State Estimation and Remaining Useful Life Prediction of PMSTM Based on a Combination of SIR and HSMM
Abstract
:1. Introduction
2. MFHI Construction
2.1. Wavelet Denoising
2.2. Filter Data with Entropy Weight Method
2.3. PCA Reduces the Correlation of the Data
 Calculate the covariance matrix $\mathit{COV}$ of ${\mathit{S}}^{\left(3\right)}$$$\mathit{COV}=\frac{1}{\sigma 1}{\left({\mathit{S}}^{\left(3\right)}\right)}^{T}\xb7{\mathit{S}}^{\left(3\right)}$$
 Calculate the eigenvalues of $\mathit{COV}$, sorting from largest to smallest to get ${\phi}_{1},{\phi}_{2},\cdots ,{\phi}_{\sigma}$, and obtain the corresponding eigenvector ${\mathit{\kappa}}_{1},{\mathit{\kappa}}_{2},\cdots ,{\mathit{\kappa}}_{\sigma}$.
 Obtain the principal components, where $s=1,2,\cdots ,\sigma $$${\mathit{K}}_{s}={\mathit{S}}^{\left(3\right)}\xb7{\mathit{\kappa}}_{s}$$
 Sort the principal components to get the cumulative contribution rate $CON$:$$CON\left(s\right)=\sum _{\iota =1}^{s}{\phi}_{\iota}/\sum _{\iota =1}^{\sigma}{\phi}_{\iota}$$
2.4. Parameters Fusion
3. State Estimation and RUL Prediction Combining SIR and HSMM
3.1. Observation Sequence Acquisition
3.2. HSMM Training
 The initial state probability distribution:$$\Pi =\left\{{\pi}_{i}\right\},{\pi}_{i}=P\left({x}_{0}=c\right),1\le c\le 5$$
 The state transition probability matrix, which represents the probability of transition between states during the operation of PMSTM:$${\mathit{A}}_{\mathbf{0}}=\left\{{a}_{ij}\right\},\phantom{\rule{1.em}{0ex}}{a}_{ij}=P\left({x}_{t+1}=j\mid {x}_{t}=i\right),1\le i<j\le 5$$
 The observed state probability matrix:$$\mathit{B}=\left\{{b}_{cq}\right\},\phantom{\rule{1.em}{0ex}}{b}_{cq}=P\left({y}_{q}\mid {x}_{t}=c\right),1\le q\le Q$$
 The dwell time distribution for each state:$$\mathsf{\Theta}=\left\{{\mathit{\theta}}_{c}\right\}$$
 Through the current model parameters $\overline{\mathit{\lambda}}$, the expectation of $P(\mathit{Y},\mathit{X}\mid \mathit{\lambda})$ under condition $P(\mathit{X}\mid \mathit{Y},\overline{\mathit{\lambda}})$ is obtained by combining the Viterbi algorithm, the forward algorithm and the backward algorithm.
