{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T04:52:47Z","timestamp":1772167967667,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,11]],"date-time":"2022-01-11T00:00:00Z","timestamp":1641859200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper is devoted to identifying parameters of fractional order noises with application to noises obtained from MEMS accelerometer. The analysis and parameters estimation will be based on the Triple Estimation algorithm, which can simultaneously estimate state, fractional order, and parameter estimates. The capability of the Triple Estimation algorithm to fractional noises estimation will be confirmed by the sets of numerical analyses for fractional constant and variable order systems with Gaussian noise input signal. For experimental data analysis, the MEMS sensor SparkFun MPU9250 Inertial Measurement Unit (IMU) was used with data obtained from the accelerometer in x, y and z-axes. The experimental results clearly show the existence of fractional noise in this MEMS\u2019 noise, which can be essential information in the design of filtering algorithms, for example, in inertial navigation.<\/jats:p>","DOI":"10.3390\/s22020527","type":"journal-article","created":{"date-parts":[[2022,1,11]],"date-time":"2022-01-11T20:33:04Z","timestamp":1641933184000},"page":"527","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["MEMS Accelerometer Noises Analysis Based on Triple Estimation Fractional Order Algorithm"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7123-7708","authenticated-orcid":false,"given":"Michal","family":"Macias","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\/0000-0002-3700-3665","authenticated-orcid":false,"given":"Dominik","family":"Sierociuk","sequence":"additional","affiliation":[{"name":"Institute of Control and Industrial Electronics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8644-2552","authenticated-orcid":false,"given":"Wiktor","family":"Malesza","sequence":"additional","affiliation":[{"name":"Institute of Control and Industrial Electronics, Warsaw University of Technology, ul. 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Mag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1039\/b211229a","article-title":"MEMS technology in analytical chemistry","volume":"128","author":"Deisingh","year":"2003","journal-title":"Analyst"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1109\/MAES.2007.4385708","article-title":"MEMs for aerospace navigation","volume":"22","author":"Leclerc","year":"2007","journal-title":"IEEE Aerosp. Electron. Syst. Mag."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Valldorf, J., and Gessner, W. (2007). MEMS gyroscopes for automotive applications. Advanced Microsystems for Automotive Applications 2007, Springer. VDI-Buch Series.","DOI":"10.1007\/978-3-540-71325-8"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1109\/MCS.2021.3092855","article-title":"An Introduction to Inertial Navigation From the Perspective of State Estimation [Focus on Education]","volume":"41","author":"Goel","year":"2021","journal-title":"IEEE Control. Syst. Mag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1640","DOI":"10.1109\/5.704269","article-title":"Micromachined inertial sensors","volume":"86","author":"Yazdi","year":"1998","journal-title":"Proc. IEEE"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"55352","DOI":"10.1109\/ACCESS.2021.3071435","article-title":"Double Recurrent Perturbation Fuzzy Neural Network Fractional-Order Sliding Mode Control of Micro Gyroscope","volume":"9","author":"Chen","year":"2021","journal-title":"IEEE Access"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Mohd-Yasin, F., Nagel, D.J., and Korman, C.E. (2010). Noise in MEMS. Meas. Sci. Technol., 21.","DOI":"10.1088\/0957-0233\/21\/1\/012001"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.cnsns.2018.12.001","article-title":"Mathematical modeling of MEMS elements subjected to external forces, temperature and noise, taking account of coupling of temperature and deformation fields as well as a nonhomogenous material structure","volume":"72","author":"Krysko","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"107941","DOI":"10.1016\/j.ymssp.2021.107941","article-title":"The impact of thermal-noise on bifurcation MEMS sensors","volume":"161","author":"Qiao","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_12","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_13","unstructured":"Samko, S., Kilbas, A., and Maritchev, O. (1987). Fractional Integrals and Derivative. Theory and Applications, Gordon & Breach Sci. Publishers."},{"key":"ref_14","unstructured":"Miller, K., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differenctial Equations, John Wiley & Sons Inc."},{"key":"ref_15","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_16","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_17","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_18","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies), Elsevier Science Inc."