A hybrid prognostic method for system degradation based on particle filter and relevance vector machine

Accepted Manuscript

A hybrid prognostic method for system degradation based on particle filter and relevance vector machine Yang Chang, Huajing Fang PII: DOI: Reference:

S0951-8320(18)30908-6 https://doi.org/10.1016/j.ress.2019.02.011 RESS 6379

To appear in:

Reliability Engineering and System Safety

Received date: Revised date: Accepted date:

23 July 2018 15 January 2019 1 February 2019

Please cite this article as: Yang Chang, Huajing Fang, A hybrid prognostic method for system degradation based on particle filter and relevance vector machine, Reliability Engineering and System Safety (2019), doi: https://doi.org/10.1016/j.ress.2019.02.011

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Highlights • The proposed prognostic method can provide accurate and stable RUL prediction; • The proposed prognostic method can construct a prediction interval to assess the prediction uncertainty; • Four types of comparative experiments are performed to verify the wide applicability of the proposed method;

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• Reliable prognostic result can be provided by proposed method to ensure the system reliability.

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A hybrid prognostic method for system degradation based on particle filter and relevance vector machine Yang Changa,b , Huajing Fanga,b,∗ a School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China Key Laboratory of Science and Technology on Multispectral Information Processing, Wuhan 430074, China

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b National

Abstract

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Prognostics of the remaining useful life has become a critical technique to ensure the reliability and safety of system, however, due to the uncertainty of system degradation, the prognostic result is usually not so satisfactory. To solve this problem, a hybrid prognostic scheme with the capability of uncertainty assessment is proposed in this paper, which combines particle filter (PF) and relevance vector machine (RVM). The prognostic result comprises a set of deterministic prediction values to represent the degradation process and a prediction interval to evaluate the prediction uncertainty. In order to to examine the performance of the proposed hybrid method, four types of comparative experiments based on two types of lithium-ion battery datasets and two degradation models are performed. The experimental results show that the proposed hybrid scheme is a reliable prognostic method which can ensure the accuracy of the deterministic prediction result and provide precise assessment for the prediction uncertainty. Keywords: Prognostics, Particle filter, Relevance vector machine, Deterministic prediction, Prediction interval, Lithium-ion battery

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1. Introduction

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Nowadays, prognostics and health management (PHM) is becoming more and more important for the maintenance of complex system, and has been successfully applied in engine [1, 2], electronic [3, 4], nuclear power plant [5], rotary machinery [6, 7], lithium-ion battery [8, 9] and the like [10, 11]. In the framework of PHM, prognostics is a major enabling technique that can be utilized to estimate the state of health (SOH) and predict the remaining useful life (RUL) of system based on available historical data or degradation model. Once the RUL is predicted, health management can be scheduled in advance to avoid malfunction and reduce maintenance cost, which can effectively enhance the reliability and safety of system. Among the various types of prognostic methods, modelbased and data-driven are two kinds of fundamental approaches [4, 9]. The model-based prognostic method relies on the pre-built parametric model to predict the degradation process of system, while the data-driven prognostic method adopts the monitored data collected from historical operations to mine the hidden information and predict the future trend of degradation process. Apart from these two basic methods, an improved approach named fusion method [12] or hybrid method [2] has become the hotspot for system prognostics, which integrates multiple ∗ Corresponding

author Email addresses: [email protected] (Yang Chang), [email protected] (Huajing Fang) Preprint submitted to Elsevier

basic methods and can make full use of information to improve the performance of prognostic result. In the past, many prognostic methods focused on point prediction, which only predicted a deterministic curve to represent the degradation process and obtained a single value as the RUL prediction result without uncertainty assessment [2, 7, 13–16]. However, for the prognostic problem, the uncertainties, including measurement noise, intrinsic randomness of the degradation process, uncertainty in the model parameters [9, 17], will impose a lot of challenges and difficulties on obtaining a reliable prognostic result, for example, the quality of historical data highly influences the performance of data-driven method, and the model uncertainty can seriously affect the reliability of model-based method. Therefore, many prognostic methods with the capability of uncertainty assessment have been proposed for degraded system. Apart from a deterministic prediction result, these methods can obtain an interval to bracket the prediction result with a prescribed probability [100(1-α)%], which can provide more information about system degradation and offer a more reliable prognostic result for decision maker to schedule repair and maintenance. The reliability functions such as Weibull distributions are the classic methods for RUL prediction and uncertainty assessment, they can describe the life distribution by identifying the parameters of function, and have been widely used in prognostics of capacitor [1], engine [1], wind turbine [18], and lithium-ion battery [19, 20]. Another widely used prognostic method with uncertainty assessment is the February 2, 2019

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To overcome these problems, a hybrid prognostic scheme is proposed in this paper and the problem of lithium-ion battery prognostics is taken as the example to demonstrate the performance of proposed hybrid scheme. Lithium-ion battery is an important energy storage device adopted in various types of electronic products and usually determines the reliability of these products [29, 30]. Similar to other systems, when we try to measure the degradation data or build a degradation model for the lithium-ion battery, the differences in manufacturing assemblies and material properties, as well as varying operating conditions and measurement noise can bring in many uncertainties [34], and these uncertainties may seriously affect the effectiveness of prognostic method and deteriorate the performance of prognostic result. In this study, the proposed hybrid scheme combines PF and RVM, and the prognostic result consists of a deterministic prediction result and an interval with prescribed probability. Based on the assumption that the PF-based prediction error follows the normal distribution with zero mean, the deterministic prediction result which represents the predicted degradation process is obtained by calculating the mean of ensemble PF-based deterministic prediction results. The interval indicates the variation range of the uncertainties in deterministic prediction result, the uncertainties can be divided into model uncertainty and measurement uncertainty, the former is obtained by calculating the variance of ensemble PF-based deterministic prediction results, the latter is estimated by RVM method. Four types of comparative experiments using PF and proposed hybrid method are conducted based on two different degradation models and two types of capacity datasets to demonstrate the reliable performance and wide range applicability of proposed hybrid method, each type of experiment is repeated ten times to test the stability of prognostic result. The experimental results show that the proposed hybrid method can provide not only more accurate and stable degradation prediction result, but also more precise interval to assess the prediction uncertainty. The remainder of this paper is organized as follows: Section 2 reviews some basic algorithms for describing the proposed hybrid method. The detailed implementation of the proposed hybrid method is presented in section 3. Section 4 provides the detailed experimental results of lithium-ion battery prognostics and corresponding analyses. Conclusions and future works are discussed in section 5.

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Wiener process, it has a great potential to capture the uncertainty in degradation process [21–23], by estimating the model parameters through maximum likelihood estimation or some other algorithms, it can obtain the final RUL prediction result with an explicit probability density function (PDF) [23]. With the development of machine learning methods, prognostic methods based on Bayesian inference have received more and more attentions, Bayesian inference is a statistical inference method and adopts the Bayes’ theorem to predict and update the probability of degradation process. The Bayesian-based methods include Gaussian process regression (GPR) [24, 25], relevance vector machine (RVM) [8, 26, 27] and particle filter (PF) [28]. In the literature, pure PF method is often classified as a model-based approach, for its application is highly dependent on the pre-established parametric model. In fact, the prognostic methods involving the use of PF are the most widely used techniques to predict RUL and evaluate prediction uncertainty [3, 4, 9, 28–35] and have been adopted for the prognostics of water tank system, bearing, light emitting diode, lithium-ion battery and so on [28]. Compared with the data-driven methods aforementioned, PF is a recursive estimation algorithm and does not require large amounts of historical data to fully train the model, which makes it more suitable for on-line applications [35]. Compared with other filter methods, PF approximates the degradation state by a set of random particles and can provide prognostic result with uncertainty assessment, for example, Xing et.al [30] compared the performance of PF with unscented Kalman filter (UKF) and non-linear least squares (NLLS) method for lithium-ion battery prognostics and concluded that PF can provide more accurate prediction result, furthermore, PF can also describe the prediction uncertainty by giving the distribution of RUL. However, for PF method, the main drawback is the particle degeneracy phenomenon, although resampling method is often adopted to avoid this problem, it will lead to particle impoverishment and reduce the diversity of particles, these disadvantages may seriously affect the reliability of prognostic result [35]. Moreover, because of the inherent randomness of PF, including the randomness of particles position at initial time, the importance sampling at each cycle, the prognostic result is unstable. Thus, if PF is applied repeatedly for the same prognostic problem, the prognostic results may not be so accurate and the prediction error may be different each time. Furthermore, in order to address the problem of particle impoverishment, artificial evolution approach is often adopted, in which an artificial evolution is added to the parameter evolution equation [36], however, it may result in the posterior variance inflation problem [9]. For PF-based prognostic result, the width of the interval for degradation prediction result is determined by the distribution of posterior PDF, which is the intersection of distributions of prior samples, measurement noise and process noise [37]. Thus, improper selection of artificial evolution may lead to wide interval in the final prognostic result.

2. Literature review In order to understand the whole work, the basic knowledge about interval is firstly discussed in this section. In addition, the framework of RVM and PF is introduced later. 2.1. Basic knowledge about interval Given a dataset D = {xk , yk }M k=1 , the relationship between the input vector x = [x1 , · · · , xM ]T and the mea3

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surement vector y = [y1 , · · · , yM ]T can be represented by a combination of an unknown nonlinear deterministic component and a stochastic component: yk = µ(xk ) + k

be treated as the weight and corresponding kernel function is 1. Based on the Eq.(1), the probability p(y|x) follows the normal distribution N (y|µ(x), σ2 ), namely, the likelihood of the complete dataset is:

(1)

where yk is the kth measured output in the dataset D, µ(xk ) is the deterministic component that represents the true output, k is the stochastic component which is assumed to follow the normal distribution with zero mean. Usually, the deterministic component is not equal to the measured output due to the existence of k , therefore, the stochastic component k can be treated as the measurement uncertainty. The objective of most prognostic methods before prediction is to construct a model based on available information to identify the deterministic component µ(xk ). However, in most cases, the constructed model yˆk = µ ˆ(xk ) is not equal to µ(xk ) due to the model uncertainty. Therefore, we have:

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p(y|ω, σ2 ) = (2πσ2 )− 2 exp{−

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where Φ = [φ(x1 ), · · · , φ(xM )]T is the M ×(M +1) ‘design’ matrix with φ(xk ) = [1, K(xk , x1 ), · · · , K(xk , xM )]T , ω = [ω0 , · · · , ωM ]T is the corresponding weight vector and σ2 is the variance of the measurement noise. Next, a Bayesian perspective is adopted and constraints are superimposed on the parameters to avoid the severe overfitting problem caused by maximum likelihood estimation of weight vector ω and variance σ2 , in which a prior normal distribution with zero mean is defined over the weight vector ω: p(ω|α) =

(2)

the prediction interval (PI) [yk − yˆk ] which assesses the uncertainty between measured output and the estimated output comprises two parts [38]: the confidence interval (CI) [µ(xk ) − µ ˆ(xk )] and k . The former qualifies the different between the true output and the estimated output, the latter indicates the measurement uncertainty. Assume that the terms [µ(xk ) − µ ˆ(xk )] and k are mutually independent, thus, the variance associated with [yk − yˆk ] comprises two terms:

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p(y|ω, σ2 )p(ω|α) p(y|α, σ2 ) (M +1) 1 (ω − m)T Σ−1 (ω − m) = (2π)− 2 |Σ|− 2 exp{− } 2

p(ω|y, α, σ2 ) =

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the marginal likelihood for the hyperparameters is given as:

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RVM is a machine learning method based on Bayesian learning theory. Nowadays, it has been widely applied for data regression and classification, in this paper, we adopt RVM for data regression. Based on the dataset D = {xk , yk }M k=1 discussed in section 2.1, the relationship between the input vector x and the true output vector µ(x) can be written as [39]: ωk K(x, xk ) + ω0

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with Σ = (σ−2 ΦT Φ + A)−1 , m = σ−2 ΣΦT y, and A =diag(α0 , · · · , αM ). By integrating out the weight parameters, Z p(y|α, σ2 ) = p(y|ω, σ2 )p(ω|α)dω (8)

2.2. Relevance vector machine

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M Y α ω2 α √ k exp(− k k ) 2 2π k=0

where α is a vector of M +1 hyperparameters and controls the sparsity property of the regression model. Based on the Bayes’ rule, the posterior distribution over the weights is obtained as follows:

where the term σµ2 k represents the model uncertainty caused by model misspecification and parameter estimation error, the term σ2k is the noise variance and indicates the measurement uncertainty. When the value of these terms is estimated correctly, the PI for the deterministic prediction result yˆk can be constructed.

µ(x) =

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[yk − yˆk ] = [µ(xk ) − µ ˆ(xk )] + k

σy2k = σµ2 k + σ2k

ky − Φωk2 } 2σ2

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p(y|α, σ2 ) = (2π)− 2 |σ2 I + ΦA−1 ΦT |− 2 2

× exp{−

yT (σ I + ΦA−1 ΦT )−1 y } 2

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Because the value of α and σ2 cannot be analytically derived by maximizing Eq.(9), an iterative re-estimation method is adopted to calculate their most-probable (MP) values:

(4) αknew =

k=1

In Eq.(4), K(x, xk ) is the kernel function and ωk is the corresponding weight, ω0 is the bias parameter which can

γk m2k

(σ2 )new =

ky − Φmk PM M − k=0 γk

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with γk = 1 − αΣk,k , where mk is the ith element of pos4

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terior mean m, Σk,k is the diagonal element of posterior covariance Σ. Having found the MP values for the hyperparameters that maximize the marginal likelihood Eq.(9), the predictive distribution can be obtained when a new input x∗ is given:

analytically deduce the PDFs due to the complex highdimensional integrals. Therefore, PF resorts to Monte Carlo simulation method with sampling a set of weighted particles to estimate the posterior PDF: p(xk |y1:k ) ≈

p(y∗ |y, αM P , σ2M P ) p(y∗ |ω, σ2M P )p(ω|y, αM P , σ2M P )dω

with =

σ2M P

The

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+ φ(x∗ ) Σφ(x∗ )

ωki δ(xik − xk )

i ω eki = ω ek−1

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p(yk |xik )p(xik |xik−1 ) q(xik |xi0:k−1 , y1:k )

2.3. Particle filter

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The PF is a kind of recursive Bayesian estimation technique based on Monte Carlo simulation and sequential importance sampling. It can approximate the posterior PDF of state through a set of random samples (particles) which are chosen using the sequential importance sampling technique, and continuously adjust the associated weights and positions according to the measurements. In general, the system proposed to demonstrate PF can be represented by following equations:  xk = f (xk−1 , wk−1 ) (13) yk = h(xk , k )

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i ω eki = ω ek−1 p(yk |xik )

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ω eki

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ω ei ωki = PN k

i=1

ω eki

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In order to avoid the problem of particle degeneracy, the resampling method is used to eliminate the particles having small weights and duplicate the ones having large weights to renormalize the distribution, the weight of each renormalized particle is set to 1/N [36]. 3. The method for prognostics Since the problem of lithium-ion battery prognostics is adopted as the example in this paper, two degradation models for lithium-ion battery are introduced firstly to show the extensive applicability of proposed hybrid method, the hybrid method is then presented based on the results of PF and RVM.

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where xk ∈ Rn represents the unobservable state, yk ∈ Rm represents the measurement. f (·) and h(·) represent the state transition equation and the measurement equation, respectively. wk−1 and k are system noise and measurement noise, which can be either Gaussian or non-Gaussian. Generally, the implementation of PF consists of two steps: prediction step and update step. In the prediction step, it is assumed that the posterior PDF p(xk−1 |y1:k−1 ) at cycle k − 1 is already obtained, then the prior PDF at cycle k can be calculated as follows: Z p(xk |y1:k−1 ) = p(xk |xk−1 )p(xk−1 |y1:k−1 )dxk−1 (14)

3.1. Degradation models

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3.1.1. Dual exponential model In the literature, many degraded systems can be characterized by exponential-type models [23], such as gas turbine [40], bearing [31, 41], lithium-ion battery [35] and electrolytic capacitor [3]. In this section, a dual exponential model based on analysis of capacity data is introduced to describe the degradation of lithium-ion battery [34]:

where y1:k−1 represents y1:k−1 = [y1 , · · · , yk−1 ] and p(xk |xk−1 ) is the one-step transition probability. When the measurement at cycle k is available, the posterior PDF can be updated via Bayesian rule: p(yk |xk )p(xk |y1:k−1 ) p(yk |xk )p(xk |y1:k−1 )dxk

ω ei ωki = PN k

where q(xik |xi0:k−1 , y1:k ) is the importance function [9]. In most cases, the standard PF adopts the transition probability as the importance function, namely q(xik |xi0:k−1 , y1:k ) = p(xik |xik−1 ), then the transition equation of weight can be simplified into:

is the variance of the predictive distribution.

p(xk |y1:k ) = R

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where δ(·) is the Dirac function, xik (i = 1, · · · , N ) is the particle sampled from importance function, and ωki is the corresponding weight determined by:

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= N (y∗ |mT φ(x∗ ), σ∗2 ) σ∗2

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Qk = a · exp(b · k) + c · exp(d · k)

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where k is the cycle number, Qk is the battery capacity at cycle k, parameters a and c are related to the internal impedance, while b and d represent the aging rate. This model is widely used in various literature [9, 13, 35] and can properly describe the degradation trend [35]. In order to perform prognostics using PF, the dual exponential

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where p(yk |xk ) is the likelihood function. Repeat the calculation of Eq.(14) and Eq.(15) alternately to form a recursive Bayesian estimation. However, it is hard to 5

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model should be rewritten into the form of Eq.(13) with considering parameter noise and measurement noise:  xk+1 = xk + wk (20) Qk = ak exp(bk · k) + ck exp(dk · k) + k

of measurement stop. Since there are no measurements available in the prediction horizon, the particles sampled at cycle Ts are adopted to compute the degradation trajectories at prediction horizon based on Eq.(20) or Eq.(23) with invariant parameters, and

where xk = [ak , bk , ck , dk ] represents the model parameter vector, both wk = [wka , wkb , wkc , wkd ] and k are the Gaussian noise with zero mean. In this paper, since the variance of parameter noise is hard to know, the artificial evolution approach is adopted, while for the variance of measurement noise, we can estimate it based on Eq.(12).

ˆk = Q

aj xk−j + wk

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where k=Ts + 1, Ts + 2, · · · is the cycle in the prediction horizon, i = 1, · · · , N stands for the particle number and Qik is the value predicted by the ith particle at cycle k, it is obviously that there are N trajectories corresponding to ˆ k is the deterministic prediction result N particles, and Q and represents the predicted degradation process. Sort the N trajectories, then the upper and lower ˆ k can be constructed by drawing the perbounds for Q centiles of the sorted trajectories. Conduct the prediction process continually until the degradation prediction result and the two trajectories of the interval are all below the predefined end-of-life (EOL) threshold. Assume that TˆeU and TˆeL represent the EOL cycles for the upper and lower bounds, Tˆe is the EOL cycle for the degradation prediction result, then the RUL prediction result at cycle Ts is:

(21)

j=1

[ RU Ls = Tˆe − Ts

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where the current state xk is obtained by p states backward xk−j (j = 1, · · · , p) and model error wk . The parameter p is the order of the AR model determined by Akaike information criterion (AIC) method, aj is the autoregressive coefficient which can be estimated by the Burg approach. Furthermore, the measurement model can be determined as: (22)

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where k is related with measurement noise and can be estimated based on Eq.(12). In order to conduct prognostics based on PF, the Eq.(21) and Eq.(22) can be rewritten as:  Xk+1 = AXk + Wk (23) Qk = CXk + k where Xk = [xk , xk−1 , · · · , xk−p+1 ]T , T [wk , 0,  · · · , 0] , C = [1, 0, · · · , 0], k = a1 a2 · · · ap 1    A= . . ..   1 0

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And the interval for the RUL prediction result is:

Interval = TˆeU − TˆeL

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The detailed process of PF-based prognostics can be found in [4, 9]. Repeat the PF-based prognostic method L times under the same conditions (initial distribution, amount of particles), then L estimated models yˆ(xjs ) and degradation ˆ j (k = Ts + 1, · · · , j = 1, · · · , L) can be prediction results Q k obtained. Due to the randomness of particles position at initial cycle, importance sampling at each cycle and some other problems, the parameters of each model yˆ(xjs ) may be different. Based on the assumption that the PF-based prediction error follows the normal distribution with zero mean, then the detailed implementation process of proposed hybrid method is described as follows:

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ωsi Qik

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3.1.2. Autoregressive model The Autoregressive model (AR) model has the properties of simple structure, convenient construction, and high efficiency, in addition, the AR model can achieve the same accuracy as moving average (MA) model and ARMA model with high order. Actually, the AR model has been adopted for the bearing prognostics based on enhanced PF method [32]. Therefore, the AR model is adopted as the state model in this study to test the performance of proposed hybrid method, which is defined as: xk =

N X

Wk = [k ] and

ˆ en . Step 1. Obtain the degradation prediction result Q k The degradation prediction result of proposed hybrid method is: L

1 X ˆj ˆ en Q Q k = L j=1 k

3.2. The proposed hybrid method Apply PF to estimate the model states and parameters based on the available measurements Q0:s = [Q0 , Q1 , · · · , Qs ], where Q0 stands for the measured capacity of a fresh battery, the subscript s represents the cycle

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ˆ j is the jth PF-based degradation predicwhere Q k ˆ en is the degradation pretion result at cycle k, Q k diction result of proposed hybrid method at cycle k. 6

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Step 2. Estimation of σy2ˆk . The variance σy2ˆk adopted to indicate the model uncertainty is calculated as:

performance of experimental results, and finally give the experimental results and corresponding analyses.