 According to the current observation sequence, the most likely hidden state sequence is obtained by the Viterbi algorithm.Calculate the local state at the initial moment:$$\begin{array}{c}{\delta}_{1}\left(i\right)={\pi}_{i}{b}_{i}\left({y}_{1}\right)\\ {\Psi}_{1}\left(i\right)=0\end{array}$$$$\begin{array}{c}{\delta}_{t}\left(j\right)={\displaystyle \underset{1\le i\le 5}{max}}\left[{\delta}_{t1}\left(i\right){a}_{ij}\left({x}_{t1}\right)\right]{b}_{j}\left({y}_{t}\right)\\ {\Psi}_{t}\left(j\right)=arg{\displaystyle \underset{1\le i\le 5}{max}}\left[{\delta}_{t1}\left(i\right){a}_{ij}\left({x}_{t1}\right)\right].\end{array}$$The maximum ${\delta}_{t}\left(i\right)$ at time t is the probability of the most likely hidden state. The auxiliary variable ${\Psi}_{t}\left(j\right)$ is used to store the optimal state of PMSTM at time $t1$ under the condition that time t is in state j. Thus:$${x}_{t}^{*}=\underset{1\le j\le 5}{argmax}\left[{\delta}_{t}\left(j\right)\right]$$Backtracking to get the sequence of hidden states:$${x}_{t1}^{*}={\Psi}_{t}\left({x}_{t}^{*}\right)$$$${d}_{t}\left(j\right)={x}_{t}^{*}\left(j\right)\xb7{x}_{t1}^{*}\left(j\right)\xb7{d}_{t1}\left(j\right)+1$$
 Calculate variables using forward and backward algorithms.Calculate the forward probability of each hidden state at the initial moment:$$\begin{array}{c}{\alpha}_{1}\left(i\right)={\pi}_{i}{b}_{i}\left({y}_{1}\right)\\ {\mathit{A}}_{{x}_{1}}=P\left({x}_{1}\right)+\left(\mathit{X}P\left({x}_{1}\right)\right)\xb7{\mathit{A}}_{0}\end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\alpha}_{t+1}\left(j\right)=\left[\sum _{i=1}^{5}{\alpha}_{t}\left(i\right){a}_{ij}\left({x}_{t}\right)\right]{b}_{j}\left({y}_{t+1}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\mathit{A}}_{{x}_{t+1}}=P\left({x}_{t+1}\right)+\left(\mathit{X}P\left({x}_{t+1}\right)\right)\xb7{\mathit{A}}_{0}\hfill \end{array}$$Calculate variables ${\beta}_{t}\left(i\right)$ using the backward algorithm$$\begin{array}{c}{\beta}_{T}\left(i\right)=1\\ {\beta}_{t}\left(i\right)=\left[\sum _{i=1}^{5}{a}_{ij}\left({x}_{t}\right){b}_{j}\left({y}_{t+1}\right)\right]{\beta}_{t+1}\left(j\right)\end{array}$$$$L(\mathit{\lambda},\overline{\mathit{\lambda}})=\sum _{\mathit{X}}P(\mathit{X}\mid \mathit{Y},\overline{\mathit{\lambda}})logP(\mathit{Y},\mathit{X}\mid \mathit{\lambda})$$
 The parameters of the model can be updated by maximizing the expected value$$\overline{\mathit{\lambda}}=arg\underset{\mathit{\lambda}}{max}\sum _{\mathit{X}}P(\mathit{X}\mid \mathit{Y},\overline{\mathit{\lambda}})logP(\mathit{Y},\mathit{X}\mid \mathit{\lambda})$$The equation for updating the model parameters can then be obtained:$${\overline{\pi}}_{i}={\gamma}_{1}\left(i\right)$$$$\phantom{\rule{1.em}{0ex}}{\overline{a}}_{ij}^{0}=\frac{\left({\sum}_{t=1}^{T1}{\xi}_{t}(i,j)\right)\odot G}{\left({\sum}_{t=1}^{T1}{\gamma}_{t}\left(i\right)\right)\odot G}$$$${\xi}_{t}(i,j)=P\left({x}_{t}=i,{x}_{t+1}=j\mid \mathit{Y},\mathit{\lambda}\right)=\frac{{\alpha}_{t}\left(i\right){a}_{ij}\left({x}_{t}\right){b}_{j}\left({y}_{t+1}\right){\beta}_{t+1}\left(j\right)}{{\sum}_{i=1}^{5}{\sum}_{j=1}^{5}{\alpha}_{t}\left(i\right){a}_{ij}\left({x}_{t}\right){b}_{j}\left({y}_{t+1}\right){\beta}_{t+1}\left(j\right)}$$$$\phantom{\rule{1.em}{0ex}}{\gamma}_{t}\left(i\right)=\frac{{\alpha}_{t}\left(i\right)}{{\sum}_{5}^{5}{\alpha}_{t}\left(i\right)}$$$${\mu}_{i,d}=\frac{{\sum}_{t=1}^{T1}{\alpha}_{t=1}\left(i\right)\left({\sum}_{j=1,j\ne