
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)11956-1
10.1016/j.heliyon.2024.e35925
e35925
Research Article
Adaptive Wiener process–based remaining useful life prediction method considering multi-source variability
Zheng Jianfei zjf302@126.com

Dong Qing 18756528162@163.com
⁎
Wang Xuanjun wangxj503@sina.com

Zhang Qingchao 568279427@qq.com

Du Dangbo ddb_effort@126.com

Rocket Force University of Engineering, Xi'an, 710025, China
⁎ Corresponding author. 18756528162@163.com
08 8 2024
30 8 2024
08 8 2024
10 16 e3592531 3 2024
9 7 2024
6 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Existing remaining useful life (RUL) prediction methods considering multi-source variability were not applicable to the situation that the uneven measurement interval distribution and inconsistent measurement frequency of degrading equipment. This type of method also has ignored the variability of adaptive drift in the future degradation process. In view of this, based on adaptive Wiener process, the paper proposes a new nonlinear degradation method of the RUL prediction. Firstly, adopting the adaptive Wiener process, we have constructed the nonlinear degradation model with multi-source variability, which randomness of the parameters in the nonlinear function. Secondly, the real-time estimation of multiple hidden states can be realized by the particle filter algorithm. It has derived the RUL distribution in the sense of first hitting time. Using monitoring data of degrading equipment, the adaptive update of model parameters was implemented by expectation maximization algorithm. Finally, the effectiveness and superiority of the proposed model are validated through numerical simulation and lithium-ion battery experiments. The results show that it can effectively improve the prediction accuracy, which has potential application value.

Graphical abstract

Image 1

Keywords

Remaining useful life
Multi-source variability
Nonlinear
Adaptive Wiener process
Particle filter
==== Body
pmc1 Introduction

With the development of advanced technology, the level of sophistication of modern industrial equipment is changing constantly. Owing to the comprehensive effect of internal stress (load change, energy consumption) or external environment (high temperature and pressure, vibration impact), the performance and health level of such equipment inevitably exhibits a trend of deterioration that leads to equipment degradation and failure. In practical engineering, it leads to the inability to complete normal tasks and functions, and even causes major economic losses [[1], [2], [3], [4], [5]]. If the remaining useful life (RUL) of stochastic equipment can be predicted at the degradation stage, and corresponding maintenance strategies scheduled, the failures can be effectively avoided. In recent years, the technology of prognostics and health management (PHM) has been widely applied, that providing effective informational and theoretical support for equipment maintenance decisions. Therefore, it is of great theoretical significance and engineering application value to accurately predict the RUL of equipment [[6], [7], [8]].

After decades of development, the RUL prediction method has achieved fruitful theoretical achievements and been widely applied in the different fields. Pecht et al. [9] divided RUL methods into two categories: mechanism-based modeling and data driven methods. Chai et al. [10] has putted forward data-driven RUL methods that had provided a feasible way of solving the control, decision-making, and optimization of such systems. It mainly includes artificial intelligence methods and statistical data driven methods. Yuan et al. [11] systematically have summarized and analyzed the PHM technology of equipment from the perspectives of physical model, data-driven, and fusion methods separately. Based on the RUL prediction problem of stochastic degradation equipment in the context of big data, Li et al. [12] had divided RUL prediction methods into failure mechanism and data driven methods. Data-driven methods mainly include statistical data driven methods and machine-learning methods. The former can be divided into failure data and degradation data methods. Compared with the failure data method, the degradation data method can establish a stochastic model describing the degraded process of equipment performance through the condition monitoring data of equipment. Then, the probability distribution of RUL can be obtained that is convenient in quantifying the variability of RUL, thus laying a foundation for health management. And it is more suitable for equipment with scarce failure data or costly access to failure data. The degradation data driven method can be subdivided into Wiener, Gamma, and inverse Gaussian processes. Compared with other methods, the Wiener process is a non-monotonic degradation process with a Gaussian independent distribution increment. Owing to the good mathematical characteristics, the Wiener process has been widely used in equipment reliability engineering and RUL prediction.

