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Analysis of iterative learning control for a class of fractional differential equations

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Abstract

In this paper, we discuss PD-type learning control law for linear differential equations of fractional order \(\alpha \in (1,2)\). We derive convergence results for open-loop and closed-loop iterative learning schemes with zero initial error and random but bounded initial error in the sense of \(\lambda \)-norm by utilizing properties of Mittag–Leffler functions. Numerical examples are presented to demonstrate the validity of the design methods.

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References

  1. Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2012)

    Book  Google Scholar 

  2. Diethelm, K., Ford, N.J.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006)

    MATH  Google Scholar 

  4. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

    MATH  Google Scholar 

  5. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  6. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  7. Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, HEP, New York (2011)

    Google Scholar 

  8. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  MATH  Google Scholar 

  9. Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits, Studies in Systems, Decision and Control. Springer, London (2015)

    Google Scholar 

  10. Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284, 339–408 (2002)

    Google Scholar 

  11. Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific Publishing Company, Singapore (2000)

    Book  MATH  Google Scholar 

  12. Debbouche, A., Baleanu, D.: Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, 1442–1450 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ahmad, B., Ntouyas, S.K.: On Hadamard fractional integrodifferential boundary value problems. J. Appl. Math. Comput. 47, 119–131 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Abbas, S., Benchohra, M., Rivero, M., Trujillo, J.J.: Existence and stability results for nonlinear fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations. Appl. Math. Comput. 247, 319–328 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Wang, Q., Lu, D., Fang, Y.: Stability analysis of impulsive fractional differential systems with delay. Appl. Math. Lett. 40, 1–6 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Stamova, I., Stamov, G.: Stability analysis of impulsive functional systems of fractional order. Commun. Nonlinear Sci. Numer. Simulat. 19, 702–709 (2014)

    Article  MathSciNet  Google Scholar 

  17. Ma, Q., Wang, J., Wang, R., Ke, X.: Study on some qualitative properties for solutions of a certain two-dimensional fractional differential system with Hadamard derivative. Appl. Math. Lett. 36, 7–13 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Han, Z., Li, Y., Sui, M.: Existence results for boundary value problems of fractional functional differential equations with delay. J. Appl. Math. Comput. (2015). doi:10.1007/s12190-015-0910-x

  19. Bien, Z., Xu, J.X.: Iterative Learning Control Analysis: Design, Integration and Applications. Springer, Berlin (1998)

    Book  Google Scholar 

  20. Chen, Y.Q., Wen, C.: Iterative Learning control: Convergence, Robustness and Applications. Springer, London (1999)

    Book  MATH  Google Scholar 

  21. Norrlof, M.: Iterative Learning Control: Analysis, Design, and Experiments, Linkoping Studies in Science and Technology, Dissertations, No. 653, Sweden, (2000)

  22. Ahn, H.S., Chen, Y.Q., Moore, K.L.: Iterative Learning Control. Springer, New York (2007)

    Book  MATH  Google Scholar 

  23. Xu, J.X., Panda, S.K., Lee, T.H.: Real-Time Iterative Learning Control: Design and Applications, Advances in Industrial Control. Springer, Berlin (2009)

    Google Scholar 

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

    Article  Google Scholar 

  25. de Wijdeven, J.V., Donkers, T., Bosgra, O.: Iterative learning control for uncertain systems: robust monotonic convergence analysis. Automatica 45, 2383–2391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sun, M., Wang, D., Wang, Y.: Varying-order iterative learning control against perturbed initial conditions. J. Frankl. Inst. 347, 1526–1549 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, X., Kong, X.: Nonlinear fuzzy model predictive iterative learning control for drum-type boiler-turbine system. J. Process Control 23, 1023–1040 (2013)

    Article  Google Scholar 

  29. Zhang, C., Li, J.: Adaptive iterative learning control for nonlinear pure-feedback systems with initial state error based on fuzzy approximation. J. Frankl. Inst. 351, 1483–1500 (2014)

    Article  MathSciNet  Google Scholar 

  30. Xia, Y.H., Chen, X., Romanovski, V.G.: On the linearization theorem of fenner and pinto. J. Math. Anal. Appl. 400, 439–451 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Romanovski, V., Xia, Y.H., Zhang, X.: Varieties of local integrability of analytic differential systems. J. Differ. Equ. 257, 3079–3101 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, Y., Chen, Y.Q., Ahn, H.S.: Fractional-order iterative learning control for fractional-order linear systems. Asian J. Control 13, 1–10 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Lan, Y.H., Zhou, Y.: \(D\)-type iterative learning control for fractional order linear time-delay systems, Asian. J. Control. 15, 669–677 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Liu, S., Wang, J., Wei, W.: Iterative learning control based on noninstantaneous impulsive fractional order system, J. Vib. Control, (2014), http://jvc.sagepub.com/content/early/2014/08/18/1077546314545638

  37. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  38. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag–Leffler Functions, Related Topics and Applications. Springer, New York (2014)

    Book  MATH  Google Scholar 

  39. Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag–Leffler functions and their applications. J. Appl. Math. 2011, 1–51 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, J., Fec̆kan, M., Zhou, Y.: Presentation of solutions of impulsive fractional Langevin equations and existence results. Eur. Phys. J. Special Top. 222, 1857–1874 (2013)

    Article  Google Scholar 

  41. Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag–Leffler function \(E_{\alpha ,\beta }(z)\) and its derivative. Fract. Calc. Appl. Anal. 5, 491–518 (2002). Correction: Fract. Calc. Appl. Anal. 6, 111–112 (2003)

  42. Ruan, X., Bien, Z.: Pulse compensation for \(\mathit{PD}\)-type iterative learning control against initial state shift. Int. J. Syst. Sci. 43, 2172–2184 (2012)

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Acknowledgments

The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from the editor too. This work is partially supported by NSFC (No.11201091;11261011) and Outstanding Scientific and Technological Innovation Talent Award of Education Department of Guizhou Province ([2014]240).

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Guizhou University, Guiyang, 550025, Guizhou, People’s Republic of China

    Shengda Liu, JinRong Wang & Wei Wei

  2. Key and Special Laboratory of System Optimization and Scientific Computing of Guizhou Province, Guiyang, 550025, Guizhou, People’s Republic of China

    JinRong Wang

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  1. Shengda Liu
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  2. JinRong Wang
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  3. Wei Wei
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Correspondence to JinRong Wang.

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Liu, S., Wang, J. & Wei, W. Analysis of iterative learning control for a class of fractional differential equations. J. Appl. Math. Comput. 53, 17–31 (2017). https://doi.org/10.1007/s12190-015-0955-x

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  • DOI: https://doi.org/10.1007/s12190-015-0955-x

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