i}^{5}{a}_{ij}\left({x}_{t}\left(i\right)\right){b}_{j}\left({y}_{t+1}\right){\beta}_{t+1}\left(j\right)\right){d}_{t}\left(i\right)}{{\sum}_{t=1}^{T1}{\alpha}_{t=1}\left(i\right)\left({\sum}_{j=1,j\ne i}^{5}{a}_{ij}\left({x}_{t}\left(i\right)\right){b}_{j}\left({y}_{t+1}\right){\beta}_{t+1}\left(j\right)\right)}$$$${\delta}_{i,d}^{2}=\frac{{\sum}_{t=1}^{T1}{\alpha}_{t=1}\left(i\right)\left({\sum}_{j=1,j\ne i}^{5}{a}_{ij}\left({d}_{t}\left(i\right)\right){b}_{j}\left({y}_{t+1}\right){\beta}_{t+1}\left(j\right)\right){\left({d}_{t}\left(i\right){\mu}_{i,d}\right)}^{2}}{{\sum}_{t=1}^{T1}{\alpha}_{t=1}\left(i\right)\left({\sum}_{j=1,j\ne i}^{5}{a}_{ij}\left({d}_{t}\left(i\right)\right){b}_{j}\left({y}_{t+1}\right){\beta}_{t+1}\left(j\right)\right)}$$$$\phantom{\rule{1.em}{0ex}}{b}_{j}\left(l\right)=\frac{{\sum}_{t=1,{y}_{t}={Y}_{l}}^{T}{\gamma}_{t}\left(j\right)}{{\sum}_{t=1}^{T}{\gamma}_{t}\left(j\right)}$$
Algorithm 1: HSMM training procedure. 

3.3. Recurrent Estimation of Current Health State
 Generate an initial particle set $\left\{{\mathit{x}}_{0}^{\left(r\right)}\right\},1\le r\le R$ according to the state probability distribution $\Pi $ at the initial moment.
 State transition (prediction): According to the particle set $\left\{{\mathit{x}}_{t1}^{\left(r\right)}\right\}$ obtained at time $t1$, the particle set $\left\{{\mathit{x}}_{t}^{\left(r\right)}\right\}$ of the state at time t is obtained through the state transition probability matrix ${\mathit{A}}_{t1}$:$${\mathit{x}}_{t}^{\left(r\right)}:P\left({\mathit{x}}_{t}^{\left(r\right)}\mid {\mathit{x}}_{t1}^{\left(r\right)}\right)\in {\mathit{A}}_{t1}$$
 Calculate particle weights (update): According to the observed value ${y}_{t}$ at time t and the observed state probability matrix $\mathit{B}$, the weight value ${\mathit{w}}_{t}^{\left(r\right)}$ of each predicted particle is obtained:$${\mathit{w}}_{t}^{\left(r\right)}=P\left({y}_{t}\mid {\mathit{x}}_{t}^{\left(r\right)}\right)\in \mathit{B}$$
 Normalize the calculated weight value of each particle:$${\widehat{\mathit{w}}}_{t}^{\left(r\right)}=\frac{{\mathit{w}}_{t}^{\left(r\right)}}{{\sum}_{r=1}^{R}{\mathit{w}}_{t}^{\left(r\right)}}$$
 State estimation: Calculate the estimated value of the current health state according to the particle set at time t and the weight, $\left\{{\mathit{x}}_{t}^{\left(r\right)},{\widehat{\mathit{w}}}_{t}^{\left(r\right)}\right\}$, of each particle:$${\widehat{\mathit{x}}}_{t}=\sum _{r=1}^{R}{\widehat{\mathit{w}}}_{t}^{\left(r\right)}\xb7{\mathit{X}}_{t}^{\left(r\right)}$$
 Resampling: Calculate the number of effective particles according to the normalized weight ${\widehat{\mathit{w}}}_{t}^{\left(r\right)}$ of each particle, and resample and update the particle set as the particle set for state estimation at the next moment. The effective particle number ${R}_{eff}$ can be calculated as:$${R}_{eff}=\frac{R}{{\sum}_{r=1}^{R}{\widehat{\mathit{w}}}_{t}^{\left(r\right)}}$$
 State transition probability matrix update: Calculate a new state transition probability matrix ${\mathit{A}}_{t}$ according to the residence time ${d}_{t}$ of each healthy state of the PMSTM:$${\mathit{A}}_{t}=P\left({x}_{t}\right)+\left(\mathit{X}P\left({x}_{t}\right)\right)\xb7{\mathit{A}}_{0}$$$${d}_{t}={\widehat{x}}_{t}\xb7{\widehat{x}}_{t1}\xb7{d}_{t1}+1$$
3.4. RUL Prediction
 Calculate the remaining time of the current state.$$d\left({x}_{t}\right)=\sum _{i=1}^{5}\left({\mu}_{{d}_{i}}{d}_{t}\left(i\right)\right)\odot {\delta}_{t}\left(i\right)$$An estimate of the remaining time of the PMSTM in this state can be obtained by weighted summation.
