Skip to main content
SpringerLink
Log in

Sliding Mode Based Control and Synchronization of Chaotic Systems in Presence of Parametric Uncertainties

  • Chapter
  • First Online:
Applications of Sliding Mode Control in Science and Engineering

Part of the book series: Studies in Computational Intelligence ((SCI,volume 709))

Abstract

This chapter deals with the control and synchronization of chaos where both of them are regarded as a case of a control problem. The proposed approach consists in using the sliding mode control theory. We first compare the pioneering OGY control method to the sliding mode control method. We next present a sliding surface design based on the Lyapunov theory. We show that for the class of chaotic systems that can be stabilized using a smooth feedback controller, a sliding manifold can be easily constructed using the Lyapunov function. Besides, we prove that the designed sliding surface is a stable manifold for the originally chaotic system. Thus, should the state behavior be confined to it, then the trajectory will slide towards the equilibrium. We also prove that the proposed controller is robust to mismatched parametric uncertainties. To diminish the effect of the unwanted chattering phenomenon resulting from high sliding gains, an adaptive sliding controller is finally designed to present a robust model independent controller that achieves stabilization of the equilibrium points as well as synchronization of two systems. All these results will be confirmed through numerical simulations on Rossler’s system.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Behjameh MR, Delavari H, Vali A (2015) Global finite time synchronization of two nonlinear chaotic gyros using high order sliding mode control. J Appl Comput Mech 01(01):26–34

    Google Scholar 

  2. Boccaletti S, Grebogi C, Lai YC, Mancini H, Maza D (2000) The control of chaos: theory and applications. Phys Rep 329:103–197

    Article  MathSciNet  Google Scholar 

  3. Chang W, Park J, Joo Y, Chen G (2003) Static output-feedback fuzzy controller for chen’s chaotic system with uncertainties. Inf Sci 151:227–244

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen G (1999) Controlling chaos and bifurcations in engineering systems. CRC-Press

    Google Scholar 

  5. Di Bernardo M, Tse C (2002) Chaos in power electronics: an overview. In: Chen G, Ueta T (eds) Chaos in circuits and systems, chap. 16, vol 11. World Scientific, New York, pp 317–340

    Google Scholar 

  6. Feki M (2003) An adaptive feedback control of linearizable chaotic systems. Chaos Solitons Fractals 15:883–890

    Article  MathSciNet  MATH  Google Scholar 

  7. Feki M (2004) Model-independent adaptive control of chua’s system with cubic nonlinearity. Int J Bifurc Chaos 14(12):4249–4263

    Article  MathSciNet  MATH  Google Scholar 

  8. Feki M (2004) Synchronization of chaotic systems with parametric uncertainties using sliding observers. Int J Bifurc Chaos 14(7):2467–2475

    Article  MathSciNet  MATH  Google Scholar 

  9. Feki M (2006) Synchronization of generalized lorenz system using adaptive controller. In: IFAC conference on analysis and control of chaotic systems CHAOS’06, CD–ROM. Reims-France (2006)

    Google Scholar 

  10. Feki M (2009) Observer-based synchronization of chaotic systems with unknown nonlinear function. Chaos Solitons Fractals 39:981–990

    Article  MathSciNet  Google Scholar 

  11. Feki M (2009) Sliding mode control and synchronization of chaotic systems with parametric uncertainties. Chaos Solitons Fractals 41:1390–1400

    Article  MathSciNet  MATH  Google Scholar 

  12. Fourati A, Feki M, Derbel N (2010) Stabilizing the unstable periodic orbits of a chaotic system using model independent adaptive time-delayed controller. Nonlinear Dyn 62(3):687–704

    Article  MathSciNet  MATH  Google Scholar 

  13. Gammoudi IE, Feki M (2013) Synchronization of integer order and fractional order chua’s systems using robust observer. Commun Nonlinear Sci Numer Simul 18(3):625–638

    Article  MathSciNet  MATH  Google Scholar 

  14. Garfinkel A, Spano ML, Ditto WL, Weiss JN (1992) Controlling cardiac chaos. Science 257:1230–1235

    Article  Google Scholar 

  15. Hwang CC, Hsieh JY, Lin RS (1997) A linear continuous feedback control of chua’s circuit. Chaos Solitons Fractals 8:1507–1515

    Article  Google Scholar 

  16. Isidori A (1995) Nonlinear control systems, 3rd edn. Springer, United Kingdom

    Book  MATH  Google Scholar 

  17. Jemaâ-Boujelben SB, Feki M (2016) Integral higher order sliding mode control for mimo uncertain systems: application to chaotic pmsm. In: Proceedings of engineering & technology (PET), pp. 632–637. Hammamet-Tunisia