In engineering practice, the equipment is easily affected by internal and external factors besides normal operation, and then its performance degradation level usually exhibits time-varying and individual difference variability, such as lithium-ion battery capacity degradation and bearing fatigue cracking. In addition, it was inevitable that measurement errors can be introduced during the measurement process, which may result in measurement variability. Therefore, it is necessary to consider the multi-variability of the degradation process, to improving the accuracy of RUL prediction, and reducing the risk of equipment failure. Ye et al. [13] established a mixed effect degradation model based on the Wiener process for the wear degradation of hard disk drives, but failed to further deduce the distribution of RUL. On this basis, Si et al. [14] established a linear stochastic degradation model based on the Wiener process, and systematically studied RUL prediction under one, two, and three levels of variability. Zheng et al. [15] and Wang et al. [16] extended the linear model to a nonlinear one, and established a nonlinear stochastic degradation model based on the Wiener process. Classical Kalman filter theory was used to predict the RUL of the degrading equipment, and it was further proved that considering multi-source variability and nonlinearity at the same time could effectively improve the accuracy of RUL prediction. Lei et al. also studied the coexistence of multi-source variability and nonlinearity, and proposed a particle filter algorithm to predict the RUL of degrading equipment [17].

Based on the above, numerous researches have been done on the RUL prediction method with multi-source variability, but the following problems must still to be solved.(1) The above research usually considers multi-source variability and nonlinearity in degradation modeling, and regards the potential degradation state and drift coefficient as double hidden states. Then, it predicts the RUL of degradation equipment by a Kalman filter, but ignores the randomness of parameters in nonlinear functions. When we consider the multi-source variability, adding the randomness of the parameters in the nonlinear function will transform the double hidden states into multiple hidden states, which the hidden states are be coupled nonlinearly. It will make the classical Kalman filter theory difficult to perform.

(2) In the existing research of degradation modeling and RUL prediction methods based on the Wiener process, it has not considered the actual case of uneven monitoring interval and inconsistent measurement frequency with historical data in equipment lifecycle. At the same time, the update of the model drift coefficient is only limited to the update of real-time monitoring data, without considering the variability of model adaptive drift in future RUL prediction.

To address the above problems, considering the influence of multi-source variability and nonlinearity, this paper proposes a nonlinear degradation modeling and RUL prediction method based on an adaptive Wiener process. Firstly, a nonlinear stochastic degradation model with multiple implicit states is established under the framework of a state space model, which not only considers the multi-source variability and nonlinear influence in the degradation process, but also breaks through the requirement of fixed measurement interval and consistent sampling frequency. It also considers the variability of drift coefficient in future RUL prediction of equipment. Then, the real-time estimation of multiple hidden states is realized based on a particle filter, and the RUL distribution in the sense of first hitting time (FHT) derived. The self-adaptive estimation and updating of model parameters are realized by using an expectation maximization (EM) algorithm. Finally, the effectiveness and superiority of the method proposed in this paper are verified by an actual case of lithium-ion batteries and numerical simulations.

Generally, the main contribution and innovation of this paper can be summarized into three aspects.(1) Considering the influence of multiple source uncertainties, we had constructed a nonlinear stochastic degradation model with multiple hidden states, and the RUL distribution is derived in the sense of FHT.

(2) It solved the limitation of the existing method in fixed interval measurement and consistent sampling frequency, and realizing the variability of the adaptive drift term in the future degradation process.

(3) In the condition of historical data or prior information, the model parameters can be estimated and updated online based on the EM algorithm.

The remainder of this paper is organized as follows. In Section 2, we present the degradation modeling of the adaptive Wiener process with multi-source variability. In Section 3, we focus on the multiplicity state estimation based on the particle filter, and derive the analytical expression of RUL distribution. Section 4 provides a model parameter estimation method based on a particle EM algorithm. Numerical simulations and a case study of lithium-ion-battery data are presented in Section 5. Section 6 concludes the whole paper.

2 Degradation modeling of the adaptive Wiener process under multi-source variability

2.1 Adaptive Wiener process–based degradation model

Letting X(t) denote the degradation process of the equipment over time, the Wiener process degradation model usually takes the following form [18](1) Xt=X0+λt+σBBt

Where, λ and σB are the drift and diffusion coefficients, respectively. B(t) denotes standard Brownian motion (BM), and σBBt∼N0,σBt2 is utilized to describe the time-varying variability of the degradation process. Letting X(0)=x0 represent the initial degradation state, without loss of generality, assumes the initial degradation state X(0)=0.

