Journal of Optimization Theory and Applications

, Volume 156, Issue 1, pp 127–140 | Cite as

A Survey on Fractional-Order Iterative Learning Control

Article

Abstract

In this paper, an overview of fractional-order iterative learning control (FOILC) is presented including main developments of this field since 2001. Many theoretical and experimental results are provided to show the advantages of FOILC such as the improvement of transient and steady-state performances. Some unique characters of fractional-order operators are illustrated to show the new features and techniques of FOILC. A number of unsolved problems are briefly presented.

Keywords

Iterative learning control Fractional calculus Convergence speed Transient and steady-state performances Digital and analog implementations 

References

  1. 1.
    Arimoto, S., Kawamura, S., Miyazaki, F.: Bettering operation of robots by learning. J. Robot. Syst. 1(2), 123–140 (1984) CrossRefGoogle Scholar
  2. 2.
    Chen, Y.Q., Moore, K.L., Ahn, H.-S.: Iterative Learning Control. Encyclopedia of the Science of Learning. Springer, New York (2012). Book chapter of Seel, N. M. (Editor in Chief) Google Scholar
  3. 3.
    Ahn, H.-S., Moore, K.L., Chen, Y.Q.: Iterative Learning Control: Robustness and Monotonic Convergence for Interval Systems. Springer, New York, London (2007) MATHGoogle Scholar
  4. 4.
    Sun, M.X., Wang, D.W.: Higher relative degree nonlinear systems with ILC using lower-order differentiations. Asian J. Control 4(1), 38–48 (2008) CrossRefGoogle Scholar
  5. 5.
    Norrlof, M.: Disturbance rejection using an ILC algorithm with iteration varying filters. Asian J. Control 6(3), 432–438 (2008) CrossRefGoogle Scholar
  6. 6.
    Lee, F.-S., Chien, C.-J., Wang, J.-C., Liu, J.-J.: Application of a model-based iterative learning technique to tracking control of a piezoelectric system. Asian J. Control 7(1), 29–37 (2008) CrossRefGoogle Scholar
  7. 7.
    Ye, Y., Wang, D.W., Zhang, B., Wang, Y.: Simple LMI based learning control design. Asian J. Control 11(1), 74–77 (2009) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Xu, J.-X.: The frontiers of iterative learning control—Part I. J. Syst. Control Inf. 46(2), 563–594 (2002) Google Scholar
  9. 9.
    Xu, J.-X.: The frontiers of iterative learning control—Part II. J. Syst. Control Inf. 46(5), 233–243 (2002) Google Scholar
  10. 10.
    Xu, J.-X., Tan, Y.: Robust optimal design and convergence properties analysis of iterative learning control approaches. Automatica 38(11), 1867–1880 (2001) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Han, C., Qu, Z.H., Kaloust, J.H.: Nonlinar iterative learning for a class of nonlinear systems based on Lyapunov’s direct method. In: Proceedings of American Control Conference, pp. 3024–3028. IEEE, New York (1995) Google Scholar
  12. 12.
    Frueh, M., Rogers, E.: Nonlinear iterative learning by an adaptive Lyapunov technique. Int. J. Control 73(10), 858–870 (2000) CrossRefGoogle Scholar
  13. 13.
    Sulikowski, B., Galkowski, K., Rogers, E., Owens, D.H.: PI Control of discrete linear repetitive processes. Automatica 42(5), 877–880 (2006) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Sulikowski, B., Galkowski, K., Rogers, E.: PI output feedback control of differential linear repetitive processes. Automatica 44(5), 1442–1445 (2008) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wang, H.B., Wang, Y.: Iterative learning control for nonlinear systems with uncertain state delay and arbitrary initial error. J. Control Theory Appl. 9(4), 541–547 (2011) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Meng, D.Y., Jia, Y.M., Du, J.P., Yuan, S.Y.: Feedback iterative learning control for time-delay systems based on 2D analysis approach. J. Control Theory Appl. 8(4), 457–463 (2010) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yin, C.K., Xu, J.X., Hou, Z.S.: A high-order internal model based iterative learning control scheme for nonlinear systems with time-iteration-varying parameters. IEEE Trans. Autom. Control 55(1), 2665–2670 (2010) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Arif, M., Ishihara, T., Inooka, H.: Incorporation of experience in iterative learning controllers using locally weighted learning. Automatica 37(6), 881–888 (2001) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Moore, K.L., Lashhab, F.: Iteration-domain closed-loop frequency response shaping for discrete-repetitive processes. In: Proceedings of American Control Conference, pp. 1284–1289. IEEE, New York (2010) Google Scholar
  20. 20.
    Fine, B.T., Mishra, S., Tomizuka, M.: Model inverse based iterative learning control using finite impulse response approximations. In: Proceedings of American Control Conference, pp. 931–936. IEEE, New York (2009) Google Scholar
  21. 21.
    Mishra, S.: Fundamental Issues in Iterative Learning Controller Design: Convergence, Robustness, and Steady State Performance. Ph.D. Dissertation, University of California, Berkeley (2008) Google Scholar
  22. 22.
    Ahn, H.-S., Chen, Y.Q., Moore, K.L.: Iterative learning control: Brief survey and categorization. IEEE Trans. Syst. Man Cybern., Part C, Appl. Rev. 37(6), 1099–1121 (2007) CrossRefGoogle Scholar
  23. 23.
    Rogers, E.: Iterative learning control—from Hilbert spaces to robotics to healthcare engineering. United Kingdom Automatic Control Conference Annual Lecture (2007). URL: http://eprints.ecs.soton.ac.uk/14560/
  24. 24.
    Le, F., Markovsky, I., Freeman, C.T., Rogers, E.: Identification of electrically stimulated muscle models of stroke patients. Control Eng. Pract. 18(4), 396–407 (2010) CrossRefGoogle Scholar
  25. 25.
    Chen, Y.Q., Moore, K.L.: On Dα-type iterative learning control. In: Proceedings of IEEE Conference on Decision and Control, vol. 5, pp. 4451–4456. IEEE, New York (2001) Google Scholar
  26. 26.
    Ye, Y., Tayebi, A., Liu, X.: All-pass filtering in iterative learning control. Automatica 45(1), 257–264 (2009) MATHCrossRefGoogle Scholar
  27. 27.
    Li, Y., Chen, Y.Q., Ahn, H.-S.: Fractional-order iterative learning control for fractional-order linear systems. Asian J. Control 13(1), 1–10 (2011) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lazarevic, M.P.: PD α-type iterative learning control for fractional LTI system. In: Proceedings of International Congress of Chemical and Process Engineering, p. 0359. Czech Society of Chemical Engineering, Praha (2004) Google Scholar
  29. 29.
    Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007) MATHGoogle Scholar
  30. 30.
    Li, Y., Chen, Y.Q., Ahn, H.-S.: A generalized fractional-order iterative learning control. In: Proceedings of IEEE Conference on Decision and Control and European Control Conference, pp. 5356–5361. IEEE, New York (2011) CrossRefGoogle Scholar
  31. 31.
    Papoulis, A.: The Fourier Integral and its Applications. McGraw-Hill, New York (1962) MATHGoogle Scholar
  32. 32.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  33. 33.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
  34. 34.
    Bien, Z., Huh, K.M.: Higher-order iterative learning control algorithm. IEE Proc., Control Theory Appl. 136(3), 105–112 (1989) MATHCrossRefGoogle Scholar
  35. 35.
    Chen, Y.Q., Wen, C.Y.: Iterative Learning Control—Convergence, Robustness and Applications. Springer, London (1999) MATHCrossRefGoogle Scholar
  36. 36.
    Ruan, X., Bien, Z., Wang, Q.: Convergence properties of iterative learning control processes in the sense of the Lebesgue-p norm. Asian J. Control 14(4), 1095–1107 (2012) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Moore, K.L., Dahleh, M., Bhattacharyya, S.P.: Iterative learning control: a survey and new results. J. Robot. Syst. 9(5), 563–594 (1992) MATHCrossRefGoogle Scholar
  38. 38.
    Moore, K.L., Chen, Y.Q., Ahn, H.-S.: Iterative learning control: A tutorial and big picture view. In: Proceedings of the 45th IEEE Conference on Decision and Control, pp. 2352–2357 (2006) CrossRefGoogle Scholar
  39. 39.
    Goh, C.J.: A frequency domain analysis of learning control. ASME J. Dyn. Syst. Meas. Control 116(4), 781–786 (1994) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Hideg, L.M., Judd, R.P.: Frequency domain analysis of learning systems. In: Proceedings of the 27th IEEE Conference on Decisions and Control, pp. 586–591 (1988) CrossRefGoogle Scholar
  41. 41.
    Lan, Y.-H., He, L.-J.: P-type iterative learning control of fractional order nonlinear time-delay systems. In: Proceedings of the 24th Chinese Control and Decision Conference, pp. 1027–1031 (2012) Google Scholar
  42. 42.
    Lazarevic, M.P.: PD α-type iterative learning control for fractional LTI system. In: Proceedings of the 4th International Carpathian Control Conference, pp. 869–872 (2003) Google Scholar
  43. 43.
    Lazarevic, M.P.: Iterative learning control for fractional linear time delay system: PI β D α type. In: Proceedings of the 17th International Congress of Chemical and Process Engineering, p. 5.19 (2006) Google Scholar
  44. 44.
    Li, Y., Chen, Y.Q., Ahn, H.-S.: Fractional order iterative learning control. In: Proceedings of the ICROS-SICE International Joint Conference, pp. 2106–3110 (2009) Google Scholar
  45. 45.
    Lazarevic, M.P.: Iterative learning feedback control for nonlinear fractional order system—PD α type. In: Proceedings of the 4th IFAC Workshop Fractional Differentiation and its Applications (2012). Paper No. 259 Google Scholar
  46. 46.
    Li, Y., Ahn, H.-S., Chen, Y.Q.: Iterative learning control of a class of fractional order nonlinear systems. In: Proceedings of the IEEE International Symposium on Intelligent Control, Part of IEEE Multi-Conference on Systems and Control, pp. 779–782 (2010) Google Scholar
  47. 47.
    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) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Li, Y., Chen, Y.Q., Ahn, H.-S.: On the PD α-type iterative learning control for the fractional-order nonlinear systems. In: Proceedings of the American Control Conference, pp. 4320–4325 (2011) Google Scholar
  49. 49.
    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
  50. 50.
    Xu, J.-X., Hou, Z.-S.: On learning control: the state of the art and perspective. Acta Autom. Sin. 31(6), 943–955 (2005) Google Scholar
  51. 51.
    Lee, H.-S., Bien, Z.: A note on convergence property of iterative learning controller with respect to sup norm. Automatica 33(8), 1591–1593 (1997) MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Li, H.S., Huang, J.C., Liu, D., Zhang, J.H., Teng, F.L.: Design of fractional order iterative learning control on frequency domain. In: Proceedings of the IEEE International Conference on Mechatronics and Automation, pp. 2056–2060 (2011) Google Scholar
  53. 53.
    Xu, M., Tan, W.: Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions. Sci. China Ser. G, Phys. Astron. 46(2), 145–157 (2003) CrossRefGoogle Scholar
  54. 54.
    Sheng, H., Chen, Y.Q., Qiu, T.S.: Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. Springer, London, New York (2012) MATHCrossRefGoogle Scholar
  55. 55.
    Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5(4), 367–386 (2002) MathSciNetMATHGoogle Scholar
  56. 56.
    Podlubny, I., Heymans, N.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45(5), 765–772 (2006) CrossRefGoogle Scholar
  57. 57.
    Zhang, F.R., Li, C.P.: Remarks on the initialization of Caputo derivative. In: Proceedings of the IEEE/ASME 8th IEEE/ASME International Conference on Mechatronics and Embedded Systems and Applications, pp. 325–329 (2012) CrossRefGoogle Scholar
  58. 58.
    Xu, J.-X., Yan, R.: On initial conditions in iterative learning control. IEEE Trans. Autom. Control 50(9), 1349–1354 (2005) MathSciNetCrossRefGoogle Scholar
  59. 59.
    Lan, Y.H.: Iterative learning control with initial state learning for fractional order nonlinear systems. Comput. Math. Appl. (2012). doi:10.1016/j.camwa.2012.03.086 Google Scholar
  60. 60.
    Song, X.N., Tejado, I., Chen, Y.Q.: Remote stabilization for fractional-order systems via communication networks. In: Proceedings of the American Control Conference, pp. 6698–6703. IEEE, New York (2010) Google Scholar
  61. 61.
    Chen, Y.Q.: Ubiquitous fractional order controls? In: Proceedings of the 2nd IFAC Symposium on Fractional Derivatives and Applications, vol. 2 (2006). doi:10.3182/20060719-3-PT-4902.00081 Google Scholar
  62. 62.
    Westerlund, S., Ekstam, L.: Capacitor theory. IEEE Trans. Dielectr. Electr. Insul. 1(5), 826–839 (1994) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Yan Li
    • 1
  • YangQuan Chen
    • 2
  • Hyo-Sung Ahn
    • 3
  • Guohui Tian
    • 1
  1. 1.School of Control Science and EngineeringShandong UniversityJinanP.R. China
  2. 2.School of EngineeringUniversity of CaliforniaMercedUSA
  3. 3.Department of MechatronicsGwangju Institute of Science and Technology (GIST)GwangjuKorea

Personalised recommendations