An Age-Dependent and State-Dependent Adaptive Prognostic Approach for Hidden Nonlinear Degrading System

Zhenan Pang, Xiaosheng Si, Changhua Hu, and Zhengxin Zhang

Abstract—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 age-dependent 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 unit-to-unit 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 age-dependent and state-dependent nonlinear degradation model taking into consideration the unit-to-unit variability and hidden degradation states. Then, the Kalman filter (KF) is utilized to update the hidden degradation states in real time, and the expectation-maximization (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 Li-ion batteries and rolling element bearings.

I. INTRODUCTION

REMAINING useful life (RUL) estimation can improve the availability, reliability and security of degrading systems through providing useful information for conditionbased maintenance (CBM), optimal inspection and spare part provision [1]–[3]. How to accurately estimate the RUL distribution has been one of the core issues of prognostic and health management (PHM), and it has attracted widespread attention from scholars in recent years [4], [5].

Because of various uncertain factors, the stochastic process model is usually used to characterize the degradation process of the degrading system. Siet al.[6] presented a comprehensive review of common stochastic degradation models utilized to estimate the RUL of the degrading system,including Wiener processes [7]–[9], Gamma processes [10],[11], and inverse Gaussian processes [12]–[14]. Among them,the Wiener process is widely used to describe the nonmonotonic continuous degradation process because of good mathematical and physical meanings, such as [15]–[17].Zhanget al.[18] further reviewed the latest advances in Wiener process-based approaches for degradation data modeling and RUL estimation and their applications in the field of PHM. These models can accurately estimate the RUL of the degrading system when the degradation process is suitable to be characterized by the model only depending on the age.

The degradation process of most degrading systems,however, is affected not only by the system age, but also by its own degradation states. A number of scholars have found that the degradation processes of fatigue cracks [19], cylinder sleeves [20], metal alloys [21] and ball bearings [22] all have this characteristic. Taking the process of the fatigue crack propagation as an example [19], its degradation rate is low in the initial stage of the crack propagation. As the crack propagates, its degradation accelerates significantly. Under this circumstance, the degradation model that depends only on the age of the system can not accurately characterize the degradation process, and a certain deviation between the estimated RUL and the actual RUL when applied to prognostics. Recently, there have been only a few studies on age- and state-dependent degradation models to handle the RUL estimation of such degrading systems. Giorgioet al.[21]studied a degradation model on the basis of the four-parameter Markov chain to characterize the state-dependence. Anet al.[23] proposed the sequential Monte Carlo (SMC) algorithm to solve the RUL estimation problem of the state-dependent model, and applied it to the degradation process of batteries and fatigue cracks. Orchardet al.[24] also used the SMC method on the basis of the state-dependent model to estimate the probability density function (PDF) of the RUL. Zhanget al.[22] established a nonlinear Wiener process model with age and state dependence for the RUL estimation of bearings.Liet al.[25] presented the degradation process simulation(DPS) method to estimate the RUL for the degrading system,where the degradation process is an age-dependent and statedependent Wiener process. On the basis of considering the age and state dependence of the degradation process, Zhanget al.[26] developed a degradation model that considers the longterm dependence, which is expressed by fractional Brownian motion (FBM)-based process. In the above studies, whether considering the univariate degradation process [22], [26] or regarding the degradation process of different individuals as the same stochastic process [21], [25], these studies did not take into account the unit-to-unit variability and hidden degradation states during the RUL estimation.

Because of different operating conditions and health states,the unit-to-unit variability is usually defined as the change in degradation rates between different individuals in the same population. [15], [27], and [28] discussed the unit-to-unit variability for the age-dependent Wiener process models and obtained analytical solutions for the RUL distribution.Considering the unit-to-unit variability in age-dependent and state-dependent models is a largely underexplored domain,and too little work has been devoted to solving such a problem, Liet al.[29] provided an RUL estimation method on the basis of the Wiener process model, and the drift coefficient was updated by particle filter, which took the unitto-unit variability into account. However, this method does not consider the measurement error in the degradation process, which is a very common problem in practical engineering and will affect the accuracy of the RUL estimation.

