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International Conference on Advanced Intelligent Systems and Informatics

AISI 2019: Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2019 pp 277–287Cite as

Controlling Chaotic System via Optimal Control

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Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1058))

Abstract

Chaos is a bounded unstable dynamic behavior that exhibits sensitive dependence on initial conditions and includes infinite unstable periodic motions. This article examines the controlling of a chaotic system via optimal control technique which is based on the Pontryagin minimum principle. A 3D chaotic system is considered to apply this scheme which have 5 equilibrium points. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed method. The simulation results illustrated the stabilized behaviour of states and control functions for different equilibrium points.

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References

  1. Azar, A.T., Vaidyanathan, S.: Advances in Chaos Theory and Intelligent Control. Springer, Berlin (2016)

    Book  Google Scholar 

  2. Azar, A.T., Ouannas, A., Singh, S.: Control of new type of fractional chaos synchronization. In: International Conference on Advanced Intelligent Systems and Informatics, pp. 47–56. Springer, Cham (2017)

    Google Scholar 

  3. Azar, A.T., Vaidyanathan, S., Ouannas, A.: Fractional order control and synchronization of chaotic systems. In: Studies in Computational Intelligence, vol. 688. Springer, Berlin (2017)

    Google Scholar 

  4. Babaei, M.: A novel text and image encryption method based on chaos theory and DNA computing. Nat. Comput. 12(1), 101–107 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bhat, M.A., Shikha: Complete synchronisation of non-identical fractionalorder hyperchaotic systems using active control. Int. J. Autom. Control 13(2), 140–157 (2019)

    Google Scholar 

  6. Bozoki, Z.: Chaos theory and power spectrum analysis in computerized cardiotocography. Eur. J. Obstet. Gynecol. Reprod. Biol. 71(2), 163–168 (1997)

    Article  Google Scholar 

  7. Çavuşoğlu, Ü., Panahi, S., Akgül, A., Jafari, S., Kaçar, S.: A new chaotic system with hidden attractor and its engineering applications: analog circuit realization and image encryption. Analog Integr. Circ. Sig. Process 98(1), 85–99 (2019)

    Article  Google Scholar 

  8. Chekan, J.A., Ali Nojoumian, M., Merat, K., Salarieh, H.: Chaos control in lateral oscillations of spinning disk via linear optimal control of discrete systems. J. Vib. Control 23(1), 103–110 (2017)

    Article  MathSciNet  Google Scholar 

  9. Chien, T.I., Liao, T.L.: Design of secure digital communication systems using chaotic modulation, cryptography and chaotic synchronization. Chaos, Solitons Fractals 24(1), 241–255 (2005)

    Article  Google Scholar 

  10. Din, Q.: Stability, bifurcation analysis and chaos control for a predator-prey system. J. Vib. Control 25(3), 612–626 (2019)

    Article  MathSciNet  Google Scholar 

  11. Edelman, M.: On stability of fixed points and chaos in fractional systems. Chaos Interdisc. J. Nonlinear Sci. 28(2), 023112 (2018). https://doi.org/10.1063/1.5016437

    Article  MathSciNet  MATH  Google Scholar 

  12. Čelikovský, S., Lynnyk, V., Chen, G.: Robust synchronization of a class of chaotic networks. J. Franklin Inst. 350(10), 2936–2948 (2013)

    Article  MathSciNet  Google Scholar 

  13. Frey, D.R.: Chaotic digital encoding: an approach to secure communication. In: IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 40, no. 10, pp. 660–666 (1993)

    Google Scholar 

  14. Garfinkel, A.: Controlling cardiac chaos. Science (1992)

    Google Scholar 

  15. Khan, A., Singh, S.: Mixed tracking and projective synchronization of 6D hyperchaotic system using active control. Int. J. Nonlinear Sci. 22(1), 44–53 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Khan, A., Singh, S.: Combination synchronization of time-delay chaotic system via robust adaptive sliding mode control. Pramana 88(6), 91 (2017)

    Article  Google Scholar 

  17. Khan, A., Singh, S.: Hybrid function projective synchronization of chaotic systems via adaptive control. Int. J. Dyn. Control 5(4), 1114–1121 (2017)

    Article  MathSciNet  Google Scholar 

  18. Khan, A., Singh, S.: Chaotic analysis and combination-combination synchronization of a novel hyperchaotic system without any equilibria. Chin. J. Phys. 56(1), 238–251 (2018)

    Article  MathSciNet  Google Scholar 

  19. Khan, A., Singh, S.: Generalization of combination-combination synchronization of n-dimensional time-delay chaotic system via robust adaptive sliding mode control. Math. Methods Appl. Sci. 41(9), 3356–3369 (2018)

    Article  MathSciNet  Google Scholar 

  20. Khan, A., Tyagi, A.: Analysis and hyper-chaos control of a new 4-D hyper-chaotic system by using optimal and adaptive control design. Int. J. Dyn. Control 5(4), 1147–1155 (2017)

    Article  MathSciNet  Google Scholar 

  21. Lau, F., Tse, C.K.: Chaos-based digital communication systems. Springer (2003)

    Google Scholar 

  22. Ott, E., Spano, M.: Controlling chaos. In: AIP Conference Proceedings, vol. 375, pp. 92–103. AIP (1996)

    Google Scholar 

  23. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)

