Optimal, adaptive and single state feedback control for a 3D chaotic system with golden proportion equilibria

Hassan Saberi Nik; Ping He; Sayyed Taha Talebian

Kybernetika (2014)

  • Volume: 50, Issue: 4, page 596-615
  • ISSN: 0023-5954

Abstract

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In this paper, the problems on purposefully controlling chaos for a three-dimensional quadratic continuous autonomous chaotic system, namely the chaotic Pehlivan-Uyaroglu system are investigated. The chaotic system, has three equilibrium points and more interestingly the equilibrium points have golden proportion values, which can generate single folded attractor. We developed an optimal control design, in order to stabilize the unstable equilibrium points of this system. Furthermore, we propose Lyapunov stability to control the Pehlivan-Uyaroglu system with unknown parameters by way of a feedback control approach and a single controller. Numerical simulations are performed to demonstrate the effectiveness of the proposed control strategies.

How to cite

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Saberi Nik, Hassan, He, Ping, and Talebian, Sayyed Taha. "Optimal, adaptive and single state feedback control for a 3D chaotic system with golden proportion equilibria." Kybernetika 50.4 (2014): 596-615. <http://eudml.org/doc/261993>.

@article{SaberiNik2014,
abstract = {In this paper, the problems on purposefully controlling chaos for a three-dimensional quadratic continuous autonomous chaotic system, namely the chaotic Pehlivan-Uyaroglu system are investigated. The chaotic system, has three equilibrium points and more interestingly the equilibrium points have golden proportion values, which can generate single folded attractor. We developed an optimal control design, in order to stabilize the unstable equilibrium points of this system. Furthermore, we propose Lyapunov stability to control the Pehlivan-Uyaroglu system with unknown parameters by way of a feedback control approach and a single controller. Numerical simulations are performed to demonstrate the effectiveness of the proposed control strategies.},
author = {Saberi Nik, Hassan, He, Ping, Talebian, Sayyed Taha},
journal = {Kybernetika},
keywords = {autonomous chaotic system; optimal control; adaptive control; single state feedback control; Pontryagin Minimum Principle; autonomous chaotic system; optimal control; adaptive control; single state feedback control; Pontryagin minimum principle},
language = {eng},
number = {4},
pages = {596-615},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Optimal, adaptive and single state feedback control for a 3D chaotic system with golden proportion equilibria},
url = {http://eudml.org/doc/261993},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Saberi Nik, Hassan
AU - He, Ping
AU - Talebian, Sayyed Taha
TI - Optimal, adaptive and single state feedback control for a 3D chaotic system with golden proportion equilibria
JO - Kybernetika
PY - 2014
PB - Institute of Information Theory and Automation AS CR
VL - 50
IS - 4
SP - 596
EP - 615
AB - In this paper, the problems on purposefully controlling chaos for a three-dimensional quadratic continuous autonomous chaotic system, namely the chaotic Pehlivan-Uyaroglu system are investigated. The chaotic system, has three equilibrium points and more interestingly the equilibrium points have golden proportion values, which can generate single folded attractor. We developed an optimal control design, in order to stabilize the unstable equilibrium points of this system. Furthermore, we propose Lyapunov stability to control the Pehlivan-Uyaroglu system with unknown parameters by way of a feedback control approach and a single controller. Numerical simulations are performed to demonstrate the effectiveness of the proposed control strategies.
LA - eng
KW - autonomous chaotic system; optimal control; adaptive control; single state feedback control; Pontryagin Minimum Principle; autonomous chaotic system; optimal control; adaptive control; single state feedback control; Pontryagin minimum principle
UR - http://eudml.org/doc/261993
ER -