    Google Scholar 

  18. Jiang GP, Chen G, Tang WKS (2002) Stabilizing unstable equilibrium points of a class of chaotic systems using a state pi regulator. IEEE Trans Circuits Syst I 49(12):1820–1826

    Article  MathSciNet  Google Scholar 

  19. Khalil HK (1992) Nonlinear systems. Macmillan, New York

    MATH  Google Scholar 

  20. Konishi K, Hirai M, Kokame H (1998) Sliding mode control for a class of chaotic systems. Phys Lett A 245:511–517

    Article  Google Scholar 

  21. Koubaâ K, Feki M (2014) Quasi-periodicity, chaos and coexistence in the time delay controlled two-cell dc-dc buck converter. Int J Bifurc Chaos 24(10): 1450124

    Google Scholar 

  22. Li TY, Yorke JA (1975) Period three implies chaos. Am Math Mon 82(10):985–992

    Article  MathSciNet  MATH  Google Scholar 

  23. Lin JS, Yan JJ, Liao TL (2006) Robust control of chaos in lorenz systems subject to mismatch uncertainties. Chaos Solitons Fractals 27:501–510

    Article  MathSciNet  MATH  Google Scholar 

  24. Miladi Y, Feki M, Derbel N (2015) Stabilizing the unstable periodic orbits of a hybrid chaotic system using optimal control. Commun Nonlinear Sci Numer Simul 20:1043–1056

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Robert B, Feki M, Iu H (2006) Control of a pwm inverter using proportional plus extended time-delayed feedback. Int J Bifurc Chaos 16(1):113–128

    Article  MATH  Google Scholar 

  27. Rössler O (1976) An equation for continuous chaos. Phys Lett A 57(5):397–398

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Sundarapandian V (2012) Global chaos control of hyperchaotic liu system via sliding control method. Int J Control Theory Appl 5(2):117–123

    Google Scholar 

  30. Sundarapandian V (2013) Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters. J Eng Sci Technol Rev 6(4):53–65

    Google Scholar 

  31. Sundarapandian V, Pehlivan I (2012) Analysis, control, synchronization, and circuit design of a novel chaotic system. Math Comput Model 55(7–8):1904–1915

    Article  MathSciNet  MATH  Google Scholar 

  32. Tian YC, Tadé MO, Levy D (2002) Constrained control of chaos. Phys Lett A 296:87–90

    Article  MathSciNet  MATH  Google Scholar 

  33. Vidyasagar M (1993) Nonlinear systems analysis, 2nd edn. Prentice Hall, New Jersey

    MATH  Google Scholar 

  34. Yang CH, Wu CL, Chen YJ, Shiao SH (2015) Reduced fuzzy controllers for lorenz-stenflo system control and synchronization. Int J Fuzzy Syst 17(2):158–169

    Article  MathSciNet  Google Scholar 

  35. Yu X, Chen G, Xia Y, Song Y, Cao Z (2001) An invariant-manifold-based method for chaos control. IEEE Trans Circuits Syst I 48(8):930–937

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhan-Shan Z, Jing Z, Gang D, Da-Kun Z (2015) Chaos synchronization of coronary artery system based on higher order sliding mode adaptive control. Acta Phys Sin 64(21): 210–508

    Google Scholar 

  37. Zhang H, Ma X-K, Li M, Zou J-L (2005) Controlling and tracking hyperchaotic rössler system via active backstepping design. Chaos Solitons Fractals 26: 353–361

    Google Scholar 

  38. Zou YL, Luo XS, Jiang P-Q, Wang BH, Chen G, Fang JQ, Quan HJ (2003) Controlling the chaotic \(n\)-scroll chua’s circuit. Int J Bifurc Chaos 13(9): 2709–2714

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. École Supérieure des Sciences et de la Technologie de Hammam Sousse, University of Sousse, Sousse, Tunisia

    Moez Feki

Authors
  1. Moez Feki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moez Feki .

Editor information

Editors and Affiliations

  1. Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

    Sundarapandian Vaidyanathan

  2. Maritime College and Department of Marine Engineering, National Kaohsiung Marine University, Kaohsiung, Taiwan

    Chang-Hua Lien

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Feki, M. (2017). Sliding Mode Based Control and Synchronization of Chaotic Systems in Presence of Parametric Uncertainties. In: Vaidyanathan, S., Lien, CH. (eds) Applications of Sliding Mode Control in Science and Engineering. Studies in Computational Intelligence, vol 709. Springer, Cham. https://doi.org/10.1007/978-3-319-55598-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-55598-0_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-55597-3

  • Online ISBN: 978-3-319-55598-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation