{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T16:38:40Z","timestamp":1772555920583,"version":"3.50.1"},"reference-count":14,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,10]],"date-time":"2019-05-10T00:00:00Z","timestamp":1557446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper deals with an approximation of a first derivative of a signal using a dynamic system of the first order. After formulating the problem, a proposition and a theorem are proven for a possible approximation structure, which consists of a dynamic system. In particular, a proposition based on a Lyapunov approach is proven to show the convergence of the approximation. The proven theorem is a constructive one and shows directly the suboptimality condition in the presence of noise. Based on these two results, an adaptive algorithm is conceived to calculate the derivative of a signal with convergence in infinite time. Results are compared with an approximation of the derivative using an adaptive Kalman filter (KF).<\/jats:p>","DOI":"10.3390\/a12050101","type":"journal-article","created":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T03:57:07Z","timestamp":1557719827000},"page":"101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["An Adaptive Derivative Estimator for Fault-Detection Using a Dynamic System with a Suboptimal Parameter"],"prefix":"10.3390","volume":"12","author":[{"given":"Manuel","family":"Schimmack","sequence":"first","affiliation":[{"name":"Institute of Product- and Processinnovation, Leuphana University of Lueneburg, Volgershall 1, D-21339 Lueneburg, Germany"},{"name":"Institute for Electrical Engineering and Information Technology, Kiel University, Kaiserstra\u00dfe 2, D-24143 Kiel, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3288-5280","authenticated-orcid":false,"given":"Paolo","family":"Mercorelli","sequence":"additional","affiliation":[{"name":"Institute of Product- and Processinnovation, Leuphana University of Lueneburg, Volgershall 1, D-21339 Lueneburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"9349","DOI":"10.3390\/s140509349","article-title":"Real-Time Algebraic Derivative Estimations Using a Novel Low-Cost Architecture Based on Reconfigurable Logic","volume":"14","year":"2014","journal-title":"Sensors"},{"key":"ref_2","first-page":"201","article-title":"A numerical procedure for filtering and efficient high-order signal differentiation","volume":"14","author":"Ibrir","year":"2004","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1080\/00207179208934355","article-title":"Output feedback stabilization of fully linearizable systems","volume":"56","author":"Estafandiari","year":"1992","journal-title":"Int. J. Control"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1080\/002071799220029","article-title":"Discrete-time implementation of high-gain observers for numerical differentiation","volume":"72","author":"Dabroom","year":"1999","journal-title":"Int. J. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1109\/TAC.2002.802740","article-title":"Time-varying high-gain observers for numerical differentiation","volume":"47","author":"Chitour","year":"2002","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1504\/IJMIC.2017.082951","article-title":"A sliding mode control using an extended Kalman filter as an observer for stimulus-responsive polymer fibres as actuator","volume":"27","author":"Schimmack","year":"2017","journal-title":"Int. J. Model. Identif. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1334","DOI":"10.1016\/j.mechatronics.2009.08.008","article-title":"Robust feedback linearization using an adaptive PD regulator for a sensorless control of a throttle valve","volume":"19","author":"Mercorelli","year":"2009","journal-title":"Mechatron. J. IFAC"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Mercorelli, P. (2012, January 11\u201313). A Decoupling Dynamic Estimator for Online Parameters Indentification of Permanent Magnet Three-Phase Synchronous Motors. Proceedings of the IFAC 16th Symposium on System Identification, Brussels, Belgium.","DOI":"10.3182\/20120711-3-BE-2027.00131"},{"key":"ref_9","unstructured":"Dabroom, A., and Khalil, H.K. (1997, January 12). Numerical differentiation using high-gain observers. Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, USA."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/S0005-1098(97)00209-4","article-title":"Robust exact differentiation via sliding mode technique","volume":"34","author":"Levant","year":"1998","journal-title":"Automatica"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1109\/9.948475","article-title":"Universal single-input-single-output (SISO) sliding-mode controllers with finite-time convergence","volume":"46","author":"Levant","year":"2001","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1049\/el:19960970","article-title":"Nonlinear derivative estimator","volume":"32","author":"Yu","year":"1996","journal-title":"Electron. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Levant, A. (2001, January 4\u20137). Higher order sliding modes and arbitrary-order exact robust differentiation. Proceedings of the 2001 European Control Conference (ECC), Porto, Portugal.","DOI":"10.23919\/ECC.2001.7076043"},{"key":"ref_14","unstructured":"Levant, A., Livne, M., and Yu, X. (2017, January 9\u201314). Sliding-Mode-Based Differentiation and Its Application. Proceedings of the 20th IFAC World Congress, Toulouse, France."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/12\/5\/101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:50:41Z","timestamp":1760187041000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/12\/5\/101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,10]]},"references-count":14,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["a12050101"],"URL":"https:\/\/doi.org\/10.3390\/a12050101","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,10]]}}}