×

Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. (English) Zbl 1219.93023

Summary: The problem of finite-time chaos synchronization between two different chaotic systems with fully unknown parameters is investigated. First, a new nonsingular terminal sliding surface is introduced and its finite-time convergence to the zero equilibrium is proved. Then, appropriate adaptive laws are derived to tackle the unknown parameters of the systems. Afterwards, based on the adaptive laws and finite-time control idea, an adaptive sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time. It is mathematically proved that the introduced sliding mode technique has finite-time convergence and stability in both reaching and sliding mode phases. Finally, some numerical simulations are presented to demonstrate the applicability and effectiveness of the proposed technique.

MSC:

93B12 Variable structure systems
34H10 Chaos control for problems involving ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93D15 Stabilization of systems by feedback
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chen, G.; Dong, X., From Chaos to Order: Methodologies, Perspectives and Applications (1998), World Scientific: World Scientific Singapore
[2] Song, Q.; Cao, J.; Liu, F., Synchronization of complex dynamical networks with nonidentical nodes, Phys. Lett. A, 374, 544-551 (2010) · Zbl 1234.05218
[3] Cao, J.; Wang, Z.; Sun, Y., Synchronization in an array of linearly stochastically coupled networks with time delays, Physica A, 385, 718-728 (2007)
[4] Lu, J.; Ho, D. W.C.; Cao, J., A unified synchronization criterion for impulsive dynamical networks, Automatica, 46, 1215-1221 (2010) · Zbl 1194.93090
[5] Grzybowski, J. M.V.; Rafikov, M.; Balthazar, J. M., Synchronization of the unified chaotic system and application in secure communication, Commun. Nonlinear Sci. Numer. Simulat., 14, 2793-2806 (2009) · Zbl 1221.94047
[6] Wang, B.; Wen, G., On the synchronization of a class of chaotic systems based on backstepping method, Phys. Lett. A, 370, 35-39 (2007) · Zbl 1209.93108
[7] Wang, F.; Liu, C., Synchronization of unified chaotic system based on passive control, Physica D, 225, 55-60 (2007) · Zbl 1119.34332
[8] Lee, S. M.; Ji, D. H.; Park, J. H.; Won, S. C., \(H_∞\) synchronization of chaotic systems via dynamic feedback approach, Phys. Lett. A, 372, 4905-4912 (2008) · Zbl 1221.93087
[9] Lin, J.; Yan, J., Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller, Nonlinear Anal. RWA, 10, 1151-1159 (2009) · Zbl 1167.37329
[10] Chang, W., PID control for chaotic synchronization using particle swarm optimization, Chaos Soliton Fract., 39, 910-917 (2009) · Zbl 1197.93118
[11] Chen, Y.; Wu, X.; Gui, Z., Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control, Appl. Math. Model., 34, 4161-4170 (2010) · Zbl 1201.93045
[12] Yau, H.; Shieh, C., Chaos synchronization using fuzzy logic controller, Nonlinear Anal. RWA, 9, 1800-1810 (2008) · Zbl 1154.34334
[13] Pourmahmood, M.; Khanmohammadi, S.; Alizadeh, G., Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller, Commun. Nonlinear Sci. Numer. Simulat. (2010)
[14] Yan, J.; Hung, M.; Liao, T., Adaptive sliding mode control for synchronization of chaotic gyros with fully unknown parameters, J. Sound Vibr., 298, 298-306 (2006) · Zbl 1243.93097
[15] Chen, Y.; Hwang, R. R.; Chang, C., Adaptive impulsive synchronization of uncertain chaotic systems, Phys. Lett. A, 374, 2254-2258 (2010) · Zbl 1237.34099
[16] Wu, X.; Guan, Z.; Wu, Z.; Li, T., Chaos synchronization between Chen system and Genesio system, Phys. Lett. A, 364, 484-487 (2007) · Zbl 1203.37066
[17] Sharma, B. B.; Kar, I. N., Contraction theory based adaptive synchronization of chaotic systems, Chaos Soliton Fract., 41, 2437-2447 (2009) · Zbl 1198.93197
[18] Yu, Y.; Zhang, S., Adaptive backstepping synchronization of uncertain chaotic system, Chaos Soliton Fract., 21, 643-649 (2004) · Zbl 1062.34053
[19] Wang, C.; Ge, S. S., Adaptive synchronization of uncertain chaotic systems via bachstepping design, Chaos Soliton Fract., 12, 1199-1206 (2001) · Zbl 1015.37052
[20] El-Gohary, A.; Sarhan, A., Optimal control and synchronization of Lorenz system with complete unknown parameters, Chaos Soliton Fract., 30, 1122-1132 (2006) · Zbl 1142.93408
