresponding author: Ju H. Park

Received  August 2019 Revised  December 2019 Published  May 2020

Fund Project: The first author is supported by Open Project of Anhui Province Key Laboratory of Special and Heavy Load Robot under Grant TZJQR005-2020 and the Excellent Youth Talent Support Program of Universities in Anhui Province under Grant GXYQZD2019021

This paper is concerned with the issue of fault-tolerant anti-synchro-nization control for chaotic switched neural networks with time delay and reaction-diffusion terms under the drive-response scheme, where the response system is assumed to be disturbed by stochastic noise. Both arbitrary switching signal and average dwell-time limited switching signal are taken into account. With the aid of the Lyapunov-Krasovskii functional approach and combining with the generalized Itô formula, sufficient conditions on the mean-square exponential stability for the anti-synchronization error system are presented. Then, by utilizing some decoupling methods, constructive design strategies on the desired fault-tolerant anti-synchronization controller are proposed. Finally, an example is given to demonstrate the effectiveness of our design strategies.

Keywords: Time delay, reaction diffusion, neural networks, fault-tolerant control, anti-synchronization.
Mathematics Subject Classification: Primary: 92B20, 34D06; Secondary: 34H10.
Citation: Jianping Zhou, Yamin Liu, Ju H. Park, Qingkai Kong, Zhen Wang. Fault-tolerant anti-synchronization control for chaotic switched neural networks with time delay and reaction diffusion. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020357
References:
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A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 91-118.  doi: 10.3934/dcdss.2015.8.91.  Google Scholar

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C. K. Ahn, Adaptive $H_{\infty}$ anti-synchronization for time-delayed chaotic neural networks, Prog. Theoretical Phys., 122 (2009), 1391-1403.  doi: 10.1143/PTP.122.1391.  Google Scholar

[3]

J. Cao, R. Rakkiyappan, K. Maheswari and A. Chandrasekar, Exponential $H_{\infty}$ filtering analysis for discrete-time switched neural networks with random delays using sojourn probabilities, Sci. China Technol. Sci., 59 (2016), 387-402.  doi: 10.1007/s11431-016-6006-5.  Google Scholar

[4]

X. Chang, R. Liu and J. H. Park, A further study on output feedback $H_{\infty}$ control for discrete-time systems, IEEE Trans. Circuits Systems II: Express Briefs, 67 (2020), 305-309.  doi: 10.1109/TCSII.2019.2904320.  Google Scholar

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N. D. Cong and T. S. Doan, On integral separation of bounded linear random differential equations, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 995-1007.  doi: 10.3934/dcdss.2016038.  Google Scholar

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Y. Fan, X. Huang, Y. Li, J. Xia and G. Chen, Aperiodically intermittent control for quasi-synchronization of delayed memristive neural networks: An interval matrix and matrix measure combined method, IEEE Trans. Systems Man Cybernetics: Systems, 49 (2019), 2254-2265.  doi: 10.1109/TSMC.2018.2850157.  Google Scholar

[7]

Y. Fan, X. Huang, H. Shen and J. Cao, Switching event-triggered control for global stabilization of delayed memristive neural networks: An exponential attenuation scheme, Neural Networks, 117 (2019), 216-224.  doi: 10.1016/j.neunet.2019.05.014.  Google Scholar

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J. Hou, Y. Huang and S. Ren, Anti-synchronization analysis and pinning control of multi-weighted coupled neural networks with and without reaction-diffusion terms, Neurocomputing, 330 (2019), 78-93.  doi: 10.1016/j.neucom.2018.10.079.  Google Scholar

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Y.-L. Huang, S.-Y. Ren, J. Wu and B.-B. Xu, Passivity and synchronization of switched coupled reaction-diffusion neural networks with non-delayed and delayed couplings, Int. J. Comput. Math., 96 (2019), 1702-1722.  doi: 10.1080/00207160.2018.1463437.  Google Scholar

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T. Jiao, J. H. Park, G. Zong, Y. Zhao and Q. Du, On stability analysis of random impulsive and switching neural networks, Neurocomputing, 350 (2019), 146-154.  doi: 10.1016/j.neucom.2019.03.039.  Google Scholar

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R. Konnur, Synchronization-based approach for estimating all model parameters of chaotic systems, Phys. Rev. E, 67 (2003), 1387-1396.  doi: 10.1103/PhysRevE.67.027204.  Google Scholar