    Article  MathSciNet  Google Scholar 

  24. Ouannas, A., Grassi, G., Azar, A.T., Singh, S.: New control schemes for fractional chaos synchronization. In: International Conference on Advanced Intelligent Systems and Informatics, pp. 52–63. Springer (2018)

    Google Scholar 

  25. Ouannas, A., Grassi, G., Azar, A.T., Gasri, A.: A new control scheme for hybrid chaos synchronization. In: Hassanien, A.E., Tolba, M.F., Shaalan, K., Azar, A.T. (eds.) Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2018, pp. 108–116. Springer International Publishing, Cham (2019)

    Google Scholar 

  26. Ouannas, A., Grassi, G., Azar, A.T.: Fractional-order control scheme for Q-S chaos synchronization. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., F Tolba, M. (eds.) The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA 2019), pp 434–441. Springer International Publishing, Cham (2020)

    Google Scholar 

  27. Ouannas, A., Grassi, G., Azar, A.T.: A new generalized synchronization scheme to control fractional chaotic systems with non-identical dimensions and different orders. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., F Tolba M. (eds.) The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019), pp. 415–424. Springer International Publishing, Cham (2020)

    Google Scholar 

  28. Schiff, S.J., Jerger, K., Duong, D.H., Chang, T., Spano, M.L., Ditto, W.L., et al.: Controlling chaos in the brain. Nature 370(6491), 615–620 (1994)

    Article  Google Scholar 

  29. Singh, S., Azar, A.T., Ouannas, A., Zhu, Q., Zhang, W., Na, J.: Sliding mode control technique for multi-switching synchronization of chaotic systems. In: 2017 9th International Conference on Modelling, pp. 880–885. Identification and Control (ICMIC). IEEE (2017)

    Google Scholar 

  30. Singh, S., Azar, A.T., Bhat, M.A., Vaidyanathan, S., Ouannas, A.: Active control for multi-switching combination synchronization of non-identical chaotic systems. In: Advances in System Dynamics and Control, pp. 129–162. IGI Global (2018)

    Google Scholar 

  31. Singh, S., Azar, A.T., Vaidyanathan, S., Ouannas, A., Bhat, M.A.: Multiswitching synchronization of commensurate fractional order hyperchaotic systems via active control. In: Mathematical Techniques of Fractional Order Systems, pp. 319–345. Elsevier (2018)

    Google Scholar 

  32. Sivakumar, B.: Chaos theory in hydrology: important issues and interpretations. J. Hydrol. 227(1–4), 1–20 (2000)

    Article  Google Scholar 

  33. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2001)

    MATH  Google Scholar 

  34. Vaidyanathan, S., Azar, A.T.: Analysis, control and synchronization of a nine-term 3-D novel chaotic system. In: Chaos Modeling and Control Systems Design. Springer, pp. 19–38 (2015)

    Google Scholar 

  35. Vaidyanathan, S., Azar, A.T.: Anti-synchronization of identical chaotic systems using sliding mode control and an application to vaidyanathan-madhavan chaotic systems. In: Azar, A.T., Zhu, Q. (eds.) Advances and Applications in Sliding Mode Control Systems, Studies in Computational Intelligence, vol. 576, pp. 527–547. Springer, Berlin (2015)

    Google Scholar 

  36. Vaidyanathan, S., Azar, A.T.: Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidyanathan chaotic systems. In: Azar, A.T., Zhu, Q. (eds.) Advances and Applications in Sliding Mode Control Systems, Studies in Computational Intelligence, vol. 576. Springer, Berlin, pp. 549–569 (2015)

    Google Scholar 

  37. Vaidyanathan, S., Idowu, B.A., Azar, A.T.: Backstepping controller design for the global chaos synchronization of sprott’s jerk systems. In: Chaos Modeling and Control Systems Design, pp. 39–58. Springer (2015)

    Google Scholar 

  38. Vaidyanathan, S., Azar, A.T., Boulkroune, A.: A novel 4-D hyperchaotic system with two quadratic nonlinearities and its adaptive synchronisation. Int. J. Autom. Control 12(1), 5–26 (2018)

    Article  Google Scholar 

  39. Xing-yuan, W., Yong-feng, G.: A switch-modulated method for chaos digital secure communication based on user-defined protocol. Commun. Commun. Nonlinear Sci. Numer. Simul. 15(1), 99–104 (2010). sI: Chaos, Complexity and Transport: Theory and Applications

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Jesus and Mary College, University of Delhi, New Delhi, India

    Shikha Singh

  2. College of Engineering, Prince Sultan University, Riyadh, Kingdom of Saudi Arabia

    Ahmad Taher Azar

  3. Faculty of Computers and Information, Benha University, Benha, Egypt

    Ahmad Taher Azar

Authors
  1. Shikha Singh
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  2. Ahmad Taher Azar
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Corresponding author

Correspondence to Ahmad Taher Azar .

Editor information

Editors and Affiliations

  1. Cairo University, Giza, Egypt

    Aboul Ella Hassanien

  2. The British University in Dubai, Dubai, United Arab Emirates

    Khaled Shaalan

  3. Ain Shams University, Cairo, Egypt

    Mohamed Fahmy Tolba

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Singh, S., Azar, A.T. (2020). Controlling Chaotic System via Optimal Control. In: Hassanien, A., Shaalan, K., Tolba, M. (eds) Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2019. AISI 2019. Advances in Intelligent Systems and Computing, vol 1058. Springer, Cham. https://doi.org/10.1007/978-3-030-31129-2_26

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