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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:
93B12Variable structure systems
34H10Chaos control (ODE)
34C28Complex behavior, chaotic systems (ODE)
34D06Synchronization
37D45Strange attractors, chaotic dynamics
93D15Stabilization of systems by feedback
References:
[1]Chen, G.; Dong, X.: From chaos to order: methodologies, perspectives and applications, (1998)
[2]Song, Q.; Cao, J.; Liu, F.: Synchronization of complex dynamical networks with nonidentical nodes, Phys. lett. A 374, 544-551 (2010)
[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 · doi:10.1016/j.automatica.2010.04.005
[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 · doi:10.1016/j.cnsns.2008.09.028
[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 · doi:10.1016/j.physleta.2007.05.030
[7]Wang, F.; Liu, C.: Synchronization of unified chaotic system based on passive control, Physica D 225, 55-60 (2007) · Zbl 1119.34332 · doi:10.1016/j.physd.2006.09.038
[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 · doi:10.1016/j.physleta.2008.05.047
[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 · doi:10.1016/j.nonrwa.2007.12.005
[10]Chang, W.: PID control for chaotic synchronization using particle swarm optimization, Chaos soliton fract. 39, 910-917 (2009) · Zbl 1197.93118 · doi:10.1016/j.chaos.2007.01.064
[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 · doi:10.1016/j.apm.2010.04.013
[12]Yau, H.; Shieh, C.: Chaos synchronization using fuzzy logic controller, Nonlinear anal. RWA 9, 1800-1810 (2008) · Zbl 1154.34334 · doi:10.1016/j.nonrwa.2007.05.009
[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)
[15]Chen, Y.; Hwang, R. R.; Chang, C.: Adaptive impulsive synchronization of uncertain chaotic systems, Phys. lett. A 374, 2254-2258 (2010)
[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 · doi:10.1016/j.physleta.2006.12.031
[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 · doi:10.1016/j.chaos.2008.09.031
[18]Yu, Y.; Zhang, S.: Adaptive backstepping synchronization of uncertain chaotic system, Chaos soliton fract. 21, 643-649 (2004) · Zbl 1062.34053 · doi:10.1016/j.chaos.2003.12.067
[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 · doi:10.1016/S0960-0779(00)00089-8
[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 · doi:10.1016/j.chaos.2005.09.025
[21]El-Gohary, A.: Optimal synchronization of Rössler system with complete uncertain parameters, Chaos soliton fract. 27, 345-355 (2006) · Zbl 1091.93025 · doi:10.1016/j.chaos.2005.03.043
[22]Lu, J.; Cao, J.: Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos 15 (2005) · Zbl 1144.37378 · doi:10.1063/1.2089207
[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 · doi:10.1016/j.physleta.2006.11.092
[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 · doi:10.1016/j.physleta.2005.10.033
[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 · doi:10.1016/j.physleta.2007.07.080
[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 · doi:10.1016/j.apm.2009.09.023
[27]Chen, H.: Stability criterion for synchronization of chaotic systems using linear feedback control, Phys. lett. A 372, 1841-1850 (2008) · Zbl 1220.93031 · doi:10.1016/j.physleta.2007.10.049
[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 · doi:10.1016/j.physleta.2008.08.022
[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 · doi:10.1016/j.physleta.2004.11.033
[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 · doi:10.1016/j.physleta.2009.03.057
[31]Yu, W.: Finite-time stabilization of three-dimensional chaotic systems based on CLF, Phys. lett. A 374, 3021-3024 (2010)
[32]Yang, X.; Cao, J.: Finite-time stochastic synchronization of complex networks, Appl. math. Model. 34, 3631-3641 (2010) · Zbl 1201.37118 · doi:10.1016/j.apm.2010.03.012
[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 · doi:10.1016/j.cnsns.2007.12.006
[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 · doi:10.1016/j.cnsns.2009.12.012
[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 · doi:10.1016/j.cnsns.2008.08.013
[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 · doi:10.1016/j.cnsns.2009.09.021
[37]Li, S.; Tian, Y.: Finite-time synchronization of chaotic systems, Chaos soliton fract. 15, 303-310 (2003) · Zbl 1038.37504 · doi:10.1016/S0960-0779(02)00100-5
[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 · doi:10.1016/j.nonrwa.2008.08.010
[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 · doi:10.1137/S0363012997321358
[41]Slotine, J.; Li, W.: Applied nonlinear control, (1991) · Zbl 0753.93036
[42]Yu, X.; Man, Z.: Multi-input uncertain linear systems with terminal sliding-mode control, Automatica 34, 389-392 (1998) · Zbl 0915.93012 · doi:10.1016/S0005-1098(97)00205-7
[43]Yu, X.; Man, Z.: Fast terminal sliding-mode control design for nonlinear dynamical systems, IEEE trans. Circuit syst. 49, 261-264 (2002)
[44]Liu, C.; Liu, T.; Liu, L.; Liu, K.: A new chaotic attractor, Chaos soliton fract. 22, 1031-1038 (2004)
[45]Lorenz, E.: Deterministic nonperiodic flow, J. atmos. Sci. 20, 130-141 (1963)
[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)