{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T22:25:18Z","timestamp":1781821518686,"version":"3.54.5"},"reference-count":28,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T00:00:00Z","timestamp":1720656000000},"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>Rubidium atomic clocks have been used extensively in various fields, with applications such as a core component of Global Navigation Satellite Systems (GNSS). However, they exhibit inherently poor long-term stability. This paper presents the development of a control system for rubidium atomic clocks. It introduces an adaptive Kalman filtering algorithm for the disciplining of a rubidium atomic clock, utilizing autocovariance least squares (ALS) to estimate the clock\u2019s noise parameters. The experimental results demonstrate that the proposed algorithm achieves a high estimation accuracy. The standard deviation of the clock error between the steered rubidium atomic clock 1 Pulse Per Second (1PPS) and Coordinated Universal Time (UTC) provided by the National Time Service Center (NTSC) is better than 2.568 nanoseconds(ns), with peak-to-peak values improving to within 11.358 ns. Notably, its frequency stability is reduced to 3.06 \u00d7 10\u221213 @100,000 s. The results for the rubidium atomic clock demonstrate that the adaptive Kalman filtering algorithm proposed herein constitutes an accurate and effective control strategy for the rubidium atomic clock discipline.<\/jats:p>","DOI":"10.3390\/s24144495","type":"journal-article","created":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T15:59:48Z","timestamp":1720713588000},"page":"4495","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Disciplining a Rubidium Atomic Clock Based on Adaptive Kalman Filter"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-1139-8627","authenticated-orcid":false,"given":"Kun","family":"Liu","sequence":"first","affiliation":[{"name":"National Time Service Center, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Xi\u2019an 710600, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaolong","family":"Guan","sequence":"additional","affiliation":[{"name":"National Time Service Center, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Xi\u2019an 710600, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoqian","family":"Ren","sequence":"additional","affiliation":[{"name":"National Time Service Center, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Xi\u2019an 710600, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianfeng","family":"Wu","sequence":"additional","affiliation":[{"name":"National Time Service Center, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Xi\u2019an 710600, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"},{"name":"Key Laboratory of Time Reference and Applications, Chinese Academy of Sciences, Xi\u2019an 710600, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Liang, Y.F., Xu, J.N., Wu, M., and Li, F.N. (2022). Analysis of the Long-Term Characteristics of BDS On-Orbit Satellite Atomic Clock: Since BDS-3 Was Officially Commissioned. Remote Sens., 14.","DOI":"10.3390\/rs14184535"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"738","DOI":"10.1109\/TUFFC.2010.1472","article-title":"The In-Orbit Performances of GIOVE Clocks","volume":"57","author":"Waller","year":"2010","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10291-020-01059-x","article-title":"Past, present and future of atomic clocks for GNSS","volume":"25","author":"Jaduszliwer","year":"2021","journal-title":"Gps Solut."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mallette, L.A., Rochat, P., and White, J. (2008, January 1\u20138). An introduction to satellite based atomic frequency standards. Proceedings of the 2008 IEEE Aerospace Conference, Big Sky, MT, USA.","DOI":"10.1109\/AERO.2008.4526366"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1007\/s10291-022-01239-x","article-title":"Advanced space rubidium atomic frequency standard for satellite navigation","volume":"26","author":"Bandi","year":"2022","journal-title":"Gps Solut."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"064101","DOI":"10.1103\/PhysRevLett.109.064101","article-title":"Self-Organized Synchronization in Decentralized Power Grids","volume":"109","author":"Rohden","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2299","DOI":"10.1109\/TPWRD.2016.2521318","article-title":"Utilization of Chip-Scale Atomic Clock for Synchrophasor Measurements","volume":"31","author":"Zhan","year":"2016","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_8","first-page":"17","article-title":"Development and application of an autonomous time synchronization sensor device using a chip scale atomic clock","volume":"219","author":"Kurata","year":"2018","journal-title":"Sens. Transducers"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1038\/d41586-020-03510-y","article-title":"Quantum engineering for optical clocks","volume":"588","author":"Lisdat","year":"2020","journal-title":"Nature"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2553","DOI":"10.1093\/mnras\/stad613","article-title":"Research on the joint timekeeping of pulsars and atomic clocks based on Vondrak\u2013Cepek filtering","volume":"521","author":"Liu","year":"2023","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Petrov, A.A., and Davydov, V.V. (2015, January 26\u201328). Improvement