{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:06:15Z","timestamp":1760148375394,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T00:00:00Z","timestamp":1682553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Norwegian University of Science and Technology, Norway","award":["FZNS-2023-0011","42-01-09\/169\/2021-5"],"award-info":[{"award-number":["FZNS-2023-0011","42-01-09\/169\/2021-5"]}]},{"name":"Sevastopol State University","award":["FZNS-2023-0011","42-01-09\/169\/2021-5"],"award-info":[{"award-number":["FZNS-2023-0011","42-01-09\/169\/2021-5"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The problem of virtual sensor design for nonlinear systems under the disturbance is investigated. Two different mathematical techniques are used to solve the problem: the algebra of functions and the logic-dynamic approach. The first one allows obtaining a general solution while the second one produces a solution for nonlinear systems by linear algebra methods. The virtual sensors are designed to be insensitive to the disturbance based on invariant functions. They estimate the prescribed function of the original system state vector. The practical example illustrates theoretical results.<\/jats:p>","DOI":"10.3390\/sym15050993","type":"journal-article","created":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T04:36:15Z","timestamp":1682656575000},"page":"993","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Virtual Sensors for Nonlinear Discrete-Time Dynamic Systems"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4270-6872","authenticated-orcid":false,"given":"Oleg","family":"Sergiyenko","sequence":"first","affiliation":[{"name":"Engineering Institute, Universidad Autonoma de Baja California, Mexicali 21280, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5927-7117","authenticated-orcid":false,"given":"Alexey","family":"Zhirabok","sequence":"additional","affiliation":[{"name":"Department of Automation and Robotics, Far Eastern Federal University, 690990 Vladivostok, Russia"},{"name":"Institute of Marine Technology Problems, 690990 Vladivostok, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1252-260X","authenticated-orcid":false,"given":"Ibrahim A.","family":"Hameed","sequence":"additional","affiliation":[{"name":"Department of ICT and Natural Sciences, Norwegian University of Science and Technology, Larsgardsvegen, 2, 6009 Alesund, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7869-6373","authenticated-orcid":false,"given":"Ahmad Taher","family":"Azar","sequence":"additional","affiliation":[{"name":"College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Faculty of Computers and Artificial Intelligence, Benha University, Benha 13518, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander","family":"Zuev","sequence":"additional","affiliation":[{"name":"Department of Automation and Robotics, Far Eastern Federal University, 690990 Vladivostok, Russia"},{"name":"Institute of Marine Technology Problems, 690990 Vladivostok, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vladimir","family":"Filaretov","sequence":"additional","affiliation":[{"name":"Institute of Automation and Control Processes, 690041 Vladivostok, Russia"},{"name":"Department of Informatics and Control in Technical Systems, Sevastopol State University, 299053 Sevastopol, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vera","family":"Tyrsa","sequence":"additional","affiliation":[{"name":"Engineering Faculty, Mexicali, Universidad Autonoma de Baja California, Mexicali 21100, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7009-3634","authenticated-orcid":false,"given":"Ibraheem Kasim","family":"Ibraheem","sequence":"additional","affiliation":[{"name":"Research Center, the University of Mashreq, Baghdad 10001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1832","DOI":"10.1109\/JSEN.2011.2105471","article-title":"Virtual sensors for automotive engine sensors fault diagnosis in second-order sliding modes","volume":"11","author":"Ahmed","year":"2011","journal-title":"IEEE Sens. 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