It is well-identified that due to the complexity of the system itself or the high cost of directly monitoring the degradation states, the states during the actual degradation process are often hidden or partially observable. Zhenget al.[30]developed a nonlinear stochastic process model to describe the three-source variability including temporal, unit-to-unit, and measurement variability. Liet al.[31] studied a generalized Wiener model that took into account the unit-to-unit variability, time-correlated structure, and measurement errors.Due to the existence of measurement errors, how to properly describe the relationship between condition monitoring (CM)observations and actual degradation states is the key to solving this problem [7]. The state space model is an effective method to describe this relationship. In addition, it can also be utilized to estimate and update the model parameters in real time according to observations. The state space model has been extensively utilized in the RUL estimation. For example,Fenget al.[32] developed a prognostic method based on the state space model for the case where hidden degradation states and observations have a general nonlinear function relationship, and derived the RUL distribution of the corresponding analysis form. Siet al.[33] proposed the RUL estimation approach for a class of stochastic degrading systems with surviving degradation paths and uncertain observations, which combined the estimation uncertainty for the degradation state and could be updated on the basis of the available surviving yet uncertain observations in real time.

None of the above-mentioned studies on age-dependent and state-dependent degradation models has taken into account the unit-to-unit variability and hidden degradation states together.In this paper, we develop a general nonlinear age-dependent and state-dependent degradation model by considering the unit-to-unit variability and hidden degradation states jointly.Considering the existence of hidden states, the state space model is presented through associating hidden states with the observations. Subsequently, the Kalman filter (KF) [34] and the expectation-maximization (EM) algorithm [35] are jointly utilized to estimate and update hidden degradation states and unknown model parameters. Then, the analytical RUL distribution is approximately derived from the concept of the first hit time (FHT). When the latest observation is obtained,we can update the model parameters and the RUL distribution to achieve real-time RUL estimation. To illustrate the accuracy and effectiveness of the proposed approach, we conducted a numerical simulation and a case study for Li-ion batteries from the Ames Prognostics Center in the National Aeronautics and Space Administration (NASA) and rolling element bearings from the Xi’an Jiaotong University and the Changxing Sumyoung Technology Co., Ltd. (XJTU-SY).

The main contributions of this paper are as follows. 1) We explicitly considered the influence of the degradation rate change among different units and developed an age-dependent and state-dependent nonlinear degradation model considering the unit-to-unit variability; 2) We incorporated the uncertainty of the hidden state estimation from the observations into the RUL estimation, and obtained the approximate analytical RUL distribution from the concept of the FHT; 3) Once the new observation is obtained, the model parameters and hidden states can be updated by the KF algorithm and EM algorithm so as to update the RUL distribution and realize real-time RUL estimation.

The remaining parts of this paper are structured as follows.Section II presents the abbreviations and notations. In Section III, an age-dependent and state-dependent nonlinear degradation model is developed, which considers the unit-tounit variability and hidden states. In Section IV, the analytical solution of the RUL distribution is derived. Section V presents an adaptive estimation framework for unknown model parameters. In Section VI, a simulation study is proposed for illustrative purposes. Two practical cases as experimental demonstrations are presented in Section VII to verify the proposed model. Conclusions are drawn in Section VIII.

II. ABBREVIATIONS AND NOTATIONS

A. Abbreviations

AIC Akaike information criterion

BM Brownian motion

CBM Condition-based maintenance

CM Condition monitoring

CRA Cumulative relative accuracy

DPS Degradation process simulation

EM Expectation-maximization

FBM Fractional Brownian motion

FHT First hitting time

KF Kalman filter

MC Monte Carlo

MLE Maximum likelihood estimation

MSE Mean squared error

PDF Probability density function

PHM Prognostics and health management

RUL Remaining useful life

SDE Stochastic differential equations

SMC Sequential Monte Carlo

SOA Score of accuracy

B. Notations

III. NONLINEAR DEGRADATION MODELING

Let {X(t),t≥0} represent the nonlinear diffusion process of the system during its working time [27], andX(t) represent the degradation state of the system at timet. The degradation process of the degrading system can be expressed as a stochastic differential equation (SDE) as follows:

where μ(X(t),t;θ) and σ(X(t),t;θ) denote the nonlinear drift coefficient function and diffusion coefficient function,respectively. θ denotes the unknown parameter vector inμ(X(t),t;θ) and σ(X(t),t;θ). {B(t),t≥0} denotes a standard Brownian motion (BM)B(t).