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T. H. Lee, C. P. Lim, S. Nahavandi and J. H. Park, Network-based synchronization of T-S fuzzy chaotic systems with asynchronous samplings, J. Franklin Inst., 355 (2018), 5736-5758.  doi: 10.1016/j.jfranklin.2018.05.023.  Google Scholar

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[17]

Y. Liu, J. H. Park and F. Fang, Global exponential stability of delayed neural networks based on a new integral inequality, IEEE Trans. Systems Man Cybernetics: Systems, 49 (2019), 2318-2325.  doi: 10.1109/TSMC.2018.2815560.  Google Scholar

[18]

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[19]

Q. Ma, S. Xu, Y. Zou and G. Shi, Synchronization of stochastic chaotic neural networks with reaction-diffusion terms, Nonlinear Dynam., 67 (2012), 2183-2196.  doi: 10.1007/s11071-011-0138-8.  Google Scholar

[20]

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[21]

S. Nakata, T. Miyata, N. Ojima and K. Yoshikawa, Self-synchronization in coupled salt-water oscillators, Phys. D, 115 (1998), 313-320.  doi: 10.1016/S0167-2789(97)00240-6.  Google Scholar

[22]

N. Ozcan, M. S. Ali, J. Yogambigai, Q. 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.  Google Scholar

[23]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.  Google Scholar

[24]

F. Ren and J. Cao, Anti-synchronization of stochastic perturbed delayed chaotic neural networks, Neural Comput. Appl., 18 (2009), 515-521.  doi: 10.1007/s00521-009-0251-5.  Google Scholar

[25]

I. Stamova, T. Stamov and X. Li, Global exponential stability of a class of impulsive cellular neural networks with supremums, Internat. J. Adapt. Control Signal Process., 28 (2014), 1227-1239.  doi: 10.1002/acs.2440.  Google Scholar

[26]

V. Sundarapandian and R. Karthikeyan, Anti-synchronization of Lü and Pan chaotic systems by adaptive nonlinear control, European J. Sci. Res., 64 (2011), 94-106.   Google Scholar

[27]

W. Tai, Q. Teng, Y. Zhou, J. Zhou and Z. Wang, Chaos synchronization of stochastic reaction-diffusion time-delay neural networks via non-fragile output-feedback control, Appl. Math. Comput., 354 (2019), 115-127.  doi: 10.1016/j.amc.2019.02.028.  Google Scholar

[28]

Z. Wang, L. Li, Y. Li and Z. Cheng, Stability and Hopf bifurcation of a three-neuron network with multiple discrete and distributed delays, Neural Process. Lett., 48 (2018), 1481-1502.  doi: 10.1007/s11063-017-9754-8.  Google Scholar

[29]

I. Wedekind and U. Parlitz, Synchronization and antisynchronization of chaotic power drop-outs and jump-ups of coupled semiconductor lasers, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.026218.  Google Scholar

[30]

J. Xia, G. Chen and W. Sun, Extended dissipative analysis of generalized Markovian switching neural networks with two delay components, Neurocomputing, 260 (2017), 275-283.  doi: 10.1016/j.neucom.2017.05.005.  Google Scholar

[31]

Z. Yan, X. Huang and J. Cao, Variable-sampling-period dependent global stabilization of delayed memristive neural networks via refined switching event-triggered control, SCIENCE CHINA Information Sciences, in progress. doi: 10.1007/s11432-019-2664-7.  Google Scholar

[32]

D. Ye and G. Yang, Adaptive fault-tolerant tracking control against actuator faults with application to flight control, IEEE Trans. Control Systems Tech, 14 (2006), 1088-1096.  doi: 10.1109/TCST.2006.883191.  Google Scholar

[33]

E. Yucel, M. S. Ali, N. Gunasekaran and S. Arik, Sampled-data filtering of Takagi–Sugeno fuzzy neural networks with interval time-varying delays, Fuzzy Sets and Systems, 316 (2017), 69-81.  doi: 10.1016/j.fss.2016.04.014.  Google Scholar

[34]

X. Zhang, X. Lv and X. Li, Sampled-data-based lag synchronization of chaotic delayed neural networks with impulsive control, Nonlinear Dynam., 90 (2017), 2199-2207.  doi: 10.1007/s11071-017-3795-4.  Google Scholar

[35]

W. Zhang, S. Yang, C. Li and Z. Li, Finite-time and fixed-time synchronization of complex networks with discontinuous nodes via quantized control, Neural Process. Lett., 50 (2019), 2073-2086.  doi: 10.1007/s11063-019-09985-9.  Google Scholar

