{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:02:50Z","timestamp":1767916970219,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T00:00:00Z","timestamp":1695945600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61640302"],"award-info":[{"award-number":["61640302"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61703224"],"award-info":[{"award-number":["61703224"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper proposes a novel robust model predictive control (RMPC) scheme for constrained linear discrete-time systems with bounded disturbance. Firstly, the adjustable error tube set, which is affected by local error and error variety rate, is introduced to overcome uncertainties and disturbances. Secondly, the auxiliary control rate associated with the cost function is designed to minimize the discrepancy between the actual system and the nominal system. Finally, a constrained deep neural network (DNN) architecture with symmetry properties is developed to address the optimal control problem (OCP) within the constrained system while conducting a thorough convergence analysis. These innovations enable more flexible adjustments of state and control tube cross-sections and significantly improve optimization speed compared to the homothetic tube MPC. Moreover, the effectiveness and practicability of the proposed optimal control strategy are illustrated by two numerical simulations. In practical terms, for 2-D systems, this approach achieves a remarkable 726.23-fold improvement in optimization speed, and for 4-D problems, it demonstrates an even more impressive 7218.07-fold enhancement.<\/jats:p>","DOI":"10.3390\/sym15101845","type":"journal-article","created":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T05:48:13Z","timestamp":1695966493000},"page":"1845","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Constrained DNN-Based Robust Model Predictive Control Scheme with Adjustable Error Tube"],"prefix":"10.3390","volume":"15","author":[{"given":"Shizhong","family":"Yang","sequence":"first","affiliation":[{"name":"College of Information and Control Engineering, Qingdao University of Technology, Qingdao 266525, China"}]},{"given":"Yanli","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Information and Control Engineering, Qingdao University of Technology, Qingdao 266525, China"}]},{"given":"Huidong","family":"Cao","sequence":"additional","affiliation":[{"name":"College of Information and Control Engineering, Qingdao University of Technology, Qingdao 266525, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"112215","DOI":"10.1016\/j.oceaneng.2022.112215","article-title":"Trajectory tracking with external disturbance of bionic underwater robot based on CPG and robust model predictive control","volume":"263","author":"Yang","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.jprocont.2018.09.003","article-title":"Robust constrained model predictive fault-tolerant control for industrial processes with partial actuator failures and interval time-varying delays","volume":"75","author":"Shi","year":"2019","journal-title":"J. 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