The above Wiener process model has three shortcomings: uneven measurement interval, inconsistent measurement frequency, and ignorance of adaptive drift variability. Thus, when the effects of multi-source variability are not considered, an adaptive Wiener process is considered to describe the equipment degradation process {X(t),t>0} [19](2) {λt=λ0+kDtXt=∫0tλτdSτ;α+σBBt

Where, λ(t) is a drift coefficient that conforms to the Wiener process and changes with time, λ0>0 is the initial drift rate, k represents the adaptive drift diffusion coefficient, D(t) is standard Brownian motion independent of B(t), and S(τ;α) serves as a nonlinear function that increases with time t, abbreviating as S(τ). When k=0 and α=1, the equipment degradation model is a linear degradation process, and the degradation model becomes a Wiener process as shown in (1).

2.2 Degradation model under multi-source variability

In engineering practice, the obtained degradation monitoring data are often inevitably affected by noise, disturbances, irrational measuring instruments, and other factors. In this case, the monitored degradation data are non-rational and can only partially reflect the degradation state of the equipment, which is usually not directly considered to be the actual degradation state. Therefore, to characterize such measurement variability, the measurement process {Y(t),t>0} can be formulated as(3) Y(t)=h(X(t);ξ)+εt

where h(X(t);ξ) is the nonlinear function of potential degradation state X(t) and unknown parameter ξ, εt represents a random measurement error subject to Gaussian distribution, and εt∼N(0,σε2) [20].

At the same time, different individuals in the same batch of equipment often hold significant differences in their degradation trajectories due to the interaction between internal factors and the external environment. To describe the degradation process of a certain individual in the same batch of equipment and reflect the variability of individual differences, the drift coefficient λ of the degradation model and the parameter α in the nonlinear function S(τ;α) are randomized, i.e., λ∼N(uλ,σλ2) and α∼N(uα,σα2). Furthermore, assuming that λ, α, εt, B(t), and D(t) are independent of each other, we let Θ denote the model parameter vector in degradation modeling; that is, Θ=[uλ,σλ2,uα,σα2,σB2,σε2].

We let stochastic processes {X(t),t>0} and {Y(t),t>0} represent the practical degradation process and measurement process of the equipment. When considering the influence of multi-source variability, the equipment degradation model is presented as,(4) {λ(t)=λ0+kD(t)X(t)=∫0tλ(τ)dS(τ;α)+σBB(t)Y(t)=h(X(t);ξ)+εt

To realize the accurate prediction of the RUL of degrading equipment, the life of the equipment is defined according to the concept of the FHT of the stochastic degradation process; that is, the actual degradation process {X(t),t>0} of the equipment reaches the preset failure threshold for the first time (the threshold is generally determined by some industrial standards, such as vibration amplitude and gyro drift coefficient), and the equipment is considered to be degraded. Therefore, the equipment life T can be defined as(5) T=inf{t:X(t)≥w|X(0)<w}

Where, w is the preset failure threshold.

Similarly, the RUL Lk of degrading equipment in the sense of FHT is defined as(6) Lk=inf{lk>0:X(lk+tk)≥w|X(tk)<w}

Correspondingly, the probability density function (PDF) and cumulative distribution function (CDF) of the RUL of the degraded equipment at time tk can be denoted fLk|Θ,Y1;k(lk|Θ,Y1;k) and FLk|Θ,Y1;k(lk|Θ,Y1;k), respectively.

