Storage Life Prediction Method of the Aerospace Electromagnetic Relays Based on Physics of Failure and Data-Driven Fusion

The storage life prediction of the aerospace electromagnetic relays(AEMRs) has become an engineering challenge due to the nonlinearity of AEMRs’ complex degradation process. As the widely used life prediction approaches, data-driven methods also failed to address this issue effectively. The main reason is their inherent flaw: the inability to effectively quantify the direct correlation between top-level failures and underlying physical or chemical changes. Therefore, based on the fusion of the physics of failure(PoF) and data-driven, a storage life prediction method for AEMR has been proposed in this manuscript. Firstly, according to the characteristics of the AEMR during storage, the degradation performance parameters were analyzed, and the accurate expression of the degradation model was established based on the failed physical process. Then, the degradation process of AEMR was modelled by fusion of the PoF and data-driven, while the model parameters were updated by an improved particle filter based on degradation data. Finally, the storage life of AEMR was predicted by the updated model. Compared with typical methods and machine-learning-based methods, on the one hand, the results show that the proposed method can restrain the fluctuation of nonlinear degradation data to some extent and make the fluctuation range of prediction error more stable, thus making the life prediction result more accurate. On the other hand, even when there is only a small sample size of observation data, it also gives a good prediction effect. Moreover, it establishes the relationship between life and underlying physical and chemical processes, which will be conducive to the optimization design of AEMR, to further improve the reliability of weapons and equipment.

Life prediction using the fusion method is currently in 103 the exploratory stage, but some academics have successfully 104 applied it in their respective domains. To explain that the 105 fusion method is more advantageous than a single model, 106 Chao et al. [9] presented a fusion method of the PoF and 107 data-driven which achieved effective remaining useful life 108 prediction for turbofan engines under actual flight conditions. 109 A fusion algorithm by merging the electric-thermal coupling 110 model with state monitoring data in [10] realized the online 111 reliability evaluation of IGBT. Chen et al. [11] proposed 112 an adaptive residual life prediction method based on the 113 combination of EM-EKF for airborne electronic equipment 114 and proved the accuracy of the prediction method. Although 115 Chen et al. chose an extended Kalman filter in their paper to 116 broaden the application scope of the Kalman filter to solve 117 nonlinear problems, linearization of nonlinear problems by 118 Taylor expansion and omission of higher-order terms may 119 easily lead to bring larger estimation errors. The particle filter 120 (PF) algorithm, theoretically, has more significant advantages 121 for treating nonlinear system problems [12], [13], [14], [15], 122 but its particle degradation problem also seriously restricts the 123 practical application effect, resulting in an unsatisfactory final 124 prediction result. Therefore, solving the particle degradation 125 problem will significantly enhance prediction accuracy. 126 In summary, using the data-driven AEMR storage life pre-127 diction method must address three problems. The first is that 128 existing models failed to capture the representative relation-129 ship between the underlying mechanism and AEMR lifetime. 130 The second is that data-driven methods require excessive 131 data, making it hard for AEMR to provide vast data. The 132 third is particle deterioration in the implementation of the 133 fusion method. Therefore, in this paper, we propose a fusion 134 method of the PoF and a data-driven to predict the storage 135 life of AEMR. Firstly, according to the storage characteristics 136 of AEMR, the physical failure process of storage degrada-137 tion is analyzed, and establish a model. Then, combining 138 the PoF model with the data-driven method, the modeled 139 degradation model, and using the improved particle filter to 140 update the model. Finally, the updated model predicts the 141 storage failure life according to the failure threshold. 142 The remainder of this article is organized as follows. 143 Section II, the PoF model of AEMR is established based on 144 stress relaxation. In section III, we proposed an improved 145 particle filter algorithm and conducted particle diversity tests. 146 In section IV, a case study is carried out to verify the storage 147 degradation data of AEMR. Section V concludes this article. 148

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The internal reeds, coils, bobbins, enameled wires, and mag-151 netic materials of AEMR are susceptible to deterioration 152 during long-term storage, which affects the movement and 153 contact properties of the AEMR itself. That will fail if it 154 exceeds the acceptable range. In the case of AEMR in storage, the main reason is the stress relaxation of the reeds, which 156 decreases the response force and therefore affects the match-157 ing of the reaction force characteristics, resulting in contact 158 failure and malfunction [16].

