Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year


IMPACT FACTOR 2017: 0.314
5-year IMPACT FACTOR: 0.462

CiteScore 2017: 0.46

SCImago Journal Rank (SJR) 2017: 0.339
Source Normalized Impact per Paper (SNIP) 2017: 0.845

Mathematical Citation Quotient (MCQ) 2017: 0.26

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 68, Issue 3

Issues

Iterative learning control with pulse compensation for fractional differential systems

Shengda Liu / JinRong Wang
  • Department of Mathematics Guizhou University Guiyang, Guizhou 550025 P. R. China
  • School of Mathematical Sciences Qufu Normal University Qufu Shandong 273165 P. R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yong Zhou
  • Department of Mathematics Xiangtan University Xiangtan, Hunan 411105 P. R. China
  • Nonlinear Analysis and Applied Mathematics (NAAM) Research Group Faculty of Science King Abdulaziz University Jeddah 21589 Saudi arabia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Michal Fečkan
  • Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and Informatics Comenius University in Bratislava Mlynská dolina 842 48 Bratislava Slovakia
  • Mathematical Institute of Slovak Academy of Sciences Štefánikova 49 814 73 Bratislava Slovakia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-05-18 | DOI: https://doi.org/10.1515/ms-2017-0125

Abstract

In this paper, we explore PD-type ILC schemes of fractional version with pulse compensation for single-input-single-output fractional differential systems. More precisely, we design a new type of pulse-based ILC schemes involving fractional derivative and sign function for a class of fractional order linear systems with initial state shift. In order to tracking discrepancy incurred by the initial state shift effectively, a new function of pulse compensation is introduced. The effectiveness of the result is illustrated by numerical simulations.

MSC 2010: Primary 26A33, 34A08; Secondary 34K35, 93B05

Keywords: P&Dα-type iterative learning control; pulse compensation; fractional differential systems; sign function

The first and second authors acknowledge the support by NNSF of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), and Foundation of Postgraduate of Guizhou Province (KYJJ2017001); the third author acknowledges the support by NNSF of China (11671339); the fourth author acknowledges the support by the Slovak Grant Agency VEGA No. 2/0153/16 and No. 1/0078/17, and by the Slovak Research and Development Agency under the contract No. APVV-14-0378.

References

  • [1]

    Ahn, H. S.—Chen, Y. Q.—Moore, K. L.: Iterative Learning Control, Springer, London, 2007.Google Scholar

  • [2]

    Arimoto, S.—Kawamura, S.—Miyazaki, F.: Bettering operation of robots by learning, J. Robotic Systems 1 (1984), 123–140.CrossrefGoogle Scholar

  • [3]

    Arimoto, S.: Mathematical theory of learning with applications to robot control. In: Adaptive and Learning Systems: Theory and Applications, (K. S. Narendra, ed.), Yale University, New Haven, Connecticut, USA, 1985, pp. 379–388.Google Scholar

  • [4]

    Bien, Z.—Xu, J. X.: Iterative Learning Control Analysis: Design, Integration and Applications, Springer, New York, 1998.Google Scholar

  • [5]

    Chen, Y. Q.,—Wen, C.: Iterative Learning Control: Convergence, Robustness and Applications, Springer-Verlag, London, 1999.Google Scholar

  • [6]

    Garh, M.—RAO, A.—Kalla, S. I.: Fractional generalization of temperature fields problems in oil strata, Mat. Bilten 30 (2006), 71–84.Google Scholar

  • [7]

    Hilfer, R.: Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.Google Scholar

  • [8]

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

  • [9]

    Kilbas, A. A.—Srivastava, H. M.—Trujillo, J. J.: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Google Scholar

  • [10]

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

  • [11]

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

  • [12]

    Li, M.—Wang, J.: Finite time stability of fractional delay differential equations, Appl. Math. Lett. 64 (2017), 170–176.Web of ScienceCrossrefGoogle Scholar

  • [13]

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

  • [14]

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

  • [15]

    Liu, S.—Debbouche, D.—Wang, J.: On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths, J. Comput. Appl. Math. 312 (2017), 47–57.Web of ScienceCrossrefGoogle Scholar

  • [16]

    Michalski, M. W.: Derivatives of Noninteger Order and their Applications, Dissertationes Math., Polska Akademia Nauk., Instytut Matematyczny, Warszawa, 1993.Google Scholar

  • [17]

    Miller, K. S.—Ross, B.: An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1992.Google Scholar

  • [18]

