{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T17:30:53Z","timestamp":1789579853432,"version":"build-2803163510"},"reference-count":45,"publisher":"Public Library of Science (PLoS)","issue":"9","license":[{"start":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T00:00:00Z","timestamp":1757462400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"Flexible spacecraft possess the ability to adapt to complex environments and use energy more efficiently, offering enhanced flexibility and stability in space missions, particularly in tasks with significant external disturbances such as deep space exploration and satellite attitude control. However, vibration suppression in flexible spacecraft remains a critical challenge. This study addresses the problem of vibration suppression in flexible spacecraft systems under external disturbances and input constraints. First, a partial differential equation (PDE) with boundary initial conditions is derived using Hamilton\u2019s principle, accurately describing the dynamic characteristics of the flexible structure. A backstepping controller based on the Nassbaum function and a disturbance observer is then designed to ensure system stability in the presence of input constraints and external disturbances. A Lyapunov function is constructed, and appropriate control parameters are selected to further guarantee system stability. Numerical simulations confirm the superiority of the proposed control method, with results showing an 80<\/mml:mn>%<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula> reduction in settling time and a 94<\/mml:mn>%<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula> decrease in peak overshoot compared to conventional PD control. The proposed scheme significantly enhances the performance and stability of flexible spacecraft systems, demonstrating its potential for improving spacecraft dynamics in challenging space environments.<\/jats:p>","DOI":"10.1371\/journal.pone.0331610","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T17:52:11Z","timestamp":1757526731000},"page":"e0331610","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":1,"title":["Active control of flexible spacecraft in orbit based on partial differential equations"],"prefix":"10.1371","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-8790-7229","authenticated-orcid":true,"given":"Bo","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ming","family":"Wen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2025,9,10]]},"reference":[{"issue":"2","key":"pone.0331610.ref001","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0094-5765(00)00163-6","article-title":"Vibration suppression of flexible spacecraft during attitude control","volume":"49","author":"G Song","year":"2001","journal-title":"Acta Astronautica."},{"issue":"2","key":"pone.0331610.ref002","doi-asserted-by":"crossref","first-page":"377","DOI":"10.2514\/1.12967","article-title":"Vibration suppression of flexible spacecraft during attitude maneuvers","volume":"28","author":"Q Hu","year":"2005","journal-title":"Journal of Guidance, Control, and Dynamics."},{"issue":"1","key":"pone.0331610.ref003","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1109\/TAES.2017.2649259","article-title":"Attitude control for flexible spacecraft with disturbance rejection","volume":"53","author":"C Zhong","year":"2017","journal-title":"IEEE Transactions on Aerospace and Electronic Systems."},{"issue":"3","key":"pone.0331610.ref004","doi-asserted-by":"crossref","DOI":"10.1063\/5.0051432","article-title":"Nonlinear vibration energy harvesting and vibration suppression technologies: designs, analysis, and applications","volume":"8","author":"T Yang","year":"2021","journal-title":"Applied Physics Reviews."},{"key":"pone.0331610.ref005","doi-asserted-by":"crossref","first-page":"109782","DOI":"10.1016\/j.rser.2020.109782","article-title":"Noise and vibration suppression in hybrid electric vehicles: state of the art and challenges","volume":"124","author":"Y Qin","year":"2020","journal-title":"Renewable and Sustainable Energy Reviews."},{"key":"pone.0331610.ref006","doi-asserted-by":"crossref","first-page":"109496","DOI":"10.1016\/j.ymssp.2022.109496","article-title":"Towards novel energy shunt inspired vibration suppression techniques: principles, designs and applications","volume":"182","author":"X Huang","year":"2023","journal-title":"Mechanical Systems and Signal Processing."},{"key":"pone.0331610.ref007","doi-asserted-by":"crossref","unstructured":"Ro K, Kuk T, Kamman J. 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