{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T16:44:17Z","timestamp":1764002657293,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:00:00Z","timestamp":1690156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"science and technology research project of the Henan province","award":["222102240014"],"award-info":[{"award-number":["222102240014"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, the safe optimal control method for continuous-time (CT) nonlinear safety-critical systems with asymmetric input constraints and unmatched disturbances based on the adaptive dynamic programming (ADP) is investigated. Initially, a new non-quadratic form function is implemented to effectively handle the asymmetric input constraints. Subsequently, the safe optimal control problem is transformed into a two-player zero-sum game (ZSG) problem to suppress the influence of unmatched disturbances, and a new Hamilton\u2013Jacobi\u2013Isaacs (HJI) equation is introduced by integrating the control barrier function (CBF) with the cost function to penalize unsafe behavior. Moreover, a damping factor is embedded in the CBF to balance safety and optimality. To obtain a safe optimal controller, only one critic neural network (CNN) is utilized to tackle the complex HJI equation, leading to a decreased computational load in contrast to the utilization of the conventional actor\u2013critic network. Then, the system state and the parameters of the CNN are uniformly ultimately bounded (UUB) through the application of the Lyapunov stability method. Lastly, two examples are presented to confirm the efficacy of the presented approach.<\/jats:p>","DOI":"10.3390\/e25071101","type":"journal-article","created":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:47:03Z","timestamp":1690159623000},"page":"1101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Critic Learning-Based Safe Optimal Control for Nonlinear Systems with Asymmetric Input Constraints and Unmatched Disturbances"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8238-5922","authenticated-orcid":false,"given":"Chunbin","family":"Qin","sequence":"first","affiliation":[{"name":"School of Artificial Intelligence, Henan University, Zhengzhou 450000, China"}]},{"given":"Kaijun","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Henan University, Zhengzhou 450000, China"}]},{"given":"Jishi","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Software, Henan University, Kaifeng 475000, China"}]},{"given":"Tianzeng","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Henan University, Zhengzhou 450000, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.neucom.2022.06.110","article-title":"Adaptive dynamic programming-based visual servoing control for quadrotor","volume":"504","author":"Yi","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"107681","DOI":"10.1016\/j.ast.2022.107681","article-title":"A methodology to integrate reliability into the conceptual design of safety-critical multirotor unmanned aerial vehicles","volume":"127","author":"Pollet","year":"2022","journal-title":"Aerosp. 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