Abstract
In this paper, a new robust adaptive finite-time synchronization method has been proposed for chaotic systems by considering model uncertainties and external disturbances. The finite-time synchronization between master and slave systems is achieved under terminal sliding mode control method. The most important innovation of this paper is designing of an adaptive law to estimate the unknown upper bound of the disturbance and uncertainty. To handle the disturbances/uncertainties with unknown upper bounds and get finite-time synchronization, the adaptive law is combined with terminal sliding mode control method. Numerical simulation results and comparative studies show the effectiveness of proposed method.
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Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics Reports, 469(3), 93–153. https://doi.org/10.1016/j.physrep.2008.09.002.
Behjameh, M. R., Delavari, H., & Vali, A. (2014). Global finite time synchronization of two nonlinear chaotic gyros using high order sliding mode control. Journal of Applied and Computational Mechanics, 1(1), 26–34. https://doi.org/10.22055/jacm.2014.10549.
Belykh, I., Jeter, R., & Belykh, V. (2017). Foot force models of crowd dynamics on a wobbly bridge. Science advances, 3(11), e1701512. https://doi.org/10.1126/sciadv.1701512.
Bhat, S. P., & Bernstein, D. S. (1997, June). Finite-time stability of homogeneous systems. In Proceedings of the 1997 American control conference (Cat. No. 97CH36041) (Vol. 4, pp. 2513–2514). IEEE. https://doi.org/10.1109/acc.1997.609245.
Bhat, S. P., & Bernstein, D. S. (2005). Geometric homogeneity with applications to finite time stability. Mathematics of Control, Signals, and Systems, 17(2), 101–127. https://doi.org/10.1007/s00498-005-0151-x.
Buscarino, A., Fortuna, L., Frasca, M., & Valentina, G. L. (2012). A chaotic circuit based on Hewlett-Packard memristor. Chaos An Interdisciplinary Journal of Nonlinear Science, 22(2), 023136. https://doi.org/10.1063/1.4729135.
Chen, G., & Lai, D. (1998). Feedback anti-control of discrete chaos. International Journal of Bifurcation and Chaos, 8(07), 1585–1590. https://doi.org/10.1142/S0218127498001236.
Chen, G., Mao, Y., & Chui, C. K. (2004). A symmetric image encryption scheme based on 3D chaotic cat maps. Chaos, Solitons and Fractals, 21(3), 749–761. https://doi.org/10.1016/j.chaos.2003.12.022.
Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and chaos, 9(07), 1465–1466. https://doi.org/10.1142/S0218127499001024.
Chenarani, H., & Binazadeh, T. (2017). Flexible structure control of unmatched uncertain nonlinear systems via passivity-based sliding mode technique. Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 41(1), 1–11. https://doi.org/10.1007/s40998-017-0012-x.
Du, H., Li, S., & Qian, C. (2011). Finite time attitude tracking control of spacecraft with application to attitude synchronization. IEEE Transactions on Automatic Control, 56(11), 2711–2717. https://doi.org/10.1109/TAC.2011.2159419.
Eckhardt, B., Ott, E., Strogatz, S. H., Abrams, D. M., & McRobie, A. (2007). Modeling walker synchronization on the Millennium Bridge. Physical Review E, 75(2), 021110. https://doi.org/10.1103/PhysRevE.75.021110.
Fang, L., Li, T., Li, Z., & Li, R. (2013). Adaptive terminal sliding mode control for anti-synchronization of uncertain chaotic systems. Nonlinear Dynamics, 74(4), 991–1002. https://doi.org/10.1007/s11071-013-1017-2.
Guan, Z. H., & Liu, N. (2010). Generating chaos for discrete time-delayed systems via impulsive control. Chaos An Interdisciplinary Journal of Nonlinear Science, 20(1), 013135. https://doi.org/10.1063/1.3266929.
Haimo, V. T. (1986). Finite time controllers. SIAM Journal on Control and Optimization, 24(4), 760–770. https://doi.org/10.1137/0324047.
Hirosawa, K., Kittaka, S., Oishi, Y., Kannari, F., & Yanagisawa, T. (2013). Phase locking in a Nd: YVO 4 waveguide laser array using Talbot cavity. Optics Express, 21(21), 24952–24961. https://doi.org/10.1364/OE.21.024952.
Karimi, H. R. (2011). Robust synchronization and fault detection of uncertain master-slave systems with mixed time-varying delays and nonlinear perturbations. International Journal of Control, Automation and Systems, 9(4), 671. https://doi.org/10.1007/s12555-011-0408-8.
