×

zbMATH — the first resource for mathematics

Reduced-order observer design for the synchronization of the generalized Lorenz chaotic systems. (English) Zbl 1250.34045
Summary: This paper is concerned with the design of the novel observer that synchronizes with the generalized Lorenz chaotic systems. A simple reduced-order observer scheme is proposed by including extra terms. The boundedness of the state variables in the original system is not necessary for the observer design. The prior knowledge on the derivative signal of the system output is not required. It is also shown that by adopting Lyapunov method and choosing appropriate design parameters, the synchronization (observation) errors can converge to zero exponentially. Finally, numerical examples are provided to illustrate the validity of the theoretical result.

MSC:
34D06 Synchronization of solutions to ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Pecora, L.M.; Caroll, T.L., Synchronization in chaotic systems, Physical review letters, 64, 821-824, (1990) · Zbl 0938.37019
[2] Zhang, Z.; Shen, H.; Li, J., Adaptive stabilization of uncertain unified chaotic systems with nonlinear input, Applied mathematics and computation, 218, 4260-4267, (2011) · Zbl 1245.65076
[3] Zhang, Z.; Wang, Y.; Du, Z., Adaptive synchronization of single-degree-of-freedom oscillators with unknown parameters, Applied mathematics and computation, 218, 6833-6840, (2012) · Zbl 1254.34082
[4] Park, J.H.; Ji, D.H.; Won, S.C.; Lee, S.M., \(H_\infty\) synchronization of time-delayed chaotic systems, Applied mathematics and computation, 204, 170-177, (2008) · Zbl 1152.93027
[5] Lee, S.M.; Kwon, O.M.; Park, J.H., Synchronization of chaotic lur’e systems with delayed feedback control using deadzone nonlinearity, Chinese physics B, 21, 010506, (2011)
[6] Ji, D.H.; Park, J.H.; Lee, S.M.; Koo, J.H.; Won, S.C., Synchronization criterion for lur’e systems via delayed PD controller, Journal of optimization theory and applications, 147, 298-317, (2010) · Zbl 1202.93138
[7] Ji, D.H.; Park, J.H.; Yoo, W.J.; Won, S.C.; Lee, S.M., Synchronization criterion for lur’e type complex dynamical networks with time-varying delay, Physics letters A, 374, 1218-1227, (2010) · Zbl 1236.05186
[8] Nijmeijer, H.; Mareels, I.M.Y., An observer looks at synchronization, IEEE transactions on circuits and systems I, 44, 882-890, (1997)
[9] Morgul, O.; Solak, E., Observer based synchronization of chaotic systems, Physical review E, 54, 4803-4811, (1996)
[10] Ushio, T., Synthesis of synchronized chaotic systems based on observers, International journal of bifurcation and chaos, 9, 541-546, (1999) · Zbl 0941.93533
[11] Kailath, T., Linear systems, (1980), Prentice-Hall Englewood Cliffs, NJ · Zbl 0458.93025
[12] Zhang, Z.; Xu, S.; Shen, H., Reduced-order observer-based output-feedback tracking control of nonlinear systems with state delay and disturbance, International journal of robust and nonlinear control, 20, 1723-1738, (2010) · Zbl 1204.93062
[13] Solak, E., A reduced-order observer for the synchronization of Lorenz systems, Physics letters A, 325, 276-278, (2004) · Zbl 1161.37361
[14] Sun, Y.J., A simple observer design of the generalized Lorenz chaotic systems, Physics letters A, 374, 933-937, (2010) · Zbl 1235.34138
[15] Sun, Y.J., On the state reconstructor design for a class of chaotic systems, Chaos, solitons and fractals, 40, 815-820, (2009) · Zbl 1197.37044
[16] Sun, Y.J., An exponential observer for the generalized rossler chaotic system, Chaos, solitons and fractals, 40, 2457-2461, (2009) · Zbl 1198.93204
[17] Boutayeb, M.; Darouach, M.; Rafaralahy, H., Generalized state-space observers for chaotic synchronization and secure communication, IEEE transactions on circuits and systems I, 49, 345-349, (2002) · Zbl 1368.94087
[18] Lorı´a, A.; Panteley, E.; Zavala-Rı´o, A., Adaptive observers with persistency of excitation for synchronization of chaotic systems, IEEE transactions on circuits and systems I, 56, 2703-2716, (2009)
[19] Sun, Y.J., Solution bounds of generalized Lorenz chaotic systems, Chaos, solitons and fractals, 40, 691-696, (2009) · Zbl 1197.37043
[20] Lü, J.; Chen, G.; Cheng, D.; Celikovsky, S., Bridge the gap between the Lorenz system and the Chen system, International journal of bifurcation and chaos, 12, 2917-2926, (2002) · Zbl 1043.37026
[21] Lorenz, E.N., Deterministic non-periods flows, Journal of the atmospheric sciences, 20, 130-141, (1963)
[22] Chen, G.; Ueta, T., Yet another chaotic attractor, International journal of bifurcation and chaos, 9, 1465-1466, (1999) · Zbl 0962.37013
[23] Lü, J.; Chen, G., A new chaotic attractor coined, International journal of bifurcation and chaos, 12, 659-661, (2002) · Zbl 1063.34510
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.