Abstract
The master-slave synchronization (MSS) issue of chaotic Lur’e systems (LSs) with time-varying delay (TVD) is explored in this work by means of feedback control. After taking the nonlinear function into full consideration, a creative Lyapunov-Krasovskii functional (LKF) is attained. Then novel criteria which ensure the synchronization of the studied systems are obtained with the aid of the improved inequality technique and the reciprocally convex combination method to cope with the derivative of LKF. The effectiveness and virtues of the synchronization conditions are testified via the notable Chua’s circuit in three different cases.
Similar content being viewed by others
References
T. Wu, L. L. Xiong, J. D. Cao, and H. Y. Zhang, “Stochastic stability and extended dissipativity analysis for uncertain neutral systems with semi-Markovian jumping parameters via novel free matrix-based integral inequality,” International Journal of Robust and Nonlinear Control, vol. 29, no. 9, pp. 2525–2545, February 2019.
A. Seuret and F. Gouaisbaut, “Wirtinger-based integral inequality: Application to time-delay systems,” Automatica, vol. 49, no. 9, pp. 2860–2866, September 2013.
H. B. Zeng, Y. He, M. Wu, and J. H. She, “Free-matrix-based integral inequality for stability analysis of systems with time-varying delay,” IEEE Transactions on Automatic Control, vol. 60, no. 10, pp. 2768–2772, February 2015.
L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, February 1990.
H. Mkaouar and O. Boubaker, “Chaos synchronization for master slave piecewise linear systems: Application to Chua’s circuit,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1292–1302, March 2012.
Q. Wang and D. L. Qi, “Synchronization for fractional order chaotic systems with uncertain parameters,” International Journal of Control, Automation, and Systems, vol. 14, no. 1, pp. 211–216, February 2016.
T. L. Liao and S. H. Tsai, “Adaptive synchronization of chaotic systems and its application to secure communications,” Chaos, Solitons and Fractals, vol. 11, no. 9, pp. 1387–1396, July 2000.
Y. Tao and L. O. Chua, “Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 44, no. 10, pp. 976–988, October 1997.
J. Wang, K. Liang, X. Huang, Z. Wang, and H. Shen, “Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback,” Applied Mathematics and Computation, vol. 328, pp. 247–262, July 2018.
C. J. Cheng, T. L. Liao, and C. C. Hwang, “Exponential synchronization of a class of chaotic neural networks,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 197–206, April 2005.
X. P. Zhang, X. H. Zhang, D. Li, and D. Yang, “Adaptive synchronization for a class of fractional order time-delay uncertain chaotic systems via fuzzy fractional order neural network,” International Journal of Control, Automation, and Systems, vol. 17, no. 5, pp. 1209–1220, May 2019.
T. Li, S. M. Fei, Q. Zhu, and S. Cong, “Exponential synchronization of chaotic neural networks with mixed delays,” Neurocomputing, vol. 71, no. 13–15, pp. 3005–3019, February 2008.
W. W. Yu and J. D. Cao, “Synchronization control of stochastic delayed neural networks,” Physica A: Statistical Mechanics and its Applications, vol. 373, pp. 252–260, January 2007.
C. Ge, C. C. Hua, and X. P. Guan, “Master-slave synchronization criteria of Lur’e systems with time-delay feedback control,” Applied Mathematics and Computation, vol. 244, pp. 895–902, October 2014.
S. H. Lee, M. J. Park, O. M. Kwon, and P. Selvaraj, “Improved synchronization criteria for chaotic neural networks with sampled-data control subject to actuator saturation,” International Journal of Control, Automation, and Systems, vol. 17, no. 9, pp. 2430–2440, May 2019.
B. Liu and X. C. Jia, “New absolute stability criteria for uncertain Lur’e systems with time-varying delays,” Journal of the Franklin Institute, vol. 355, no. 9, pp. 4015–4031, April 2018.
