Nonlinear Degradation Modeling and Online Prediction of Remaining Life for Equipment with Measurement Errors

doi:10.3969/j.issn.1004-132X.2026.01.016

Abstract

Abstract:

The existing online prediction methods for remaining life typically updated the drift parameters of stochastic degradation models based on Bayesian theory， while did not update the diffusion parameters. So a new method was proposed to simultaneously update both drift parameters and diffusion parameters. A stochastic degradation model was established considering multiple degradation modes， and the probability density functions of lifetime and remaining life were derived based on the first-passage-time principle. The initial parameters of the model were estimated offline by maximum likelihood method. Subsequently， the drift parameters and diffusion parameters were updated online by integrating Bayesian theory and expectation maximization algorithm. The effectiveness of the proposed method was validated by capacitor degradation data， gyroscope drift data， and aluminum alloy components crack growth data.

Key words: degradation modeling, parameter estimation, parameter updating, remaining life prediction

摘要：

现有的剩余寿命在线预测方法通常基于贝叶斯理论更新随机退化模型的漂移参数，但未更新扩散参数，为此，提出一种同时更新漂移与扩散参数的新方法。建立了考虑多种退化模式的随机退化模型，并依据首达时间原理推导出寿命及剩余寿命概率密度函数。先采用极大似然法离线估计模型的初始参数，再结合贝叶斯原理与期望最大化算法在线更新漂移参数与扩散参数。电容退化数据、陀螺仪漂移数据、铝合金构件裂纹增长数据验证了所提方法的有效性。

关键词: 退化建模, 参数估计, 参数更新, 剩余寿命预测

CLC Number:

TP17

PENG Caihua, LI Jianhua, REN Lina, JIA Shilin. Nonlinear Degradation Modeling and Online Prediction of Remaining Life for Equipment with Measurement Errors[J]. China Mechanical Engineering, 2026, 37(1): 147-161.

彭才华, 李建华, 任丽娜, 贾世琳. 带测量误差的设备非线性退化建模与剩余寿命在线预测[J]. 中国机械工程, 2026, 37(1): 147-161.

Figures/Tables 39

Fig.1 Degradation data of capacitors

Fig.2 Probability density function curves for capacitor remaining life predicted by model M1

Fig.3 Remaining life curves of the capacitor predicted by model M1

Fig.4 Mean squared error for capacitor remaining life predicted by model M1

Tab.1 Prediction errors for capacitor remaining life from model M1

方法	E_TMSE/月²	E_RMSE/月	E_MAE/月
方法Ⅰ	326.2281	24.4025	4.5532
方法Ⅱ	73.3883	0.7862	0.6248
方法Ⅲ	35.8910	0.3720	0.4600
方法Ⅳ	0.5570	0.2364	0.3793

Fig.5 Probability density function curves for capacitor remaining life predicted by model M2

Fig.6 Remaining life curves of the capacitor predicted by model M2

Fig.7 Mean squared error for capacitor remaining life predicted by model M2

Tab.2 Prediction errors for capacitor remaining life from model M2

方法	E_TMSE/月²	E_RMSE/月	E_MAE/月
方法Ⅰ	379.5832	24.6906	4.6608
方法Ⅱ	79.0720	0.9876	0.7294
方法Ⅲ	38.6987	0.4655	0.5657
方法Ⅳ	0.8733	0.2751	0.4847

Fig.8 Probability density function curves for capacitor remaining life predicted by model M3

Fig.9 Remaining life curves of the capacitor predicted by model M3

Fig.10 Mean squared error for capacitor remaining life predicted by model M3

Tab.3 Prediction errors for capacitor remaining life from model M3

方法	E_TMSE/月²	E_RMSE/月	E_MAE/月
方法Ⅰ	484.0462	42.5474	5.4540
方法Ⅱ	263.7029	3.0340	1.2960
方法Ⅲ	156.0063	1.8165	1.2115
方法Ⅳ	144.7175	1.5137	0.9737

Fig.11 Gyroscope drift data

Fig.12 Probability density function curves for gyroscope remaining life predicted by model M1

Fig.13 Remaining life curves of the gyroscope predicted by model M1

Fig.14 Mean squared error for gyroscope remaining life predicted by model M1

Tab.4 Prediction errors for gyroscope remaining life from model M1

方法	E_TMSE/h²	E_RMSE/h	E_MAE/h
方法Ⅰ	2.9603×10³	329.3539	12.9184
方法Ⅱ	2.9055×10³	306.3711	12.1036
方法Ⅲ	2.0938×10³	283.1639	11.2551
方法Ⅳ	1.5650×10³	162.3579	10.8203

Fig.15 Probability density function curves for gyroscope remaining life predicted by model M2

Fig.16 Remaining life curves of the gyroscope predicted by model M2

Fig.17 Mean squared error for gyroscope remaining life predicted by model M2

Tab.5 Prediction errors for gyroscope remaining life from model M2

方法	E_TMSE/h²	E_RMSE/h	E_MAE/h
方法Ⅰ	1.6222×10³	167.0967	10.2399
方法Ⅱ	1.6061×10³	163.4507	10.1967
方法Ⅲ	1.5426×10³	158.1633	10.0415
方法Ⅳ	1.3300×10³	142.1238	9.9852

