{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T20:46:57Z","timestamp":1769633217130,"version":"3.49.0"},"reference-count":42,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,1]],"date-time":"2025-05-01T00:00:00Z","timestamp":1746057600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62203151"],"award-info":[{"award-number":["62203151"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["252102240132"],"award-info":[{"award-number":["252102240132"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021BS084"],"award-info":[{"award-number":["2021BS084"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific and Technological Project of Henan Province of China","award":["62203151"],"award-info":[{"award-number":["62203151"]}]},{"name":"Scientific and Technological Project of Henan Province of China","award":["252102240132"],"award-info":[{"award-number":["252102240132"]}]},{"name":"Scientific and Technological Project of Henan Province of China","award":["2021BS084"],"award-info":[{"award-number":["2021BS084"]}]},{"name":"High-level Talents Fund Project of Henan University of Technology","award":["62203151"],"award-info":[{"award-number":["62203151"]}]},{"name":"High-level Talents Fund Project of Henan University of Technology","award":["252102240132"],"award-info":[{"award-number":["252102240132"]}]},{"name":"High-level Talents Fund Project of Henan University of Technology","award":["2021BS084"],"award-info":[{"award-number":["2021BS084"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, a PD-type iterative learning control algorithm with a forgetting factor is developed for MIMO nonlinear systems with randomly varying iteration lengths, initial state shifts and disturbances. Firstly, considering the randomly varying iteration lengths, a modified tracking error is designed. Secondly, for the initial state shift and disturbances, a PD-type iterative learning control algorithm with a forgetting factor (PDILCFF) method is proposed. A contraction mapping method is exploited to obtain the convergence property of the proposed control scheme, which can guarantee that the tracking error is bounded. Considering the iteration-varying trial lengths, the proposed PDILCFF algorithm is closely related to symmetry. Symmetry can provide prior information about the system\u2019s structure and characteristics for the proposed PDILCFF method, which is helpful for designing more efficient control algorithms. On the other hand, the proposed PDILCFF method exploits system symmetries across different intervals through iterative processes to achieve accurate control and performance optimization of MIMO unknown nonlinear systems. Finally, two simulations are presented, one with a subway train tracking control system and the other with a two-degree-of-freedom robot manipulator system, to verify the effectiveness of the theoretical studies.<\/jats:p>","DOI":"10.3390\/sym17050694","type":"journal-article","created":{"date-parts":[[2025,5,4]],"date-time":"2025-05-04T20:42:37Z","timestamp":1746391357000},"page":"694","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Iterative Learning Control with Forgetting Factor for MIMO Nonlinear Systems with Randomly Varying Iteration Lengths and Disturbances"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7855-5909","authenticated-orcid":false,"given":"Genfeng","family":"Liu","sequence":"first","affiliation":[{"name":"College of Electrical Engineering, Henan University of Technology, Zhengzhou 450001, China"}]},{"given":"Yangyang","family":"Wang","sequence":"additional","affiliation":[{"name":"Information Engineering University, Zhengzhou 450001, China"}]},{"given":"Jinhao","family":"Li","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Henan University of Technology, Zhengzhou 450001, China"}]},{"given":"Qinghe","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Henan University of Technology, Zhengzhou 450001, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,1]]},"reference":[{"key":"ref_1","first-page":"123","article-title":"Bettering operation of robots by learning","volume":"1","author":"Arimoto","year":"1984","journal-title":"J. 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