IEEE/CAA Journal of Automatica Sinica
Citation:  Z. N. Pang, X. S. Si, C. H. Hu, and Z. X. Zhang, “An agedependent and statedependent adaptive prognostic approach for hidden nonlinear degrading system,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 5, pp. 907–921, May 2022. doi: 10.1109/JAS.2021.1003859 
Remaining useful life (RUL) estimation approaches on the basis of the degradation data have been greatly developed, and significant advances have been witnessed. Establishing an applicable degradation model of the system is the foundation and key to accurately estimating its RUL. Most current researches focus on agedependent degradation models, but it has been found that some degradation processes in engineering are also related to the degradation states themselves. In addition, due to different working conditions and complex environments in engineering, the problems of the unittounit variability in the degradation process of the same batch of systems and actual degradation states cannot be directly observed will affect the estimation accuracy of the RUL. In order to solve the above issues jointly, we develop an agedependent and statedependent nonlinear degradation model taking into consideration the unittounit variability and hidden degradation states. Then, the Kalman filter (KF) is utilized to update the hidden degradation states in real time, and the expectationmaximization (EM) algorithm is applied to adaptively estimate the unknown model parameters. Besides, the approximate analytical RUL distribution can be obtained from the concept of the first hitting time. Once the new observation is available, the RUL distribution can be updated adaptively on the basis of the updated degradation states and model parameters. The effectiveness and accuracy of the proposed approach are shown by a numerical simulation and case studies for Liion batteries and rolling element bearings.
[1] 
M. Pecht, Prognostics and Health Management of Electronics. New York, NY, USA: John Wiley, 2008.

[2] 
X. S. Si, T. M. Li, and Q. Zhang, “Optimal replacement of degrading components: A controllimit policy,” SCIENCE CHINA Information Sciences, vol. 64, no. 10, p. 209205, 2021.

[3] 
X. S. Si, C. H. Hu, T. M. Li, and Q. Zhang, “A joint orderreplacement policy for deteriorating components with reliability constraint,” SCIENCE CHINA Information Sciences, vol. 64, no. 8, p. 189203, 2021.

[4] 
A. Lorton, M. Fouladirad, and A. Grall, “Methodology for probabilistic modelbased prognosis,” Eur. J. Oper. Res., vol. 225, pp. 443–454, 2013. doi: 10.1016/j.ejor.2012.10.025

[5] 
X. S. Si, T. M. Li, Q. Zhang, and X. Hu, “An optimal conditionbased replacement method for systems with observed degradation signals,” IEEE Trans. Rel., vol. 67, no. 3, pp. 1281–1293, 2018. doi: 10.1109/TR.2018.2830188

[6] 
X. S. Si, W. B. Wang, C. H. Hu, and D. H. Zhou, “Remaining useful life estimation–A review on the statistical data driven approaches,” Eur. J. Oper. Res., vol. 213, no. 1, pp. 1–14, 2011. doi: 10.1016/j.ejor.2010.11.018

[7] 
P. P. Wang, Y. C. Tang, S. J. Bae, and Y. He, “Bayesian analysis of twophase degradation data based on changepoint Wiener process,” Rel. Eng. Syst. Saf., vol. 170, pp. 244–256, 2018. doi: 10.1016/j.ress.2017.09.027

[8] 
X. S. Si, T. M. Li, Q. Zhang, and C. H. Hu. “Prognostics for linear stochastic degrading systems with survival measurements,” IEEE Trans. Ind. Electron., DOI: 10.1109/TIE.2019.2908617, 2019.

[9] 
A. C. Xu, L. J. Chen, B. X. Wang, and Y. C. Tang, “On modeling bivariate Wiener degradation process,” IEEE Trans. Rel., vol. 67, no. 3, pp. 897–906, 2018. doi: 10.1109/TR.2018.2791616

[10] 
M. E. Cholette, H. Y. Yu, P. Borghesani, M. Lin, and K. Geoff, “Degradation modeling and conditionbased maintenance of boiler heat exchangers using Gamma processes,” Rel. Eng. Syst. Saf., vol. 183, pp. 184–196, 2019. doi: 10.1016/j.ress.2018.11.023

[11] 
P. H. Jiang, X. W. Bing, and T. W. Fang, “Inference for constantstress accelerated degradation test based on Gamma process,” Applied Mathematical Modelling, vol. 67, pp. 123–134, 2019. doi: 10.1016/j.apm.2018.10.017

[12] 
W. W. Peng, Y. F. Li, Y. J. Yang, J. H. Mi, and H. Z. Huang, “Bayesian degradation analysis with inverse Gaussian process models under timevarying degradation rates,” IEEE Trans. Rel., vol. 66, no. 1, pp. 84–96, 2017. doi: 10.1109/TR.2016.2635149

[13] 
J. B. Guo, C. X. Wang, J. Cabrera, and E. A. Elsayed, “Improved inverse Gaussian process and bootstrap: Degradation and reliability metrics,” Reliability Engineering &System Safety, vol. 178, pp. 269–277, 2018.