Karimi, H. R. (2012). A sliding mode approach to H∞ synchronization of master–slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties. Journal of the Franklin Institute, 349(4), 1480–1496. https://doi.org/10.1016/j.jfranklin.2011.09.015.
Kengne, R., Tchitnga, R., Mezatio, A., Fomethe, A., & Litak, G. (2017). Finite time synchronization of fractional-order simplest two-component chaotic oscillators. The European Physical Journal B, 90(5), 88. https://doi.org/10.1140/epjb/e2017-70470-8.
Kiliç, R., Saraçoğlu, Ö. G., & Yildirim, F. (2006). A new nonautonomous version of Chua’s circuit: Experimental observations. Journal of the Franklin Institute, 343(2), 191–203. https://doi.org/10.1016/j.jfranklin.2005.12.001.
Li, C., & Zhang, J. (2016). Synchronisation of a fractional-order chaotic system using finite time input-to-state stability. International Journal of Systems Science, 47(10), 2440–2448. https://doi.org/10.1080/00207721.2014.998741.
Liu, Y. (2012). Analysis of global dynamics in an unusual 3D chaotic system. Nonlinear Dynamics, 70(3), 2203–2212. https://doi.org/10.1007/s11071-012-0610-0.
Liu, N., & Guan, Z. H. (2011). Chaotification for a class of cellular neural networks with distributed delays. Physics Letters A, 375(3), 463–467. https://doi.org/10.1016/j.physleta.2010.07.039.
Liu, Y., Li, L., & Feng, Y. (2016). Finite time synchronization for high-dimensional chaotic systems and its application to secure communication. Journal of Computational and Nonlinear Dynamics, 11(5), 051028. https://doi.org/10.1115/1.4033686.
Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141. https://doi.org/10.1175/1520-0469(1963)020%3c0130:DNF%3e2.0.CO;2.
Lü, J., Chen, G., Cheng, D., & Celikovsky, S. (2002). Bridge the gap between the Lorenz system and the Chen system. International Journal of Bifurcation and Chaos, 12(12), 2917–2926. https://doi.org/10.1142/S021812740200631X.
Mobayen, S. (2016). Finite-time stabilization of a class of chaotic systems with matched and unmatched uncertainties: An LMI approach. Complexity, 21(5), 14–19. https://doi.org/10.1002/cplx.21624.
Mohammadpour, S., & Binazadeh, T. (2018). Robust finite time synchronization of uncertain chaotic systems: Application on Duffing-Holmes system and chaos gyros. Systems Science and Control Engineering, 6(1), 28–36. https://doi.org/10.1080/21642583.2018.1428695.
Néda, Z., Ravasz, E., Brechet, Y., Vicsek, T., & Barabási, A. L. (2000). The sound of many hands clapping. Nature, 403(6772), 849–850. https://doi.org/10.1038/35002660.
Oliva, R. A., & Strogatz, S. H. (2001). Dynamics of a large array of globally coupled lasers with distributed frequencies. International journal of Bifurcation and Chaos, 11(09), 2359–2374. https://doi.org/10.1142/S0218127401003450.
Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64(8), 821. https://doi.org/10.1103/PhysRevLett.64.821.
Pecora, L. M., Carroll, T. L., Johnson, G. A., Mar, D. J., & Heagy, J. F. (1997). Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos An Interdisciplinary Journal of Nonlinear Science, 7(4), 520–543. https://doi.org/10.1063/1.166278.
Pikovsky, A., & Rosenblum, M. (2001). Comment on “Intermittency in chaotic rotations”. Physical Review E, 64(5), 058203. https://doi.org/10.1103/PhysRevE.64.058203.
Ren, H., Karimi, H. R., Lu, R., & Wu, Y. (2019). Synchronization of network systems via aperiodic sampled-data control with constant delay and application to unmanned ground vehicles. IEEE Transactions on Industrial Electronics, 67(6), 4980–4990. https://doi.org/10.1109/TIE.2019.2928241.
Rong, C. G., & Xiaoning, D. (1998). From chaos to order: Methodologies, perspectives and applications (Vol. 24). World Scientific.
Rössler, O. E. (1976). An equation for continuous chaos. Physics Letters A, 57(5), 397–398. https://doi.org/10.1016/0375-9601(76)90101-8.
Rössler, O. E. (1979). Continuous chaos—four prototype equations. Annals of the New York Academy of Sciences, 316(1), 376–392. https://doi.org/10.1111/j.1749-6632.1979.tb29482.x.
Schiff, S. J., Jerger, K., Duong, D. H., Chang, T., Spano, M. L., & Ditto, W. L. (1994). Controlling chaos in the brain. Nature, 370(6491), 615. https://doi.org/10.1038/370615a0.