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1 X ˆj 2 ˆ en (Q − Q k ) L − 1 j=1 k

4.1. Capacity data

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In this paper, four capacity datasets which can be categorized into two types are adopted to perform the comparative experiments. Figure 2 displays the degradation trend of each dataset.

σˆ2k .

Step 3. Estimation of Based on Eq.(12), the variance of measurement noise k is: 2 T σˆ2k = σM P + φ(lk ) Σφ(lk )

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A1 A2

Capacity(Ah)

Capacity(Ah)

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where lk represents the input of RVM [8]. Step 4. Construct the PI. Through the computations above, the standard deviation (SD) associated with the prediction uncertainty of proposed hybrid method is: q (30) S = σy2ˆk + σˆ2k

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Consequently, the 100(1-α)% PI with upper and lower bounds for the deterministic prediction result of proposed hybrid method is [42]: S ˆ en α Q k ± √ t 2 (L − 1) L

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where t α2 (L − 1) represents the α2 quantile of t−distribution function with the degrees of freedom L − 1.

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Figure 2 displays the detailed flowchart of proposed hybrid method.

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Qˆ k1

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ĂĂ

V H2ˆ

k

¦

Qˆ ken

V y2ˆ

k

Qˆ kL

1.6 1.4

0

50

100 Cycles

150

(b)

Figure 2: The capacity datasets. (a) Datasets of type A. (b) Datasets of type B.

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Qˆ k2

250

B5 B7

1.8

1 L ˆj ¦ Qk L j1

1 L ˆj ¦ (Qk  Qken )2 L 1 j 1

S Qˆ ken r tD /2 ( L  1) L

Figure 1: The flowchart of proposed hybrid method

4. Experiments and results In order to demonstrate the effectiveness of proposed hybrid method, four types of comparative experiments labeled as #A, #B, #C and #D are conducted using PF and proposed hybrid method. In this section, we first briefly introduce the datasets used for comparative experiments, then define some evaluation indicators to assess the 7

The datasets of type A were obtained from the Center for Advanced Life Cycle Engineering (CALCE) at University of Maryland. The lithium-ion batteries were tested by the Arbin BT2000 Battery Test system under room temperature with the rated capacity of 0.9Ah. Charging was conducted in a standard constant current (CC)/constant voltage (CV) protocol with a CC rate of 0.5C (0.45A) until the voltage reached 4.2V, then the CV started with invariant voltage until the charging current dropped to 0.05A. Discharging was carried out at a CC rate of 0.45A with a cutoff voltage of 2.7V. With repeated charging and discharging, the energy storage capability of the lithium-ion battery declines gradually. In most studies about lithiumion battery prognostics based on this type of capacity datasets, the lithium-ion battery is considered as failure when the capacity of a fully charged battery fades to 80% of the rated capacity [34], therefore, the EOL threshold for this type of datasets is set to 0.72Ah. The two datasets of type B were collected from NASA Prognostic Center of Excellence (PCoE). The life-cycle tests were conducted on 18650-size lithium-ion batteries under room temperature, and the rated capacity of the two batteries is 2 Ah. Charging was conducted with a CC rate of 1.5A until the voltage reached 4.2V and then the voltage was sustained until the charging current decreased to 0.02A. Discharging was conducted with a CC rate of 2A until the voltage dropped to 2.7V for B5 and 2.2V for B7, respectively. The EOL threshold for each lithium-ion battery is different, in this study, the EOL threshold for dataset B5 is set to 70% of its initial capacity, while for dataset B7, the EOL threshold is 75% of its initial capacity [29].

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4.2. Evaluation indicators

Table 1: Indicators for the estimated SDs (experiments #A).

In order to show the accuracy of deterministic prediction result and the quality of uncertainty assessment, three evaluation indicators are defined as follows: [ |RU Lk − RU Lk | RE = RU Lk v u N u1 X t b k )2 RM SE = (Qk − Q N

No.

Maximum (×10−3 ) 3.5954 5.3634 15.365 9.1128

A1 A2 B5 B7

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Minimum (×10−3 ) 3.5510 5.2768 15.078 8.8959

Mean (×10−3 ) 3.5639 5.2984 15.155 8.9505

Range (×10−3 ) 0.0444 0.0866 0.2870 0.2169

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k=1

[ [ sup(RU Lk ) − inf(RU Lk ) RW = RU Lk

as constant and are equal to the mean, which rarely affects the parameter estimation and PI construction. In order to test the normality of the PF-based prediction error, the Quantile-Quantile (Q-Q) plot technique is utilized in this paper. For the comparative experiments, since there are 30 PF-based deterministic prediction results adopted for the proposed hybrid method, the PF is repeated 30 times to calculate the prediction errors. If the error points basically fit the standard line in the Q-Q plot, the distribution of prediction error is indeed normal distribution, then the slope of the standard line is an estimate of the SD and the intercept is an estimate of the mean [43]. Figure 3 shows the Q-Q plots for the prediction errors based on the four datasets in experiments #A at random cycles.

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[ where RU Lk is the true RUL value at cycle k and RU Lk represents the predicted RUL at cycle k. Qk is the meab k is the predicted capacity at sured capacity at cycle k, Q [ [ cycle k. sup(RU Lk ) and inf(RU Lk ) represent the upper bound and lower bound of the interval for RUL prediction result, the probability that the interval brackets the degradation prediction result is 95%. For the prognostic result, the smaller the RE value is, the more accurate the RUL prediction result is, meanwhile, a smaller RMSE value indicates a more accurate degradation prediction result and a smaller RW value indicates a more precise interval. 4.3. Experimental validation

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0.2 0.15 Sample Quantiles

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In this section, the first three types of experiments are conducted based on the dual exponential model, the experiments #D are performed based on the AR model. The MATLAB of version 2012a is adopted and it is installed on a PC with 2.40-GHz processor and 6 GB of RAM.

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4.3.1. Comparative experiments #A In the first set of comparative experiments, the endof-measurement (EOM) threshold for capacity fade data collected from CALCE is set to 90% of the rated capacity, when the value of degradation data drops below it, the measurement process stops and the prediction process starts. As to the capacity datasets of type B, the EOM threshold is 80% of the initial capacity for B5 and 85% for B7, respectively. Based on the measured capacity data, the number of particles for PF-based prognostic method is 200, the number of PF-based deterministic prediction results involved in proposed hybrid method is L = 30, and each PF-based deterministic prediction result is the weighted mean of N = 200 particles. According to the results in section 2.1 and section 2.2, we assume that the noise in the measured dataset is subjected to normal distribution with zero mean. Table 1 lists the maximum, minimum, mean and range of the estimated SDs obtained by RVM method. It can be seen from the indicators in table 1 that the variation range of SDs is very small, which can be negligible compared with the capacity data and the mean, thus, the SDs of the noise in each measured dataset can be treated

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Figure 3: Q-Q plots (experiments #A). (a) The Q-Q plot for dataset A1 at k = 189. (b) The Q-Q plot for dataset A2 at k = 188. (c) The Q-Q plot for dataset B5 at k = 131. (d) The Q-Q plot for dataset B7 at k = 145.

