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

Jianping Zhou; Yamin Liu; Ju H. Park; Qingkai Kong; Zhen Wang

doi:10.3934/dcdss.2020357

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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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[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
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[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
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[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
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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
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[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

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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2018 Impact Factor: 0.545

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2018 Impact Factor: 0.545

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2018 Impact Factor: 0.545

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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