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Das, S. (2011). Introduction to Fractional Calculus. Functional Fractional Calculus, Springer.","DOI":"10.1007\/978-3-642-20545-3"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Anastassiou, G.A. (2021). Generalized Fractional Calculus, Springer.","DOI":"10.1007\/978-3-030-56962-4"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yang, X.J. (2019). General Fractional Derivatives: Theory, Methods and Applications, Chapman and Hall\/CRC.","DOI":"10.1201\/9780429284083"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Tarasov, V.E. (2018). Generalized Memory: Fractional Calculus Approach. Fractal Fract., 2.","DOI":"10.3390\/fractalfract2040023"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","unstructured":"Babiarz, A., Czornik, A., Klamka, J., and Niezabitowski, M. (2017). Modeling Heat Transfer Process in Grid-Holes Structure Changed in Time Using Fractional Variable Order Calculus. Theory and Applications of Non-Integer Order Systems, Springer International Publishing.","DOI":"10.1007\/978-3-319-45474-0"},{"key":"ref_25","unstructured":"Reyes-Melo, M., Martinez-Vega, J., Guerrero-Salazar, C., and Ortiz-Mendez, U. (2004, January 5\u20139). Application of fractional calculus to modelling of relaxation phenomena of organic dielectric materials. Proceedings of the 2004 IEEE International Conference on Solid Dielectrics, Toulouse, France."},{"key":"ref_26","first-page":"583","article-title":"Some applications of fractional order calculus","volume":"58","author":"Dzielinski","year":"2010","journal-title":"Bull. Pol. Acad. Sci. Tech. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ortigueira, M.D., and Val\u00e9rio, D. (2020). Fractional Signals and Systems, De Gruyter.","DOI":"10.1515\/9783110624588"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Sheng, H., Chen, Y., and Qiu, T. (2012). Fractional Processes and Fractional-Order Signal Processing, Springer.","DOI":"10.1007\/978-1-4471-2233-3"},{"key":"ref_29","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_30","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_31","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_32","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 ICFDA\u201914 International Conference on Fractional Differentiation and Its Applications, Catania, Italy.","DOI":"10.1109\/ICFDA.2014.6967452"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/978-3-319-09900-2_27","article-title":"Variable order fractional Kalman filters for estimation over lossy network","volume":"320","author":"Sierociuk","year":"2015","journal-title":"Lect. Notes Electr. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ziubinski, P., and Sierociuk, D. (2014, January 23\u201325). Improved fractional Kalman filter for variable order systems. Proceedings of the ICFDA\u201914 International Conference on Fractional Differentiation and Its Applications, Catania, Italy.","DOI":"10.1109\/ICFDA.2014.6967454"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/BF00665755","article-title":"Fractional noise","volume":"4","author":"Wyss","year":"1991","journal-title":"Found. Phys. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1177\/1077546307087438","article-title":"Fractional order signal processing of electrochemical noises","volume":"14","author":"Chen","year":"2008","journal-title":"J. Vib. Control."},{"key":"ref_37","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_38","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1016\/j.isatra.2020.01.030","article-title":"Adaptive filtering for MEMS gyroscope with dynamic noise model","volume":"101","author":"Bai","year":"2020","journal-title":"ISA Trans."},{"key":"ref_39","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_40","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., and Macias, M. (2021). Triple Estimation of Fractional Variable Order, Parameters, and State Variables Based on the Unscented Fractional Order Kalman Filter. Sensors, 21.","DOI":"10.3390\/s21238159"},{"key":"ref_41","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","volume":"48","author":"Sierociuk","year":"2017","journal-title":"Int. J. Syst. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Haykin, S. (2001). Kalman Filtering and Neural Networks, John Wiley & Sons, Inc.","DOI":"10.1002\/0471221546"},{"key":"ref_43","unstructured":"Sierociuk, D. (2021, December 10). Fractional Variable Order Derivative Simulink Toolkit. MATLAB Central File Exchange. Available online: http:\/\/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\/22\/2\/527\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:27:07Z","timestamp":1760362027000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/2\/527"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,11]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["s22020527"],"URL":"https:\/\/doi.org\/10.3390\/s22020527","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,11]]}}}