Estimating the Probability Density Function of Remaining Useful Life for Wiener Degradation Process with Uncertain Parameters

  • Guo XieEmail author
  • Xin Li
  • Xi Peng
  • Fucai Qian
  • Xinhong Hei


The effective prediction of remaining useful life is essential to realize system failure diagnosis and health management. The existing researches often assume that the degradation model is constant or the degradation process is measurable. The accurate degradation model, however, usually can not be established, and the parametric variation and measurement error of the degradation process are unavoidable, which makes it hard to obtain the exact value for predicting the remaining useful life. Regarding this problem, on basis of the concept of first failure time, a real-time probability density function is derived for the Wiener degradation process with the uncertainty of parameters, the stochasticity of degradation process and the randomness of measurement error. The main steps are as follows: firstly, the degradation model with three kinds of uncertainties is established, and then the stochastic degradation state and the parameters of the uncertainty model are estimated by fusion Kalman/UFIR filter; then, the analytical expression of the probability density function of remaining useful life is deduced. Finally, the correctness and effectiveness of the proposed method are verified by a group of comparison experiments and Monte Carlo simulations.


Probability density function remaining useful life stochastic system Wiener degradation process 


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  1. [1]
    M. G. Pecht and R. Jaai, “A prognostics and health management roadmap for information and electronics-rich system,” Microelectronics Reliability, vol. 50, no. 3, pp. 317–323, 2010.CrossRefGoogle Scholar
  2. [2]
    M. G. Pecht, “Prognostics and Health Management of Electronics,” Encyclopedia of Structural Health Monitoring & John Wiley, Ltd 2009.Google Scholar
  3. [3]
    H. Yang, F. Qian, J. Huang, and S. Gao, “The complete statistical characterization control for a class of stochastic systems,” Control Theory & Applications, vol. 33, no. 5, pp. 669–675, 2016.zbMATHGoogle Scholar
  4. [4]
    H. Wang, Y. Zhao, and X. Ma, “Remaining useful life prediction using a novel two-stage Wiener process with stage correlation,” IEEE Access, vol. 6, pp. 65227–65238, 2018.CrossRefGoogle Scholar
  5. [5]
    J. Noortwijk, J. Weide, M. J. Kallen, and M. D. Pandey, “Gamma processes and peaks-over-threshold distributions for time-dependent reliability,” Reliability Engineering & System Safety, vol. 92, no. 12, pp. 1651–1658, 2007.CrossRefGoogle Scholar
  6. [6]
    S. Tseng, N. Balakrishnan, and C. Tsai, “Optimal stepstress accelerated degradation test plan for gamma degradation processes,” IEEE Transactions on Reliability, vol. 58, no. 4, pp. 611–618, 2009.CrossRefGoogle Scholar
  7. [7]
    X. Wang and D. Xu, “An inverse Gaussian process model for degradation data,” Technometrics, vol. 52, no. 2, pp. 188–197, 2010.MathSciNetCrossRefGoogle Scholar
  8. [8]
    H. Sun, D. Cao, Z. D. Zhao, and X. Kang, “A hybrid approach to cutting tool remaining useful life prediction based on the Wiener process,” IEEE Transactions on Relibility, vol. 67, no. 3, pp. 1–10, 2018.CrossRefGoogle Scholar
  9. [9]
    N. P. Li, Y. G. Lei, T. Yan, N. B. Li, and T. Y. Han “A Wiener process model-based method for remaining useful life prediction considering unit-to-unit variability,” IEEE Transactions on Industrial Electronics, vol. 66, no. 3, pp. 1–1, 2018.Google Scholar
  10. [10]
    B. Peng, J. Zhou, and Z. Pan, “Bayesian method for reliability assessment of products with wiener process degradation,” Systems Engineering-Theory & Practice, vol. 30, no. 3, pp. 543–549, 2010.Google Scholar
  11. [11]
    Q. Zhai and Z. Ye, “RUL prediction of deteriorating products using an adaptiveWiener process model,” IEEE Transactions on Industrial Informatics,, vol. 13, no. 6, pp. 2911–2921, 2017.CrossRefGoogle Scholar
  12. [12]
    C. Peng and S. Tseng, “Mis-specification analysis of linear degradation models,” IEEE Transactions on Reliability, vol. 58, no. 3, pp. 444–455, 2009.CrossRefGoogle Scholar
  13. [13]
