Skip to main content
SpringerLink
Log in

Fractional-order iterative learning control with initial state learning design

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

Abstract

In this paper, we present a fractional-order iterative learning control (ILC) framework with initial state learning for the tracking problems of linear time-varying systems. Both open-loop and closed-loop \(D^{\alpha }\)-type iterative learning updating laws are considered. To design ILC scheme for practical control systems, initialization assumption, i.e., the system initial states should be the same at each repetition, is removed by using an initial state learning scheme together with the \(D^{\alpha }\)-type ILC updating law. Sufficient conditions of convergence to the desired trajectory is theoretically proved for a linear time-varying mechanical system. Numerical simulation results are presented to illustrate effectiveness of the control strategies. Moreover, we also show that the proposed learning scheme can be applied to the motion control of robot manipulators under some reasonable conditions.

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

Similar content being viewed by others

References

  1. Arimoto, S., Kawamura, S., Miyazaki, F.: Bettering operation of robots by learning. J. Robot. Syst. 1(2), 123–140 (1984)

    Article  Google Scholar 

  2. Xu, J.X.: A survey on iterative learning control for nonlinear systems. Int. J. Control 84(7), 1275–1294 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bristow, D.A., Tharayil, M., Alleyne, A.G.: A survey of iterative learning control. IEEE Control Syst. Mag. 26(3), 96–114 (2006)

    Article  Google Scholar 

  4. Ahn, H.-S., Chen, Y.Q., Moore, K.L.: Iterative learning control: survey and categorization from 1998 to 2004. IEEE Trans. Syst. Man Cybern. Part C 37(6), 1099–1121 (2007)

    Article  Google Scholar 

  5. Jin, X.: Adaptive iterative learning control for high-order nonlinear multi-agent systems consensus tracking. Syst. Control Lett. 89, 16–23 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hamamoto, K., Sugie, T.: Iterative learning control for robot manipulators using the finite dimensional input subspace. IEEE Trans. Robot. Autom. 18(4), 632–635 (2002)

    Article  Google Scholar 

  7. Chen, Y., Moore, K.L., Yu, J., Zhang, T.: Iterative learning control and repetitive control in hard disk drive industry-a tutorial. Int. J. Adapt. Control Signal Process. 22(4), 325–343 (2008)

    Article  MATH  Google Scholar 

  8. Hou, Z., Xu, J., Yan, J.: An iterative learning approach for density control of freeway traffic flow via ramp metering. Transp. Res. Part C 16(1), 71–97 (2008)

    Article  Google Scholar 

  9. Meng, D., Jia, Y., Du, J., Zhang, J.: On iterative learning algorithms for the formation control of nonlinear multi-agent systems. Automatica 50(1), 291–295 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Shi, J., He, X., Zhou, D.: Iterative learning based estimation of periodically occurring faults. IET Control Theory Appl. 10(2), 244–251 (2016)

    Article  MathSciNet  Google Scholar 

  11. Huang, D., Xu, J.X., Li, X., Xu, C., Yu, M.: D-type anticipatory iterative learning control for a class of inhomogeneous heat equations. Automatica 49(8), 2397–2408 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Podlubny, I.: Fractional Differential Equations. Academie Press, New York (1999)

    MATH  Google Scholar 

  13. Hilfer, R.: Applications of Fractional Calculus in Physics. World Science Publishing, Singapore (2000)

    Book  MATH  Google Scholar 

  14. Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus. Springer, Dordrecht (2007)

    Book  MATH  Google Scholar 

  15. Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, Berlin (2014)

    MATH  Google Scholar 

  16. Machado, J.A.T.: Fractional order modelling of fractional-order holds. Nonlinear Dyn. 70(1), 789–796 (2012)

    Article  MathSciNet  Google Scholar 

  17. Victor, S., Malti, R., Garnier, H., Oustaloup, A.: Parameter and differentiation order estimation in fractional models. Automatica 49(4), 926–935 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gabano, J.D., Poinot, T.: Fractional modelling and identification of thermal systems. Signal Process. 91(3), 531–541 (2011)