 Calculate the remaining time of the subsequent state.Calculate the next state according to the initial state transition probability matrix ${\mathit{A}}_{0}$ until the failure state. The probability that the next state of the PMSTM may appear is defined as:$${\delta}_{\mathrm{next}\phantom{\rule{4.pt}{0ex}}}={\left[{\delta}_{t+\tilde{d}}\left(i\right)\right]}_{1\le i\le 5}={\left({\mathit{A}}_{0}\right)}^{T}\xb7{\delta}_{t}\left(i\right)$$The highest probability is the state that may appear at the next moment:$${x}_{next}={x}_{t+\tilde{d}}=\underset{1\le i\le 5}{argmax}{\delta}_{t+\tilde{d}}\left(i\right)$$If ${x}_{t+\tilde{d}}$ reaches a failure state, the PMSTM will fail when the dwell time is reached in that state. Calculate the remaining time in each state:$$d\left({x}_{t+\tilde{d}}\right)=\sum _{i=1}^{5}{\mu}_{{d}_{i}}\odot {\delta}_{t+\tilde{d}}\left(i\right)$$
4. Proposed Method
5. Experimental Details and Analysis of Results
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
DCNN  Deep convolutional neural network 
GMM  Gaussian mixture model 
HI  Health index 
HMM  Hidden Markov model 
HSMM  Hidden SemiMarkov model 
ISOMAP  Isometric mapping 
LPP  Lifetime prediction performance 
MFHI  Multiparameter fusion health index 
PCA  Principal component analysis 
PMSTM  Permanent magnet synchronous traction motor 
PSO  Particle swarm optimization 
RMSE  Root mean square error 
RNN  Recurrent neural network 
RUL  Remaining useful life 
SIR  Sample importance resampling 
SNR  Signaltonoise ratio 
SVM  Support vector machine 
URT  Urban rail transit 
References
 Li, X.; Peter, E.D. Love. Procuring urban rail transit infrastructure by integrating land value capture and publicprivate partnerships: Learning from the cities of Delhi and Hong Kong. Cities 2022, 122, 103545. [Google Scholar] [CrossRef]
 Zhang, C.; Lu, D.; Xiao, X.; Wang, Y. Modeling and analysis of global energy consumption process of urban rail transit system based on Petri net. J. Rail Transp. Plan. Manag. 2022, 21, 100293. [Google Scholar]
 Jia, Z.; Wu, L.; Chen, W.; Yu, L.; Cao, Y.; Jia, H. Optimization of Transverse Flux Permanent Magnet Machine with Double OmegaHoop Stator. In Proceedings of the 2019 IEEE International Electric Machines Drives Conference (IEMDC), San Diego, CA, USA, 11–15 May 2019; pp. 1925–1928. [Google Scholar]
 Wang, X.; Fang, X.; Lin, F.; Yang, Z. Predictive current control of permanentmagnet synchronous motors for rail transit including quasi sixstep operation. In Proceedings of the 2017 IEEE Transportation Electrification Conference and Expo, AsiaPacific (ITEC AsiaPacific), Harbin, China, 7–10 August 2017; Volume 122, pp. 1–6. [Google Scholar]
 Zhang, C.; Zhang, Y.; Dui, H.; Wang, S.; Tomovic, M.M. Importance measurebased maintenance strategy considering maintenance costs. Eksploat. Niezawodn. Maint. Reliab. 2022, 24, 15–24. [Google Scholar] [CrossRef]
 Yang, L.; Wang, F.; Zhang, J.; Ren, W. Remaining useful life prediction of ultrasonic motor based on Elman neural network with improved particle swarm optimization. Measurement 2019, 143, 27–38. [Google Scholar] [CrossRef]