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Stress relaxation [17] is a phenomenon when a part or mate-161 rial is subjected to deformation, transitions from elastic to 162 plastic strain and stress decrease with time. When the reed 163 material is processed through deformation and failure process 164 treatment, the uneven distribution of the intergranular type 165 II stress and macroscopic stress in the material leads to an 166 increase in the elastic strain energy of system, making the 167 whole system, which is a stable state, destabilized. Then the 168 free energy of a system can be expressed as: where dG denote the entropy change in system, dU is internal 171 energy, dG is Gibbs free energy, T is absolute temperature.

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When the reed is subjected to bending stress, we have 173 dE = A · f · dp. dp denote the load micro-element on the reed, Since during stress relaxation, the total strain ε t is the sum of 187 plastic strain ε p and elastic strain ε e : where σ 0 is initial stress, σ is residual stress, E c is elastics 190 modulus. Take the derivative of both sides of (2): According to Orowan equation [18], the plastic strain rate 193 can be expressed as: where is the geometric parameter, ρ m is the movable 196 dislocation density, b is burgers vector and v is the average 197 dislocation rate. According to the Johnston-Gilman formula, 198 the average dislocation rate can be expressed as: where B and m * are the material constants at a given temper-201 ature. Simultaneous (3) to (5) are obtained: Further, it can be obtained: The equation (7) is Li's equation, where σ * is the effective 206 According to (7), after the stress relaxation of the spring 209 occurs, the reaction force F s of the AEMR meets the follow-210 ing relationship: The relationship between the density of movable disloca-213 tions and time is considered in the [19]: Substituted (9) into (6) can be solved: The equation (10) can be established under the condition 218 that dislocation density is unevenly distributed. In the equa-219 If the 221 distribution is uniform, the Li equation can be used directly. 222 Therefore, according to (10), the degradation formula of 223 the reaction force can be expressed as: By combining the structural features of the AEMR with the 228 storage degradation mechanism analysis, [16] concluded that 229 the increase in the release time during storage is mainly 230 caused by the reduction of the reed reaction force. However, 231 the reed is sealed inside the relay, and it is not easy to directly 232 open and test the stress relaxation state of the reed during 233 actual storage. Therefore, finite element simulation can be 234 used to obtain the relationship between the initial force of the 235 reed and the release time, and the results are shown in Fig. 1. 236 The approximately linear relationship between the release 237 time T release and the initial force F initial of the underlying per-238 formance parameter as the reaction force of the reed decreases 239 in a certain range. Therefore, this section will express the 240 relationship between the two as a linear function: where a and b represent model parameters related to design 243 and manufacturing process parameters, characterizing indi-244 vidual heterogeneity among AEMRs of the same batch.

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From the analysis of mechanism, it can be seen that the 246 basic reason for the reduction of reaction force is caused by 247 VOLUME 10, 2022  (12): To make (13) 267 where x k is the predicted value, y k is the observed value, 268 q k−1 is the predicted noise and r k is the observed noise 269 respectively.

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If the initial probability density function (PDF) of the 271 known state is f (x 0 |y 0 ) = f (x 0 ), the above nonlinear dynamic 272 system can be regarded as a hidden Markov process, which 273 satisfies the following two equations:

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According to the Bayesian theory, the conditional proba-  The values obtained by the estimation criteria can be obtained 280 in this CDF. Therefore, the filtering problem is transformed 281 into the issue of solving the distribution. Under the framework 282 of Bayesian filtering, the CDF can be obtained in two stages: 283 prediction and update. Then the state prediction equation can 284 be written: the state update equation can be written: It is evident from the above formula that solving distribu-289 tion may face dimensional catastrophe, so the particle filter 290 (PF) algorithm is proposed to solve the integral by using 291 particle approximation distribution. Its core idea is based on 292 the sequential Monte Carlo sampling method. However, sam-293 pling is usually difficult to sample the target PDF. Therefore, 294 the importance density function, which is easy to sample, 295 is selected for sampling, and samples are given weight to 296 complete the approximation of the target PDF through the 297 weighted samples. Therefore, the PDF can be expressed as: 298 where δ(·) is Dirac function, H (·) is the importance density 301 function, and ω i j is the connection weight.

