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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 5157267, 10 pages
Research Article

Lebesgue-p Norm Convergence Analysis of PDα-Type Iterative Learning Control for Fractional-Order Nonlinear Systems

Department of Applied Mathematics, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

Correspondence should be addressed to Lei Li; moc.621@7836il_iel

Received 28 September 2017; Revised 2 January 2018; Accepted 4 February 2018; Published 1 March 2018

Academic Editor: Qin Sheng

Copyright © 2018 Lei Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The first-order and second-order PDα-type iterative learning control (ILC) schemes are considered for a class of Caputo-type fractional-order nonlinear systems. Due to the imperfection of the -norm, the Lebesgue-p () norm is adopted to overcome the disadvantage. First, a generalization of the Gronwall integral inequality with singularity is established. Next, according to the reached generalized Gronwall integral inequality and the generalized Young inequality, the monotonic convergence of the first-order PDα-type ILC is investigated, while the convergence of the second-order PDα-type ILC is analyzed. The resultant condition shows that both the learning gains and the system dynamics affect the convergence. Finally, numerical simulations are exploited to verify the results.