{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T06:09:46Z","timestamp":1774418986485,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2021,12,6]],"date-time":"2021-12-06T00:00:00Z","timestamp":1638748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Council for Automatic Control, Electronics, and Electrical Engineering, Warsaw University of Technology","award":["grant admitted in 2020"],"award-info":[{"award-number":["grant admitted in 2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In this paper, a method for states, parameters, and fractional order estimation is presented. The proposed method is an extension of the traditional dual estimation method and uses three blocks of filters with appropriate data interconnections. As the main part of the estimation algorithm, the Fractional Unscented Kalman Filter was used. The proposed Triple Estimation algorithm might be treated as a convenient tool for estimation and analysis of a wide range of dynamical systems with fractional constants or variable order nature, especially when knowledge about the identified system is very restricted and both order and system parameters are unknown. In order to show the performance of the proposed algorithm, sets of numerical results are presented.<\/jats:p>","DOI":"10.3390\/s21238159","type":"journal-article","created":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T02:48:13Z","timestamp":1638845293000},"page":"8159","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Triple Estimation of Fractional Variable Order, Parameters, and State Variables Based on the Unscented Fractional Order Kalman Filter"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3700-3665","authenticated-orcid":false,"given":"Dominik","family":"Sierociuk","sequence":"first","affiliation":[{"name":"Institute of Control and Industrial Electronics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7123-7708","authenticated-orcid":false,"given":"Michal","family":"Macias","sequence":"additional","affiliation":[{"name":"Institute of Control and Industrial Electronics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,6]]},"reference":[{"key":"ref_1","unstructured":"Miller, K., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differenctial Equations, John Wiley & Sons Inc."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., and Feliu, V. (2010). Fractional-Order Systems and Controls, Springer.","DOI":"10.1007\/978-1-84996-335-0"},{"key":"ref_3","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.sigpro.2010.08.003","article-title":"On the fractional signals and systems","volume":"91","author":"Magin","year":"2011","journal-title":"Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Guvenc, Z.B., and Machado, J.A.T. (2010). Fractional Order Model of Beam Heating Process and Its Experimental Verification. New Trends in Nanotechnology and Fractional Calculus Applications, Springer.","DOI":"10.1007\/978-90-481-3293-5"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"20120146","DOI":"10.1098\/rsta.2012.0146","article-title":"Modelling heat transfer in heterogeneous media using fractional calculus","volume":"371","author":"Sierociuk","year":"2013","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Sakrajda, P., and Wiraszka, M.S. (2018, January 28\u201331). Fractional variable-order model of heat transfer in time-varying fractal media. Proceedings of the IEEE 2018 19th International Carpathian Control Conference (ICCC), Szilvasvarad, Hungary.","DOI":"10.1109\/CarpathianCC.2018.8399691"},{"key":"ref_8","first-page":"294","article-title":"Switching Energy Loss in Fractional-Order Time-Varying Heat Diffusion Model","volume":"Volume 559","author":"Malinowska","year":"2018","journal-title":"Advances in Non-Integer Order Calculus and Its Applications. RRNR 2018. Lecture Notes in Electrical Engineering"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sakrajda, P., and S\u0142awomir Wiraszka, M. (2018, January 16\u201318). Fractional-order diffusion model for social networks. Proceedings of the International Conference on Fractional Differentiation and Its Applications (ICFDA), Amman, Jordan.","DOI":"10.2139\/ssrn.3271330"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Sheng, H., Chen, Y., and Qiu, T. (2012). Signal Processing Fractional Processes and Fractional-Order Signal Processing, Springer.","DOI":"10.1007\/978-1-4471-2233-3"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2055","DOI":"10.1007\/s00034-016-0255-1","article-title":"Dual Estimation of Fractional Variable Order Based on the Unscented Fractional Order Kalman Filter for Direct and Networked Measurements","volume":"35","author":"Sierociuk","year":"2016","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ziubinski, P., and Sierociuk, D. (2014, January 2\u20135). Improved Fractional Kalman Filter for Variable Order Systems with lossy and delayed network. Proceedings of the 2014 19th International Conference on Methods and Models in Automation and Robotics (MMAR), Midzyzdroje, Poland.","DOI":"10.1109\/MMAR.2014.6957342"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1016\/j.sigpro.2010.03.014","article-title":"Improved fractional Kalman Filter and its application to estimation over lossy networks","volume":"91","author":"Sierociuk","year":"2011","journal-title":"Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3861","DOI":"10.1007\/s00034-014-9835-0","article-title":"Fractional order estimation schemes for fractional and integer order systems with constant and variable fractional order colored noise","volume":"33","author":"Sierociuk","year":"2014","journal-title":"Circuits, Syst. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ziubinski, P., and Sierociuk, D. (2015, January 24\u201327). Fractional order noise identification with application to temperature sensor data. Proceedings of the 2015 IEEE International Symposium on Circuits and Systems (ISCAS), Lisbon, Portugal.","DOI":"10.1109\/ISCAS.2015.7169151"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"199","DOI":"10.3846\/1392-6292.2009.14.199-209","article-title":"Application of fractional sensor fusion algorithms for inertial MEMS sensing","volume":"14","author":"Romanovas","year":"2009","journal-title":"Math. Model. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Muresan, C.I., Birs, I.R., Dulf, E.H., Copot, D., and Miclea, L. (2021). A Review of Recent Advances in Fractional-Order Sensing and Filtering Techniques. Sensors, 21.","DOI":"10.3390\/s21175920"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1624","DOI":"10.3390\/e15051624","article-title":"Genetic Algorithm-Based Identification of Fractional-Order Systems","volume":"15","author":"Zhou","year":"2013","journal-title":"Entropy"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.cnsns.2018.12.003","article-title":"Variable order fractional systems","volume":"71","author":"Ortigueira","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1016\/j.sigpro.2010.04.006","article-title":"Variable-order fractional derivatives and their numerical approximations","volume":"91","author":"Valerio","year":"2011","journal-title":"Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1023\/A:1016586905654","article-title":"Variable order and distributed order fractional operators","volume":"29","author":"Lorenzo","year":"2002","journal-title":"Nonlinear Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3876","DOI":"10.1016\/j.apm.2014.12.009","article-title":"Derivation, interpretation, and analog modelling of fractional variable order derivative definition","volume":"39","author":"Sierociuk","year":"2015","journal-title":"Appl. Math. Model."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1007\/s00034-014-9895-1","article-title":"On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling","volume":"34","author":"Sierociuk","year":"2015","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Macias, M., and Sierociuk, D. (2014, January 23\u201325). An alternative recursive fractional variable-order derivative definition and its analog validation. Proceedings of the International Conference on Fractional Differentiation and its Applications, Catania, Italy.","DOI":"10.1109\/ICFDA.2014.6967452"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., and Macias, M. (2013, January 17\u201319). Equivalent switching strategy and analog validation of the fractional variable order derivative definition. Proceedings of the European Control Conference 2013 (ECC\u20192013), Zurich, Switzerland.","DOI":"10.23919\/ECC.2013.6669416"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., and Macias, M. (2013, January 10\u201313). Switching scheme, equivalence, and analog validation of the alternative fractional variable-order derivative definition. Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy.","DOI":"10.1109\/CDC.2013.6760481"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., and Macias, M. (2013, January 26\u201329). On a new definition of fractional variable-order derivative. Proceedings of the 14th International Carpathian Control Conference (ICCC), Rytro, Poland.","DOI":"10.1109\/CarpathianCC.2013.6560566"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3098","DOI":"10.1080\/00207721.2017.1365969","article-title":"Fractional variable order discrete-time systems, their solutions and properties Int","volume":"48","author":"Sierociuk","year":"2017","journal-title":"J. Syst. Sci."},{"key":"ref_29","first-page":"793","article-title":"Dual Kalman filtering methods for nonlinear prediction, smoothing, and estimation","volume":"Volume 9","author":"Mozer","year":"1997","journal-title":"Advances in Neural Information Processing Systems 9: Proceedings of the 1996 Conference"},{"key":"ref_30","first-page":"666","article-title":"Dual estimation and the unscented transformation","volume":"Volume 12","author":"Solla","year":"1999","journal-title":"Advances in Neural Information Processing Systems 12"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Haykin, S. (2001). Kalman Filtering and Neural Networks, John Wiley & Sons Inc.","DOI":"10.1002\/0471221546"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., and Macias, M. (2015, January 24\u201327). Practical analog realization of multiple order switching for recursive fractional variable order derivative. Proceedings of the 20th International Conference on Methods and Models in Automation and Robotics (MMAR), Mi\u0119dzyzdroje, Poland.","DOI":"10.1109\/MMAR.2015.7283938"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., and Macias, M. (2015). Numerical schemes for initialized constant and variable fractional-order derivatives: Matrix approach and its analog verification. J. Vib. Control.","DOI":"10.1177\/1077546314565438"},{"key":"ref_34","unstructured":"Sierociuk, D. (2021, October 10). Fractional Variable Order Derivative Simulink Toolkit. Available online: https:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/38801-fractional-variable-order-derivative-simulink-toolkit."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/23\/8159\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:40:20Z","timestamp":1760168420000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/23\/8159"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,6]]},"references-count":34,"journal-issue":{"issue":"23","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["s21238159"],"URL":"https:\/\/doi.org\/10.3390\/s21238159","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,6]]}}}