IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 64, NO. 7, JULY 2017
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Remaining Useful Life Prediction Based on a General Expression of Stochastic Process Models Naipeng Li, Yaguo Lei, Member, IEEE, Liang Guo, Tao Yan, and Jing Lin
Abstract—In remaining useful life (RUL) prediction, stochastic process models are widely used to describe the degradation processes of systems. For age-dependent stochastic process models, the RUL probability density function (PDF) can be calculated using a closed-form solution. For state-dependent models, however, it is difficult to calculate such a closed-form solution. Therefore, the RUL is always approximately estimated using a sequential Monte Carlo-based method, but this method has some limitations. First, it only provides a numerical approximation result whose accuracy highly relies on the quality and quantity of the simulated degradation trajectories. Second, the time interval is unable to be adjusted during the state transition process, resulting in too few discrete probability densities in the result near the end-of-life. This paper describes the degradation processes using a general expression of ageand state-dependent models. The analytical solution of the RUL PDF is derived from the general expression. After that, a new RUL prediction method is proposed. In this method, a series of degradation trajectories are generated through degradation process simulation. The RUL PDF is estimated by inputting the state values of the degradation trajectories into the analytical solution. The validity of the proposed method is verified using fatigue-crack-growth data. Index Terms—Degradation process simulation (DPS), remaining useful life (RUL) prediction, sequential Monte Carlo, stochastic process model.
I. INTRODUCTION ITH recent advancements in health maintenance strategies, the predictive maintenance strategy has attracted more and more attentions [1]–[3]. The principle of predictive
W
Manuscript received October 1, 2016; revised December 24, 2016; accepted February 15, 2017. Date of publication March 2, 2017; date of current version June 9, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 51475355 and Grant 61673311, in part by the Young Talent Support Plan of the Central Organization Department, and in part by the Visiting Scholar Foundation of the State Key Laboratory of Traction Power at Southwest Jiaotong University under Grant TPL1703. (Corresponding author: Yaguo Lei.) The authors are with the State Key Laboratory for Manufacturing Systems Engineering and the Shaanxi Key Laboratory of Mechanical Product Quality Assurance and Diagnostics, Xi’an Jiaotong University, Xi’an 710049, China (e-mail:
[email protected]; yaguolei@ mail.xjtu.edu.cn;
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2017.2677334
maintenance is to predict the remaining useful life (RUL) based on condition monitoring information and make optimal maintenance decisions before the breakdown of equipment. The RUL is defined as the time left before the degradation processes of systems exceed a failure threshold (FT) [3]–[7]. Since the degradation processes of systems are generally stochastic, it is reasonable to describe the degradation processes using stochastic process models [3]. In most studies, the degradation rates are assumed to be physically dependent on only the system age [8]–[13]. Under this assumption, the degradation models can be expressed using the following age-dependent function: t t μ(τ )dτ + σ(τ )dB(τ ) (1) s(t) = sk + tk
tk
where sk is the state at time tk , s(t) is the state at t with t ≥ tk . The state is defined as the health state of a component, which directly represents the deterioration severity, such as the fatigue crack length, the flank wear value of a machine tool or the wear area of a bearing or a gear. μ(τ ) is the drift coefficient function representing the degradation rate, σ(τ ) is the diffusion coefficient function, and {B(τ ), τ ≥ tk } denotes a standard Brownian motion (BM) process. Considering the stochasticity of the degradation processes, the RUL is always represented by a random variable lk = inf {l : s(l + tk ) ≥ λ|sk }
(2)
where lk is the RUL at tk , inf { · } is the inferior limit of a variable, s(l + tk ) is the degradation process after tk , and λ is the FT. The RUL has the following relationship with the end-of-life (EoL) tEoL : lk = tEoL − tk . The major task of RUL prediction is to estimate the probability density function (PDF) of lk or tEoL conditioned on sk , i.e., f (l|sk ) or f (t|sk ). Many existing publications have derived the closed-form PDF of the RUL or EoL based on various age-dependent models. Gebraeel et al. [8] derived the closed-form PDF of the EoL based on an exponential model, and applied it into the RUL prediction of rolling element bearings. Peng and Tseng [11] derived the analytical solution of the EoL PDF based on a linear degradation model, and predicted the RUL of a laser device. Park and Padgett [12] constructed degradation models based on the geometric BM and the gamma process, and derived the closed-form expression of the EoL PDF. Si et al. [13] described the degradation of systems using a
