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Model reference control of hyperchaotic systems. (English) Zbl 1263.93178

Summary: We apply a famous engineering method, called model reference control, to control hyperchaos. We have proposed a general description of the hyperchaotic system and its reference system. By using the Lyapunov stability theorem, we obtain the expression of the controller. Four examples for both certain case and the uncertain case show that our method is very effective for controlling hyperchaotic systems with certain parameters and uncertain parameters.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34H10 Chaos control for problems involving ordinary differential equations
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[1] C. Li, X. Liao, and K.-W. Wong, “Lag synchronization of hyperchaos with application to secure communications,” Chaos, Solitons & Fractals, vol. 23, no. 1, pp. 183-193, 2005. · Zbl 1068.94004
[2] T. Gao and Z. Chen, “A new image encryption algorithm based on hyper-chaos,” Physics Letters A, vol. 372, no. 4, pp. 394-400, 2008. · Zbl 1217.94096
[3] J. Peng, D. Zhang, and X. Liao, “A digital image encryption algorithm based on hyper-chaotic cellular neural network,” Fundamenta Informaticae, vol. 90, no. 3, pp. 269-282, 2009.
[4] Q. T. Yang and T. G. Gao, “One-way hash function based on hyper-chaotic cellular neural network,” Chinese Physics B, vol. 17, no. 7, pp. 2388-2393, 2008.
[5] M. Eiswirtha, T. Kruelb, G. Ertla, and F. W. Schneider, “Hyperchaos in a chemical reaction,” Chemical Physics Letters, vol. 193, no. 4, pp. 305-310, 1992.
[6] G. Gandhi, “An improved Chua’s circuit and its use in hyperchaotic circuit,” Analog Integrated Circuits and Signal Processing, vol. 46, no. 2, pp. 173-178, 2005.
[7] K. Thamilmaran, M. Lakshmanan, and A. Venkatesan, “Hyperchaos in a modified canonical Chua’s circuit,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 1, pp. 221-243, 2004. · Zbl 1067.94597
[8] J. M. V. Grzybowski, M. Rafikov, and J. M. Balthazar, “Synchronization of the unified chaotic system and application in secure communication,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 6, pp. 2793-2806, 2009. · Zbl 1221.94047
[9] H. Wang, Z. Z. Han, and Z. Mo, “Synchronization of hyperchaotic systems via linear control,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1910-1920, 2010. · Zbl 1222.93191
[10] M. Rafikov, J. M. Balthazar, and H. F. von Bremen, “Mathematical modeling and control of population systems: applications in biological pest control,” Applied Mathematics and Computation, vol. 200, no. 2, pp. 557-573, 2008. · Zbl 1139.92022
[11] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196-1199, 1990. · Zbl 0964.37501
[12] G. P. Jiang and W. X. Zheng, “An LMI criterion for linear-state-feedback based chaos synchronization of a class of chaotic systems,” Chaos, Solitons and Fractals, vol. 26, pp. 437-443, 2005. · Zbl 1153.93390
[13] J. H. Park, O. M. Kwon, and S. M. Lee, “LMI optimization approach to stabilization of Genesio-Tesi chaotic system via dynamic controller,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 200-206, 2008. · Zbl 1245.93115
[14] H. Salarieh and A. Alasty, “Control of stochastic chaos using sliding mode method,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 135-145, 2009. · Zbl 1162.65062
[15] R.-A. Tang, Y.-L. Liu, and J.-K. Xue, “An extended active control for chaos synchronization,” Physics Letters A, vol. 373, no. 16, pp. 1449-1454, 2009. · Zbl 1228.34078
[16] M. Rafikov and J. M. Balthazar, “Optimal linear and nonlinear control of design for chaotic systems,” in Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC ’05), pp. 24-28, Long Beach, Calif, USA, September 2005.