It is observed from the subfigures that most of the error points lie around the standard line, although there are some points deviating away from the standard line due to the fluctuation of the degradation process, the prediction error points still fit the trend of the standard line. Moreover, the intercept of the standard line in each subfigure 8

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Figure 4: Prognostic results for dataset A1 (experiments #A1). (a) PF method. (b) Proposed hybrid method.

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Figure 5: Prognostic results for dataset A2 (experiments #A1). (a) PF method. (b) Proposed hybrid method.

is close to zero. Therefore, it is reasonable to conclude that the distribution of PF-based prediction error in experiments #A is close to the normal distribution with zero mean. Figure 4—figure 7 show the degradation prediction results and corresponding intervals for the experiments numbered #A1. Tables 2 and 3 list the prognostic results and corresponding evaluation indicators. It can be found in figure 4—figure 7 that the degradation prediction results obtained by proposed hybrid method are closer to the true capacity fade trajectories, the RUL prediction results for the four capacity datasets based on proposed hybrid method are also closer to the true RULs. Furthermore, the 95% degradation intervals with upper and lower bounds predicted by proposed hybrid scheme are narrower. Therefore, it can be preliminarily concluded

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Figure 6: Prognostic results for dataset B5 (experiments #A1). (a) PF method. (b) Proposed hybrid method.

that the proposed hybrid method can achieve more reliable prognostic result. Further comparisons can be made based on the results in table 2 and table 3.

Table 2: Prognostic results for the capacity datasets of type A (experiments #A).

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A1 (Ts = 140, RU Ls = 68) [ [ Method No. RU Ls Interval RE RMSE RW Time(s) RU Ls #A1 60 [49,89] 0.1176 0.0116 0.5882 4.47 55 #A2 64 [51,74] 0.0588 0.0093 0.3382 4.03 52 #A3 84 [72,113] 0.2353 0.0086 0.6029 4.94 69 #A4 82 [47,118] 0.2059 0.0069 0.5294 4.98 45 #A5 73 [55,101] 0.0735 0.0057 0.6765 4.99 57 PF #A6 79 [51,102] 0.1618 0.0062 0.7500 4.98 49 #A7 58 [46,68] 0.1471 0.0131 0.3235 4.51 69 #A8 82 [55,96] 0.2059 0.0072 0.6029 4.13 88 #A9 79 [69,90] 0.1618 0.0062 0.3088 4.42 82 #A10 75 [53,114] 0.1029 0.0055 0.8971 4.20 49 #A1 70 [65,79] 0.0294 0.0061 0.2059 131.11 61 #A2 72 [65,80] 0.0588 0.0058 0.2206 131.97 59 #A3 66 [59,75] 0.0294 0.0080 0.2353 131.69 62 #A4 73 [65,84] 0.0735 0.0055 0.2794 130.65 61 #A5 70 [64,79] 0.0294 0.0060 0.2206 131.81 67 Hybrid #A6 73 [66,84] 0.0735 0.0053 0.2647 128.89 60 #A7 72 [65,82] 0.0588 0.0059 0.2500 128.01 63 #A8 69 [63,76] 0.0147 0.0056 0.2131 130.70 60 #A9 70 [63,82] 0.0294 0.0061 0.2794 128.16 59 #A10 74 [68,84] 0.0882 0.0055 0.2353 128.05 60 9

A2 (Ts = 124, RU Ls = 64) Interval RE RMSE RW Time(s) [40,73] 0.1406 0.0103 0.5156 4.22 [38,68] 0.1875 0.0130 0.4688 3.72 [44,102] 0.0781 0.0070 0.9063 3.69 [34,61] 0.2969 0.0175 0.4219 3.74 [44,78] 0.1094 0.0080 0.5313 3.68 [37,89] 0.2344 0.0187 0.8125 3.72 [44,102] 0.0781 0.0070 0.9063 3.80 [56,150] 0.3750 0.0184 1.4688 3.78 [41,152] 0.2813 0.0172 1.7344 3.77 [38,65] 0.2344 0.0159 0.4219 4.50 [56,69] 0.0469 0.0038 0.2031 104.24 [54,68] 0.0781 0.0050 0.2188 104.97 [55,71] 0.0313 0.0039 0.2500 104.31 [55,70] 0.0625 0.0042 0.2343 104.02 [61,77] 0.0469 0.0056 0.2500 104.75 [55,68] 0.0625 0.0048 0.2031 103.72 [57,71] 0.0156 0.0036 0.2188 104.35 [54,68] 0.0625 0.0045 0.2188 104.26 [53,66] 0.0781 0.0058 0.2031 103.87 [54,67] 0.0625 0.0046 0.2031 103.01

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Table 3: Prognostic results for the capacity datasets of type B (experiments #A).

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B7 (Ts = 84, RU Ls = 75) Interval RE RMSE RW Time(s) [60,104] 0.1600 0.0188 0.5867 3.86 [53,85] 0.1333 0.0207 0.4267 3.52 [51,108] 0.0800 0.0168 0.7600 3.53 [54,84] 0.1067 0.0239 0.4000 3.53 [43,79] 0.2800 0.0369 0.4800 3.57 [50,83] 0.1467 0.0243 0.4400 3.57 [47,139] 0.1867 0.0307 1.2267 3.55 [45,70] 0.3067 0.0604 0.3333 3.59 [54,112] 0.0533 0.0322 0.7733 3.59 [51,69] 0.2267 0.0336 0.2400 3.56 [69,90] 0.0400 0.0156 0.2800 83.77 [67,90] 0.0267 0.0154 0.3067 82.88 [64,80] 0.0533 0.0178 0.2133 82.63 [66,83] 0.0267 0.0162 0.2267 82.67 [64,84] 0.0400 0.0169 0.2667 84.05 [60,79] 0.0800 0.0199 0.2533 83.81 [73,92] 0.0933 0.0168 0.2533 84.86 [69,89] 0 0.0157 0.2667 82.74 [65,86] 0.0133 0.0161 0.2800 84.33 [64,84] 0.0400 0.0169 0.2667 84.00

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B5 (Ts = 100, RU Ls = 61) [ Interval RE RMSE RW Time(s) RU Ls [56,95] 0.1148 0.0210 0.6393 3.94 87 [32,67] 0.3770 0.0385 0.5738 3.38 65 [48,82] 0.0492 0.0172 0.5574 3.01 81 [43,101] 0.2459 0.0290 0.9508 3.60 67 [35,74] 0.0820 0.0159 0.6393 3.67 54 [45,64] 0.1311 0.0152 0.3115 3.68 64 [52,128] 0.1967 0.0254 1.2459 3.66 61 [43,91] 0.0328 0.0207 0.7869 3.66 52 [35,58] 0.2295 0.0266 0.3770 3.65 71 [37,70] 0.2623 0.0286 0.5410 3.66 58 [50,68] 0.0492 0.0117 0.2951 83.34 78 [53,69] 0.0164 0.0123 0.2623 83.97 77 [52,66] 0.0328 0.0125 0.2295 83.36 71 [50,63] 0.0656 0.0122 0.2131 83.93 73 [50,65] 0.0656 0.0120 0.2459 84.09 72 [53,69] 0 0.0117 0.2623 83.69 69 [49,66] 0.0656 0.0119 0.2787 83.02 82 [54,74] 0.0328 0.0144 0.3279 83.62 75 [53,69] 0.0164 0.0126 0.2623 83.82 74 [50,65] 0.0820 0.0121 0.2459 83.52 72

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Figure 7: Prognostic results for dataset B7 (experiments #A1). (a) PF method. (b) Proposed hybrid method.