Without loss of generality, it is assumed that, whent=0,the degradation stateX(0)=0 in this paper. The model (1)adopts a more general diffusion process, which is different from the general model which only depends on the degradation time. The drift and diffusion coefficient functions of the model (1) not only depend on the timet, but also on its degradation stateX(t).

Furthermore, it is difficult to measure the hidden degradation state perfectly. Under the circumstances, the measurement data can only describe the partial information about the hidden degradation state. The relationship between observations and hidden degradation states at timetcan be

IV. RUL DISTRIBUTION WITH ANALYTICAL SOLUTION

To derive the RUL distribution of the degrading system with age-dependent and state-dependent, the Lamperti transformation proposed in [36] and [37] can be employed to transform the degradation process {X(t),t≥0} characterized by (1) into a degradation process with constant diffusion coefficient. There are two benefits of performing the Lamperti transformation. First, some conclusions in [27] can be used to derive an analytical RUL distribution approximately after this transformation. Second, the stochastic process after the Lamperti transformation has a closed-form of approximate state transition density function, which is convenient for estimating unknown model parameters. The specific form of the Lamperti transformation can be referred to [22].Obviously, the transformed stochastic process model is described as follows, which still depends on age and state:

where the drift coefficient function μ(X(t),t;θ) is obtained by Ito’s formula, which has been extensively applied (see[36]–[38] and references therein). σBis the constant diffusion coefficient reflecting the uncertainty of the degradation process. Without loss of generality, for the sake of simplicity,in the following sections, we focus on such stochastic degradation model {X(t),t≥0} in (5).

Each system may experience various working conditions,thus showing different degradation rates. As a result, the consideration of the unit-to-unit variability in the degradation process can make the degradation model closer to the actual situation. Considering the drift coefficient function μ (x,t;θ) in stochastic process (5) is dependent on state and age, we assume μ(x,t;θ)=ax+bh(t;ξ) , where θ=(a,b,ξ) , andh(t,ξ)is a function of timetand the unknown parameter vector ξ. In this case, we take the parameteraas a fixed constant, and the parameterbas the random effect representing individual differences. For simplicity, we assume thatb～N(μb,σ2b), and it is independent of {B(t),t≥0}, which is similar to existing literature in [16], [27], [28]. To obtain the RUL of the system at the observation timetk, we introduce the following lemma.

Lemma 1 [22]:For the degradation process{X(t),t≥0}characterized by (5), the PDF of the RUL distribution attkis equivalent to the FHT of the degradation process{U(lk),lk≥0} over a defined threshold ωk=ω?xk, whereU(lk)=X(lk+tk)?xk, andU(0)=0 . The process{U(lk),lk≥0}ght}can be described as

Algorithm 1 KF for the estimation of and ?sk|k Pk|k]]1: Initialize:i=1,2,...,k ?s0|0 =[ 0μb , P0|0 =2: For 3: State estimation:?si|i?1 = Ak?si?1|i?1[ 0 0 0 σ2b 4:Pi|i?1 = AiPi?1|i?1 ATi +Qi 5:6:K(i)=Pi|i?1 CT ■CPk|k?1 CT+σ2ε■?1 ?si|i =?si|i?1 +K(i)(yi ?C?si|i?1 7:8: Covariance update:Pi|i =Pi|i?1 ?K(i)CPi|i?1)9:10: End for 11: Obtain the estimation of and ?sk|k Pk|k

According to the above results, the RUL PDF of the monitored system can be estimated and updated through Theorem 1, when the latest degradation observation is obtained. However, several unknown initial parameters of the model (12) need to be estimated from observations. The procedure of estimating these parameters is the focus of the next section.