3 The RUL prediction of equipment under multi-source variability

3.1 The multiple hidden state estimation based on particle filter algorithm

First, the equipment degradation process is reconstructed based on the state-space model framework, and a nonlinear adaptive Wiener process–based degradation model considering multi-source uncertainties is established.(7) {λk=λk−1+ηxk=xk−1+λk−1ΔSk+γkyk=h(xk;ζ)+εkαk=αk−1

Where, λk represents the drift coefficient, and η=kΔDk, η∼N(0,k2Δtk). xk represents the potential degradation state, ΔSk denotes the nonlinear function at time interval Δtk, Δtk=tk−tk−1. γk=k∫tk−1tkDτdSτ+σBΔBtk, according to the BM property, it is further γk∼N(0,φ(tk)), φ(tk)=k2∫tk−1tk(ΔS(tk)−ΔS(τ))2dτ+σB2Δtk. yk represents the practical observation data, measurement errors εk obeys the normal distribution N(0,σε2). αk is a parameter of nonlinear function, and the initial random parameter α0=α∼N(uα,σα2).

Because of considering the randomness of parameter α in the nonlinear function S(τ;α) in this paper, Kalman filtering cannot be applied to the joint estimation of potential degradation state and random parameters. The particle filter technique is used to estimate the multiple hidden states of the model in real time. The potential degradation state xk, as well as the random parameters λk and αk, are regarded as multiple hidden states. Letting zk=[xkλkαk]T, based on the acquired degradation monitoring data Y1:k, the posterior distribution p(zk|Θ,Y1:k) of multiple hidden states can be obtained. According to the particle filter technique [[21], [22], [23]], the posterior distribution at time tk has the following form,(8) p(zk|Θ,Y1:k)≈∑i=1NWkiδ(zk−zki)

Where, N is the number of sampled particles, δ(⋅) is the Dirac delta function, zki=[xkiλkiαki] represents the sampled particle, Wki is the normalized weight, and ∑i−1NWki=1.

To calculate Eq. (8), it is necessary to use the prior distribution p(zk|zk−1i,Θ) to sample the particle zki. According to the discrete state-space model in (7), the prior distribution p(zk|zk−1i,Θ) is as follows,(9) {λki=λk−1i=λ0i∼N(uλ,σλ2)αki=αk−1i=α0i∼N(uα,σα2)p(xk|xk−1i,Θ)=12πφ(tk)×exp{−(xk−xk−1−λk−1ΔSk)22φ(tk)}

Where, φ(tk)=k2∫tk−1tk(ΔS(tk)−ΔS(τ))2dτ+σB2Δtk, uλ=λ0 and σλ2=k2Δtk.

Then, the importance weight wki corresponding to particle zki can be calculated in the following recursive form:(10) wki∝wk−1ipyk|zki,Θ=wk−1i12πσε2exp−yk−hxki;ζ22σε2

Furthermore, the normalized importance weight Wki is,(11) Wki=wki∑i=1Nwki

As the number of operations increases, only a small number of particles have relatively large importance weights, and most particles have small importance weights. Therefore, the variance of the particle importance weight becomes larger with increasing runtime, and the number of effective particles that play a role in the posterior distribution in the state space is relatively small, which makes most of the calculation time wasted on updating the particles with small importance weights, resulting in a decline in the overall estimation performance. Therefore, to assess the degree of particle degradation, an effective sampling scale is introduced and defined as(12) Neff=1∑i=1N(Wki)2

Where, the smaller the value of Neff, the more serious the particle degradation.

Usually, an effective method of solving the above problem is to use the resampling algorithm to resample the particles in order to remove the particles with small importance weight, and ‘reproduce’ the particles with large importance weight, thereby reducing the particle degradation. In general, the resampling threshold Nth is set, and, when the effective sampling scale Neff<Nth, resampling is performed, as shown in Fig. 1.Fig. 1 Resampling diagram.

Fig. 1

As shown in Fig. 1, the original weighted sample set {zki,Wki}i=1N obtains a new equal weight sample set {z˜ki,1/N}i=1N with weight 1/N by the resampling algorithm. Therefore, (10) can be written as(13) Wki∝p(yk|zki,Θ)

3.2 The RUL prediction under multi-source variability

Under the stochastic degradation modeling framework, the RUL Lk of the equipment is defined as(14) Lk=inf{lk>0:X(lk+tk)≥w}