W. H. Chen, D. Wei, and X. M. Lu, “Global exponential synchronization of nonlinear time-delay Lur’e systems via delayed impulsive control,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 9, pp. 3298–3312, September 2014.
C. K. Zhang, L. Jiang, Y. He, Q. H. Wu, and M. Wu, “Asymptotical synchronization for chaotic Lur’e systems using sampled-data control,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 10, pp. 2743–2751, October 2013.
S. H. He, Y. Q. Wu, and Y. Z. Li, “Asymptotic stability of master-slave systems using sampled-data controller with time-delay,” International Journal of Control, Automation, and Systems, vol. 17, no. 6, pp. 1473–1482, May 2019.
C. Ge, W. W. Zhang, W. Li, and X. C. Sun, “Improved stability criteria for synchronization of chaotic Lur’e systems using sampled-data control,” Neurocomputing, vol. 151, part 1, pp. 215–222, March 2015.
H. Zhang, R. Y. Ye, J. D. Cao, and A. Alsaedi, “Delay-independent stability of Riemann-Liouville fractional neutral-type delayed neural networks,” Neural Processing Letters, vol. 47, no. 2, pp. 427–442, April 2018.
T. Li, G. B. Zhang, S. M. Fei, and S. J. Zhang, “Further criteria on master-slave synchronizationin chaotic Lur’e systems using delay feedback control,” Circuits Systems and Signal Processing, vol. 35, no. 8, pp. 2992–3014, August 2016.
H. Zhang, R. Y. Ye, J. D. Cao, A. Alsaedi, X. D. Li, and Y. Wan, “Lyapunov functional approach to stability analysis of Riemann-Liouville fractional neural networks with time-varying delays,” Asian Journal of Control, vol. 20, no. 6, pp. 1–14, November 2018.
W. H. Chen, Z. P. Wang, and X. M. Lu, “On sampled-data control for master-slave synchronization of chaotic Lur’e systems,” IEEE Transactions on Circuits and Systems, vol. 59, no. 8, pp. 515–519, August 2012.
Q. H. Fu, J. Y. Cai, S. M. Zhong, and Y. B. Yu, “Pinning impulsive synchronization of stochastic memristor-based neural networks with time-varying delays,” International Journal of Control, Automation, and Systems, vol. 17, no. 1, pp. 243–252, January 2019.
W. I. Lee and P. G. Park, “Second-order reciprocally convex approach to stability of systems with interval timevarying delays,” Applied Mathematics and Computation, vol. 229, pp. 245–253, February 2014.
T. Li, J. J. Yu, and Z. Wang, “Delay-range-dependent synchronization criterion for Lur’e systems with delay feedback control,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1796–1803, May 2009.
X. F. Ji, M. W. Ren, and H. Y. Su, “On designing time-delay feedback control for master-slave synchronization of Lur’e system,” Asian Journal of Control, vol. 16, no. 1, pp. 308–312, January 2014.
Q. L. Han, “New delay-dependent synchronization criteria for Lur’e systems using time delay feedback control,” Physics Letters A, vol. 360, no. 4–5, pp. 563–569, January 2007.
T. Li, A. G. Song, and S. M. Fei, “Master-slave synchronization for delayed Lur’e systems using time-delay feedback control,” Asian Journal of Control, vol. 13, no. 6, pp. 879–892, November 2011.
Q. L. Han, “On designing time-varying delay feedback controllers for master-slave synchronization of Lur’e systems,” IEEE Transactions on Circuits and Systems, vol. 54, no. 7, pp. 1573–1583, July 2007.
Y. He, G. L. Wen, and Q. G. Wang, “Delay-dependent synchronization criterion for Lur’e systems with delay feedback control,” International Journal of Bifurcation and Chaos, vol. 16, no. 10, pp. 3087–3091, October 2006.
