{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:52:53Z","timestamp":1760244773513,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"24","license":[{"start":{"date-parts":[[2022,12,7]],"date-time":"2022-12-07T00:00:00Z","timestamp":1670371200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"FCT, through IDMEC, under LAETA","doi-asserted-by":"publisher","award":["UIDB\/50022\/2020"],"award-info":[{"award-number":["UIDB\/50022\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Reset control is a simple non-linear control technique that can help overcome the structural limitations of linear control. Fractional control uses the concept of fractional derivatives to expand the range of possibilities when modeling a controller, making it more robust. Fractional reset control merges the advantages of both areas and is the object of this paper. Fractional-order versions of different reset controllers were implemented, namely a fractional Clegg integrator, a fractional generalized first-order reset element, a fractional generalized second-order reset element, and fractional \u201cconstant in gain lead in phase\u201d controllers with first- and second-order reset elements. These were computed directly from a numerical implementation of the Gr\u00fcnwald\u2013Letnikov definition of fractional derivatives, and their performances were analyzed.<\/jats:p>","DOI":"10.3390\/math10244630","type":"journal-article","created":{"date-parts":[[2022,12,7]],"date-time":"2022-12-07T05:50:52Z","timestamp":1670392252000},"page":"4630","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Generalization of Reset Controllers to Fractional Orders"],"prefix":"10.3390","volume":"10","author":[{"given":"Henrique","family":"Paz","sequence":"first","affiliation":[{"name":"IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9388-4308","authenticated-orcid":false,"given":"Duarte","family":"Val\u00e9rio","sequence":"additional","affiliation":[{"name":"IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/S0947-3580(00)70906-X","article-title":"Limitations on Control System Performance","volume":"6","year":"2000","journal-title":"Eur. J. Control"},{"key":"ref_2","unstructured":"Goodwin, G.C., Graebe, S.F., and Salgado, M.E. (2001). Control System Design, Prentice-Hall."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Seron, M.M., Braslavsky, J.H., and Goodwin, G.C. (1997). Fundamental Limitations in Filtering and Control, Springer.","DOI":"10.1007\/978-1-4471-0965-5"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Beker, O., Hollot, C.V., and Chait, Y. (2001, January 25\u201327). Plant with integrator: An example of reset control overcoming limitations of linear feedback. Proceedings of the 2001 American Control Conference, Arlington, VA, USA.","DOI":"10.1109\/ACC.2001.945941"},{"key":"ref_5","unstructured":"\u00c5str\u00f6m, K.J., and Murray, R.M. (2008). Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press."},{"key":"ref_6","first-page":"41","article-title":"A nonlinear integrator for Servomechanisms","volume":"77","author":"Clegg","year":"1958","journal-title":"Trans. Am. Inst. Electr. Eng. Part II Appl. Ind."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1446","DOI":"10.1109\/TCST.2008.2009066","article-title":"Frequency-domain properties of reset systems with application in hard-disk-drive systems","volume":"17","author":"Guo","year":"2009","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_8","unstructured":"Hazeleger, L., Heertjes, M., and Nijmeijer, H. (2006, January 14\u201316). Second-order reset elements for Stage Control Design. Proceedings of the 2006 American Control Conference (ACC), Minneapolis, MN, USA."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1176","DOI":"10.1109\/TMECH.2019.2909082","article-title":"\u201cConstant in gain lead in phase\u201d element\u2014Application in Precision Motion Control","volume":"24","author":"Saikumar","year":"2019","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Val\u00e9rio, D., and S\u00e1 da Costa, J. (2013). An Introduction to Fractional Control, Institution of Engineering and Technology.","DOI":"10.1049\/PBCE091E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/s11760-012-0336-y","article-title":"Fractional reset control","volume":"6","year":"2012","journal-title":"Signal Image Video Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1016\/j.mechatronics.2013.03.005","article-title":"Fractional-order Reset Control: Application to a servomotor","volume":"23","author":"HosseinNia","year":"2013","journal-title":"Mechatronics"},{"key":"ref_13","unstructured":"Monje Concepci\u00f3n, A. (2010). Fractional-Order Systems and Controls: Fundamentals and Applications, Springer."},{"key":"ref_14","unstructured":"Saikumar, N., and HosseinNia, H. (2017, January 21\u201330). Generalized Fractional Order Reset Element (GFrORE). Proceedings of the 9th European Nonlinear Dynamics Conference, Budapest, Hungary."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1404","DOI":"10.1109\/TCST.2019.2913534","article-title":"Development of Robust Fractional-Order Reset Control","volume":"28","author":"Chen","year":"2020","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s11071-020-06138-9","article-title":"Fractional-order single state reset element","volume":"104","author":"Karbasizadeh","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_17","unstructured":"Oustaloup, A., Mathieu, B., and Melchior, P.L. (1991). Commande CRONE: Commande Robuste d\u2019Ordre Non Entier, Herm\u00e8s."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Val\u00e9rio, D., Ortigueira, M.D., and Lopes, A.M. (2022). How Many Fractional Derivatives Are There?. Mathematics, 10.","DOI":"10.3390\/math10050737"},{"key":"ref_19","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_20","unstructured":"Miller, K., and Ross, B.A. (1993). Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.jcp.2014.07.019","article-title":"What is a fractional derivative?","volume":"293","author":"Ortigueira","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ortigueira, M.D., and Tenreiro Machado, J.A. (2017). Which Derivative?. Fractal Fract., 1.","DOI":"10.3390\/fractalfract1010003"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.matcom.2018.05.003","article-title":"Recent progress in impulsive control systems","volume":"155","author":"Yang","year":"2017","journal-title":"Math. Comput. Simul."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/10\/24\/4630\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:35:40Z","timestamp":1760146540000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/10\/24\/4630"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,7]]},"references-count":23,"journal-issue":{"issue":"24","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["math10244630"],"URL":"https:\/\/doi.org\/10.3390\/math10244630","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2022,12,7]]}}}