Owing to the randomness of BM, the lifetime T is a random variable with inverse Gaussian distribution. According to Ref. [19], the PDF of the lifetime distribution of the adaptive Wiener process is(15) fT|λ(t|λ)=12πφ(t)[w−λS(t)φ(t)φ′(t)+λS′(t)]exp{−(w−λS(t))22φ(t)}

where φ(t)=k2∫0t(ΔS(t)−ΔS(τ))2dτ+σB2t, φ′(t)=dφ(t)dt.Lemma 1 [24]: Given the potential degradation state xk and random parameter λk at time tk, based on degradation data Y1:k, we have(16) fLk|xk,λk,Y1:k(lk|xk,λk,Y1:k)=fLk|xk,λk,(lk|xk,λk)

Therefore, considering multi-source variability, the PDF of the RUL can be calculated as follows:(17) fLk|Θ,Y1:k(lk|Θ,Y1:k)=∫−∞∞fLk|zk,Θ,Y1:k(lk|zk,Θ,Y1:k)p(zk|Θ,Y1:k)dzk=Εzk|Θ,Y1:k[fLk|zk,Θ,Y1:k(lk|zk,Θ,Y1:k)]≈∑i=1NWki{12πφ(lk)(w−xk−λkiS˜(tk)φ(lk)φ'(lk)+λkiS˜'(tk))exp{−(w−xk−λkiS˜(tk))22φ(lk)}}

where S˜(tk)=S(tk+lk)−S(tk) and S˜′(tk)=dS˜(tk)dtk.

Proof: According to lemma 1, we have(18) fLk|Θ,Y1:k(lk|Θ,Y1:k)=fLk|zk,Θ(lk|zk,Θ)

Based on (15) and (16), there are(19) fLk|zk,Θ,Y1:k(lk|zk,Θ,Y1:k)=12πφ(lk)(w−xk−λkiS˜(tk)φ(lk)φ'(lk)+λkiS˜'(tk))exp{−(w−xk−λkiS˜(tk))22φ(lk)}

Then, using the particle filter algorithm, we obtain(20) fLk|Θ,Y1:k(lk|Θ,Y1:k)=∫−∞∞fLk|zk,Θ,Y1:k(lk|zk,Θ,Y1:k)p(zk|Θ,Y1:k)dzk=∑i=1NWki∫−∞∞fLk|zk,Θ,Y1:k(lk|zk,Θ,Y1:k)δ(zk−zki)dzk

Further, (19) is brought into (20) to obtain the PDF of the RUL distribution of the equipment under multi-source variability.□

4 Parameter estimation

To estimate the model parameters, we first set Θ=[uλ,σλ2,uα,σα2,σB2,σε2]. Since the state-space model zk contains three hidden states (potential degradation state xk and random parameters λk and αk), according to the maximum-likelihood estimation (MLE) method, the logarithmic likelihood function of the equipment monitoring data Y1:k with respect to the model parameter vector Θ is(21) l(Θ)=lnp(Y1:k|Θ)

where p(Y1:k|Θ) is the joint PDF. The maximum-likelihood function value Θk of the model parameter vector Θ can be obtained by maximizing (21)(22) Θk=argmaxΘlk(Θ)

Owing to the existence of multiple hidden states zk in the degradation model, the model likelihood function cannot be directly maximized. Therefore, it is necessary to estimate the model parameter vector Θ based on the observable degradation data Y1:k. In this paper, the estimation of unknown parameters will be realized under the framework of the EM algorithm. According to the principle of the EM algorithm [25], the specific implementation process can be iterated by the following two steps:

E step:(23) l(Θ|Θk(q))=Εzk|Θk(q),Y1:k{lnp(zk,Y1:k|Θ)}

M step:(24) Θk(q)==argmaxΘ{l(Θ|Θk(q))}

By continuously iterating the E-step and M-step until a certain convergence condition is satisfied, the optimal parameter estimation results are obtained. According to the state-space model, based on the Bayesian theorem and Markov property, the log joint likelihood function of the model parameter vector Θ is(25) lnp(zk,Y1:k|Θ)=lnp(Y1:k|zk,Θ)+lnp(zk|Θ)=lnp(z1|z0,Θ)+∑m=1k−1lnp(zm+1|zm,Θ)+∑m=1klnp(ym|zm,Θ)

We find the conditional mathematical expectation of (20) by the E-step(26) l(Θ|Θk(q))=Λ1+Λ2+Λ3