    Norrlöf, M.: Iterative Learning Control: Analysis, Design, and Experiments. Linkoping Studies in Science and Technology, Dissertations, No. 653, Sweden, 2000.Google Scholar

  • [19]

    Porter, B.—Mohamed, S. S.: Iterative learning control of partially irregular multivariable plants with initial state shifting, Internat. J. Systems Sci. 22 (1991), 229–235.CrossrefGoogle Scholar

  • [20]

    Ruan, X.—Bien, Z.: Pulse compensation for PD-type iterative learning control against initial state shift, Internat. J. Systems Sci. 43 (2012), 2172–2184.CrossrefGoogle Scholar

  • [21]

    Ruan, X.—Bien, Z. Z.—Wang, Q.: Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue-p norm, IET Control Theory Appl. 6 (2012), 707–714.Web of ScienceCrossrefGoogle Scholar

  • [22]

    Ruan, X.—Zhao, J.: Convergence monotonicity and speed comparison of iterative learning control algorithms for nonlinear systems, IMA J. Math. Control Inform. 30 (2013), 473–486.CrossrefGoogle Scholar

  • [23]

    Uchiyama, M.: Formulation of high-speed motion pattern of a mechanical arm by trial, Trans. Soc. Instrum. Contr. Eng. 14 (1978), 706–712.CrossrefGoogle Scholar

  • [24]

    Wang, J.—Fečkan, M.—Zhou, Y.: Presentation of solutions of impulsive fractional Langevin equations and existence results, Eur. Phys. J. Special Topics 222 (2013), 1857–1874.CrossrefWeb of ScienceGoogle Scholar

  • [25]

    Wang, J.—Fečkan, M.—Zhou, Y.: A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal. 19 (2016), 806–831.Google Scholar

  • [26]

    Wang, J.—Fečkan, M.—Zhou, Y.: Center stable manifold for planar fractional damped equations, Appl. Math. Comput. 296 (2017), 257–269.Web of ScienceGoogle Scholar

  • [27]

    Wang, J.—Ibrahim, A. G.—Fečkan, M.: Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces, Appl. Math. Comput. 257 (2015), 103–118.Web of ScienceGoogle Scholar

  • [28]

    Wang, J.—Li, X.—Fečkan, M.—Zhou, Y.: Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal. 92 (2013), 2241–2253.Web of ScienceCrossrefGoogle Scholar

  • [29]

    Wang, J.—Lv, L.—Zhou, Y.: New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 2530–2538.CrossrefWeb of ScienceGoogle Scholar

  • [30]

    Wang, J.—Zhang, Y.: On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett. 39 (2015), 85–90.Web of ScienceCrossrefGoogle Scholar

  • [31]

    Wang, J.—Zhou, Y.—Wei, W.: Fractional Schrödinger equations with potential and optimal controls, Nonlinear Anal. Real World Appl. 13 (2012), 2755–2766.CrossrefGoogle Scholar

  • [32]

    Wang, J.—Zhou, Y.—Wei, W.—Xu, H.: Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls, Comput. Math. Appl. 62 (2011), 1427–1441.CrossrefGoogle Scholar

  • [33]

    Wang, Y.—Gao, F.—Doyle III, F. J.: Survey on iterative learning control, repetitive control, and run-to-run control, J. Process Control 19 (2009), 1589–1600.CrossrefWeb of ScienceGoogle Scholar

  • [34]

    De Wijdeven, J. V.—Donkers, T.—Bosgra, O.: Iterative learning control for uncertain systems: Robust monotonic convergence analysis, Automatica 45 (2009), 2383–2391.CrossrefWeb of ScienceGoogle Scholar

  • [35]

    Xu, J. X.: Analysis of iterative learning control for a class of nonlinear discrete-time systems, Automatica 33 (1997), 1905–1907.Google Scholar

  • [36]

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

  • [37]

    Xu, J. X.—Panda, S. K.—Lee, T. H.: Real-Time Iterative Learning Control: Design and Applications. Adv. Indust. Control, Springer, London, 2009.Google Scholar

About the article


Received: 2016-05-12

Accepted: 2017-03-01

Published Online: 2018-05-18

Published in Print: 2018-06-26


Citation Information: Mathematica Slovaca, Volume 68, Issue 3, Pages 563–574, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0125.

Export Citation

© 2018 Mathematical Institute Slovak Academy of Sciences.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Jing Zhao, Sen Jiang, Fei Xie, Zhen He, and Jian Fu
Mathematical Problems in Engineering, 2018, Volume 2018, Page 1

Comments (0)

Please log in or register to comment.
Log in