Singh, P. P., Singh, J. P., & Roy, B. K. (2014). Synchronization and anti-synchronization of Lu and Bhalekar-Gejji chaotic systems using nonlinear active control. Chaos, Solitons and Fractals, 69, 31–39. https://doi.org/10.1016/j.chaos.2014.09.005.
Strogatz, S. H. (2001). Exploring complex networks. Nature, 410(6825), 268. https://doi.org/10.1038/35065725.
Strogatz, S. (2003). Sync: The emerging science of spontaneous. New York: Order.
Strogatz, S. H., Abrams, D. M., McRobie, A., Eckhardt, B., & Ott, E. (2005). Crowd synchrony on the Millennium Bridge. Nature, 438(7064), 43–44. https://doi.org/10.1038/438043a.
Sun, Y., Wu, X., Bai, L., Wei, Z., & Sun, G. (2016). Finite time synchronization control and parameter identification of uncertain permanent magnet synchronous motor. Neurocomputing, 207, 511–518. https://doi.org/10.1016/j.neucom.2016.05.036.
Wah, W. C. (2007). Synchronization in complex networks of nonlinear dynamical systems. World Scientific.
Wang, X. F., & Chen, G. (2002). Synchronization in small-world dynamical networks. International Journal of Bifurcation and chaos, 12(01), 187–192. https://doi.org/10.1142/S0218127402004292.
Wang, H., Han, Z. Z., Xie, Q. Y., & Zhang, W. (2009). Finite time synchronization of uncertain unified chaotic systems based on CLF. Nonlinear Analysis Real World Applications, 10(5), 2842–2849. https://doi.org/10.1016/j.nonrwa.2008.08.010.
Wang, S., Kuang, J., Li, J., Luo, Y., Lu, H., & Hu, G. (2002). Chaos-based secure communications in a large community. Physical Review E, 66(6), 065202. https://doi.org/10.1103/PhysRevE.66.065202.
Wang, J., Tang, W. S., & Zhang, J. X. (2012). Approximate synchronization of two non-linear systems via impulsive control. Proceedings of the Institution of Mechanical Engineers, Part I Journal of Systems and Control Engineering, 226(3), 338–347. https://doi.org/10.1177/0959651811420329.
Wiesenfeld, K., Colet, P., & Strogatz, S. H. (1996). Synchronization transitions in a disordered Josephson series array. Physical Review Letters, 76(3), 404. https://doi.org/10.1103/PhysRevLett.76.404.
Winful, H. G., & Rahman, L. (1990). Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers. Physical Review Letters, 65(13), 1575. https://doi.org/10.1103/PhysRevLett.65.1575.
Wu, X., Lu, J. A., Chi, K. T., Wang, J., & Liu, J. (2007). Impulsive control and synchronization of the Lorenz systems family. Chaos, Solitons and Fractals, 31(3), 631–638. https://doi.org/10.1016/j.chaos.2005.10.017.
Wu, Y., Lu, Y., He, S., & Lu, R. (2019). Synchronization control for unreliable network systems in intelligent robots. IEEE/ASME Transactions on Mechatronics, 24(6), 2641–2651. https://doi.org/10.1109/TMECH.2019.2939416.
Yousefi, M., & Binazadeh, T. (2018). Delay-independent sliding mode control of time-delay linear fractional order systems. Transactions of the Institute of Measurement and Control, 40(4), 1212–1222. https://doi.org/10.1177/0142331216678059.
Yu, W. (2010). Finite time stabilization of three-dimensional chaotic systems based on CLF. Physics Letters A, 374(30), 3021–3024. https://doi.org/10.1016/j.physleta.2010.05.040.
Zhang, Q., Lu, J., Lu, J., & Chi, K. T. (2008). Adaptive feedback synchronization of a general complex dynamical network with delayed nodes. IEEE Transactions on Circuits and Systems II: Express Briefs, 55(2), 183–187. https://doi.org/10.1109/TCSII.2007.911813.
Zhang, W., Wong, K. W., Yu, H., & Zhu, Z. L. (2013). An image encryption scheme using reverse 2-dimensional chaotic map and dependent diffusion. Communications in Nonlinear Science and Numerical Simulation, 18(8), 2066–2080. https://doi.org/10.1016/j.cnsns.2012.12.012.
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Li, W., Bai, G. & Imani Marrani, H. A New Robust Finite-Time Synchronization and Anti-Synchronization Method for Uncertain Chaotic Systems by Using Adaptive Estimator and Terminal Sliding Mode Approaches. J Control Autom Electr Syst 31, 1375–1385 (2020). https://doi.org/10.1007/s40313-020-00650-4
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DOI: https://doi.org/10.1007/s40313-020-00650-4