Where,(27) {Λ1=∫lnp(z1|z0,Θ)p(zk|Θk(q),Y1:k)dz1Λ2=∑m=1k−1∫∫lnp(zm+1|zm,Θ)p(zm+1,zm|Θk(q),Y1:k)dzmdzm+1Λ3=∑m=1k∫lnp(ym|zm,Θ)p(zm|Θk(q),Y1:k)dzm

Since the degradation model is nonlinear, it is difficult to obtain the analytical solution of (27). In this paper, a particle smoothing algorithm is used to solve (27) numerically. The specific steps of the algorithm are the following.Step 1: Obtain a weighted sample set {zmi,Wmi}i=1N,m=1,2,…,k, by particle filtering.

Step 2: Initialize Let m=k, the particle smoothing importance weight Wk|ki=Wki,i=1,2,…N, and start backward iteration from m=k−1.

Step 3: Calculate the importance weight for particle smoothing:

Wm|ki=Wmi∑j=1NWm+1|kjp(zm+1j|zmj,Θ)∑l=1NWmlp(zm+1j|zmj,Θ)

Step 4: Calculate the smoothing density:

p(zm|Θ,Y1:k)≈∑i=1NWm|kiδ(zm−zmi)

Step 5: Iterative update. Let m=m−1, and, if m>0, return to Step 3; otherwise, terminate the algorithm.

Based on the E-step, the conditional mathematical expectation l(Θ|Θk(q)) is obtained, and the parameter estimation Θk(q+1) of the M-step is obtained by maximizing l(Θ|Θk(q)) with respect to the model parameter vector Θ. Since in this paper we use nonlinear degradation modeling, it is difficult to obtain the specific analytical solution of Θk(q+1) in the M-step, and thus a gradient-based search technique is used to solve the problem, as shown in the following equation:(28) ∂l(Θ|Θk(q))∂Θ=∂Λˆ1∂Θ+∂Λˆ2∂Θ+∂Λˆ3∂Θ

The E-step and M-step are executed alternately until the estimated result satisfies the preset algorithm termination condition; that is, the maximum number of iterations or parameter convergence. When new condition monitoring data are encountered, the previously estimated parameters are used as the initial values of the EM algorithm, and the EM algorithm is executed based on the new monitoring data to obtain the latest estimated values of the relevant parameters.

Based on the above results, the adaptive Wiener process RUL prediction algorithm proposed in this paper under multi-source uncertainty is summarized as algorithm 1.Algorithm 1 (The adaptive Wiener process RUL prediction algorithm under multi-source uncertainty)

(1) Initialization. tk(k=1) serves as the initial moment, q=0 denotes the number of iterations, and the model parameter vector Θˆk(0) is initialized.	
(2) E-step and M-step calculation. The particle smoothing calculation yields l(Θ|Θˆk(q)) in Eq. (23); according to Eq. (28), we can calculate,
Θˆk(q+1)=argmaxΘ{l(Θ|Θˆk(q))}	
(3) Iteration. Iterating the E-step and M-step until a certain convergence condition is satisfied.	
(4) RUL prediction. Based on parameter estimates Θˆk(q+1) of M-step, we had computed the posterior distribution of multiple implicit state zk, i.e. p(zk|Θˆk(q+1),Y1:k)≈∑i=1NWkiδ(zk−zki), and calculated the RUL prediction of the equipment at time tk through Eq. (17).
(5) Adaptive. When the next time of degradation data was obtained, we had made the current parameter estimate Θˆk(q+1) as the new initial value, and k=k+1,q=0, returning (2).	

5 Experiments

The Monte Carlo method is introduced for numerical simulations to verify the superiority of the proposed model. Then, lithium-ion-battery capacity degradation is adopted for example verification.

5.1 Numerical simulation

Owing to the complexity and variability of the equipment operating environment, the obtained degradation data may have a certain contingency, resulting in a large deviation between the predicted results and the real life. To eliminate the influence of this contingency, a set of data is randomly generated by the Monte Carlo simulation method. For convenience, the method proposed in this paper is denoted M0, and the adaptive Wiener process model proposed in Ref. [19] is denoted M1. These two methods are used to predict the RUL, and the accuracy of the RUL prediction results of the two models is compared.

