{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T10:04:15Z","timestamp":1771063455378,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T00:00:00Z","timestamp":1751932800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"\u201cFunda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia\u201d","award":["LAETA\u2014UIDB\/50022\/2020\u2014"],"award-info":[{"award-number":["LAETA\u2014UIDB\/50022\/2020\u2014"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Actuators"],"abstract":"<jats:p>Underwater exploration is vital for advancing scientific understanding of marine ecosystems, biodiversity, and oceanic processes. Autonomous underwater vehicles and sensor platforms play a crucial role in continuous monitoring, but their operational endurance is often limited by energy constraints. Various control strategies have been proposed to enhance energy efficiency, including robust and optimal controllers, energy-optimal model predictive control, and disturbance-aware strategies. Recent work introduced a variable structure depth controller for a sensor platform with a variable buoyancy module, resulting in a 22% reduction in energy consumption. This paper extends that work by providing a formal stability proof for the proposed switching controller, ensuring safe and reliable operation in dynamic underwater environments. In contrast to the conventional approach used in controller stability proofs for switched systems\u2014which typically relies on the existence of multiple Lyapunov functions\u2014the method developed in this paper adopts a different strategy. Specifically, the stability proof is based on a novel analysis of the system\u2019s trajectory in the net buoyancy force-versus-depth error plane. The findings were applied to a depth-controlled sensor platform previously developed by the authors, using a well-established system model and considering physical constraints. Despite adopting a conservative approach, the results demonstrate that the control law can be implemented while ensuring formal system stability. Moreover, the study highlights how stability regions are affected by different controller parameter choices and mission requirements, namely, by determining how these aspects affect the bounds of the switching control action. The results provide valuable guidance for selecting the appropriate controller parameters for specific mission scenarios.<\/jats:p>","DOI":"10.3390\/act14070340","type":"journal-article","created":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T11:58:07Z","timestamp":1751975887000},"page":"340","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Variable Structure Depth Controller for Energy Savings in an Underwater Device: Proof of Stability"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5439-0329","authenticated-orcid":false,"given":"Jo\u00e3o","family":"Bravo Pinto","sequence":"first","affiliation":[{"name":"Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal"}]},{"given":"Jo\u00e3o","family":"Falc\u00e3o Carneiro","sequence":"additional","affiliation":[{"name":"Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal"},{"name":"Instituto de Ci\u00eancia e Inova\u00e7\u00e3o em Engenharia Mec\u00e2nica e Engenharia Industrial, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias s\/n, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8573-967X","authenticated-orcid":false,"given":"Fernando","family":"Gomes de Almeida","sequence":"additional","affiliation":[{"name":"Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal"},{"name":"Instituto de Ci\u00eancia e Inova\u00e7\u00e3o em Engenharia Mec\u00e2nica e Engenharia Industrial, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias s\/n, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6365-9492","authenticated-orcid":false,"given":"Nuno A.","family":"Cruz","sequence":"additional","affiliation":[{"name":"Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal"},{"name":"Institute for Systems and Computer Engineering, Technology and Science, INESC TEC, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Camus, L., Andrade, H., Aniceto, A.S., Aune, M., Bandara, K., Basedow, S.L., Christensen, K.H., Cook, J., Daase, M., and Dunlop, K. 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