{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T18:39:40Z","timestamp":1775155180900,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,17]],"date-time":"2023-03-17T00:00:00Z","timestamp":1679011200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In the context of reaching the best way to control the movement of autonomous cars linearly and angularly, making them more stable and balanced on different roads and ensuring that they avoid road obstacles, this manuscript chiefly aims to reach the optimal approach for a fractional-order PID controller (or PI\u03b3D\u03c1-controller) instead of the already classical one used to provide smooth automatic parking for electrical autonomous cars. The fractional-order PI\u03b3D\u03c1-controller is based on the particle swarm optimization (PSO) algorithm for its design, with two different approximations: Oustaloup\u2019s approximation and the continued fractional expansion (CFE) approximation. Our approaches to the fractional-order PID using the results of the PSO algorithm are compared with the classical PID that was designed using the results of the Cohen\u2013Coon, Ziegler\u2013Nichols and bacteria foraging algorithms. The scheme represented by the proposed PI\u03b3D\u03c1-controller can provide the system of the autonomous vehicle with more stable results than that of the PID controller.<\/jats:p>","DOI":"10.3390\/axioms12030306","type":"journal-article","created":{"date-parts":[[2023,3,17]],"date-time":"2023-03-17T05:36:01Z","timestamp":1679031361000},"page":"306","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["A Numerical Implementation of Fractional-Order PID Controllers for Autonomous Vehicles"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8443-8848","authenticated-orcid":false,"given":"Iqbal M.","family":"Batiha","sequence":"first","affiliation":[{"name":"Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan"},{"name":"Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman P.O. Box 346, United Arab Emirates"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7614-4727","authenticated-orcid":false,"given":"Osama Y.","family":"Ababneh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zarqa University, Zarqa 11831, Jordan"}]},{"given":"Abeer A.","family":"Al-Nana","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia"}]},{"given":"Waseem G.","family":"Alshanti","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3313-506X","authenticated-orcid":false,"given":"Shameseddin","family":"Alshorm","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan"}]},{"given":"Shaher","family":"Momani","sequence":"additional","affiliation":[{"name":"Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman P.O. Box 346, United Arab Emirates"},{"name":"Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R.F., and Tolba, M. (2020). Advances in Intelligent Systems and Computing, Proceedings of the International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019), Cairo, Egypt, 28\u201330 March 2019, Springer International Publishing.","DOI":"10.1007\/978-3-030-14118-9"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hassanien, A.E., Tolba, M.F., Elhoseny, M., and Mostafa, M. (2018). 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