First, the degradation predicted by the two methods is given. It can be observed from Fig. 2 that the degradation path of model M0 is closer to the degradation monitoring data of numerical simulation than model M1, considering the influence of multi-source variability.Fig. 2 Numerical simulation degradation data.

Fig. 2

In order to verify the performance of the proposed algorithm, based on the numerical simulation data in Fig. 2, we have compared the algorithm proposed with the MLE algorithm. The parameter estimation of the two algorithms can be obtained on the same simulation data, as shown in Table 1. As can be seen from Table 1, the algorithm used in this paper is more accurate in parameter estimation, and its performance is more effective.Table 1 Estimation of model parameters.

Table 1Parameter	True value	M0	MLE	
μλ	0.05	0.0572	0.2406	
σB	0.4	0.3864	0.2396	
μα	2	1.9203	1.5594	

By comparing the prediction performance of the two methods, the RUL distribution at different monitoring points is given, as shown in Fig. 3. It can be seen from the figure that, compared with model M1, model M0 considers the influence of multi-source variability of monitoring degradation data and parameter randomness of the nonlinear function on RUL prediction. The RUL distribution of the former is narrow and sharp, the variability of the corresponding prediction results smaller, and the prediction results more accurate.Fig. 3 RUL of two models (M0 and M1) at different monitoring points.

Fig. 3

To further prove the effectiveness of the proposed method, i.e., M0, the absolute error (AE) of the RUL prediction results of the two methods is given for performance evaluation. It can be seen from Fig. 4 that the AE of the RUL prediction results of the two methods decreases with the continuous operation of the equipment, but that of the prediction of model M0 is significantly smaller than that of model M1, and the prediction accuracy of the former is higher.Fig. 4 Absolute error of RUL prediction results.

Fig. 4

5.2 Lithium-ion-battery data

As a new product of energy savings and environmental protection, lithium-ion batteries have advantages of long-life, high-energy density, high utilization rate, and low pollution. They are widely used in new energy, aerospace, automobile, and other fields. At the same time, lithium-ion batteries are widely used in missile weapons and equipment to provide electrical energy for various pieces of equipment to ensure their smooth operation. Therefore, accurate prediction of lithium-ion battery RUL is of great significance to maintain equipment performance.

In view of this, we selected the National Aeronautics and Space Administration (NASA) lithium-ion-battery capacity degradation data [26] for example verification. Under normal room-temperature conditions, the dataset is obtained by charge-discharge experiments, and the changes of battery state information (including battery capacity information) with charge–discharge cycles recorded. Owing to the loss of battery material caused by chemical reactions, the capacity of lithium-ion battery will decrease with the charge–discharge cycle. Fig. 5 shows the capacity degradation data of four lithium-ion batteries (# 5, # 6, # 7, and # 18).Fig. 5 Lithium-ion battery capacity degradation data.

Fig. 5

It can be seen from Fig. 5 that, with the battery charge–discharge cycle, the battery capacity degradation data show a nonlinear downward trend as a whole, and the magnitude of the fluctuation is determined by the time-varying variability in the degradation process and the measurement variability in the monitoring data. Therefore, to be more in line with the actual situation and improve the accuracy of RUL prediction, it is necessary to consider both multi-source variability and nonlinearity in degradation modeling. The particle filter technique and EM algorithm proposed in the preceding section were used to identify the model parameters.

It can be seen from Fig. 6 that, when the EM algorithm iterates for a certain number of steps, the estimated value of the model parameters will converge to the corresponding value, indicating the effectiveness of the parameter identification method proposed in this paper.Fig. 6 Model parameter identification process.

Fig. 6

In engineering practice, it is generally believed that, when the degradation capacity of a lithium-ion battery is attenuated by 30 %, it is considered that the use demand is not met; that is, the fault is considered to be invalid. Battery #5 was selected for verification. Through the corresponding transformation, the real and predicted degradation data of a lithium-ion battery conforming to the Wiener process can be obtained, as shown in Fig. 6. The results in Fig. 7 show that the proposed method has a good fitting effect between the predicted and real data, and can accurately predict the capacity degradation process of lithium-ion batteries.Fig. 7 Battery-capacity degradation process.