Frequency Stability of Caesium Atomic Clock for Satellite Communication System. Proceedings of the 15th International Conference on Next-Generation Wired\/Wireless Advanced Networks and Systems (NEW2AN) and 8th Conference on Internet of Things and Smart Spaces (ruSMART), St Petersburg, Russia.","DOI":"10.1007\/978-3-319-23126-6_68"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Schiller, S., Tino, G., Lemonde, P., Sterr, U., Lisdat, C., G\u00f6rlitz, A., Poli, N., Nevsky, A., and Salomon, C. (2017, January 20). The space optical clocks project. Proceedings of the International Conference on Space Optics\u2014ICSO 2010, Rhodes Island, Greece.","DOI":"10.1117\/12.2309166"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1049\/el.2011.0389","article-title":"High-performance laser-pumped rubidium frequency standard for satellite navigation","volume":"47","author":"Bandi","year":"2011","journal-title":"Electron. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5530\/bems.9.1.1","article-title":"A comprehensive overview of atomic clocks and their applications","volume":"9","author":"Bandi","year":"2023","journal-title":"Biol. Eng. Med. Sci. Rep."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wu, Y.Y., Qin, F.J., Ding, Z.C., Xu, R., and Li, D.Y. (2024). Research on the Frequency Stabilization System of an External Cavity Diode Laser Based on Rubidium Atomic Modulation Transfer Spectroscopy Technology. Photonics, 11.","DOI":"10.3390\/photonics11040298"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1500507","DOI":"10.1109\/TIM.2023.3348883","article-title":"Realization of a Rubidium Atomic Frequency Standard with Short-Term Stability in 10\u201314 \u03c4-1\/2 Level","volume":"73","author":"Cui","year":"2024","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Allan, D.W., and Weiss, M.A. (1980, January 28\u201330). Accurate time and frequency transfer during common-view of a GPS satellite. Proceedings of the 34th Annual Symposium on Frequency Control, Philadelphia, PA, USA.","DOI":"10.1109\/FREQ.1980.200424"},{"key":"ref_18","first-page":"980","article-title":"Research on rubidium atomic clock disciplined method based on GPS receiver","volume":"27","author":"Fan","year":"2013","journal-title":"J. Electron. Meas. Instrum."},{"key":"ref_19","unstructured":"Stein, S. (June, January 31). Kalman filter analysis of precision clocks with real-time parameter estimation. Proceedings of the 43rd Annual Symposium on Frequency Control, Denver, CO, USA."},{"key":"ref_20","unstructured":"Hutsell, S.T. (1995, January 29). Relating the Hadamard variance to MCS Kalman filter clock estimation. Proceedings of the 27th Annual Precise Time and Time Interval Systems and Applications Meeting, San Diego, CA, USA."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1109\/TUFFC.2005.1509784","article-title":"Enhancements to GPS operations and clock evaluations using a \u201ctotal\u201d Hadamard deviation","volume":"52","author":"Howe","year":"2005","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_22","first-page":"90","article-title":"An Algorithm for the Rubidium Atomic Clock Control Based on the Kalman Filter","volume":"36","author":"Fan","year":"2015","journal-title":"J. Astronaut."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Akhlaghi, S., Zhou, N., and Huang, Z. (2017, January 16\u201320). Adaptive adjustment of noise covariance in Kalman filter for dynamic state estimation. Proceedings of the 2017 IEEE Power & Energy Society General Meeting, Chicago, IL, USA.","DOI":"10.1109\/PESGM.2017.8273755"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.automatica.2005.09.006","article-title":"A new autocovariance least-squares method for estimating noise covariances","volume":"42","author":"Odelson","year":"2006","journal-title":"Automatica"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1016\/j.jprocont.2007.11.003","article-title":"A generalized autocovariance least-squares method for Kalman filter tuning","volume":"18","author":"Poulsen","year":"2008","journal-title":"J. Process Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1048","DOI":"10.1109\/PROC.1978.11080","article-title":"Characterization of phase and frequency instabilities in precision frequency sources: Fifteen years of progress","volume":"66","author":"Rutman","year":"1978","journal-title":"Proc. IEEE"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"080201","DOI":"10.7498\/aps.64.080201","article-title":"A new noise covariance matrix estimation method of Kalman filter for satellite clock errors","volume":"64","author":"Lin","year":"2015","journal-title":"Acta Phys. Sin."},{"key":"ref_28","unstructured":"Updegraff, R. (1989, January 10\u201313). Precise time and frequency control using the Navstar GPS and a disciplined rubidium oscillator. Proceedings of the 1989 Second International Conference on Frequency Control and Synthesis, Leicester, UK."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/14\/4495\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:15:18Z","timestamp":1760109318000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/14\/4495"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,11]]},"references-count":28,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["s24144495"],"URL":"https:\/\/doi.org\/10.3390\/s24144495","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,11]]}}}