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nonlinear stochastic process model and derived the closed-form PDF of the RUL. These methods are able to provide precise estimation results for the RUL when the degradation processes are suitable to be described using age-dependent models. However, most degradation processes are influenced by not only the system age but also the system state. Taking the fatigue crack growth for example [14], the degradation rates are low in the initial period of fatigue crack. With the growth of the fatigue crack, the stress intensity will increase gradually, resulting in the increment of the degradation rates. In such cases, the degradation models only depending on the system age fail to describe the degradation processes. Consequently, the RUL prediction results acquired from the age-dependent models will have large discrepancy with the ground truth RUL. To reveal the relationship between the degradation rates and the system state, some state-dependent degradation models have been constructed. An et al. [15] used state-dependent models to describe the battery degradation process and the fatigue crack growth, and predicted the RUL using the sequential Monte Carlo (SMC) algorithm. Orchard et al. [16] developed a state-dependent model for the lithium-ion battery degradation, and then used the SMC-based method to estimate the PDF of the RUL. Lei et al. [17] employed the Paris–Erdogan model to describe the degradation processes of rolling element bearings, and estimated the RUL using the SMC algorithm. Zio and Peloni, [18] used a state-dependent model to describe the fatigue crack growth, and developed a RUL prediction method based on the SMC algorithm. Liu et al. [19] described the state-dependent degradation processes of batteries using neural networks and neural fuzzy systems, and predicted the RUL with the combination of the SMC algorithm. For the state-dependent models, the RUL PDF is difficult to be expressed using a closed-form solution. Therefore, these above studies try to approximate the RUL PDF using a numerical solution with the help of the SMC algorithm [20]. The basic idea of the SMC algorithm is to approximate the RUL PDF using discrete probability densities of the RULs observed from large numbers of simulated degradation trajectories [21]. The SMC-based method is suitable to deal with the issue of the RUL prediction for the state-dependent models. However, it still has the following limitations. 1) This method only provides a numerical approximation result for the RUL PDF, and the accuracy of the result highly depends on the quality and quantity of the simulated degradation trajectories. 2) The time interval is unable to be adjusted during the state transition process, resulting in too few discrete probability densities in the estimation result near the EoL, which fails to describe the continuous RUL PDF. In this paper, the degradation processes of systems are described using a general expression of stochastic process models depending on both the system age and the system state. The analytical solution of the RUL PDF is derived from the general expression. Based on the general expression and the analytical solution, a new RUL prediction method is proposed. At first, the degradation processes are simulated through step-by-step state transition, and a series of degradation trajectories are generated. Then, the RUL probability density of each degradation trajectory is calculated by inputting the state values of the
trajectory into the analytical solution of the RUL PDF. Finally, the RUL PDF is approximated by integrating the discrete RUL probability densities from all of the simulated degradation trajectories. The major contributions of this paper are as follows. 1) A straightforward analytical solution of the RUL PDF is derived from a general expression of age- and statedependent models. 2) A new RUL prediction method based on the general expression is proposed by inheriting the basic idea of SMC and taking advantage of the exactness of the analytical solution. 3) To keep the diversity of the discrete probability densities in the estimation result of the RUL PDF, a self-adjustment strategy is developed to adjust the time interval during the state transition process. The remaining parts of this paper are organized as follows. Section II presents the general expression of the age- and statedependent degradation models and derives the analytical solution of the RUL PDF. Section III describes the proposed RUL prediction method. In Section IV, a practical case study of the fatigue-crack-growth is used to demonstrate the effectiveness of the proposed method. Conclusions are drawn in Section V. II. MODEL CONSTRUCTION AND RUL ANALYTICAL SOLUTION Equation (1) gives the general expression of the agedependent degradation models. By replacing its drift coefficient function with an age- and state-dependent function, the general expression is changed to be t t μ(s(τ ), τ )dτ + σ(τ )dB(τ ) (3) s(t) = sk + tk
tk
where μ(s(τ ), τ ) is the drift coefficient function depending on both the age and state. Fig. 1(a) gives an example of the degradation processes described by (3). It is seen that the degradation trajectory is deterministic before the current life, and becomes stochastic after the current life. There are countless possible degradation trajectories from the current life to the EoL. Influenced by the stochasticity of the degradation process, the RUL is uncertain and therefore represented by a random variable, as shown in (2). The major task is to derive the analytical solution of the RUL PDF f (l|sk ) from the general expression of the stochastic process models in (3). In this paper, the RUL PDF is derived through three-step transformations: translation transformation, standard transformation and inverse transformation. The flowchart of the threestep transformations is shown in Fig. 1. First, the general degradation process is moved to the origin of coordinates through the translation transformation. Then, the translational degradation process is changed to be a standard BM process through the standard transformation. Finally, the PDF of the standard BM process crossing a time-varying threshold is transformed into the PDF of the general degradation process crossing a constant threshold through the inverse transformation. More details about the three-step transformations are explained as follows.