Y. M. Wang, L. L. Xiong, X. Z. Liu, and H. Y. Zhang, “New delay-dependent synchronization criteria for uncertain Lur’e systems via time-varying delayed feedback control,” Journal of Nonlinear Sciences and Applications, vol. 10, pp. 1927–1940, April 2017.
K. B. Shi, X. Z. Liu, H. Zhu, S. M. Zhong, Y. Zeng, and C. Yin, “Novel delay-dependent master-slave synchronization criteria of chaotic Lur’e systems with time-varying-delay feedback control,” Applied Mathematics and Computation, vol. 282, pp. 137–154, May 2016.
H. M. Zhang, J. D. Cao, and L. L. Xiong, “Novel synchronization conditions for time-varying delayed Lur’e systems with parametric uncertainty,” Applied Mathematics and Computation, vol. 350, pp. 224–236, June 2019.
M. E. Yalcin, J. A. K. Suykens, and J. Vandewalle, “Master-slave synchronization of Lur’e systems with time-delay,” International Journal of Bifurcation and Chaos, vol. 11, no. 6, pp. 1707–1722, June 2001.
J. Xiang, Y. J. Li and W. Wei, “An improved condition for master-slave synchronization of Lur’e systems with time-delay,” Physics Letters A, vol. 362, no. 2–3, pp. 154–158, July 2006.
P. G. Park, J. W. Ko, and C. K. Jeong, “Reciprocally convex approach to stability of systems with time-delay delays,” Automatica, vol. 47, pp. no. 1, 235–238, January 2011.
K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Birkhauser, Basel, January 2003.
C. K. Zhang, Y. He, L. Jiang, and M. Wu, “Notes on stability of time-delay systems: Bounding inequalities and augmented Lyapunov-Krasovskii functions,” IEEE Transactions on Automatic Control, vol. 62, no. 10, pp. 5331–5336, October 2017.
N. Liu, J. Fang, W. Deng, Z. J. Wu, and G. Q. Ding, “Synchronization for a class of fractional-order linear complex networks via impulsive control,” International Journal of Control, Automation, and Systems, vol. 16, no. 6, pp. 2839–2844, October 2018.
P. G. Park, W. I. Lee, and S. Y. Lee, “Auxiliary function-based intergal inequalities for quadratic functions and their applicatons to time-delay systems,” Journal of the Franklin Institute, vol. 352, no. 4, pp. 1378–1396, January 2015.
J. Chen, S. Y. Xu, W. M. Chen, B. Y. Zhang, Q. Ma, and Y. Zou, “Two general integral inequalities and their applications to stability analysis for systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 26, no. 18, pp. 4088–4103, December 2016.
Y. Z. Zhu, W. X. Zheng, and D. H. Zhou, “Quasi-synchronization of discrete-time Lur’e-type switched systems with parameter mismatches and relaxed PDT constraints,” IEEE Transactions on Cybernetics, vol. 50, no. 5, pp. 2026–2037, 2020.
X. M. Zhang, Q. L. Han, and Z. G. Zeng, “Hierarchical type stability criteria for delayed neural networks via canonical Bessel-Legendre inequalities,” IEEE Transactions on Cybernetics, vol. 48, no. 5, pp. 1660–1671, May 2018.
D. Zhang, Y. P. Shen, S. Q. Zhou, X. W. Dong, and L. Yu, “Distributed secure platoon control of connected vehicles subject to DoS attack: Theory and application,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020. DOI: https://doi.org/10.1109/TSMC.2020.2968606
D. Zhang, Z. H. Xu, G. Feng, and H. Y. Li, “Asynchronous resilient output consensus of switched heterogeneous linear multivehicle systems with communication delay,” IEEE/ASME Transactions on Mechatronics, vol. 24, no. 6, pp. 2627–2640, December 2019.
Y. Wang, H. R. Karimi, H. K. Lam, and H. Yan, “Fuzzy output tracking control and filtering for nonlinear discrete-time descriptor systems under unreliable communication links,” IEEE Transactions on Cybernetics, vol. 50, no 6, pp. 2369–2379, 2020.