[14] 
W. W. Peng, S. P. Zhu, and L. J. Shen, “The transformed inverse Gaussian process as an ageand statedependent degradation model,” Applied Mathematical Modelling, vol. 75, pp. 837–852, 2019. doi: 10.1016/j.apm.2019.07.004

[15] 
Z. S. Ye, N. Chen, and Y. Shen, “A new class of Wiener process models for degradation analysis,” Rel. Eng. Syst. Safety, vol. 138, pp. 58–67, 2015.

[16] 
J. X. Zhang, C. H. Hu, X. He, X. S. Si, Y. Liu, and D. H. Zhou, “Lifetime prognostics for furnace wall degradation with timevarying random jumps,” Rel. Eng. Syst. Safety, vol. 167, pp. 338–350, 2017. doi: 10.1016/j.ress.2017.05.047

[17] 
N. P. Li, N. Gebraeel, Y. G. Lei, L. K. Bian, and X. S. Si, “Remaining useful life prediction of machinery under timevarying operating conditions based on a twofactor state space model,” Rel. Eng. Syst. Safety, vol. 186, pp. 88–100, 2019. doi: 10.1016/j.ress.2019.02.017

[18] 
Z. X. Zhang, X. S. Si, C. H. Hu, and Y. G. Lei, “Degradation data analysis and remaining useful life estimation: A review on Wienerprocessbased methods,” Eur. J. Oper. Res., vol. 271, no. 3, pp. 775–796, 2018. doi: 10.1016/j.ejor.2018.02.033

[19] 
P. C. Paris and F. Erdogan, “A critical analysis of crack propagation laws,” Journal of Fluids Engineering, vol. 85, no. 4, pp. 528–533, 1963.

[20] 
M. Giorgio, M. Guida, and G. Pulcini, “A parametric Markov chain to model ageand statedependent wear processes”, in Complex Data Modelling and Computationally Intensive Statistical Methods”, Milan, Italy: Springer, 2010, pp. 85–97.

[21] 
M. Giorgio, M. Guida, and G. Pulcini, “An ageand statedependent Markov model for degradation processes,” IIE Transactions, vol. 43, pp. 621–632, 2011.

[22] 
Z. X. Zhang, X. S. Si, and C. H. Hu, “An ageand statedependent nonlinear prognostic model for degrading systems,” IEEE Trans. Rel., vol. 64, no. 4, pp. 1214–1228, 2015. doi: 10.1109/TR.2015.2419220

[23] 
D. An, J. H. Choi, and N. H. Kim, “Prognostics 101: A tutorial for particle filterbased prognostics algorithm using Matlab,” Reliability Engineering &System Safety, vol. 115, pp. 161–169, 2013.

[24] 
M. E. Orchard, P. HeviaKoch, B. Zhang, and L. Tang, “Risk measures for particlefilteringbased stateofcharge prognosis in lithiumion batteries,” IEEE Trans. Industrial Electronics, vol. 60, no. 11, pp. 5260–5269, Nov. 2013. doi: 10.1109/TIE.2012.2224079

[25] 
N. P. Li, Y. G. Lei, L. Guo, T. Yan, and J. Lin, “Remaining useful life prediction based on a general expression of stochastic process models,” IEEE Trans. Industrial Electronics, vol. 64, pp. 5709–5718, Jul. 2017. doi: 10.1109/TIE.2017.2677334

[26] 
H. W. Zhang, D. H. Zhou, M. Y. Chen, and X. P. Xi, “Predicting remaining useful life based on a generalized degradation with fractional Brownian motion,” Mech. Syst. Signal Process, vol. 115, pp. 736–752, 2019. doi: 10.1016/j.ymssp.2018.06.029

[27] 
X. S. Si, W. Wang, C. H. Hu, D. H. Zhou, and M. G. Pecht, “Remaining useful life estimation based on a nonlinear diffusion degradation process,” IEEE Trans. Rel., vol. 61, no. 1, pp. 50–67, 2012. doi: 10.1109/TR.2011.2182221

[28] 
J. X. Zhang, C. H. Hu, X. He, X. S. Si, Y. Liu, and D. H. Zhou, “A novel lifetime estimation method for twophase degrading systems,” IEEE Trans. Rel., vol. 68, no. 2, pp. 689–709, 2018.

[29] 
N. P. Li, Y. G. Lei, T. Yan, N. B. Li, and T. Y. Han, “A Wiener process modelbased method for remaining useful life prediction considering unittounit variability,” IEEE Trans. Industrial Electronics, vol. 66, no. 3, pp. 2092–2101, 2019. doi: 10.1109/TIE.2018.2838078

[30] 
J. F. Zheng, X. S. Si, C. H. Hu, Z. X. Zhang, and W. Jiang, “A nonlinear prognostic model for degrading systems with threesource variability,” IEEE Trans. Rel., vol. 65, no. 2, pp. 736–750, 2016. doi: 10.1109/TR.2015.2513044

[31] 
J. X. Li, Z. H. Wang, Y. B. Zhang, H. M. Fu, C. R. Liu, and S. Krishnaswamy, “Degradation data analysis based on a generalized Wiener process subject to measurement error,” Mech. Syst. Signal Process,