V. ADAPTIVE PARAMETERS ESTIMATION

Algorithm 2 RTS smoothing algorithm 1: Forward filtering by Algorithm 1 to get and .i=k, k ?1,...,1 ?sk|k Pk|k 2: For 3: Smoothing iteration backwards:Ji?1=Pi?1|i?1 AT 4:i|i?1P?1 i|i?1 5:?si?1|k=?si?1|i?1+Ji?1(?si|k ??si|i?1)Pi?1|k=Pi?1|i?1+Ji?1(Pi|k ?Pi|i?1)JTi?1 6:7: Initialization:

8:9: Update the covariance matrix:Mi|k =Pk|k JTi?1+Ji Mk|k =[I ?K(k)C]Ak?1Pk?1|k?1)JTi?1 10:11: End for ?si|k Pi|k Mi|k(Mi+1|k ?AiPi|i 12: Obtain the estimation of , , and .

Fig. 1. The flowchart of the proposed method.

According to the above the derivation, the flowchart of proposed method is shown in Fig.1. The state space model is used to describe the degradation process of the system. When observation data are available, the hidden state including degradation state and the random-effect parameter at the present time can be estimated by using the KF algorithm. The RTS smoothing algorithm can obtain the conditional expectations needed in the EM algorithm by using the KF estimation results. The unknown parameters of the model can be estimated and updated by the EM algorithm. According to the estimated model parameters and the above KF estimation results, the RUL distribution can be updated and the RUL estimation can be realized in real time.

VI. SIMULATION STUDY

Fig. 2. Comparisons of the RUL PDFs obtained by Monte Carlo method and the three models.

section mainly compares the RUL estimation accuracy of models from the perspective of taking into consideration the unit-to-unit variability and measurement errors.

VII. EXPERIMENTAL DEMONSTRATION

In addition to the modelsM1andM2mentioned in the previous section, in the experimental demonstration, the modelM4in [29] is introduced. The modelM4is also an agedependent and state-dependent degradation model, which considers the unit-to-unit variability, but ignores the influence of hidden degradation states, which means that measurement errors are not considered.

To further evaluate our proposed method, we illustrate two real-data examples for Li-ion batteries degradation data from the NASA Ames Prognostics Center [41] used in [42] and[43], and rolling element bearings degradation data from XJTU-SY used in [44]. Therefore, in this section, we will use these two sets of data to compare the accuracy and effectiveness of the modelsM1,M2, andM4.

A. Li-ion Batteries

Compared with lead-acid batteries, nickel-cadmium batteries, and nickel-hydrogen batteries, Li-ion batteries have many advantages of high power, long life, and light weight.Therefore, Li-ion batteries are widely considered as an alternative energy in many fields such as aircrafts, electric automobiles and satellites. The performance of Li-ion batteries, however, inevitably undergoes the degradation process with cycling and aging, leading to failures and even catastrophic accidents. This degradation is manifested in the reduction of battery capacity during the charging cycle. In other words, battery capacity can be regarded as a health indicator. At present, scholars and engineers usually evaluate the health status of Li-ion batteries through capacity analysis.

Here, the data set includes degradation data of four batteries(#5, #6, #7 and #18) under three different operation modes at room temperature. The cycle life of a battery is usually defined as the number of times that the battery can be charged before its capacity exceeds the acceptable limit. The detailed experimental process of battery cycle life can be referred to[45]. Fig. 3 shows the capacity degradation path of four batteries based on the cycle time. It can be seen from Fig. 3 that these four batteries had a rated capacity of 2 Ahr. When the fully charged capacity is reduced to less than 70% of its rated capacity, it is considered as the end of its life, and the battery needs to be replaced. Hence, the failure threshold can be set to ω =1.4 Ahr. Obviously, the degradation trends of the four batteries were similar, so taking the #5 battery as an example for the subsequent verification.

Fig. 3. The degradation paths of four batteries.

