Skip to main content
SpringerLink
Log in

Parameters identification of nonlinear Lorenz chaotic system for high-precision model reference synchronization

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The Lorenz chaotic system synchronization has been a popular research topic in the last two decades. Most of the studies focus on the design of model reference adaptive control (MRAC) synchronization schemes. In the existing MRAC schemes, adaptive laws are designed to estimate the system parameters online. However, due to the system parameters being unknown, arbitrary selection results in a longer estimation period. Although applying large values of adaptive gains can increase the estimation convergence speed, it usually induces obvious estimation oscillations and thereby causes large control efforts. On the contrary, small adaptive gains result in smooth but sluggish transient estimations. None of the studies addressed the importance of parameters estimation for high precise synchronization. To solve this issue, two system identification schemes are presented. The first applied scheme is called the observer/Kalman filter identification (OKID). The second one is called the bilinear transform discretization (BTD). The related detail derivations for the Lorenz chaotic system parameter identification will be presented in this paper. Results show that the proposed BTD identification algorithm is much simpler and computationally efficient. Moreover, highly precise parameter estimations can be achieved as well. Nevertheless, due to the complex nonlinearity of the chaotic system, it will be illustrated that even extremely small parameter deviations could lead to dramatic mismatches for the chaotic system model output prediction. To further solve this issue, a MRAC is further designed in which the initial guess of the system parameters is obtained through the proposed BTD identification algorithm. Based on the contribution of the developed method, the identified parameters are already very close to the true values. As a consequence, applying smaller values of adaptive gains is able to achieve fast convergence of the synchronization of chaos. With the aid of precise parameter identification, the transient dynamics and the convergence performance of the MARC are both improved significantly. Simulations demonstrate the effectiveness of the proposed scheme.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Alstrom, R.B.: Controlling the chaotic motions of an airfoil with a nonlinear stiffness using closed-loop harmonic parametric excitation. Fluids 5(4), 165 (2020)

    Article  Google Scholar 

  2. Wang, C.C., Chen, C.L., Yau, H.T.: Bifurcation and chaotic analysis of aeroelastic systems. J Computat Nonlin Dyn 9(2), 021004 (2013)

    Article  Google Scholar 

  3. Ahmad, S., Ullah, A., Akgül, A.: Investigating the complex behaviour of multi-scroll chaotic system with Caputo fractal-fractional operator. Chaos, Solitons Fractals 146, 110900 (2021)

    Article  MathSciNet  Google Scholar 

  4. Akgül, A., Ahmad, S., Ullah, A., Baleanu, D., Akgül, E.K.: A novel method for analysing the fractal fractional integrator circuit. Alexandria Eng J 60(4), 3721–3729 (2021)

    Article  Google Scholar 

  5. Gu, F., Yau, H., Chen, H.: Application of chaos synchronization technique and pattern clustering for diagnosis analysis of partial discharge in power cables. IEEE Access 7, 76185–76193 (2019)

    Article  Google Scholar 

  6. Jian, B.L., Su, X.Y., Yau, H.T.: Bearing fault diagnosis based on chaotic dynamic errors in key components. IEEE Access 9, 53509–53517 (2021)

    Article  Google Scholar 

  7. Yang, S.K., Chen, C.L., Yau, H.T.: Control of chaos in lorenz system. Chaos, Solitons Fractals 13(4), 767–780 (2002)

    Article  Google Scholar 

  8. Liu, L., Pu, J., Song, X., Fu, Z., Wang, X.: Adaptive sliding mode control of uncertain chaotic systems with input nonlinearity. Nonlin Dyn 76(4), 1857–1865 (2014)

    Article  MathSciNet  Google Scholar 

  9. Li, J., Li, W., Li, Q.: Sliding mode control for uncertain chaotic systems with input nonlinearity. Communicat Nonlin Sci Numer Simulat 17(1), 341–348 (2012)

    Article  MathSciNet  Google Scholar 

  10. Yau, H.T., Chen, C.L.: Chaos control of Lorenz systems using adaptive controller with input saturation. Chaos, Solitons Fractals 34(5), 1567–1574 (2007)

    Article  Google Scholar 

  11. Yau, H.T., Yan, J.J.: Design of sliding mode controller for Lorenz chaotic system with nonlinear input. Chaos, Solitons Fractals 19(4), 891–898 (2004)

    Article  MathSciNet  Google Scholar 

  12. Chen, C.L., Peng, C.C., Yau, H.T.: High-order sliding mode controller with backstepping design for aeroelastic systems. Communicat. Nonlin. Sci. Numerical Simulat. 17(4), 1813–1823 (2012)

    Article  MathSciNet  Google Scholar 

  13. Chen, C.L., Peng, C.C.: Control of a perturbed chaotic system by using a trajectory trapping strategy. Nonlin. Dyn. 69(4), 2105–2115 (2012)

    Article  MathSciNet  Google Scholar 

  14. Yau, H.T.: Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos, Solitons Fractals 22(2), 341–347 (2004)

    Article  MathSciNet  Google Scholar 

  15. Yau, H.T., Chen, C.L.: Chattering-free fuzzy sliding-mode control strategy for uncertain chaotic systems. Chaos, Solitons Fractals 30(3), 709–718 (2006)

