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In order to improve the performance of the proposed controller, design parameters of the sliding surface are adjusted using the adaptation laws, based on the gradient descent technique. Eventually, a new version of particle swarm optimization is implemented to optimize the control gains with respect to the integral of time-weighted absolute errors as the objective function, and a constraint on the maximum control effort value. The introduced idea is put into practice for the under-actuated inverted pendulum system, and the procedure is described with details step by step. The time responses of cart position, pendulum angle, and control effort of the proposed method is then compared with those of several recently published references so as to evaluate the effectiveness of this scenario. <\/jats:p>","DOI":"10.1177\/0142331219868589","type":"journal-article","created":{"date-parts":[[2019,8,29]],"date-time":"2019-08-29T11:14:06Z","timestamp":1567077246000},"page":"285-294","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":22,"title":["An optimal adaptive hybrid controller for a fourth-order under-actuated nonlinear inverted pendulum system"],"prefix":"10.1177","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4249-8623","authenticated-orcid":false,"given":"Mohammad Javad","family":"Mahmoodabadi","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Sirjan University of Technology, Iran"}]},{"given":"Hamed","family":"Khoobroo Haghbayan","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Sirjan University of Technology, Iran"}]}],"member":"179","published-online":{"date-parts":[[2019,8,29]]},"reference":[{"key":"bibr1-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2010.07.015"},{"key":"bibr2-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1177\/0142331213504946"},{"volume-title":"Adaptive Control","year":"2008","author":"\u00c5str\u00f6m KJ","key":"bibr3-0142331219868589"},{"key":"bibr4-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1016\/j.isatra.2014.05.013"},{"issue":"3","key":"bibr5-0142331219868589","first-page":"1","volume":"2","author":"Bisheban M","year":"2013","journal-title":"Applied and Computational Mathematics"},{"key":"bibr6-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2012.02.172"},{"key":"bibr7-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1109\/3CA.2010.5533559"},{"key":"bibr8-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2009.03.005"},{"key":"bibr9-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-014-1682-9"},{"key":"bibr10-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2011.04.032"},{"key":"bibr11-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1109\/MHS.1995.494215"},{"key":"bibr12-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2015.02.012"},{"issue":"1","key":"bibr13-0142331219868589","first-page":"1","volume":"10","author":"Gao XZ","year":"2009","journal-title":"Computer Simulation"},{"key":"bibr14-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2014.12.071"},{"key":"bibr15-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2014.05.006"},{"key":"bibr16-0142331219868589","doi-asserted-by":"publisher","DOI":"10.1049\/ip-cta:20030014"},{"key":"bibr17-0142331219868589","unstructured":"Holland JH (1968) Hierarchical descriptions of universal spaces and adaptive systems. 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