4.3.2. Comparative experiments #B In order to study the effects of data size on the performance of prognostic result and demonstrate the reliability of proposed hybrid method, the EOM threshold value for datasets of type A is increased to 92.2% of the rated capacity, while for datasets of type B, the EOM threshold value is changed to 85% of initial capacity for dataset B5 and 90% for B7, respectively. The number of particles for PFbased prognostic method is 200, the value of parameters for proposed hybrid method is N = 200 and L = 30. As before, we first estimate the SDs of noise in each measured dataset for the comparative experiments #B based on RVM method. Table 4 shows the indicators for the estimated SDs. Table 4: Indicators for the estimated SDs (experiments #B).

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In the table 2 and table 3, most of the degradation prediction results obtained by proposed hybrid method have smaller RE and RMSE values than that obtained by PF. Concerning to the uncertainty assessment for RUL prediction result, it is obviously that the intervals obtained by proposed hybrid method have smaller RW values, besides, different from some PF-based prognostic results, where their intervals do not bracket the true RULs (which are highlighted in bold), the intervals obtained by proposed hybrid method can bracket the true RULs precisely. Furthermore, by comparing the ten experimental results of each method, it can be found that the fluctuation of the prognostic result obtained by proposed hybrid method, including the RUL prediction result as well as interval, is less severe.

No. A1 A2 B5 B7

Maximum (×10−3 ) 3.3377 5.7802 12.560 10.352

Minimum (×10−3 ) 3.2829 5.6532 12.179 9.9256

Mean (×10−3 ) 3.2970 5.6818 12.260 10.006

Range (×10−3 ) 0.055 0.127 0.381 0.426

Since the range of the SDs is very small compared with the mean of SDs and capacity data, the SDs of the noise in each measured dataset are treated as constant and are equal to the mean. It is worth noting that due to the weight uncertainty in RVM method [39], the SDs of measurement noise in 10

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Table 5: Prognostic results for the capacity datasets of type A (experiments #B).

A1 (Ts = 118, RU Ls = 90) A2 (Ts = 100, RU Ls = 88) [ Interval RE RMSE RW Time(s) RU Ls Interval RE RMSE RW Time(s) [62,115] 0.1556 0.0142 0.5889 5.74 70 [52,84] 0.2045 0.0233 0.3636 4.99 [54,112] 0.1889 0.0181 0.6444 5.10 71 [59,122] 0.1932 0.0226 0.7159 4.80 [67,103] 0.0667 0.0102 0.4000 6.21 73 [62,90] 0.1705 0.0188 0.3182 4.71 [63,122] 0.1333 0.0140 0.6556 6.28 94 [74,128] 0.0682 0.0047 0.6136 4.68 [56,130] 0.0333 0.0092 0.8222 6.35 116 [94,176] 0.3182 0.0109 0.9318 4.70 [59,90] 0.2444 0.0223 0.3444 6.37 96 [74,120] 0.0909 0.0056 0.5227 4.69 [82,122] 0.1111 0.0067 0.4444 6.43 65 [52,83] 0.2614 0.0265 0.3523 4.68 [50,104] 0.2111 0.0193 0.6000 6.52 87 [73,111] 0.0114 0.0076 0.4318 4.73 [61,90] 0.2111 0.0189 0.3222 6.59 100 [79,119] 0.1364 0.0054 0.4545 4.69 [69,113] 0.0333 0.0062 0.4889 6.77 76 [50,155] 0.1364 0.0172 1.1932 4.80 [85,109] 0.0556 0.0053 0.2667 128.90 93 [83,108] 0.0568 0.0034 0.2841 103.66 [85,118] 0.0778 0.0054 0.3667 120.77 93 [82,111] 0.0568 0.0038 0.3295 104.69 [84,109] 0.0444 0.0054 0.2778 121.69 94 [83,113] 0.0682 0.0037 0.3409 103.01 [88,115] 0.1000 0.0064 0.3000 120.74 98 [86,112] 0.1136 0.0045 0.2955 105.23 [76,106] 0.0222 0.0072 0.3333 121.77 84 [76,97] 0.0455 0.0096 0.2386 105.43 [82,118] 0.0556 0.0053 0.4000 120.70 83 [73,100] 0.0568 0.0104 0.3068 104.69 [71,101] 0.0889 0.0115 0.3333 121.95 94 [83,110] 0.0682 0.0046 0.3068 105.57 [88,107] 0.0667 0.0054 0.2111 121.30 91 [82,102] 0.0341 0.0057 0.2273 105.82 [88,111] 0.0778 0.0055 0.2556 119.90 95 [85,110] 0.0795 0.0040 0.2841 104.70 [76,99] 0.0556 0.0096 0.2556 117.08 86 [76,100] 0.0227 0.0082 0.2727 102.92

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Figure 8: Q-Q plots (experiments #B). (a) The Q-Q plot for dataset A1 at k = 208. (b) The Q-Q plot for dataset A2 at k = 156. (c) The Q-Q plot for dataset B5 at k = 155. (d) The Q-Q plot for dataset B7 at k = 159.

mality because most of the error points are located around the standard line. The intercept of the standard line in each subfigure is no longer equal to zero, nevertheless, the intercept is small compared to the capacity data and can be ignored in this study. Thus, the distribution of PFbased prediction error in experiments #B is close to the normal distribution with zero mean. The experimental results and corresponding evaluation indicators are shown in table 5 and table 6. Compare the experimental results of each method in table 5–table 6, it is once again verified that the proposed hybrid method can obtain more reliable prognostic result. However, due to the increased EOM threshold, there are fewer data points adopted for prognostics, compared with the results in experiments #A, it can be found that the accuracy of the RUL prediction results and degradation prediction results obtained by the two methods is reduced by analyzing the values of RE and RMSE. Nevertheless, the performance of prognostic result obtained by proposed hybrid method deteriorates much slower. Therefore, the proposed hybrid method can achieve more robust prognostic result. 4.3.3. Comparative experiments #C According to Eq.(16) and Eq.(27), the deterministic prediction result of proposed hybrid method at each cycle is the mean of (L × N ) particles (A PF-based deterministic prediction result is the weighed mean of N particles, and the prediction result of proposed hybrid method is calculated using L PF-based deterministic prediction results).

experiments #B are slightly different from the ones in experiments #A. Figure 8 shows the Q-Q plots for the prediction errors based on the four datasets in experiments #B at random cycles. In the subfigures, each Q-Q plot shows a good nor11

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Table 6: Prognostic results for the capacity datasets of type B (experiments #B).