 Chen, Y.; Peng, G.; Zhu, Z.; Li, S. A novel deep learning method based on attention mechanism for bearing remaining useful life prediction. Appl. Soft Comput. 2020, 86, 105919. [Google Scholar] [CrossRef]
 Liu, Y.; Hu, Z.; Todd, M.; Hu, C. DataDriven Remaining Useful Life Estimation Using Gaussian Mixture Models. In Proceedings of the AIAA Scitech 2021 Forum, Virtual Event, 11–21 January 2021. [Google Scholar]
 Downey, A.; Lui, Y.; Hu, C.; Laflamme, S.; Hu, S. Physicsbased prognostics of lithiumion battery using nonlinear least squares with dynamic bounds. Reliab. Eng. Syst. Saf. 2019, 182, 1–12. [Google Scholar] [CrossRef]
 Guo, Q.; Shi, J.; Wang, S.; Zhang, C. Deep Degradation Feature Extraction and RUL Estimation for Switching Power Unit. In Proceedings of the 2019 Prognostics and System Health Management Conference (PHMQingdao), Qingdao, China, 25–27 October 2019; pp. 1–5. [Google Scholar]
 Yang, F.; Habibullah, M.S.; Shen, Y. Remaining useful life prediction of induction motors using nonlinear degradation of health index. Mech. Syst. Signal Process. 2021, 148, 107183. [Google Scholar] [CrossRef]
 Zhu, J.; Chen, N.; Peng, W. Estimation of Bearing Remaining Useful Life Based on Multiscale Convolutional Neural Network. IEEE Trans. Ind. Electron. 2019, 66, 3208–3216. [Google Scholar] [CrossRef]
 Shifat, T.A.; JangWook, H. Remaining Useful Life Estimation of BLDC Motor Considering Voltage Degradation and AttentionBased Neural Network. IEEE Access 2020, 8, 168414–168428. [Google Scholar] [CrossRef]
 Zhang, Y.; Xiong, R.; He, H.; Pecht, M.G. Long ShortTerm Memory Recurrent Neural Network for Remaining Useful Life Prediction of LithiumIon Batteries. IEEE Trans. Veh. Technol. 2018, 67, 5695–5705. [Google Scholar] [CrossRef]
 Li, X.; Ding, Q.; Sun, J. Remaining useful life estimation in prognostics using deep convolution neural networks. Reliab. Eng. Syst. Saf. 2018, 172, 1–11. [Google Scholar] [CrossRef]
 Sateesh Babu, G.; Zhao, P.; Li, X. Deep Convolutional Neural Network Based Regression Approach for Estimation of Remaining Useful Life. In Proceedings of the International Conference on Database Systems for Advanced Applications, Cham, Dallas, TX, USA, 16–19 April 2016; pp. 214–228. [Google Scholar]
 Chen, W.; Chen, W.; Liu, H.; Wang, Y.; Bi, C.; Gu, Y. A RUL Prediction Method of Small Sample Equipment Based on DCNNBiLSTM and Domain Adaptation. Mathematics 2022, 10, 1022. [Google Scholar] [CrossRef]
 Gougam, F.; Rahmoune, C.; Benazzouz, D.; Varnier, C.; Nicod, J.M. Health Monitoring Approach of Bearing: Application of Adaptive Neuro Fuzzy Inference System (ANFIS) for RULEstimation and Autogram Analysis for FaultLocalization. In Proceedings of the 2020 Prognostics and Health Management Conference (PHMBesançon), Besancon, France, 4–7 May 2020; pp. 200–206. [Google Scholar]
 Kewalramani, R.; Ram, A. Estimation of Remaining Useful Life of Electric Motor using supervised deep learning methods. In Proceedings of the 2019 IEEE Transportation Electrification Conference (ITECIndia), Bengaluru, India, 17–19 December 2019; pp. 1–4. [Google Scholar]