References

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  1. Yagasaki, K., 10.1007/BF00044981, Nonlinear Dyn. 6 (1994), 125-142. DOI10.1007/BF00044981
  2. Han, S. K., Kerrer, C., Kuramoto, Y., 10.1103/PhysRevLett.75.3190, Phys. Rev. Lett. 75 (1995), 3190-3193. DOI10.1103/PhysRevLett.75.3190
  3. Cuomo, K. M., Oppenheim, A. V., 10.1103/PhysRevLett.71.65, Phys. Rev. Lett. 71 (1993), 65-68. DOI10.1103/PhysRevLett.71.65
  4. Nik, H. Saberi, Gorder, R. A. Van, Competitive modes for the Baier-Sahle hyperchaotic flow in arbitrary dimensions., Nonlinear Dyn. 74 (2013), 3, 581-590. MR3117644
  5. Ott, E., Grebogi, C., Yorke, J. A., 10.1103/PhysRevLett.64.1196, Phys. Rev. Lett. 64 (1990), 1196-1199. Zbl0964.37502MR1041523DOI10.1103/PhysRevLett.64.1196
  6. Azhmyakov, V., Basin, M. V., Gil-Garcia, A. E., Optimal control processes associated with a class of discontinuous control systems: applications to sliding mode dynamics., Kybernetika 50 (2014), 1, 5-18. MR3195001
  7. Jiang, Y., Dai, J., Robust control of chaos in modified FitzHugh-Nagumo neuron model under external electrical stimulation based on internal model principle., Kybernetika 47 (2011), 4, 612-629. Zbl1227.93033MR2884864
  8. Wang, H., Han, Z Z., Xie, Q. Y., Zhang, W., Finite-time chaos control of unified chaotic systems with uncertain parameters., Nonlinear Dyn. 55 (2009), 323-328. Zbl1170.70401MR2472222
  9. Lynnyk, V., Čelikovský, S., On the anti-synchronization detection for the generalized Lorenz system and its applications to secure encryption., Kybernetika 46 (2010), 1, 1-18. Zbl1190.93038MR2666891
  10. Sun, K., Liu, X., Zhu, C., Sprott, J. C., Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system., Nonlinear Dyn. 69 (2012), 1383-1391. MR2943392
  11. Wei, Z., Wang, Z., Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium., Kybernetika 49 (2013), 2, 359-374. Zbl1276.34043MR3085401
  12. Effati, S., Nik, H. Saberi, Jajarmi, A., Hyperchaos control of the hyperchaotic Chen system by optimal control design., Nonlinear Dyn. 73 (2013), 499-508. MR3080686
  13. Effati, S., Saberi-Nadjafi, J., Nik, H. Saberi, 10.1016/j.apm.2013.06.025, Appl. Math. Modell. 38 (2014), 759-774. MR3141658DOI10.1016/j.apm.2013.06.025
  14. Nik, H. Saberi, Golchaman, M., Chaos control of a bounded 4D chaotic system., Neural Comput. Appl. (2013). 
  15. Yu, S., Lu, J., Yu, X., Chen, G., 10.1109/TCSI.2011.2180429, IEEE Trans. Circuits Syst. 59-I (2012), 5, 1015-1028. MR2924533DOI10.1109/TCSI.2011.2180429
  16. Lu, J., Yu, S., Leung, H., Chen, G., 10.1109/TCSI.2005.854412, IEEE Trans. Circuits Syst. 53 (2006), 1, 149-165. DOI10.1109/TCSI.2005.854412
  17. Wang, Z. H., Sun, Y. X., Qi, G. Y., Wyk, B. J., The effects of fractional order on a 3-D quadratic autonomous system with four-wing attractor., Nonlinear Dyn. 62 (2010), 139-150. Zbl1209.34060MR2736983
  18. Cam, U., 10.1016/j.compeleceng.2004.06.001, Comput. Electr. Engrg. 30 (2004), 4, 281-290. Zbl1057.94518DOI10.1016/j.compeleceng.2004.06.001
  19. Yu, S., Lü, J., Tang, W., Chen, G., 10.1063/1.2336739, Chaos 16 (2006), 033126. Zbl1151.94432DOI10.1063/1.2336739
  20. Pehlivan, I., Uyaroglu, Y., 10.3923/jas.2007.232.236, J. Appl. Sci. 7 (2007), 7, 232-236. DOI10.3923/jas.2007.232.236
  21. Pehlivan, I., Uyaroglu, Y., 10.1016/j.compeleceng.2012.08.007, Comput. Electr. Engrg. 38 (2012), 6, 1777-1784. DOI10.1016/j.compeleceng.2012.08.007
  22. Kirk, D. E., Optimal Control Theory: An Introduction., Prentice-Hall, 1970. 
  23. Zhong, W., Stefanovski, J., Dimirovski, G., Zhao, J., Decentralized control and synchronization of time-varying complex dynamical network., Kybernetika 45 (2009), 151-167. Zbl1158.34332MR2489586
  24. Chen, Y., Lu, J., Lin, Z., 10.1016/j.automatica.2013.02.021, Automatica 49 (2013), 6, 1768-1775. MR3049226DOI10.1016/j.automatica.2013.02.021
  25. Lu, J., Chen, G., 10.1109/TAC.2005.849233, IEEE Trans. Automat. Control 50 (2005), 6, 841-846. MR2142000DOI10.1109/TAC.2005.849233
  26. Zhou, J., Lu, J., Lu, J., 10.1016/j.automatica.2007.08.016, Automatica 44 (2008), 4, 996-1003. Zbl1158.93339MR2530942DOI10.1016/j.automatica.2007.08.016
  27. He, P., Ma, S. H., Fan, T., 10.1063/1.4773005, Chaos 22 (2012), 4, 043151. DOI10.1063/1.4773005
  28. He, P., Jing, C. G., Fan, T., Chen, C. Z., 10.1002/cplx.21472, Complexity 19 (2014), 10-26. MR3158431DOI10.1002/cplx.21472
  29. He, P., Jing, C. G., Chen, C. Z., Fan, T., Nik, H. Saberi, Synchronization of general complex networks via adaptive control schemes., J. Phys. 82 (2014), 3, 499-514. 

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