B7 (Ts = 65, RU Ls = 94) Interval RE RMSE RW Time(s) [68,100] 0.1489 0.0200 0.3404 3.57 [63,101] 0.1596 0.0214 0.4043 2.98 [82,150] 0.2447 0.0424 0.7234 2.99 [74,152] 0.0213 0.0213 0.8298 3.02 [75,186] 0.2660 0.0417 1.1809 3.62 [88,150] 0.1064 0.0281 0.6596 3.66 [63,117] 0.0745 0.0208 0.5745 3.63 [63,95] 0.2128 0.0280 0.3404 3.75 [61,114] 0.1277 0.0266 0.5638 3.65 [64,99] 0.1596 0.0267 0.3723 3.75 [82,101] 0.0319 0.0171 0.2021 85.78 [79,113] 0.0106 0.0175 0.3617 83.88 [77,102] 0.0638 0.0170 0.2660 84.87 [73,104] 0.0851 0.0176 0.3298 84.65 [76,95] 0.1064 0.0177 0.2021 84.33 [81,115] 0.0106 0.0190 0.3617 85.30 [79,100] 0.0638 0.0171 0.2234 84.43 [74,95] 0.1170 0.0175 0.2234 84.67 [74,102] 0.0851 0.0177 0.2979 89.18 [77,96] 0.0957 0.0170 0.2021 82.83

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B5 (Ts = 78, RU Ls = 83) [ Interval RE RMSE RW Times(s) RU Ls [59,114] 0.1566 0.0166 0.6627 3.92 80 [53,94] 0.1807 0.0271 0.4940 3.67 79 [67,165] 0.3133 0.0496 1.1807 3.66 117 [59,101] 0.0361 0.0170 0.5060 3.63 96 [46,73] 0.3494 0.0617 0.3253 3.63 119 [57,110] 0.0241 0.0207 0.6386 3.63 104 [63,141] 0.1205 0.0302 0.9398 3.67 87 [54,99] 0.0843 0.0168 0.5422 3.15 74 [39,71] 0.4578 0.1531 0.3855 3.00 82 [50,104] 0.0482 0.0218 0.6506 3.64 79 [74,98] 0.0241 0.0204 0.2892 85.44 91 [69,100] 0.0120 0.0173 0.3735 90.49 93 [71,95] 0.0241 0.0172 0.2892 84.29 88 [78,111] 0.1084 0.0288 0.3976 84.74 86 [68,86] 0.0843 0.0173 0.2169 86.37 84 [67,91] 0.0723 0.0165 0.2892 85.66 95 [74,97] 0.0120 0.0201 0.2771 84.80 88 [78,105] 0.0843 0.0262 0.3253 84.88 83 [71,87] 0.0602 0.0166 0.1928 85.99 86 [66,91] 0.0843 0.0172 0.3012 85.07 85

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Table 7: Prognostic results of PF for the capacity datasets of type A (experiments #C).

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A1 (Ts = 140, RU Ls = 68) [ Interval RE RMSE RW Time(s) RU Ls [48,113] 0.1029 0.0056 0.9559 138.12 57 [45,106] 0.0882 0.0059 0.8971 138.09 61 [41,112] 0.1471 0.0062 1.0441 137.04 72 [38,91] 0.0882 0.0107 0.7794 137.01 53 [45,89] 0.0588 0.0095 0.6471 138.02 51 [47,116] 0.0735 0.0055 1.0147 135.29 82 [44,127] 0.1176 0.0057 1.2206 132.86 56 [36,87] 0.1176 0.0123 0.7500 138.12 63 [47,111] 0.1324 0.0060 0.9412 138.43 67 [42,88] 0.0882 0.0109 0.6765 138.55 53

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[ Method No. RU Ls #C1 75 #C2 74 #C3 78 #C4 62 #C5 64 PF #C6 73 #C7 76 #C8 60 #C9 77 #C10 62

Compared with the prognostic results obtained by proposed hybrid method in experiments #A, the computation time is almost the same, however, the intervals predicted by PF in experiments #C are much wider. Besides, the RUL prediction results are less accurate and the RMSE values are larger in experiments #C. Moreover, although the fluctuation of the PF-based prognostic result can be reduced by increasing the number of the particles, it is still more severe than the prognostic result obtained by proposed hybrid method in experiments #A.

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In order to further verify the effectiveness and reliability of the proposed hybrid method, the experiments using PF with 6000 particles are conducted. The EOM threshold of each dataset is equal to the one in experiments #A. The detailed experimental results and evaluation indicators are listed in table 7 and table 8.

A2 (Ts = 124, RU Ls = 64) Interval RE RMSE RW Time(s) [41,87] 0.1094 0.0103 0.7188 105.10 [41,102] 0.0469 0.0078 0.9531 105.31 [47,135] 0.1250 0.0096 1.3750 105.87 [38,84] 0.1719 0.0130 0.7188 105.80 [35,86] 0.2031 0.0154 0.7969 105.87 [49,155] 0.2813 0.0151 1.6563 106.21 [40,84] 0.1250 0.0161 0.6875 106.02 [39,88] 0.0156 0.0045 0.7656 105.94 [46,115] 0.0469 0.0050 1.0781 107.71 [38,84] 0.1719 0.0130 0.7188 101.89

Compared with the PF-based prognostic results in experiments #A, because of the increased number of particles, the RUL prediction results in experiments #C become more accurate, the RMSE values of the prediction results are relatively smaller, the intervals can assess the prediction uncertainty more precisely. The fluctuation of the prognostic result, including RUL prediction result and interval, becomes less severe.

4.3.4. Comparative experiments #D In order to verify the wide range applicability of the proposed hybrid method, comparative experiments based on 12

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Table 8: Prognostic results of PF for the capacity datasets of type B (experiments #C).

B5 (Ts = 100, RU Ls = 61) [ Interval RE RMSE RW Time(s) RU Ls [35,88] 0.1639 0.0155 0.8689 83.75 64 [36,94] 0.1311 0.0142 0.9508 84.73 67 [37,111] 0.1639 0.0161 1.2131 85.32 72 [35,106] 0.0492 0.0119 1.1639 85.29 66 [34,90] 0.2131 0.0197 0.9180 84.31 68 [38,110] 0.0656 0.0118 1.1803 83.98 64 [33,125] 0.1639 0.0158 1.5082 85.53 65 [39,120] 0.0328 0.0147 1.3279 84.65 71 [38,93] 0.0820 0.0122 0.9016 84.73 66 [38,134] 0.0492 0.0121 1.5738 84.35 63

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AR model are conducted. The datasets adopted in experiments #D are datasets A1 and B5, the EOM threshold for each dataset is equal to the one in experiments #A, therefore, the regression results based on RVM are also equal to the ones in experiments #A, the number of particles for PF-based prognostic method is 6000, the value of the parameters for proposed hybrid method is N = 200 and L = 30, the order of AR model is p = 8 for A1 and p = 13 for B5. The figure 9 shows the Q-Q plots for the prediction errors using PF with N = 200 particles. In each Q-Q plot, most of the error points are located around the standard line, and the intercept of standard line is close to zero, thus, the PF-based prediction error

B7 (Ts = 84, RU Ls = 75) Interval RE RMSE RW Time(s) [46,105] 0.1467 0.0239 0.7867 82.17 [44,114] 0.1067 0.0221 0.9333 81.36 [46,99] 0.0400 0.0172 0.7067 83.93 [45,154] 0.1200 0.0223 1.4533 81.02 [48,114] 0.0933 0.0207 0.8800 82.47 [43,134] 0.1467 0.0240 1.2133 81.69 [41,96] 0.1333 0.0234 0.7333 84.74 [49,101] 0.0533 0.0191 0.6933 81.28 [43,116] 0.1200 0.0229 0.9733 81.12 [45,108] 0.1600 0.0253 0.8400 80.97

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[ Method No. RU Ls #C1 51 #C2 53 #C3 51 #C4 58 #C5 48 PF #C6 57 #C7 51 #C8 63 #C9 56 #C10 58

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Figure 9: Q-Q plots (experiments #D). (a) The Q-Q plot for dataset A1 at k = 208. (b) The Q-Q plot for dataset B5 at k = 150.