[36]

D. Zhang, L. Yu, Q. G. Wang and C. J. Ong, Estimator design for discrete-time switched neural networks with asynchronous switching and time-varying delay, IEEE Trans. Neural Networks Learning Systems, 23 (2012), 827-834.  doi: 10.1109/TNNLS.2012.2186824.  Google Scholar

[37]

J. Zhou, Y. Wang, X. Zheng, Z. Wang and H. Shen, Weighted $H_{\infty}$ consensus design for stochastic multi-agent systems subject to external disturbances and ADT switching topologies, Nonlinear Dyn., 96 (2019), 853-868.  doi: 10.1007/s11071-019-04826-9.  Google Scholar

[38]

Y. Zhou, J. Xia, H. Shen, J. Zhou and Z. Wang, Extended dissipative learning of time-delay recurrent neural networks, J. Franklin Inst., 356 (2019), 8745-8769.  doi: 10.1016/j.jfranklin.2019.08.003.  Google Scholar

[39]

J. Zhou, S. Xu, H. Shen and B. Zhang, Passivity analysis for uncertain BAM neural networks with time delays and reaction-diffusions, Internat. J. Systems Sci., 44 (2013), 1494-1503.  doi: 10.1080/00207721.2012.659693.  Google Scholar

[40]

K. Zhou and P. P. Khargonekar, Robust stabilization of linear systems with norm-bounded time-varying uncertainty, Systems Control Lett., 10 (1988), 17-20.  doi: 10.1016/0167-6911(88)90034-5.  Google Scholar

[41]

J. Zhou, J. H. Park and Q. Ma, Non-fragile observer-based $H_{\infty}$ control for stochastic time-delay systems, Appl. Math. Comput., 291 (2016), 69-83.  doi: 10.1016/j.amc.2016.06.024.  Google Scholar

[42]

G. Zhuang, Q. Ma, B. Zhang, S. Xu and J. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time delays, Systems Control Lett., 114 (2018), 1-10.  doi: 10.1016/j.sysconle.2018.02.004.  Google Scholar

show all references

References:
[1]

A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 91-118.  doi: 10.3934/dcdss.2015.8.91.  Google Scholar

[2]

C. K. Ahn, Adaptive $H_{\infty}$ anti-synchronization for time-delayed chaotic neural networks, Prog. Theoretical Phys., 122 (2009), 1391-1403.  doi: 10.1143/PTP.122.1391.  Google Scholar

[3]

J. Cao, R. Rakkiyappan, K. Maheswari and A. Chandrasekar, Exponential $H_{\infty}$ filtering analysis for discrete-time switched neural networks with random delays using sojourn probabilities, Sci. China Technol. Sci., 59 (2016), 387-402.  doi: 10.1007/s11431-016-6006-5.  Google Scholar

[4]

X. Chang, R. Liu and J. H. Park, A further study on output feedback $H_{\infty}$ control for discrete-time systems, IEEE Trans. Circuits Systems II: Express Briefs, 67 (2020), 305-309.  doi: 10.1109/TCSII.2019.2904320.  Google Scholar

[5]

N. D. Cong and T. S. Doan, On integral separation of bounded linear random differential equations, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 995-1007.  doi: 10.3934/dcdss.2016038.  Google Scholar

[6]

Y. Fan, X. Huang, Y. Li, J. Xia and G. Chen, Aperiodically intermittent control for quasi-synchronization of delayed memristive neural networks: An interval matrix and matrix measure combined method, IEEE Trans. Systems Man Cybernetics: Systems, 49 (2019), 2254-2265.  doi: 10.1109/TSMC.2018.2850157.  Google Scholar

[7]

Y. Fan, X. Huang, H. Shen and J. Cao, Switching event-triggered control for global stabilization of delayed memristive neural networks: An exponential attenuation scheme, Neural Networks, 117 (2019), 216-224.  doi: 10.1016/j.neunet.2019.05.014.  Google Scholar

[8]

J. Fell and N. Axmacher, The role of phase synchronization in memory processes, Nature Rev. Neurosci., 12 (2011), 105-118.  doi: 10.1038/nrn2979.  Google Scholar

[9]

J. P. Hespanha and A. S. Morse, Stability of switched systems with average dwell-time, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, 1999, 2655–2660. doi: 10.1109/CDC.1999.831330.  Google Scholar

