{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:36:12Z","timestamp":1760142972498,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,11]],"date-time":"2024-01-11T00:00:00Z","timestamp":1704931200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Foundation of Zhejiang University of Science and Technology","award":["F701101K06","2023QN046","2022C04012"],"award-info":[{"award-number":["F701101K06","2023QN046","2022C04012"]}]},{"name":"Science Foundation for Young Scholars of Zhejiang University of Science and Technology","award":["F701101K06","2023QN046","2022C04012"],"award-info":[{"award-number":["F701101K06","2023QN046","2022C04012"]}]},{"name":"Pioneer\u201d and \u201cLeading Goose","award":["F701101K06","2023QN046","2022C04012"],"award-info":[{"award-number":["F701101K06","2023QN046","2022C04012"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper addresses the robust stabilization problem of a cart\u2013pole system. The controlled dynamics of this interconnected system are deduced by following the analytic framework of Lagrangian mechanics, and the residual terms are formulated as a bias depending on the angle and angular velocity. A geometric definition of Proportional\u2013Integral\u2013Derivative (PID) control algorithm is proposed, and a Lyapunov function is explicitly constructed through two stages of variable change. Local exponential stability of the stable equilibrium is proved, and a criterion for parameter tuning is provided by ensuring an exponential decrease in the Lyapunov function. Enlarging the control parameters to infinity allows for the extension of attraction region almost to the half circle. The effectiveness of geometric PID controller and the local exponential stability of the resulting close system are verified by simulating a numerical example.<\/jats:p>","DOI":"10.3390\/sym16010094","type":"journal-article","created":{"date-parts":[[2024,1,11]],"date-time":"2024-01-11T12:56:33Z","timestamp":1704977793000},"page":"94","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Robust and Exponential Stabilization of a Cart\u2013Pendulum System via Geometric PID Control"],"prefix":"10.3390","volume":"16","author":[{"given":"Zhifei","family":"Zhang","sequence":"first","affiliation":[{"name":"Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China"},{"name":"School of Automation and Electrical Engineering, Zhejiang University of Science and Technology, Hangzhou 310023, China"}]},{"given":"Miaoxu","family":"Fang","sequence":"additional","affiliation":[{"name":"School of Automation and Electrical Engineering, Zhejiang University of Science and Technology, Hangzhou 310023, China"}]},{"given":"Minrui","family":"Fei","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China"}]},{"given":"Jinrong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Automation and Electrical Engineering, Zhejiang University of Science and Technology, Hangzhou 310023, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1016\/0005-1098(94)00148-C","article-title":"Nonlinear control of a swinging pendulum","volume":"31","author":"Chung","year":"1995","journal-title":"Automatica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1016\/j.automatica.2007.10.040","article-title":"A family of smooth controllers for swinging up a pendulum","volume":"44","author":"Aracil","year":"2008","journal-title":"Automatica"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/S0005-1098(99)00140-5","article-title":"Swinging up a pendulum by energy control","volume":"36","author":"Furuta","year":"2000","journal-title":"Automatica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1218","DOI":"10.1109\/TAC.2002.800770","article-title":"Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment","volume":"47","author":"Ortega","year":"2002","journal-title":"IEEE Trans. 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