Note that the proposed degradation model is on the basis of the increasing trend of degradation process. To deal with the downward trend of degradation data, the original data can be converted by subtracting the capacity in the degradation process from the initial capacity. Therefore, subsequent verification results are based on the transformed degradation data. According to the failure threshold, the lifetime of #5 battery is about 125 cycles. Fig. 4 illustrates the actual degradation path and the one-step estimation path of the proposed modelM1for the capacity of #5 battery.

Fig. 4. The degradation and estimated paths of battery #5.

It is not difficult to find that the estimated path can not only track the actual degradation path effectively, but also keep a small gap between them. This indicates that the modelM1can effectively estimate the degradation path of #5 battery capacity. Meanwhile, with the accumulation of observation data, the unknown parameter vectors Θ can be estimated adaptively at each cycle time. The corresponding estimation process can be illustrated in Fig. 5.

Fig. 5. The process of parameter estimation for battery #5.

It can be seen from Fig. 5 that the estimated parameters of the proposed model can converge rapidly with the passage of cycle time. Moreover, it is not difficult to find that when the degradation path of #5 battery gives rise to some fluctuations,the parameters such asc, σ2B, and σ2αcan reflect this situation well. These results demonstrate that the proposed parameter estimation method can accurately estimate unknown parameters, and has a good adaptive ability.

The RUL estimation results of modelsM1,M2, andM4are compared in Fig. 6. It can be clearly seen that the estimated RULs of these three models keep the same trend, but the estimation accuracy ofM1is better than those ofM2andM4,andM1has the fastest convergence speed. Due to the fluctuation of degradation data, the estimations of RUL also fluctuate. However, with the accumulation of degradation data, the parameter estimation converges rapidly to ensure the accuracy of the RUL estimation.

Fig. 6. Comparison of the RULs with three models.

From another perspective, the RUL estimation accuracy of the three models is compared by utilizing performance metrics such as mean square error (MSE) [16], [46] and the score of accuracy (SOA) of estimated RUL [47]. MSE measures the estimated performance of RUL in PDF form. The smaller the value of MSE, the smaller the uncertainty of RUL estimation.SOA mainly evaluates the performance of the RUL estimation from the perspective of point estimation, which can distinguish between underestimation and overestimation. The larger the value of SOA, the better performance of the RUL estimation.

The underestimation of the RUL will only result in relatively conservative decisions on the maintenance and replacement ordering of degrading systems, while the overestimation of the RUL may lead to serious disasters and incalculable losses due to unexpected system failures. In this case, evaluating the performance of the RUL estimation model depends on whether the RUL can be estimated in advance without over-underestimation.

In order to compare the trends of MSE and SOA more clearly, we take the average of MSE and SOA every five cycle times, and plot them in Fig. 7.

Fig. 7. Comparison of MSEs and SOAs of the RUL with three models.

The results illustrated in Fig. 7 demonstrate thatM1has less uncertainty thanM2andM4in the PDF distributions of RUL at almost all cycle times from the values of MSE. In addition,it can be seen from the SOA values of all cycle times thatM1has more accurate RUL estimations than the other two models. BecauseM4considers the unit-to-unit variability in the degradation process, it can be seen from the values of MSE and SOA thatM4has better estimation results thanM2.

B. Rolling Element Bearings

The bearing datasets from XJTU-SY are provided by the Institute of Design Science and Basic Component at Xi’an Jiaotong University (XJTU), Shaanxi, China (http://gr.xjtu.edu.cn/web/yaguolei) and the Changxing Sumyoung Technology Co., Ltd. (SY), Zhejiang, China (https://www.sumyoungtech.com.cn), which are also used in [44]. This dataset contains fifteen rolling element bearings, which are LDK UER204 and are tested under three different operating conditions. The sampling frequency is 25.6 kHz, and 32 768 samples are recorded every 1 min.

Similarly to [44], the maximum amplitude (MA) of the horizontal vibration signals, that is, the maximum value of samples acquired at each sampling time, is extracted from the original vibration signals as the degradation data to monitor degradation processes of the tested bearings. The failure time of the tested bearing is defined as the time when the amplitude of the vibration signal exceeds 20 g. We choose Bearing 1-2,Bearing 2-5, and Bearing 3-1 as specific research objects under three different operating conditions.

