{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T21:00:49Z","timestamp":1765486849230,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T00:00:00Z","timestamp":1672790400000},"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":["62163035","2021D01D10","XJDX1401","ZYYD2022A05"],"award-info":[{"award-number":["62163035","2021D01D10","XJDX1401","ZYYD2022A05"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Project of Natural Science Foundation of Xinjiang","award":["62163035","2021D01D10","XJDX1401","ZYYD2022A05"],"award-info":[{"award-number":["62163035","2021D01D10","XJDX1401","ZYYD2022A05"]}]},{"name":"Xinjiang Key Laboratory of Applied Mathematics","award":["62163035","2021D01D10","XJDX1401","ZYYD2022A05"],"award-info":[{"award-number":["62163035","2021D01D10","XJDX1401","ZYYD2022A05"]}]},{"name":"Special Project for Local Science and Technology Development","award":["62163035","2021D01D10","XJDX1401","ZYYD2022A05"],"award-info":[{"award-number":["62163035","2021D01D10","XJDX1401","ZYYD2022A05"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Due to the widespread presence of disturbances in practical engineering and widespread applications of high-order systems, this paper first pays attention to a class of high-order strict-feedback nonlinear systems subject to bounded disturbance and investigates the prescribed performance tracking control and anti-disturbance control problems. A novel composite control protocol using the technique of a disturbance observer\u2014prescribed performance control\u2014is designed using the back-stepping method. The disturbance observer is introduced for estimating and compensating for unknown disturbances in each step, and the prescribed performance specifications guarantee both transient and steady-state performance of the tracking error to improve the control performance and result in better disturbance rejection. Moreover, the technique of adding a power integrator is modified to tackle controller design problems for the high-order systems. The Lyapunov function method is utilized for rigorous stability analysis. It is revealed that while the control performance completely remains in the prescribed bound, all states in the closed-loop system are input-to-state stable, and the tracking error and the disturbances estimating error asymptotically converge to zero simultaneously. Then, the feasibility and effectiveness of the proposed control protocol are verified by a simulation result.<\/jats:p>","DOI":"10.3390\/e25010103","type":"journal-article","created":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T04:08:07Z","timestamp":1672805287000},"page":"103","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Prescribed Performance Back-Stepping Tracking Control for a Class of High-Order Nonlinear Systems via a Disturbance Observer"],"prefix":"10.3390","volume":"25","author":[{"given":"Xinrui","family":"Tang","sequence":"first","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haijun","family":"Jiang","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"26","DOI":"10.2514\/2.4029","article-title":"Nonlinear flight control using neural networks","volume":"20","author":"Kim","year":"1997","journal-title":"J. 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