 Ali, M.U.; Zafar, A.; Nengroo, S.H.; Hussain, S.; Park, G.S.; Kim, H.J. Online Remaining Useful Life Prediction for LithiumIon Batteries Using Partial Discharge Data Features. Energies 2019, 12, 4366. [Google Scholar] [CrossRef] [Green Version]
 García Nieto, P.J.; GarcíaGonzalo, E.; Sánchez Lasheras, F.; de Cos Juez, F.J. Hybrid PSO–SVMbased method for forecasting of the remaining useful life for aircraft engines and evaluation of its reliability. Reliab. Eng. Syst. Saf. 2015, 138, 219–231. [Google Scholar] [CrossRef]
 Le Son, K.; Fouladirad, M.; Barros, A.; Levrat, E.; Iung, B. Remaining useful life estimation based on stochastic deterioration models: A comparative study. Reliab. Eng. Syst. Saf. 2013, 112, 165–175. [Google Scholar] [CrossRef]
 Chen, R.; Zhang, C.; Wang, S.; Hong, L. BivariateDependent Reliability Estimation Model Based on Inverse Gaussian Processes and Copulas Fusing Multisource Information. Aerospace 2022, 9, 392. [Google Scholar] [CrossRef]
 Gao, Z.; Li, J.; Wang, R. Prognostics uncertainty reduction by righttime prediction of remaining useful life based on hidden Markov model and proportional hazard model. Eksploat.Niezawodn.Maint. Reliab. 2021, 23, 154–164. [Google Scholar]
 Liu, T.; Zhu, K.; Zeng, L. Diagnosis and Prognosis of Degradation Process via Hidden SemiMarkov Model. IEEE/ASME Trans. Mechatronics 2018, 23, 1456–1466. [Google Scholar] [CrossRef]
 Xiao, Q.; Fang, Y.; Liu, Q.; Zhou, S. Online machine health prognostics based on modified durationdependent hidden semiMarkov model and highorder particle filtering. Int. J. Adv. Manuf. Technol. 2018, 94, 1283–1297. [Google Scholar] [CrossRef]
 Ma, Y.; Jia, X.; Hu, Q.; Bai, H.; Guo, C.; Wang, S. A New State Recognition and Prognosis Method Based on a Sparse Representation Feature and the Hidden SemiMarkov Model. IEEE Access 2020, 8, 119405–119420. [Google Scholar] [CrossRef]
 Zhu, K.; Liu, T. Online Tool Wear Monitoring Via Hidden SemiMarkov Model With Dependent Durations. IEEE Trans. Ind. Inform. 2018, 14, 69–78. [Google Scholar] [CrossRef]
 Cui, L.; Wang, X.; Xu, Y.; Jiang, H.; Zhou, J. A novel Switching Unscented Kalman Filter method for remaining useful life prediction of rolling bearing. Measurement 2019, 135, 678–684. [Google Scholar] [CrossRef]
 Shifat, T.A.; Yasmin, R.; Hur, J. A Data Driven RUL Estimation Framework of Electric Motor Using Deep Electrical Feature Learning from Current Harmonics and Apparent Power. Energies 2021, 14, 3156. [Google Scholar] [CrossRef]
 Wang, Y.; Peng, Y.; Zi, Y.; Jin, X.; Tsui, K. A TwoStage DataDrivenBased Prognostic Approach for Bearing Degradation Problem. IEEE Trans. Ind. Inform. 2016, 12, 924–932. [Google Scholar] [CrossRef]
 Jouin, M.; Gouriveau, R.; Hissel, D.; Péra, M.; Zerhouni, N. Particle filterbased prognostics: Review, discussion and perspectives. Mech. Syst. Signal Process. 2016, 72–73, 2–31. [Google Scholar] [CrossRef]
 Lee, Y.; Kim, I.; Choi, S.; Oh, J.; Kim, N. Remaining useful life prediction for PMSM under radial load using particle filter. Smart Struct. Syst. 2022, 29, 799–805. [Google Scholar]