Fig. 7

To verify the effectiveness and superiority of the proposed method, on the basis of the preceding section, was added the M2 method [16] for comparative study, that the RUL prediction method of the Wiener process considering triple variability.

The probability distribution of the RUL prediction for the three methods is illustrated in Fig. 8, Fig. 9. It can be observed from figure that, compared with model M1, the proposed method considers the influence of multi-source variability and the randomness of nonlinear parameters. The PDF curves of the RUL predicted by the proposed method cover the actual RUL, and as the monitoring time increases the PDF curves become increasingly sharper, which means that the predicted RUL results are increasingly accurate and the variability is increasingly lower. According to Fig. 9 the proposed method M0 and model M2 consider the influence of multiple uncertainties at the same time. However, the proposed method can overcome the uneven distribution of measurement intervals and inconsistent measurement frequencies of stochastic degradation equipment and consider the variability of adaptive drift in the future degradation process. Therefore, the prediction results of model M0 are more accurate and have higher prediction accuracy.Fig. 8 The RUL distribution of models M0 and M1.

Fig. 8

Fig. 9 The RUL distribution of models M0 and M2.

Fig. 9

Fig. 10 gives the mathematical expectation of the RUL prediction of the three methods. As shown in the figure, compared with models M1 and M2, the RUL prediction result of the proposed method, i.e., M0, is closer to the actual RUL of the equipment.Fig. 10 The RUL prediction results.

Fig. 10

To further quantify the accuracy of the RUL prediction results, the mean square error (MSE) was introduced for performance evaluation. The MSE of the equipment is defined as follows [27,28].(29) MSEk=∫0+∞(lk−l˜k)2fLk|Θ,Y1;k(lk|Θ,Y1;k)dlk

Where, l˜k is the real RUL of time tk, and fLk|Θ,Y1:k(lk|Θ,Y1:k) represents the RUL distribution at time tk.

The MSE comparison results of the three methods at all monitoring time points are illustrated in Fig. 11. It can be observed from the figure that with the continuous accumulation of degradation data, the MSE of RUL prediction is decreasing. Among the three methods M0, M1, and M2, the MSE corresponding to the proposed method, i.e., M0, is the smallest, which further proves that the proposed method has better model fitting than M1 and M2, and can effectively reduce the variability of prediction.Fig. 11 The mean square error of different monitoring points.

Fig. 11

The above experimental results indicate that the proposed method can significantly improve the accuracy of RUL prediction, reduce the variability of prediction, and have better prediction performance in practical applications.

6 Conclusions

Aiming at the problem of multi-source variability in practical engineering, an RUL prediction method based on the nonlinear adaptive Wiener process is proposed in this paper. Based on nonlinear adaptive Wiener process, we had constructed a degradation model with multiple hidden states. It not only considered the multi-source variability and nonlinearity, but also considered the variability of adaptive drift in the future degradation process. The analytical expression of RUL distribution was derived in the sense of FHT, and the model parameter identification and online updating are realized by particle filter technology and EM algorithm. Finally, the validity of the proposed method was verified by numerical simulation and lithium-ion battery degradation case, and holding certain potential application value.

In this paper, we mainly studied the single-stage degradation process of equipment. In the future, the research will be considered the RUL prediction problem under multi-stage degradation and multi-source variability.

Data availability statement

This paper used the practical data from NASA's publicly available lithium-ion battery dataset [26].

CRediT authorship contribution statement

Jianfei Zheng: Funding acquisition, Conceptualization. Qing Dong: Writing – review & editing, Writing – original draft, Methodology, Conceptualization. Xuanjun Wang: Formal analysis, Data curation. Qingchao Zhang: Writing – review & editing, Funding acquisition. Dangbo Du: Writing – review & editing, Validation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is partially supported by 10.13039/501100001809 National Natural Science Foundation of China under, Grant 62103433 ,Grant 62227814 , and Grant 62203462 ; in part by University Association for Science and Technology Young Talents Promotion Plan of Shaanxi Province under Grant 20210408 .
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