[10]

J. Hou, Y. Huang and S. Ren, Anti-synchronization analysis and pinning control of multi-weighted coupled neural networks with and without reaction-diffusion terms, Neurocomputing, 330 (2019), 78-93.  doi: 10.1016/j.neucom.2018.10.079.  Google Scholar

[11]

Y.-L. Huang, S.-Y. Ren, J. Wu and B.-B. Xu, Passivity and synchronization of switched coupled reaction-diffusion neural networks with non-delayed and delayed couplings, Int. J. Comput. Math., 96 (2019), 1702-1722.  doi: 10.1080/00207160.2018.1463437.  Google Scholar

[12]

T. Jiao, J. H. Park, G. Zong, Y. Zhao and Q. Du, On stability analysis of random impulsive and switching neural networks, Neurocomputing, 350 (2019), 146-154.  doi: 10.1016/j.neucom.2019.03.039.  Google Scholar

[13]

R. Konnur, Synchronization-based approach for estimating all model parameters of chaotic systems, Phys. Rev. E, 67 (2003), 1387-1396.  doi: 10.1103/PhysRevE.67.027204.  Google Scholar

[14]

T. H. Lee, C. P. Lim, S. Nahavandi and J. H. Park, Network-based synchronization of T-S fuzzy chaotic systems with asynchronous samplings, J. Franklin Inst., 355 (2018), 5736-5758.  doi: 10.1016/j.jfranklin.2018.05.023.  Google Scholar

[15]

X. Li, M. Bohner and C. K. Wang, Impulsive differential equations: Periodic solutions and applications, Automatica J. IFAC, 52 (2015), 173-178.  doi: 10.1016/j.automatica.2014.11.009.  Google Scholar

[16]

T. L. Liao and N. S. Huang, An observer-based approach for chaotic synchronization with applications to secure communications, IEEE Trans. Circuits Systems I: Fundamental Theory Appl., 46 (1999), 1144-1151.  doi: 10.1109/81.788817.  Google Scholar

[17]

Y. Liu, J. H. Park and F. Fang, Global exponential stability of delayed neural networks based on a new integral inequality, IEEE Trans. Systems Man Cybernetics: Systems, 49 (2019), 2318-2325.  doi: 10.1109/TSMC.2018.2815560.  Google Scholar

[18]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[19]

Q. Ma, S. Xu, Y. Zou and G. Shi, Synchronization of stochastic chaotic neural networks with reaction-diffusion terms, Nonlinear Dynam., 67 (2012), 2183-2196.  doi: 10.1007/s11071-011-0138-8.  Google Scholar

[20]

X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Process. Appl., 65 (1996), 233-250.  doi: 10.1016/S0304-4149(96)00109-3.  Google Scholar

[21]

S. Nakata, T. Miyata, N. Ojima and K. Yoshikawa, Self-synchronization in coupled salt-water oscillators, Phys. D, 115 (1998), 313-320.  doi: 10.1016/S0167-2789(97)00240-6.  Google Scholar

[22]

N. Ozcan, M. S. Ali, J. Yogambigai, Q. 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.  Google Scholar

[23]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.  Google Scholar

[24]

F. Ren and J. Cao, Anti-synchronization of stochastic perturbed delayed chaotic neural networks, Neural Comput. Appl., 18 (2009), 515-521.  doi: 10.1007/s00521-009-0251-5.  Google Scholar

[25]

I. Stamova, T. Stamov and X. Li, Global exponential stability of a class of impulsive cellular neural networks with supremums, Internat. J. Adapt. Control Signal Process., 28 (2014), 1227-1239.  doi: 10.1002/acs.2440.  Google Scholar

[26]

V. Sundarapandian and R. Karthikeyan, Anti-synchronization of Lü and Pan chaotic systems by adaptive nonlinear control, European J. Sci. Res., 64 (2011), 94-106.   Google Scholar

[27]

W. Tai, Q. Teng, Y. Zhou, J. Zhou and Z. Wang, Chaos synchronization of stochastic reaction-diffusion time-delay neural networks via non-fragile output-feedback control, Appl. Math. Comput., 354 (2019), 115-127.  doi: 10.1016/j.amc.2019.02.028.  Google Scholar

[28]

Z. Wang, L. Li, Y. Li and Z. Cheng, Stability and Hopf bifurcation of a three-neuron network with multiple discrete and distributed delays, Neural Process. Lett., 48 (2018), 1481-1502.  doi: 10.1007/s11063-017-9754-8.  Google Scholar