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In this paper, when using the particle filtering algorithm 303 for life prediction of AEMRs, the nonlinear characteristics 304 presented by the degradation process of aerospace relays lead 305 to a particle degradation problem in the solution process, 306 thereby failing to obtain good approximation results. There-307 fore, resampling is required. Generally, a random number 308 u i ∈ (0, 1] is selected, and when Equation (19) is met, 309 the particles corresponding to ω i j are selected as resampling 310 samples.
However, the traditional re-sampling method only repli-313 cates the particles with higher weights and abandons the 314 particles with lower weights. As a result, some high-weight 315 particles may be sampled multiple times, and even worse, 316 in extreme cases, all the sampling operations are carried out 317 around only one high-weight particle, which will seriously 318 lose the diversity of particles, fail to cover the region of 319 the posterior distribution, and seriously affect the subsequent 320 prediction update [22], [23]. Therefore, it is necessary to 321 introduce an effective re-sampling strategy to solve the prob-322 lem of particle scarcity in the estimation process, given the 323 nonlinear degradation characteristics of the objects studied 324 in the storage process.

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To address the problem of particle degradation in this paper, 327 we proposed a method of combining an intelligent particle  The high-weight group is reserved for elite particles, and 344 the low-weight group is common for crossover and mutation 345 operation.
where P L is the set of particles with low weight, P H is the 348 set of particles with high weight, x elite k is elite particle belong 349 to P H , and is the weight used to distinguish particles. Crossover: as shown in (22), particles with lower weights 355 after crossover are represented as x l kS . x kL is particles ran-356 domly taken from P L .
where, l = 1, 2, · · · L, L is the number of particles in P L . 359 For each x l kS , it is matched by particle x l kL randomly selected 360 from P L , and α is the cross coefficient of particles randomly 361 selected from [0, 1].

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Mutation: in order to improve the diversity of particles, the 363 new particle x l kN is obtained by mutation operation on the 364 crossed particle x l kS .
where x l kN is the mutated particle, ξ ∈ [0, 1] is the random 367 variation coefficient, 1 − (Neff /N ) is the adaptive variation 368 probability, and N is the total number of particles.

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Judgment: the weights of the mutated particle are calcu-370 lated and compared with the x elitle k , and the elite particle is 371 updated if they have a higher value.

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This part is the test of particle diversity. The actual storage 374 degradation data of AEMR is used to verify this method, 375 where the particle number is set to 100. The particle distribu-376 tions at storage time points K = 20, 60 and 100 were selected, 377 respectively. The particle distribution is shown in the figure 378 below.

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As can be seen from Fig. 3-Fig. 5, since the re-sampling 380 only replicates the particle with high weights in the PF and 381 then performs the state estimation, this leads to the distribu-382 tion of particles on a few state values, which results in the 383 phenomenon of sample depletion and is not conducive to the 384 VOLUME 10, 2022   overall state estimation. In this paper, the particle distribution

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(3) Calculate the predicted value at the next moment: (4) Neff th is used to determine whether re-sampling is 416 needed, and if not, go to step 5 to update the output. If nec-417 essary, re-sample the particles using the method mentioned 418 above.

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Update step: 420 (5) The updated valueT k (t,θ k ) release is obtained, where the 421 latest model parameter isθ k = (â 1 ∼â 5 ). 422 (6) By substituted model parameters into (13) and com-423 bined with failure threshold, the latest storage life prediction 424 results can be obtained. This part will take AEMR as the object to verify the method 429 proposed in this paper. The degradation data acquired from 430 the storage acceleration experiment used the developed time 431 parameter test system. The overall block diagram of the test 432 system is shown in Fig. 7. The testing system comprises 433 a thermostat, relay switching circuit, time parameter test-434 ing circuit, lower computer and upper computer software. 435 The relays to be tested are connected to the test circuit by 436 switching circuit for time parameter testing. In addition to 437 controlling the temperature and humidity of the thermostat 438 and the switching state of the switching circuit, the lower 439 computer is also responsible for sending the test data into 440