    Article  Google Scholar 

  16. Peng, C.C., Chen, C.L.: Robust chaotic control of lorenz system by backstepping design. Chaos, Solitons Fractals 37(2), 598–608 (2008)

    Article  MathSciNet  Google Scholar 

  17. Chen, L.H., Peng, C.C.: Extended backstepping sliding controller design for chattering attenuation and its application for servo motor control. Appl. Sci. 7(3), 220 (2017)

    Article  Google Scholar 

  18. Gao, S., Wang, Y., Dong, H., Ning, B., Wang, H.: Controlling uncertain Genesio-Tesi chaotic system using adaptive dynamic surface and nonlinear feedback. Chaos, Solitons Fractals 105, 180–188 (2017)

    Article  MathSciNet  Google Scholar 

  19. Feki, M.: An adaptive feedback control of linearizable chaotic systems. Chaos, Solitons Fractals 15(5), 883–890 (2003)

    Article  MathSciNet  Google Scholar 

  20. Haghighatdar, F., Ataei, M.: Adaptive set-point tracking of the Lorenz chaotic system using non-linear feedback. Chaos, Solitons Fractals 4(4), 1938–1945 (2009)

    Article  Google Scholar 

  21. Hua, C., Guan, X.: Adaptive control for chaotic systems. Chaos, Solitons Fractals 22(1), 55–60 (2004)

    Article  MathSciNet  Google Scholar 

  22. Yang, T., Yang, C.M., Yang, L.B.: A detailed study of adaptive control of chaotic systems with unknown parameters. Dyn. Control 8(3), 255–267 (1998)

    Article  MathSciNet  Google Scholar 

  23. Gao, R.: A novel track control for Lorenz system with single state feedback. Chaos, Solitons Fractals 122, 236–244 (2019)

    Article  MathSciNet  Google Scholar 

  24. Liu M., Wang B., Liu D.: 2020 "A Digital Twin Modeling Method for Turbofan Engine Real-time Test Data Analysis and Performance Monitoring," In 2020 11th International Conference on Prognostics and System Health Management (PHM-2020 Jinan), 2020, pp. 444–449

  25. Tu, Y.H., Peng, C.C.: An ARMA-based digital twin for MEMS gyroscope drift dynamics modeling and real-time compensation. IEEE Sens. J. 21(3), 2712–2724 (2021)

    Article  Google Scholar 

  26. Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366(1), 1–101 (2002)

    Article  MathSciNet  Google Scholar 

  27. Wagg, D.J.: Adaptive control of nonlinear dynamical systems using a model reference approach. Meccanica 38(2), 227–238 (2003)

    Article  MathSciNet  Google Scholar 

  28. Fu, G., Li, Z.: Adaptive synchronization of novel hyperchaotic Lorenz system via passive control. IEEE ICCA 2010, 1601–1605 (2010)

    Google Scholar 

  29. Araujo, A.D., Singh, S.N.: Output feedback adaptive variable structure control of chaos in lorenz system. Int J Bifurcat. Chaos 12(3), 571–582 (2002)

    Article  Google Scholar 

  30. Shao, K., Zhou, L., Guo, H., Xu, Z., Chen, R.: “Finite-time synchronization and parameter identification of fractional-order Lorenz chaotic system.” Chinese Cont. Conf. (CCC) 2019, 1120–1124 (2019)

    Google Scholar 

  31. Juang, J.-N., Phan, M., Horta, L.G., Longman, R.W.: Identification of observer/kalman filter markov parameters-theory and experiments. J. Guid. Control. Dyn. 16(2), 320–329 (1993)

    Article  Google Scholar 

  32. Tiano, A., Sutton, R., Lozowicki, A., Naeem, W.: Observer Kalman filter identification of an autonomous underwater vehicle. Control. Eng. Pract. 15(6), 727–739 (2007)

    Article  Google Scholar 

  33. Chien, T.H., Chen, Y.C.: An on-line tracker for a stochastic chaotic system using observer/kalman filter identification combined with digital redesign method. Algorithms 10(1), 25 (2017)

    Article  MathSciNet  Google Scholar 

  34. Juang, J.N., Phan, M.Q.: Identification and control of mechanical systems. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  35. Peng C.C., Huang J.J., Lee H.Y.: "Design of an Embedded Icosahedron Mechatronics for Robust Iterative IMU Calibration," IEEE/ASME Transactions on Mechatronics, pp. 1–1, 2021

  36. Deniz H.I., Taskiran Z.G.C., Sedef H.: "Chaotic Lorenz Synchronization Circuit Design for Secure Communication," In 2018 6th International conference on control engineering & information technology (CEIT), 2018, pp. 1–6

Download references

Funding

This work was supported in part by the Ministry of Science and Technology under Grant No. MOST 110–2221-E-006–095 and MOST 108–2923-E- 006–005-MY3.

Author information

Authors and Affiliations

  1. Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan

    Chao-Chung Peng & Yang-Rui Li

Authors
  1. Chao-Chung Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang-Rui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chao-Chung Peng.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peng, CC., Li, YR. Parameters identification of nonlinear Lorenz chaotic system for high-precision model reference synchronization. Nonlinear Dyn 108, 1733–1754 (2022). https://doi.org/10.1007/s11071-021-07156-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-07156-x

Keywords

Search

Navigation