    P. Wang, Y. Tang, S. Bae, and A. Xu, “Bayesian approach for two-phase degradation data based on changepoint Wiener process with measurement,” IEEE Transactions on Reliability, vol. 67, no. 2, pp. 688–700, 2018.CrossRefGoogle Scholar
  14. [14]
    X. Wang, B. Guo, and Z. Cheng, “Reliability assessment of products with wiener process degradation by fusing multiple information,” Acta Electronica Sinica, vol. 40, no. 5, pp. 977–982, 2012.Google Scholar
  15. [15]
    X. Wang, S. Lin, S. Wang, Z. M. He, and C. Zhang, “Remaining useful life prediction based on the Wiener process for an aviation axial piston pump,” Chinese Journal of Aeronautics, vol. 29, no. 3, pp. 779–788, 2016.CrossRefGoogle Scholar
  16. [16]
    L. Tang, G. Kacprzynski, K. Goebel, and G. Vachtsevanos, “Methodologies for uncertainty management in prognostics,” Proceedings of the IEEE Aerospace Conference, 2009.Google Scholar
  17. [17]
    B. Peng, J. Zhou, J. Feng, and X. Li, “Residual lifetime prediction of metallized film pulse capacitors,” Acta Electronica Sinica, vol. 39, no. 11, pp. 2674–2679, 2011.Google Scholar
  18. [18]
    X. Si, W. Wang, C. Hu, M. Chen, and D. Zhou, “A Wiener process-based degradation model with a recursive filter algorithm for remaining useful life estimation,” Mechanical Systems & Signal Processing, vol. 35, no 1–2, pp. 219–237, 2012.Google Scholar
  19. [19]
    C. Peng and S. T. Tseng, “Mis-specification analysis of linear degradation models,” IEEE Transactions on Reliability, vol. 58, no. 3, pp. 444–445, 2009.CrossRefGoogle Scholar
  20. [20]
    X. Si, C. Hu, Q. Zhang, H. He, and T. Zhou, “Estimating remaining useful life under uncertain degradation measurements,” Acta Electronica Sinica, vol. 43, no. 1, pp. 30–35, 2015.Google Scholar
  21. [21]
    X. Si, C. Hu, J. Li, G. Sun, and Q. Zhang, “Remaining useful life prediction of nonlinear stochastic degrading systems subject to uncertain measurements,” Journal of Shanghai Jiaotong University, vol. 49, no. 6, pp. 855–860, 2015.MathSciNetGoogle Scholar
  22. [22]
    X. Si, W. Wang, C. Hu, D. Zhou, and M. G. Pecht, “Remaining useful life estimation based on a nonlinear diffusion degradation process,” IEEE Transactions on Reliability,, vol. 61, no. 1, pp. 50–67, 2012.CrossRefGoogle Scholar
  23. [23]
    L. Feng, H. Wang, X. Si, and H. Zou, “A state-space-based prognostic model for hidden and age-dependent nonlinear degradation process,” IEEE Transactions on Automation Science & Engineering, vol. 10, no. 4, pp. 1072–1086, 2013.CrossRefGoogle Scholar
  24. [24]
    J. Zheng, C. Hu, X. Si, Z. Zhang, and X. Zhang, “Residual life estimation of nonlinear stochastic systems with uncertainties and individual differences,” Acta Automatica Sinica, vol. 43, no. 2, pp. 259–270, 2017.Google Scholar
  25. [25]
    B. Gao, G. Hu, S. Gao, Y. Zhong, and C. Gu, “Multi-sensor optimal data fusion for INS/GNSS/CNS integration based on unscented Kalman filter,” International Journal of Control, Automation and Systems, vol. 16, no. 1, pp. 129–140, 2018.CrossRefGoogle Scholar
  26. [26]
    S. Y. Zhao, Y. Shmaliy, P. Shi, and C. K. Ahn, “Fusion Kalman/UFIR filter for state estimation with uncertain parameters and noise statistics,” IEEE Transactions on Industrial Electronics, vol. 64, no. 4, pp. 3075–3083, 2017.CrossRefGoogle Scholar
  27. [27]
    J. M. Pak, C. K. Ahn, Y. S. Shmaliy, and M. T. Lim, “Improving reliability of particle filter-based localization in Wireless sensor networks via hybrid particle/FIR filtering,” IEEE Transactions on Industrial Informatics, vol. 11, no. 5, pp. 1089–1098, 2015.CrossRefGoogle Scholar
  28. [28]
    I. C. Moon, K. Song, S.-H. Kim, and H.-L. Choi, “State prediction of high-speed ballistic vehicles with Gaussian process,” International Journal of Control, Automation and Systems, vol. 16, no. 3, pp. 1282–1292, 2018.CrossRefGoogle Scholar

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© ICROS, KIEE and Springer 2019

Authors and Affiliations

  1. 1.School of Automation and Information EngineeringXi’an University of TechnologyBeilin District, Xi’anChina
  2. 2.Faculty of Computer Science and EngineeringXi’an University of TechnologyBeilin District, Xi’anChina

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