Usually, the degradation process of bearing includes the normal operation stage and the degradation stage. In the degradation stage, the amplitude of bearing vibration signal increases with the extension of running time, which contains abundant degradation information of bearings. Therefore, the RUL estimation is performed when the tested bearings start to degrade. The adaptive degradation detection method in [48] is used to detect the beginning point of the degradation stage.Specifically, the degradation beginning points of Bearing 1-2,Bearing 2-5, and Bearing 3-1 areT0=35 min,T0=146 min,andT0=2361 min, respectively.

According to the degradation beginning points of three bearings, the MA of three bearings and the corresponding onestep estimated paths obtained from the proposed modelM1are shown in Fig. 8.

Fig. 8. The degradation paths and the estimated paths of three bearings.

It is not difficult to find from Fig. 8 that the estimated paths can effectively track the actual degradation paths, which indicates that the modelM1can effectively estimate the hidden degradation state of the MA of the three bearings.According to the time from degradation beginning point to the failure point of each bearing, the interval of the RUL estimation is set to ?t=2 min, ?t=2 min, and ?t=6 min,respectively. Figs. 9–11 present the RUL estimation results of Bearing 1-2, Bearing 2-5, and Bearing 3-1, respectively.

Fig. 9. RUL estimation results using three models for Bearing 1-2.

Fig. 10. RUL estimation results using three models for Bearing 2-5.

Fig. 11. RUL estimation results using three models for Bearing 3-1.

As can be seen from Figs. 9–11, the RUL estimation results of the three models have certain deviation from the actual RUL in the initial stage. As time goes by, the estimated RUL gradually converges to the actual RUL. However,M1andM4have faster convergence speed because they consider the unitto-unit variability in the degradation process. Moreover,M1has higher accuracy of the RUL estimation because of considering the influence of hidden degradation states.

To further evaluate the estimation performance of these three models, the cumulative relative accuracy (CRA) and convergence (Con) criteria [49] are used to quantitatively evaluate the estimation results. CRA quantifies the accuracy of the estimation model in a specific time index from the perspective of the relative accuracy of the RUL estimation.The higher the value of CRA, the more accurate the estimation result. Con quantifies the speed at which the estimated RUL is approaching the actual RUL. The smaller the value of Con, the faster the convergence speed. The CRA and Con values of the three models are shown in Figs. 12 and 13.

The results illustrated in Fig. 12 demonstrate that CRA values increase with the order of magnitude ofM2,M4, andM1, which means that the proposed modelM1acquires more accurate estimations than the other two models. Similarly, as can be seen from Fig. 13, for each of the three bearings, the Con value ofM1is the smallest, followed byM4, andM2.This implies that the RUL estimation results ofM1converge to the actual RUL faster than the other two models.

Fig. 12. CRA values of the three models for the three bearings.

Fig. 13. Con values of the three models for the three bearings.

Through the comparative study above, it is found that the proposed approach can effectively improve the accuracy of the RUL estimation. This implies that it is necessary to consider the unit-to-unit variability and hidden degradation states when modeling the nonlinear degradation process of age- and state-dependent degrading systems.

VIII. CONCLUSIONS

This paper proposes an age-dependent and state-de pendent adaptive prognostic approach for the hidden nonlinear degrading system. In this approach, the age-dependent and state-dependent framework on the basis of the nonlinear diffusion process model is specially designed to characterize the degradation process. The coefficient of the age-dependent drift coefficient function is treated as a random variable denoting the unit-to-unit variability, and the state space model is established to characterize the relationship between observations and hidden degradation states considering measurement errors. The degradation states and the unknown model parameters are estimated by utilizing the KF algorithm and the EM algorithm jointly. Besides, the approximate analytical RUL distribution is derived from the concept of the FHT. Once a new observation is available, model parameters and system degradation states can be updated in real time,which makes it possible to estimate RUL adaptively. The effectiveness of the proposed approach is verified by a simulation study and two practical cases which show that the proposed model can obtain a more accurate RUL estimation than the model that does not consider the unit-to-unit variability and hidden degradation states.