[29]

I. Wedekind and U. Parlitz, Synchronization and antisynchronization of chaotic power drop-outs and jump-ups of coupled semiconductor lasers, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.026218.  Google Scholar

[30]

J. Xia, G. Chen and W. Sun, Extended dissipative analysis of generalized Markovian switching neural networks with two delay components, Neurocomputing, 260 (2017), 275-283.  doi: 10.1016/j.neucom.2017.05.005.  Google Scholar

[31]

Z. Yan, X. Huang and J. Cao, Variable-sampling-period dependent global stabilization of delayed memristive neural networks via refined switching event-triggered control, SCIENCE CHINA Information Sciences, in progress. doi: 10.1007/s11432-019-2664-7.  Google Scholar

[32]

D. Ye and G. Yang, Adaptive fault-tolerant tracking control against actuator faults with application to flight control, IEEE Trans. Control Systems Tech, 14 (2006), 1088-1096.  doi: 10.1109/TCST.2006.883191.  Google Scholar

[33]

E. Yucel, M. S. Ali, N. Gunasekaran and S. Arik, Sampled-data filtering of Takagi–Sugeno fuzzy neural networks with interval time-varying delays, Fuzzy Sets and Systems, 316 (2017), 69-81.  doi: 10.1016/j.fss.2016.04.014.  Google Scholar

[34]

X. Zhang, X. Lv and X. Li, Sampled-data-based lag synchronization of chaotic delayed neural networks with impulsive control, Nonlinear Dynam., 90 (2017), 2199-2207.  doi: 10.1007/s11071-017-3795-4.  Google Scholar

[35]

W. Zhang, S. Yang, C. Li and Z. Li, Finite-time and fixed-time synchronization of complex networks with discontinuous nodes via quantized control, Neural Process. Lett., 50 (2019), 2073-2086.  doi: 10.1007/s11063-019-09985-9.  Google Scholar

[36]

D. Zhang, L. Yu, Q. G. Wang and C. J. Ong, Estimator design for discrete-time switched neural networks with asynchronous switching and time-varying delay, IEEE Trans. Neural Networks Learning Systems, 23 (2012), 827-834.  doi: 10.1109/TNNLS.2012.2186824.  Google Scholar

[37]

J. Zhou, Y. Wang, X. Zheng, Z. Wang and H. Shen, Weighted $H_{\infty}$ consensus design for stochastic multi-agent systems subject to external disturbances and ADT switching topologies, Nonlinear Dyn., 96 (2019), 853-868.  doi: 10.1007/s11071-019-04826-9.  Google Scholar

[38]

Y. Zhou, J. Xia, H. Shen, J. Zhou and Z. Wang, Extended dissipative learning of time-delay recurrent neural networks, J. Franklin Inst., 356 (2019), 8745-8769.  doi: 10.1016/j.jfranklin.2019.08.003.  Google Scholar

[39]

J. Zhou, S. Xu, H. Shen and B. Zhang, Passivity analysis for uncertain BAM neural networks with time delays and reaction-diffusions, Internat. J. Systems Sci., 44 (2013), 1494-1503.  doi: 10.1080/00207721.2012.659693.  Google Scholar

[40]

K. Zhou and P. P. Khargonekar, Robust stabilization of linear systems with norm-bounded time-varying uncertainty, Systems Control Lett., 10 (1988), 17-20.  doi: 10.1016/0167-6911(88)90034-5.  Google Scholar

[41]

J. Zhou, J. H. Park and Q. Ma, Non-fragile observer-based $H_{\infty}$ control for stochastic time-delay systems, Appl. Math. Comput., 291 (2016), 69-83.  doi: 10.1016/j.amc.2016.06.024.  Google Scholar

[42]

G. Zhuang, Q. Ma, B. Zhang, S. Xu and J. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time delays, Systems Control Lett., 114 (2018), 1-10.  doi: 10.1016/j.sysconle.2018.02.004.  Google Scholar

Figure 1.  ADT switching signal
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Figure 2.  Phase plane plot of the drive system at $ x = 0.5 $
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Figure 3.  State evolution of the unforced drive-response systems at $ x = 0.5 $
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Figure 4.  State evolution of the unforced error system
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Figure 5.  State evolution of the controlled drive-response systems at $ x = 0.5 $
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Figure 6.  State evolution of the error system under control
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