Fig.18 Probability density function curves for gyroscope remaining life predicted by model M3

Fig.19 Remaining life curves of the gyroscope predicted by model M3

Fig.20 Mean squared error for gyroscope remaining life predicted by model M3

Tab.6 Prediction errors for gyroscope remaining life from model M3

方法	E_TMSE/h²	E_RMSE/h	E_MAE/h
方法Ⅰ	286.5832	37.4123	6.0974
方法Ⅱ	282.4722	37.3894	6.0955
方法Ⅲ	146.5512	17.0075	4.0955
方法Ⅳ	83.4353	11.4387	1.0973

Fig.21 Fatigue crack growth data for aluminum alloy components

Fig.22 Probability density function curves for aluminum alloy components remaining life predicted by model M1

Fig.23 Remaining life prediction curves of aluminum alloy components by model M1

Fig.24 Mean squared error for aluminum alloy components remaining life predicted by model M1

Tab.7 Prediction errors for aluminum alloy components remaining life from model M1

方法	E_TMSE/Cycle²	E_RMSE/Cycle	E_MAE/Cycle
方法Ⅰ	2.8915	0.0186	0.1040
方法Ⅱ	1.9668	0.0072	0.0843
方法Ⅲ	1.0147	7.8324×10^-4	0.0280
方法Ⅳ	0.5192	3.7198×10^-4	0.0157

Fig.25 Probability density function curves for aluminum alloy components remaining life predicted by model M2

Fig.26 Remaining life prediction curves of aluminum alloy components by model M2

Fig.27 Mean squared error for aluminum alloy components remaining life predicted by model M2

Tab.8 Prediction errors for aluminum alloy components remaining life from model M2

方法	E_TMSE/Cycle²	E_RMSE/Cycle	E_MAE/Cycle
方法Ⅰ	0.4010	0.0015	0.0264
方法Ⅱ	0.1926	3.7219×10^-4	0.0157
方法Ⅲ	0.1075	1.5755×10^-4	0.0115
方法Ⅳ	0.0595	6.4055×10^-5	0.0080

Fig.28 Probability density function curves for aluminum alloy components remaining life predicted by model M3

Fig. 29 Remaining life prediction curves of aluminum alloy components by model M3

Fig.30 Mean squared error for aluminum alloy components remaining life predicted by model M3

Tab.9 Prediction errors for aluminum alloy components remaining life from model M3

方法	E_TMSE/Cycle²	E_RMSE/Cycle	E_MAE/Cycle
方法Ⅰ	3.0789	0.0557	0.2016
方法Ⅱ	2.9413	0.0545	0.1961
方法Ⅲ	2.2155	0.0463	0.1781
方法Ⅳ	1.7311	0.0320	0.1508