    Article  MATH  Google Scholar 

  19. Bai, J., Feng, X.C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)

    Article  MathSciNet  Google Scholar 

  20. Sun, H.G., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A Stat. Mech. Appl. 388(21), 4586–4592 (2009)

    Article  Google Scholar 

  21. Cugnet, M., Sabatier, J., Laruelle, S., Grugeon, S., Sahut, B., Oustaloup, A., Tarascon, J.M.: On lead-acid-battery resistance and cranking-capability estimation. IEEE Trans. Ind. Electron. 57(3), 909–917 (2010)

    Article  Google Scholar 

  22. Sommacal, L., Melchior, P., Cabelguen, J.-M., Oustaloup, A., Ijspeert, A.J.: Fractional multi-models of the gastrocnemius frog muscle. Fract. Differ. Appl. 2(1), 254–259 (2006)

    MATH  Google Scholar 

  23. Zhao, Y., Li, Y., Chen, Y.Q.: Complete parametric identification of fractional order Hammerstein systems. In: Proceedings of International Conference on Fractional Differentiation and Its Applications, IEEE, pp. 1–6 (2014)

  24. Chen, Y.Q., Moore, K.L.: Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dyn. 29(1), 191–200 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Ma, C., Hori, Y.: Fractional-order control: theory and applications in motion control. IEEE Ind. Electron. Mag. 1(4), 6–16 (2007)

    Article  Google Scholar 

  26. Cost, A., Jos, S.D.A.: An Introduction to Fractional Control. Institution of Engineering and Technology, Stevenage (2013)

    Google Scholar 

  27. Li, Y., Chen, Y.Q., Ahn, H.S.: Convergence analysis of fractional-order iterative learning control. Control Theory Appl. 29(8), 1027–1031 (2012)

    Google Scholar 

  28. Li, Y., Chen, Y.Q., Ahn, H.S., Tian, G.: A survey on fractional-order iterative learning control. J. Optim. Theory Appl. 156(1), 127–140 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Chen, Y.Q., Moore, K.L.: On D-type iterative learning control. In: Proceedings of the 40th IEEE Conference on Decision and Control, pp. 4451–4456. USA (2001)

  30. Li, H., Huang, J., Liu, D., Zhang, J.: Design of fractional order iterative learning control on frequency domain. In: Proceedings of IEEE International Conference on Mechatronics and Automation, pp. 2056–2060 (2011)

  31. Li, Y., Chen, Y.Q., Ahn, H.S.: Fractional-order iterative learning control for fractional-order linear systems. Asian J. Control 13(1), 54–63 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Li, Y., Chen, Y.Q., Ahn, H.S.: A generalized fractional-order iterative learning control. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, pp. 5356–5361 (2011)

  33. Li, Y., Ahn, H.S., Chen, Y.Q.: Iterative learning control of a class of fractional order nonlinear systems. In: Proceedings of IEEE International Symposium on Intelligent Control, pp. 779–782. Japan (2010)

  34. Lan, Y.H.: Iterative learning control with initial state learning for fractional order nonlinear systems. Comput. Math. Appl. 64(10), 3210–3216 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yan, L., Wei, J.: Fractional order nonlinear systems with delay in iterative learning control. Appl. Math. Comput. 257, 546–552 (2015)

    MATH  MathSciNet  Google Scholar 

  36. Podlubny, I.: Fractional-order systems and \(\text{ PI }^\lambda \text{ D }^\mu \)-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)

    Article  MathSciNet  Google Scholar 

  37. Kawamura, S., Miyazaki, F., Arimoto, S.: Realization of robot motion based on a learning method. IEEE Trans. Syst. Man Cybern. 18(1), 126–134 (1988)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (61375084, 61374101) and the Natural Science Foundation of Shandong Province (ZR2015QZ08).

Author information

Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, 250061, Shandong, People’s Republic of China

    Yang Zhao, Fengyu Zhou, Yugang Wang & Yan Li

Authors
  1. Yang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fengyu Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yugang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fengyu Zhou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, Y., Zhou, F., Wang, Y. et al. Fractional-order iterative learning control with initial state learning design. Nonlinear Dyn 90, 1257–1268 (2017). https://doi.org/10.1007/s11071-017-3724-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3724-6

Keywords

Search

Navigation