 Povey, D.; Burget, L.; Agarwal, M.; Akyazi, P.; Feng, K.; Ghoshal, A.; Glembek, O.; Goel, N.K.; Karafiát, M.; Rastrow, A.; et al. Subspace Gaussian Mixture Models for speech recognition. In Proceedings of the 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, Dallas, TX, USA, 14–19 March 2010; pp. 4330–4333. [Google Scholar]
 Hu, Z.; Mahadevan, S. Probability models for dataDriven global sensitivity analysis. Reliab. Eng. Syst. Saf. 2019, 187, 40–57. [Google Scholar] [CrossRef]
 Liu, K.; Gebraeel, N.Z.; Shi, J. A DataLevel Fusion Model for Developing Composite Health Indices for Degradation Modeling and Prognostic Analysis. IEEE Trans. Autom. Sci. Eng. 2013, 10, 652–664. [Google Scholar] [CrossRef]
 Ahmad, W.; Khan, S.A.; Manjurul Islam, M.M.; Kim, J. A reliable technique for remaining useful life estimation of rolling element bearings using dynamic regression models. Reliab. Eng. Syst. Saf. 2019, 184, 67–76. [Google Scholar] [CrossRef]
 Ahmad, W.; Khan, S.A.; Kim, J. A Hybrid Prognostics Technique for Rolling Element Bearings Using Adaptive Predictive Models. IEEE Trans. Ind. Electron. 2018, 65, 1577–1584. [Google Scholar] [CrossRef]
 Liu, J.; Lei, F.; Pan, C.; Hu, D.; Zuo, H. Prediction of remaining useful life of multistage aeroengine based on clustering and LSTM fusion. Reliab. Eng. Syst. Saf. 2021, 214, 107807. [Google Scholar] [CrossRef]
 Chui, K.T.; Gupta, B.B.; Vasant, P. A Genetic Algorithm Optimized RNNLSTM Model for Remaining Useful Life Prediction of Turbofan Engine. Electronics 2021, 10, 285. [Google Scholar] [CrossRef]
State  Normal  Mild Degradation  Moderate Degradation  Severe Degradation  Failure 

Label  1  2  3  4  5 
Signal  The Hard Threshold  The Soft Threshold  The Fixed Threshold  

Number  SNR (dB)  RMSE  SNR (dB)  RMSE  SNR (dB)  RMSE 
1  22.6290  1.7373  22.6290  1.7373  22.6290  1.7373 
2  17.7301  72.3660  17.7301  72.3660  17.9301  72.3662 
3  18.7183  5.4896  18.7183  5.4896  18.7183  5.4899 
4  28.8169  295.2581  28.9579  290.4806  28.9879  289.4699 
5  58.7933  2.7439  58.7933  2.7439  58.7933  2.7437 
6  17.5773  69.3978  17.5773  69.3978  17.5773  69.3973 
7  21.9337  3.1351  21.9337  3.1351  21.9335  3.1351 
8  17.7781  181.0096  18.5190  166.2539  18.8581  159.9033 
Method  Index  1  2  3  4  5  6 

HSMM+SIR  LPP  95.16%  93.33%  90.00%  96.95%  95.16%  96.48% 
RMSE  3.2484  2.7805  4.9477  1.8166  4.9333  2.6907  
HSMM  LPP  95.00%  92.33%  90.00%  95.86%  95.16%  91.79% 
RMSE  3.7523  3.7951  5.5617  2.2050  4.1889  3.1714 
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Tian, G.; Wang, S.; Shi, J.; Qiao, Y. State Estimation and Remaining Useful Life Prediction of PMSTM Based on a Combination of SIR and HSMM. Sustainability 2022, 14, 16810. https://doi.org/10.3390/su142416810
Tian G, Wang S, Shi J, Qiao Y. State Estimation and Remaining Useful Life Prediction of PMSTM Based on a Combination of SIR and HSMM. Sustainability. 2022; 14(24):16810. https://doi.org/10.3390/su142416810
Chicago/Turabian StyleTian, Guishuang, Shaoping Wang, Jian Shi, and Yajing Qiao. 2022. "State Estimation and Remaining Useful Life Prediction of PMSTM Based on a Combination of SIR and HSMM" Sustainability 14, no. 24: 16810. https://doi.org/10.3390/su142416810