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The MRAC-FOPID controller is developed by aligning the changing of FOPID controller\u2019s five parameters with a cost function\u2019s negative gradient direction. Numerical simulations are presented and show that the proposed MRAC-FOPID controller uses 49% more control effort to achieve 72% and 40% more reductions in the relative integral squared error (RISE) and relative integral time absolute error (RITAE), respectively, compared to an optimized model reference adaptive proportional integral derivative (MRAC-PID) controller. Furthermore, the MRAC-FOPID controller is found to have better robustness than other two previously published optimal controllers, with only a 3.07% relative change in RISE when facing variable disturbance. This work also highlights a novel Matlab\/Simulink based implementation of the adaptively-varying-order derivative operators, which cannot only be extended to the design of other adaptive fractional-order controllers but may also facilitate hardware realization of the MRAC-FOPID controller in real idle speed systems.<\/jats:p>","DOI":"10.1177\/09596518241266670","type":"journal-article","created":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T02:29:40Z","timestamp":1723084180000},"page":"339-355","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal model reference adaptive fractional-order proportional integral derivative control of idle speed system under varying disturbances"],"prefix":"10.1177","volume":"239","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4170-099X","authenticated-orcid":false,"given":"Yi","family":"Yang","sequence":"first","affiliation":[{"name":"School of Engineering Technology, Purdue University, West Lafayette, IN, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haiyan Henry","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Engineering Technology, Purdue University, West Lafayette, IN, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2024,8,7]]},"reference":[{"key":"bibr1-09596518241266670","doi-asserted-by":"publisher","DOI":"10.23919\/ASCC56756.2022.9828128"},{"key":"bibr2-09596518241266670","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-810434-7.00006-5"},{"key":"bibr3-09596518241266670","doi-asserted-by":"publisher","DOI":"10.1243\/09544070360692078"},{"key":"bibr4-09596518241266670","doi-asserted-by":"publisher","DOI":"10.1109\/UKSim.2013.138"},{"key":"bibr5-09596518241266670","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-10775-7"},{"key":"bibr6-09596518241266670","doi-asserted-by":"publisher","DOI":"10.5772\/INTECHOPEN.91666"},{"key":"bibr7-09596518241266670","first-page":"533","volume":"224","author":"The SH and Nassirharand A","year":"2009","journal-title":"Proc IMechE, Part D: J Automobile Engineering"},{"key":"bibr8-09596518241266670","unstructured":"Hu Y, Gong X, Li J, et al. 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