{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:03:03Z","timestamp":1760241783037,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,9,9]],"date-time":"2018-09-09T00:00:00Z","timestamp":1536451200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002848","name":"Comisi\u00f3n Nacional de Investigaci\u00f3n Cient\u00edfica y Tecnol\u00f3gica","doi-asserted-by":"publisher","award":["Programa Basal AMTC FB0809; FONDECYT grants 1150488 and 11170154."],"award-info":[{"award-number":["Programa Basal AMTC FB0809; FONDECYT grants 1150488 and 11170154."]}],"id":[{"id":"10.13039\/501100002848","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Adaptive and non-adaptive minimal realization (MR) fractional order observers (FOO) for linear time-invariant systems (LTIS) of a possibly different derivation order (mixed order observers, MOO) are studied in this paper. Conditions on the convergence and robustness are provided using a general framework which allows observing systems defined with any type of fractional order derivative (FOD). A qualitative discussion is presented to show that the derivation orders of the observer structure and for the parameter adjustment are relevant degrees of freedom for performance optimization. A control problem is developed to illustrate the application of the proposed observers.<\/jats:p>","DOI":"10.3390\/a11090136","type":"journal-article","created":{"date-parts":[[2018,9,10]],"date-time":"2018-09-10T10:28:57Z","timestamp":1536575337000},"page":"136","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2367-4553","authenticated-orcid":false,"given":"Manuel A.","family":"Duarte-Mermoud","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, University of Chile, Av. Tupper, Santiago 2007, Chile"},{"name":"Advanced Mining Technology Center, University of Chile, Av. Tupper, Santiago 2007, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7123-4179","authenticated-orcid":false,"given":"Javier A.","family":"Gallegos","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, University of Chile, Av. Tupper, Santiago 2007, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4514-0037","authenticated-orcid":false,"given":"Norelys","family":"Aguila-Camacho","sequence":"additional","affiliation":[{"name":"Advanced Mining Technology Center, University of Chile, Av. Tupper, Santiago 2007, Chile"},{"name":"Department of Electricity, Universidad Tenol\u00f3gica Metropolitana, Av. Jos\u00e9 Pedro Alessandri, Santiago 1242, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9227-4505","authenticated-orcid":false,"given":"Rafael","family":"Castro-Linares","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, CINVESTAV, Av. IPN, M\u00e9xico DF 2508, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1109\/TAC.1966.1098323","article-title":"Observers for multivariable systems","volume":"AC-11","author":"Luenberger","year":"1966","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_2","first-page":"952","article-title":"Some results on controllability and observability of finite-dimensional fractional differential systems","volume":"2","author":"Matignon","year":"1996","journal-title":"Comput. Eng. Syst. 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