Y. Wang, B. Jiang, Z. Wu, S. Xie, and Y. Peng, “Adaptive sliding mode fault-tolerant fuzzy tracking control with application to unmanned marine vehicles,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020. DOI: https://doi.org/10.1109/TSMC.2020.2964808
Y. Wang, X. Yang, and H. Yan, “Reliable fuzzy tracking control of near-space hypersonic vehicle using aperiodic measurement information,” IEEE Transactions on Industrial Electronics, vol. 66, no. 12, pp. 9439–9447, December 2019.
D. Zhang and G. Feng, “A new switched system approach to leader-follower consensus of heterogeneous linear multiagent systems with DoS attack,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019. DOI: https://doi.org/10.1109/TSMC.2019.2895097
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Yajuan Liu under the direction of Editor Jessie (Ju H.) Park.
The authors thank the editors and the reviewers for their valuable comments of this paper.
This work was supported by the National Natural Science Foundation of China under Grant No.12061088.
Yefan Wu received her B.Sc. degree in mathematics and applied mathematics from Huaibei Normal University, Huaibei, China, in 2006. She received her M.Sc. in basic mathematics from Yunnan Minzu University, Yunnan, China, in 2020. Her research interests include nonlinear control and neutral dynamical systems.
Lianglin Xiong obtained his bachelor’s degree in mathematics and applied mathematics from Neijiang Normal University, Sichuan, China, in 2004. He got his Master and Doctoral degrees in applied mathematics from University of Electronic Science and Technology of China, Sichuan, China, in 2007 and 2009, respectively. He is a professor of the School of Mathematics and Computer Science, Yunnan Minzu University. He has reviewed for many journals, such as Journal of Computational and Applied Mathematics, Automatica, Journal of the Franklin Institute, International Journal of Systems Science, Neurocomputing, International Journal of Control, Automation, and Systems and so on. His research interests include stability analysis and controller design for functional differential systems such as neutral delayed systems, neural networks, Lur’e systems, switched systems, and Markovian jump systems.
Guisheng Zhai received his B.S. degree from Fudan University, China, in 1988, and received his M.E. and Ph.D. degrees, both in system science, from Kobe University, Japan, in 1993 and 1996, respectively. After two years of industrial experience, Dr. Zhai moved to Wakayama University, Japan, in 1998, and then to Osaka Prefecture University, Japan, in 2004. He held visiting professor positions at University of Notre Dame, Purdue University, Taiyuan University of Technology, Hubei University of Technology, and Fujian Normal University, etc. In April 2010, he joined Shibaura Institute of Technology, Japan, where he currently is a full Professor of Mathematical Sciences. His research interests include large scale and decentralized control systems, robust control, switched systems and switching control, networked control and multi-agent systems, neural networks and signal processing, etc. Dr. Zhai is on the editorial board of several academic journals including IET Control Theory and Applications, International Journal of Applied Mathematics and Computer Science, Journal of Control and Decision, Frontiers of Mechanical Engineering, Science Nature, Data Analytics and Applied Mathematics, etc. He is a Senior Member of IEEE, a member of ISCIE, SICE, JSST and JSME.
Tao Wu received his B.Sc. degree in mathematics and applied mathematics from Yunnan Minzu University, Yunnan, China, in 2016. He received his M.Sc. degree in basic mathematics from Yunnan Minzu University, Yunnan, China, in 2019. His research interests include nonlinear control, neutral dynamical systems, and complex networks.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wu, Y., Xiong, L., Zhai, G. et al. Improved Synchronization Analysis for Delayed Lur’e Systems Using Improved Technique. Int. J. Control Autom. Syst. 19, 1480–1490 (2021). https://doi.org/10.1007/s12555-020-0111-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-020-0111-8