{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T10:06:26Z","timestamp":1784196386092,"version":"3.55.0"},"reference-count":37,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T00:00:00Z","timestamp":1784160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Consejo Nacional de Ciencia y Tecnolog\u00eda (CONACYT), Paraguay","award":["POSG01-14"],"award-info":[{"award-number":["POSG01-14"]}]},{"name":"the Incentive Program for the Training of Researchers in National Graduate Programs","award":["BCAS02"],"award-info":[{"award-number":["BCAS02"]}]},{"name":"the SISNI program"},{"name":"the Strategic Project","award":["ESTR01-3"],"award-info":[{"award-number":["ESTR01-3"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>High-performance sensorless operation in multiphase electric drives requires speed estimation techniques capable of providing fast dynamic response, reduced oscillatory behavior, and low implementation complexity. In this context, a sliding-mode observer (SMO) based on an exponential reaching law (ERL) is proposed for rotor speed estimation in asymmetrical six-phase induction machines operating under indirect rotor field-oriented control. Unlike conventional SMO implementations, the proposed approach avoids auxiliary low-pass filtering (LPF) stages by employing an ERL-based adaptive gain mechanism, thereby preventing the phase delay and bandwidth reduction commonly associated with LPF-based observers. As a result, the proposed observer preserves fast transient dynamics, attenuates chattering near the sliding surface, and improves the smoothness of the estimated signals. The proposed technique is particularly suitable for multiphase drive applications, where sensorless operation reduces hardware complexity and improves system reliability by eliminating mechanical speed sensors and associated wiring. A Lyapunov-based stability analysis is presented to demonstrate the convergence properties of the observer and discuss the influence of the ERL parameters on the estimation dynamics. Simulation and experimental results obtained on a real-time test bench validate the digital implementation of the proposed SMO + ERL, demonstrating improved transient tracking, smoother estimated signals, stable low-speed operation, satisfactory speed reversal performance, and effective operation under loaded conditions.<\/jats:p>","DOI":"10.3390\/s26144513","type":"journal-article","created":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T09:23:14Z","timestamp":1784193794000},"page":"4513","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sliding Mode Observer with Exponential Reaching Law for Speed Estimation of a Six-Phase Induction Machine"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8969-4402","authenticated-orcid":false,"given":"Larizza","family":"Delorme","sequence":"first","affiliation":[{"name":"Centro de Investigaci\u00f3n en Tecnolog\u00edas Hidroel\u00e9ctricas y Energ\u00eda Distribuida\u2014CITHED, Department of Electronics and Mechatronics Engineering, Facultad de Ingenier\u00eda, Universidad Nacional de Asunci\u00f3n, Luque 110948, Paraguay"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6326-8571","authenticated-orcid":false,"given":"Magno","family":"Ayala","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n en Tecnolog\u00edas Hidroel\u00e9ctricas y Energ\u00eda Distribuida\u2014CITHED, Department of Electronics and Mechatronics Engineering, Facultad de Ingenier\u00eda, Universidad Nacional de Asunci\u00f3n, Luque 110948, Paraguay"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2191-9636","authenticated-orcid":false,"given":"Osvaldo","family":"Gonzalez","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n en Tecnolog\u00edas Hidroel\u00e9ctricas y Energ\u00eda Distribuida\u2014CITHED, Department of Electronics and Mechatronics Engineering, Facultad de Ingenier\u00eda, Universidad Nacional de Asunci\u00f3n, Luque 110948, Paraguay"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9732-704X","authenticated-orcid":false,"given":"Jorge","family":"Rodas","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n en Tecnolog\u00edas Hidroel\u00e9ctricas y Energ\u00eda Distribuida\u2014CITHED, Department of Electronics and Mechatronics Engineering, Facultad de Ingenier\u00eda, Universidad Nacional de Asunci\u00f3n, Luque 110948, Paraguay"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8020-1456","authenticated-orcid":false,"given":"Ariel","family":"Fleitas","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n en Tecnolog\u00edas Hidroel\u00e9ctricas y Energ\u00eda Distribuida\u2014CITHED, Department of Electronics and Mechatronics Engineering, Facultad de Ingenier\u00eda, Universidad Nacional de Asunci\u00f3n, Luque 110948, Paraguay"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7717-4100","authenticated-orcid":false,"given":"Ra\u00fal","family":"Gregor","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n en Tecnolog\u00edas Hidroel\u00e9ctricas y Energ\u00eda Distribuida\u2014CITHED, Department of Electronics and Mechatronics Engineering, Facultad de Ingenier\u00eda, Universidad Nacional de Asunci\u00f3n, Luque 110948, Paraguay"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9117-1689","authenticated-orcid":false,"given":"Jesus C.","family":"Hernandez","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, University of Jaen, Campus Lagunillas s\/n, Edificio A3, 23071 Jaen, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1109\/TIE.2015.2434999","article-title":"Advances in Converter Control and Innovative Exploitation of Additional Degrees of Freedom for Multiphase Machines","volume":"63","author":"Levi","year":"2016","journal-title":"IEEE Trans. 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