Table 9: Prognostic results for the capacity datasets A1 and B5. (experiments #D).

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A1 (Ts = 140, RU Ls = 68) [ [ Method No. RU Ls Interval RE RMSE RW Time(s) RU Ls #D1 61 [42,92] 0.1029 0.0091 0.7352 143.48 54 #D2 63 [42,98] 0.0735 0.0076 0.8235 141.35 58 #D3 57 [38,94] 0.1618 0.0129 0.8235 143.63 65 #D4 62 [42,91] 0.0882 0.0081 0.7206 143,25 68 #D5 64 [41,119] 0.0588 0.0069 1.1471 143.93 55 PF #D6 59 [41,100] 0.1324 0.0118 0.8676 143.01 56 #D7 61 [40,84] 0.1029 0.0091 0.6471 143.53 53 #D8 61 [37,88] 0.1029 0.0091 0.7500 142.36 69 #D9 62 [42,85] 0.0882 0.0083 0.6324 143.23 50 #D10 59 [40,83] 0.1324 0.0111 0.6324 143.45 65 #D1 65 [57,78] 0.0441 0.0063 0.3088 143.71 57 #D2 62 [54,72] 0.0882 0.0086 0.2647 144.46 64 #D3 66 [57,81] 0.0294 0.0059 0.3529 143.67 52 #D4 62 [53,77] 0.0882 0.0085 0.3529 143.11 56 #D5 63 [55,75] 0.0735 0.0079 0.2941 142.98 64 Hybrid #D6 64 [56,75] 0.0588 0.0073 0.2794 143.84 54 #D7 62 [54,74] 0.0882 0.0086 0.2941 142.64 64 #D8 64 [54,80] 0.0588 0.0073 0.3824 143.48 53 #D9 63 [56,73] 0.0735 0.0080 0.2500 143.97 65 #D10 64 [56,77] 0.0588 0.0072 0.3088 143.99 67 13

B5 (Ts = 100, RU Ls = 61) Interval RE RMSE RW Time(s) [35,94] 0.1148 0.0140 0.9672 104.13 [39,108] 0.0492 0.0120 1.1311 104.39 [39,123] 0.0656 0.0119 1.3770 104.49 [37,127] 0.1148 0.0142 1.4754 104.63 [36,112] 0.0984 0.0139 1.2459 103.45 [38,118] 0.0820 0.0123 1.3115 104.50 [35,84] 0.1311 0.0139 0.8033 104.29 [36,126] 0.1311 0.0136 1.4754 106.07 [35,98] 0.1803 0.0170 1.0328 104.41 [37,137] 0.0656 0.0121 1.6393 103.39 [47,73] 0.0656 0.0127 0.4262 104.33 [52,72] 0.0492 0.0173 0.3279 104.81 [46,65] 0.1475 0.0150 0.3115 104.76 [45,71] 0.0820 0.0139 0.4262 103.28 [52,69] 0.0492 0.0166 0.2787 103.04 [48,67] 0.1148 0.0137 0.3115 103.22 [50,70] 0.0492 0.0157 0.3279 104.37 [44,69] 0.1311 0.0142 0.4098 104.65 [53,72] 0.0656 0.0192 0.3115 104.88 [60,82] 0.0984 0.0250 0.3607 104.87

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in experiments #D is close to norm distribution with zero mean. The experimental results and corresponding evaluation indicators are shown in table 9. Compare the experimental results of each method in table 9, the hybrid method can achieve more stable and accurate prognostic results in the same computation time. Compare the PF-based prognostic results with the ones in experiments #C, the performance is basically the same without significant difference. Compare the prognostic results of proposed hybrid method with the corresponding ones in experiments #A, the most apparent difference is that the width of the PIs in experiments #D is larger. The accuracy and stability of the prognostic results for dataset A1 are not changed too much, while for dataset B5, the accuracy of RUL prediction results in experiments #D deteriorates slightly. Generally, the performance of the prognostic result based on AR model using proposed hybrid method is acceptable.

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the parameter estimation and PI construction more convenient. (4).The proposed hybrid method was tested by the exponential-type model and AR model, which shows the wide range applicability of the proposed hybrid method. For the parameters of the proposed hybrid method (N and L), only one set of values is considered(N = 200, L = 30) in this paper, in the future, we plan to study the influences of different N and L values on prognostic performance, and find the optimal parameters values to maintain the prognostic performance and reduce the computation time. Another issue needs to be studied in the future is to extend the proposed hybrid method to the prognostic problem under varying operating conditions, for this, two critical tasks need to be settled, first, the health indicator needs to be constructed to characterize the health state of system; second, the multi-state model needs to be built to describe the degradation process of system under varying operating conditions.

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6. Acknowledge

5. Conclusion

This work is supported by National Natural Science Foundation of China (Grant No. 61473127). We would like to thank the CALCE of University of Maryland and PCoE at NASA for providing the experimental datasets.

In this paper, a novel hybrid method combining PF and RVM is proposed for system prognostics. This hybrid method can obtain the prediction uncertainty while predicting the degradation process. In order to show the performance of the proposed hybrid method, four comparative experiments between proposed hybrid method and PF are conducted based on two types of lithium-ion battery datasets and two different degradation models. By comparing and analyzing the experimental results, it can be found that: (1). The proposed hybrid method can obtain more accurate RUL prediction result, and more precise PI to bracket the true RUL, the prognostic result obtained by proposed hybrid method is also more stable compared with the one obtained by PF method, therefore,the proposed hybrid method can achieve more reliable prognostic result. (2). The exponential-type models have been adopted by many kinds of degraded systems, in addition, the construction of AR model depends only on the degradation data and is not limited by the type of prognostic problem, therefore, the proposed hybrid method has wide range applicability. (3). The computation time is directly related to the number of particles, however, if the L PFs adopted by the proposed hybrid method can be run in parallel, the computation time can be greatly reduced. For the hybrid method proposed in this paper, we believe that there are four main contributions: (1).The proposed hybrid method can obtain more reliable prognostic result, which can effectively reduce maintenance cost and improve system reliability and safety when it is adopted for system maintenance. (2). We discussed the difference between CI and PI, and provided a theoretical way to construct PI, which can assess the prediction uncertainty more reasonably and precisely. (3). We adopted the RVM to estimate the variance of measurement noise, which makes

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