{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T17:17:25Z","timestamp":1782926245056,"version":"3.54.5"},"reference-count":31,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,3,26]],"date-time":"2019-03-26T00:00:00Z","timestamp":1553558400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61025014, 61573365, 61174030, 61374126, and 61573366"],"award-info":[{"award-number":["61025014, 61573365, 61174030, 61374126, and 61573366"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Young Elite Scientists Sponsorship Program","award":["2016QNRC001"],"award-info":[{"award-number":["2016QNRC001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Owing to operating condition changing, physical mutation, and sudden shocks, degradation trajectories usually exhibit multi-phase features, and the abrupt jump often appears at the changing time, which makes the traditional methods of lifetime estimation unavailable. In this paper, we mainly focus on how to estimate the lifetime of the multi-phase degradation process with abrupt jumps at the change points under the concept of the first passage time (FPT). Firstly, a multi-phase degradation model with jumps based on the Wiener process is formulated to describe the multi-phase degradation pattern. Then, we attain the lifetime\u2019s closed-form expression for the two-phase model with fixed jump relying on the distribution of the degradation state at the change point. Furthermore, we continue to investigate the lifetime estimation of the degradation process with random effect caused by unit-to-unit variability and the multi-phase degradation process. We extend the results of the two-phase case with fixed parameters to these two cases. For better implementation, a model identification method with off-line and on-line parts based on Expectation Maximization (EM) algorithm and Bayesian rule is proposed. Finally, a numerical case study and a practical example of gyro are provided for illustration.<\/jats:p>","DOI":"10.3390\/s19061472","type":"journal-article","created":{"date-parts":[[2019,3,27]],"date-time":"2019-03-27T05:03:12Z","timestamp":1553662992000},"page":"1472","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Lifetime Estimation for Multi-Phase Deteriorating Process with Random Abrupt Jumps"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6678-8297","authenticated-orcid":false,"given":"Jianxun","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Automation, Xi\u2019an Research Institute of High-Tech, Xi\u2019an 710025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaosheng","family":"Si","sequence":"additional","affiliation":[{"name":"Department of Automation, Xi\u2019an Research Institute of High-Tech, Xi\u2019an 710025, China"},{"name":"The Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2452-3538","authenticated-orcid":false,"given":"Dangbo","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Automation, Xi\u2019an Research Institute of High-Tech, Xi\u2019an 710025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chen","family":"Hu","sequence":"additional","affiliation":[{"name":"Department of Automation, Xi\u2019an Research Institute of High-Tech, Xi\u2019an 710025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Changhua","family":"Hu","sequence":"additional","affiliation":[{"name":"Department of Automation, Xi\u2019an Research Institute of High-Tech, Xi\u2019an 710025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Pecht, M. 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