[21] El-Gohary, A., Optimal synchronization of Rössler system with complete uncertain parameters, Chaos Soliton Fract., 27, 345-355 (2006) · Zbl 1091.93025
[22] Lu, J.; Cao, J., Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos, 15 (2005), art. no. 043901 · Zbl 1144.37378
[23] Chen, X.; Lu, J., Adaptive synchronization of different chaotic systems with fully unknown parameters, Phys. Lett. A, 364, 123-128 (2007) · Zbl 1203.93161
[24] Zhang, H.; Huang, W.; Wang, Z.; Chai, T., Adaptive synchronization between two different chaotic systems with unknown parameters, Phys. Lett. A, 350, 363-366 (2006) · Zbl 1195.93121
[25] Zhang, G.; Liu, Z.; Zhang, J., Adaptive synchronization of a class of continuous chaotic systems with uncertain parameters, Phys. Lett. A, 372, 447-450 (2008) · Zbl 1217.37036
[26] Mu, X.; Pei, L., Synchronization of the near-identical chaotic systems with the unknown parameters, Appl. Math. Model., 34, 1788-1797 (2010) · Zbl 1193.37046
[27] Chen, H., Stability criterion for synchronization of chaotic systems using linear feedback control, Phys. Lett. A, 372, 1841-1850 (2008) · Zbl 1220.93031
[28] Zhang, L.; Huang, L.; Zhang, Z.; Wang, Z., Fuzzy adaptive synchronization of uncertain chaotic systems via delayed feedback control, Phys. Lett. A, 372, 6082-6086 (2008) · Zbl 1223.93050
[29] Kim, J.; Park, C.; Kim, E.; Park, M., Fuzzy adaptive synchronization of uncertain chaotic systems, Phys. Lett. A, 334, 295-305 (2005) · Zbl 1123.37307
[30] Hwang, E.; Hyun, C.; Kim, E.; Park, M., Fuzzy model based adaptive synchronization of uncertain chaotic systems: robust tracking control approach, Phys. Lett. A, 373, 1935-1939 (2009) · Zbl 1229.34080
[31] Yu, W., Finite-time stabilization of three-dimensional chaotic systems based on CLF, Phys. Lett. A, 374, 3021-3024 (2010) · Zbl 1237.34093
[32] Yang, X.; Cao, J., Finite-time stochastic synchronization of complex networks, Appl. Math. Model., 34, 3631-3641 (2010) · Zbl 1201.37118
[33] Wang, H.; Han, Z.; Xie, Qi.; Zhang, W., Sliding mode control for chaotic systems based on LMI, Commun. Nonlinear Sci Numer. Simulat., 14, 1410-1417 (2009) · Zbl 1221.93049
[34] Xiang, W.; Huangpu, Y., Second-order terminal sliding mode controller for a class of chaotic systems with unmatched uncertainties, Commun. Nonlinear Sci. Numer. Simulat., 15, 3241-3247 (2010) · Zbl 1222.93045
[35] Wang, H.; Han, Z.; Xie, Q.; Zhang, W., Finite-time chaos control via nonsingular terminal sliding mode control, Commun. Nonlinear Sci. Numer. Simulat., 14, 2728-2733 (2009) · Zbl 1221.37225
[36] Jianwen, F.; Ling, H.; Chen, X.; Austin, F.; Geng, W., Synchronizing the noise-perturbed Genesio chaotic system by sliding mode control, Commun. Nonlinear Sci. Numer. Simulat., 15, 2546-2551 (2010) · Zbl 1222.93121
[37] Li, S.; Tian, Y., Finite-time synchronization of chaotic systems, Chaos Soliton Fract., 15, 303-310 (2003) · Zbl 1038.37504
[38] Wang, H.; Han, Z.; Xie, Q.; Zhang, W., Finite-time synchronization of uncertain unified chaotic systems basd on CLF, Nonlinear Anal. RWA, 10, 2842-2849 (2009) · Zbl 1183.34072
[39] Wang, H.; Han, Z.; Xie, Q.; Zhang, W., Finite-time chaos synchronization of unified chaotic system with uncertain parameters, Commun. Nonlinear Sci. Numer. Simulat., 14, 2239-2247 (2009)
[40] Bhat, S. P.; Bernstein, D. S., Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38, 751-766 (2000) · Zbl 0945.34039
[41] Slotine, J.; Li, W., Applied Nonlinear Control (1991), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey
[42] Yu, X.; Man, Z., Multi-input uncertain linear systems with terminal sliding-mode control, Automatica, 34, 389-392 (1998) · Zbl 0915.93012
[43] Yu, X.; Man, Z., Fast terminal sliding-mode control design for nonlinear dynamical systems, IEEE Trans. Circuit Syst., 49, 261-264 (2002) · Zbl 1368.93213
[44] Liu, C.; Liu, T.; Liu, L.; Liu, K., A new chaotic attractor, Chaos Soliton Fract., 22, 1031-1038 (2004) · Zbl 1060.37027
[45] Lorenz, E., Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141 (1963) · Zbl 1417.37129
[46] Hemati, N., Starnge attractors in brushless dc motors, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., 41, 40-45 (1994)
[47] Gao, Y.; Chau, K. T., Design of permanent-magnets to avoid chaos in PM synchronous machines, IEEE Trans. Magnet., 39, 2995-2997 (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.