References 24

[1]	刘小峰，亢莹莹，柏林. 轴承自驱式独立退化轨迹构建与剩余寿命灰色预测［J］. 中国机械工程， 2024， 35（9）： 1613-1621.
	LIU Xiaofeng， KANG Yingying， BO Lin. Self-driven Independent Degradation Trajectory Construction and Remaining Life Gray Prediction for Bearings［J］. China Mechanical Engineering， 2024， 35（9）： 1613-1621.
[2]	王宇，刘秋发，彭一真. 非均匀监测条件下滚动轴承剩余寿命预测方法［J］. 机械工程学报， 2023， 59（23）： 96-104.
	WANG Yu， LIU Qiufa， PENG Yizhen. A Remaining Useful Life Prediction Approach with Nonuniform Monitoring Conditions for Rolling Bearings［J］. Journal of Mechanical Engineering， 2023， 59（23）： 96-104.
[3]	WANG Chen， ZHANG Liming， CHEN Ling， et al. Remaining Useful Life Prediction of Nuclear Reactor Control Rod Drive Mechanism Based on Dynamic Temporal Convolutional Network［J］. Reliability Engineering & System Safety， 2025， 253： 110580.
[4]	ZHOU Jianghong， QIN Yi. A Continuous Remaining Useful Life Prediction Method with Multistage Attention Convolutional Neural Network and Knowledge Weight Constraint［J］. IEEE Transactions on Neural Networks and Learning Systems， 2025， 36（7）： 11847-11860.
[5]	MEEKER W Q， HAMADA M. Statistical Tools for the Rapid Development and Evaluation of High-reliability Products［J］. IEEE Transactions on Reliability， 1995， 44（2）： 187-198.
[6]	LU C， MEEKER W， ESCOBAR L. A Comparison of Degradation and Failure-time Analysis Methods for Estimating a Time-to-failure Distribution［J］. Statistica Sinica， 1996， 6（3）： 531-546.
[7]	KAHLE W， LEHMANN A. The Wiener Process as a Degradation Model： Modeling and Parameter Estimation［M］.Boston：Birkhäuser， 2010.
[8]	厉海涛，金光，周经伦，等. 动量轮维纳过程退化建模与寿命预测［J］. 航空动力学报， 2011， 26（3）： 622-627.
	LI Haitao， JIN Guang， ZHOU Jinglun， et al. Momentum Wheel Wiener Process Degradation Modeling and Life Prediction［J］. Journal of Aerospace Power， 2011， 26（3）： 622-627.
[9]	SI Xiaosheng， CHEN Maoyin， WANG Wenbin， et al. Specifying Measurement Errors for Required Lifetime Estimation Performance［J］. European Journal of Operational Research， 2013， 231（3）： 631-644.
[10]	PENG C Y， TSENG S T. Statistical Lifetime Inference with Skew-Wiener Linear Degradation Models［J］. IEEE Transactions on Reliability， 2013， 62（2）： 338-350.
[11]	GUAN Qingluan， WEI Xiukun， BAI Wenfei， et al. Two-stage Degradation Modeling for Remaining Useful Life Prediction Based on the Wiener Process with Measurement Errors［J］. Quality and Reliability Engineering International， 2022， 38（7）： 3485-3512.
[12]	SI Xiaosheng， WANG Wenbin， HU Changhua， et al. Remaining Useful Life Estimation Based on a Nonlinear Diffusion Degradation Process［J］. IEEE Transactions on Reliability， 2012， 61（1）： 50-67.
[13]	彭才华，钱富才，杜许龙. 新型非线性退化建模与剩余寿命预测［J］. 计算机集成制造系统， 2019， 25（7）： 1647-1654.
	PENG Caihua， QIAN Fucai， DU Xulong. New Nonlinear Degradation Modeling and Residual Life Prediction［J］. Computer Integrated Manufacturing Systems， 2019， 25（7）： 1647-1654.
[14]	YU Wennian， TU Wenbing， KIM I Y， et al. A Nonlinear-drift-driven Wiener Process Model for Remaining Useful Life Estimation Considering Three Sources of Variability［J］. Reliability Engineering & System Safety， 2021， 212： 107631.
[15]	LIN Jingdong， LIAO Guobo， CHEN Min， et al. Two-phase Degradation Modeling and Remaining Useful Life Prediction Using Nonlinear Wiener Process［J］. Computers & Industrial Engineering， 2021， 160： 107533.
[16]	杨升. 基于贝叶斯更新的数控车床刀具剩余寿命预测研究［D］. 长春：吉林大学， 2023.
	YANG Sheng. Research on cutting tool remaining useful life prediction of CNC lathe based on Bayesian updating［D］. Changchun： Jilin University， 2023.
[17]	任宏宇，余瑶怡，杜雄，等. 基于优化长短期记忆神经网络的IGBT寿命预测模型［J］. 电工技术学报， 2024， 39（4）： 1074-1086.
	REN Hongyu， YU Yaoyi， DU Xiong， et al. IGBT Lifetime Prediction Model Based on Optimized Long Short-term Memory Neural Network［J］. Transactions of China Electrotechnical Society， 2024， 39（4）： 1074-1086.
[18]	XU Xiaodong， TANG Shengjin， YU Chuanqiang， et al. Remaining Useful Life Prediction of Lithium-ion Batteries Based on Wiener Process under Time-varying Temperature Condition［J］. Reliability Engineering & System Safety， 2021， 214： 107675.
[19]	FAN Mengfei， ZENG Zhiguo， ZIO E， et al. A Sequential Bayesian Approach for Remaining Useful Life Prediction of Dependent Competing Failure Processes［J］. IEEE Transactions on Reliability， 2019， 68（1）： 317-329.
[20]	YE Zhisheng， WANG Yu， TSUI K L， et al. Degradation Data Analysis Using Wiener Processes with Measurement Errors［J］. IEEE Transactions on Reliability， 2013， 62（4）： 772-780.
[21]	司小胜，胡昌华，周东华. 带测量误差的非线性退化过程建模与剩余寿命估计［J］. 自动化学报， 2013， 39（5）： 530-541.
	SI Xiaosheng， HU Changhua， ZHOU Donghua. Nonlinear Degradation Process Modeling and Remaining Useful Life Estimation Subject to Measurement Error［J］. Acta Automatica Sinica， 2013， 39（5）： 530-541.
[22]	PARK C， PADGETT W J. Stochastic Degradation Models with Several Accelerating Variables［J］. IEEE Transactions on Reliability， 2006， 55（2）： 379-390.
[23]	叶海福，姚永和，于成龙. 高压脉冲电容器电容量漂移的研究［J］. 高电压技术， 2007， 33（6）： 37-41.
	YE Haifu， YAO Yonghe， YU Chenglong. Capacitance Variability of High Voltage Pulse Capacitors［J］. High Voltage Engineering， 2007， 33（6）： 37-41.
[24]	王泽洲，陈云翔，蔡忠义，等. 考虑随机失效阈值的设备剩余寿命在线预测［J］. 系统工程与电子技术， 2019， 41（5）： 1162-1168.
	WANG Zezhou， CHEN Yunxiang， CAI Zhongyi， et al. Real-time Prediction of Remaining Useful Lifetime for Equipment with Random Failure Threshold［J］. Systems Engineering and Electronics， 2019， 41（5）： 1162-1168.