[17] M. Rafikov and J. M. Balthazar, “On an optimal control design for Rössler system,” Physics Letters A, vol. 333, no. 3-4, pp. 241-245, 2004. · Zbl 1123.49300
[18] C.-I. Mor\uarescu and B. Brogliato, “Passivity-based switching control of flexible-joint complementarity mechanical systems,” Automatica, vol. 46, no. 1, pp. 160-166, 2010. · Zbl 1214.93040
[19] D. H. Ji, J. H. Koo, S. C. Won, S. M. Lee, and J. H. Park, “Passivity-based control for Hopfield neural networks using convex representation,” Applied Mathematics and Computation, vol. 217, no. 13, pp. 6168-6175, 2011. · Zbl 1209.93056
[20] C. Yang, C. H. Tao, and P. Wang, “Comparison of feedback control methods for a hyperchaotic Lorenz system,” Physics Letters A, vol. 374, no. 5, pp. 729-732, 2010. · Zbl 1235.93111
[21] M. Rafikov and J. M. Balthazar, “On control and synchronization in chaotic and hyperchaotic systems via linear feedback control,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 7, pp. 1246-1255, 2008. · Zbl 1221.93230
[22] M. M. Al-Sawalha and M. S. M. Noorani, “Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 1036-1047, 2010. · Zbl 1221.93123
[23] H. Zhang, X. Ma k, M. Li, and J. Zou -l, “Controlling and tracking hyperchaotic Rössler system via active backstepping design,” Chaos, Solitons and Fractals, vol. 26, pp. 353-361, 2005. · Zbl 1153.93381
[24] Y. Li, X. Liu, and H. Zhang, “Dynamical analysis and impulsive control of a new hyperchaotic system,” Mathematical and Computer Modelling, vol. 42, no. 11-12, pp. 1359-1374, 2005. · Zbl 1121.37031
[25] A. Ekrekli and D. J. Brookfield, “The practical implementation of model reference robot control,” Mechatronics, vol. 7, pp. 549-564, 1997.
[26] N. Hovakimyan, R. Rysdyk, and A. J. Calise, “Dynamic neural networks for output feedback control,” in Proceedings of the 38th IEEE Conference on Decision and Control (CDC ’99), vol. 2, pp. 1685-1690, December 1999. · Zbl 0969.93014
[27] P. F. Zhao, C. Liu, and X. Feng, “Model reference control for an economic growth cycle model,” Journal of Applied Mathematics, vol. 2012, Article ID 384732, 13 pages, 2012. · Zbl 1251.91044
[28] S. N. Chow and Y. Li, “Model reference control for sirs models,” Discrete and Continuous Dynamical Systems, vol. 24, no. 3, pp. 675-697, 2009. · Zbl 1163.93325
[29] O. E. Rössler, “An equation for hyperchaos,” Physics Letters A, vol. 71, no. 2-3, pp. 155-157, 1979. · Zbl 0996.37502
[30] T. Gao, Z. Chen, Q. Gu, and Z. Yuan, “A new hyper-chaos generated from generalized Lorenz system via nonlinear feedback,” Chaos, Solitons and Fractals, vol. 35, pp. 390-397, 2008.
[31] J. H. Park, “Adaptive modified projective synchronization of a unified chaotic system with uncertain parameter,” Chaos Solitons and Fractals, vol. 34, pp. 1552-1559, 2007. · Zbl 1152.93407
[32] R. Z. Luo and Z. M. Wei, “Adaptive function projective synchronization of unified chaotic with uncertain parameters,” Chaos, Solitions and Fractals, vol. 42, pp. 1266-1272, 2009. · Zbl 1198.93018
[33] X. Chen and J. Lu, “Adaptive synchronization of different chaotic systems with fully unknown parameters,” Physics Letters A, vol. 364, no. 2, pp. 123-128, 2007. · Zbl 1203.93161
[34] C. Li, “Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 405-413, 2012. · Zbl 1239.93033
[35] W. L. Li, Z. H. Liu, and J. Miao, “Adaptive synchronization for a unified chaotic system with uncertainty,” Communication in Nonlinear Science and Numerical Simulation, vol. 15, pp. 3015-3021, 2010.
[36] H. Du, Q. Zeng, and C. Wang, “Function projective synchronization of different chaotic systems with uncertain parameters,” Physics Letters A, vol. 372, no. 33, pp. 5402-5410, 2008. · Zbl 1223.34077
[37] J. H. Park, “Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters,” Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 288-293, 2008. · Zbl 1137.93035
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