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This section will use the method proposed in this paper to 457 predict the storage life of AEMR. Firstly, the initial model 458 parameters were obtained as a 1 = 7.108, a 2 = 1.117, 459 a 3 = 38.104, a 4 = −1.399, and a 5 = −0.069 using the 460 established PoF to fit the degradation data from the first 80 461 acquisitions. Then, once the new storage degradation data 462 were observed, the model parameters were updated using the 463 IPF for the AEMR. Finally, the latest degradation model was 464 used to predict the storage life of the AEMR. Fig. 11-Fig. 13 465 respectively shows the degradation model curves of AEMR 466 when 40, 80 and 120 data points are updated, and Fig.14     PoF model established based on stress relaxation in this paper. 483 Still, the parameter update algorithm adopts an ordinary PF 484 and the traditional approach of copying high-weight particles 485 for re-sampling. M3 is the method proposed in this paper. 486 Fig. 19 shows that the early life prediction results of M1 487 have a significant deviation, mainly because the classical 488 model fails to describe the degradation process well, espe-489 cially in the early transition stage of rapid and slow degrada-490 tion. Fig. 15-Fig. 18 shows the fitting curve results of model 491 superiority verification by sampling different AEMR sam-492 ples. The blue curve is based on the PoF model, and the green 493 curve is the classical logarithmic model. It is evident from the 494 figure that the PoF model can describe the degradation pro-495 cess more accurately. For M2, due to the PoF model adopted, 496 the prediction result of early life is more accurate than M1. 497 However, the overall prediction result differs dramatically 498 from the actual failure life. The main reason is that for severe 499 nonlinear problems if the re-sampling algorithm of PF is still 500 used, it will cause severe particle degradation, which will 501 directly lead to the accuracy of prediction results.

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As for the method in this paper, it can be seen from the 503 mean absolute error(MAE) curve in Fig. 20 that the errors 504 are smaller than M1 and M2, and the MAE are significantly 505 reduced to 63.4% and 51.6%, respectively. At the same time, 506 the curve changes more gently, which shows that the storage 507 life prediction method in this paper has a better suppression 508 effect on the fluctuation of nonlinear degradation data. How-509 ever, with the continuous update of data, the MAE curve will 510 also have a certain upward trend in the later period. Machine learning (ML) is the core of artificial intelligence, 514 and its main function is to enable computers to simulate or 515 implement human learning behavior by acquiring new infor-516 mation and continuously training the model to improve its 517 generalization ability [26]. ML is usually divided into surface 518 and deep learning methods, which have achieved good results 519 when applied to life span prediction. Therefore, to further 520    method is used for prediction in this paper, the data before 527 the prediction start point shown in the previous Fig. 11 is 528 first used to train the model for practice, and then when new 529 data is observed, it is fed into the completed training model to 530 predict it, where the model is trained iteratively as the data is 531 observed. A CNN-LSTM-based approach to lifetime predic-532 tion, a deep learning approach, is described in literature [28]. 533 In order to adequately train the model, a combination of 534 repetitive segmentation and sliding time windows are used to 535 generate training samples, where the window length is 80, and 536 the step size is 1. The predicted lifetime results are obtained 537 by repetitively generating samples and combining them with 538 a failure threshold based on the first arrival time. The life 539 prediction results and error curves are shown in Fig. 21  As can be seen in Fig. 21 and Fig. 22, when comparing 543 the proposed method with the ML-based method, M3 is not 544 inferior to each other in terms of lifetime prediction accuracy, 545 especially in the first half of the degradation process. As more 546 and more observations are available, ML1 and ML2 become 547 more and more effective. It shows that ML has its unique 548    Based on this, a fusion of PoF and data-driven life predic-588 tion method is further constructed. In addition, an improved 589 particle resampling strategy is proposed to improve particle 590 diversity. Thus, the fusion method's prediction accuracy is 591 improved to address the particle degradation problem caused 592 by particle filtering to update the model parameters when   In end, there are two issues needed to be further studied. The authors would like to thank the editor, associate editor, 634 and anonymous referees for their insightful suggestions and 635 constructive comments, which improved the quality of the 636 article, and also would like to thank Yigang Lin for helpful 637 suggestions.