[32] 
L. Feng, H. L. Wang, X. S. Si, and H. X. Zou, “A state spacebased prognostic model for hidden and agedependent nonlinear degradation process,” IEEE Trans. Automation Science and Engineering, vol. 10, no. 4, pp. 1072–1086, 2013. doi: 10.1109/TASE.2012.2227960

[33] 
X. S. Si, T. M. Li, and Q. Zhang, “A general stochastic degradation modelling approach for prognostics of degrading systems with surviving and uncertain measurements,” IEEE Trans. Reliability, vol. 68, no. 3, pp. 1080–1100, 2019. doi: 10.1109/TR.2019.2908492

[34] 
H. Z. Fang, N. Tian, Y. B. Wang, M. C. Zhou, and M. A. Haile, “Nonlinear Bayesian estimation: From Kalman filtering to a broader horizon,” IEEE/CAA J. Autom. Sinica, vol. 5, no. 2, pp. 401–417, 2018. doi: 10.1109/JAS.2017.7510808

[35] 
Z. X. Zhang, X. S. Si, C. H. Hu, X. X. Hu, and G. X. Sun, “An adaptive prognostic approach incorporating inspection influence for deteriorating systems,” IEEE Trans. Rel., vol. 68, no. 1, pp. 302–316, 2018.

[36] 
Y. AïtSahalia, “Maximumlikelihood estimation of discretely sampled diffusions: A closedform approach,” Econometrica, vol. 70, no. 1, pp. 223–262, 2002. doi: 10.1111/14680262.00274

[37] 
A. V. Egorov, H. Li, and Y. Xu, “Maximum likelihood estimation of timeinhomogeneous diffusions,” J. Econometr., vol. 114, pp. 107–139, 2003. doi: 10.1016/S03044076(02)00221X

[38] 
Y. AïtSahalia, “Closedform likelihood expansions for multivariate diffusions,” Ann. Statisti., vol. 36, no. 2, pp. 906–937, 2008. doi: 10.1214/009053607000000622

[39] 
J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the neldermead simplex method in low dimensions,” SIAM Journal of Optimization, vol. 9, no. 1, pp. 112–147, 1998. doi: 10.1137/S1052623496303470

[40] 
P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. New York, NY, USA: Springer, 1995.

[41] 
B. Saha and K. Goebel. Battery Data Set, in NASA Ames Prognostics Data Repository, National Aeronautics and Space Administration (NASA)’s Ames Research Center: Moffett Field, CA, USA, 2007. [Online]. Available: http://ti.arc.nasa.gov/tech/dash/pcoe/prognosticdatarepository/, Accessed on: Jan. 22, 2014.

[42] 
X. S. Si, “An adaptive prognostic approach via nonlinear degradation modeling: Application to battery data,” IEEE Trans. Ind. Electron., vol. 62, no. 8, pp. 5082–5096, 2015. doi: 10.1109/TIE.2015.2393840

[43] 
G. Dong, Z. Chen, J. Wei, and Q. Ling, “Battery health prognosis using Brownian motion modeling and particle filtering,” IEEE Trans. Ind. Electron., vol. 65, no. 11, pp. 8646–8655, 2018. doi: 10.1109/TIE.2018.2813964

[44] 
B. Wang, Y. G. Lei, N. P. Li, and N. B. Li, “A hybrid prognostics approach for estimating remaining useful life of rolling element bearings,” IEEE Trans. Ind. Electron., vol. 69, no. 1, pp. 401–412, 2018.

[45] 
J. Feng, P. Kvam, and Y. Tang, “Remaining useful lifetime prediction based on the damagemarker bivariate degradation model: A case study on lithiumion batteries used in electric vehicles,” Eng. Failure Anal.,vol. 70, pp. 323–342, 2016.

[46] 
T. Wang, M. Hu, and Y. L. Zhao, “Consensus control with a constant gain for discretetime binaryvalued multiagent systems based on a projected empirical measure method,” IEEE/CAA J. Autom. Sinica, vol. 6, no. 4, pp. 1052–1059, 2019. doi: 10.1109/JAS.2019.1911594

[47] 
K. L. Son, Mi. Fouladirad, A. Barros, E. Levrat, and B. Lung, “Remaining useful life estimation based on stochastic deterioration models: A comparative study,” Rel. Eng. Syst. Safety, vol. 112, pp. 165–175, 2013. doi: 10.1016/j.ress.2012.11.022

[48] 
N. Li, Y. Lei, J. Lin, and S. X. Ding, “An improved exponential model for predicting remaining useful life of rolling element bearings,” IEEE Trans. Ind. Electron., vol. 62, no. 12, pp. 7762–7773, 2015. doi: 10.1109/TIE.2015.2455055

[49] 
A. Saxena, J. Celaya, B. Saha, S. Saha, and K. Goebel, “Metrics for offline evaluation of prognostic performance,” Int. J. Prognostics and Health Management, vol. 1, pp. 1–20, 2010.