LI et al.: REMAINING USEFUL LIFE PREDICTION BASED ON A GENERAL EXPRESSION OF STOCHASTIC PROCESS MODELS
Fig. 1.
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Flowchart of the RUL PDF derivation through three-step transformations.
A. Translation Transformation Theorem 1: The degradation process {s(t), t ≥ tk } shown in (3), which starts from tk and sk , is transformed into a new degradation process {˜ s(l), l ≥ 0} starting from l0 = 0 and s˜0 = 0 through the following transformation: s˜(l) = s(t) − sk l = t − tk 0
l
μ ˜(˜ s(τ ), τ )dτ +
0
l
σ ˜ (τ )dB(τ )
s˜ 0
dw − σ ˜ (τ )
0
s˜
d dτ
1 dw σ ˜ (τ ) (7)
then there exists a transformation
(5)
⎧ l s˜ dw ⎪ 1/ 2 ⎪ ˜l) = ϕ(˜ ⎪ B( s , l) = (k ) exp − c (τ )dτ 1 2 ⎪ ⎪ ˜ (l) ⎪ l0 0 σ ⎪ ⎪ τ l ⎨ − (k1 )1/ 2 c1 (τ ) exp − c2 (v)dv dτ + k2 ⎪ l l ⎪ 1 0 ⎪ ⎪ τ l ⎪ ⎪ ⎪ ˜ ⎪ exp −2 c2 (v)dv dτ + k3 ⎩ l = ψ(l) = k1
and μ ˜(˜ s(τ ), τ ) = μ(s(τ + tk ), τ + tk ) . σ ˜ (τ ) = σ(τ + tk )
μ ˜(˜ s, τ ) = c1 (τ ) + c2 (τ ) σ ˜ (τ )
(4)
where s˜(l) =
Lemma 1: For the degradation process in (5), if there are two time-dependent functions c1 (τ ) and c2 (τ ) satisfying
(6)
The proof of Theorem 1 is given in the Appendix. After the translation transformation, {s(t), t ≥ tk } is transformed into {˜ s(l), l ≥ 0}, which is a degradation process starting from l0 = 0 and s˜0 = 0. B. Standard Transformation The degradation process {˜ s(l), l ≥ 0} is further transformed into a standard BM process through the following lemma [22].
l2
l0
(8) which transforms the degradation process {˜ s(l), l ≥ 0} into a standard BM process {B(˜l), ˜l ≥ 0}. In (8), li ∈ [0, ∞), i = 0, 1, 2, k1 > 0, k2 and k3 are arbitrary constants. Until now, the general degradation process in (3) is transformed into a standard BM process. Consequently, the problem of a general degradation process crossing a constant threshold can be treated as a problem of a standard BM process crossing a time-varying threshold. The following lemma [23] has given the PDF of the standard BM process crossing a time-varying threshold.
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Lemma 2: If a standard BM process {B(˜l), ˜l ≥ 0} with a covariance function ρ(ν, τ ), 0 ≤ ν ≤ τ ≤ ˜l, and a time-varying threshold T (˜l) satisfy the following assumptions: 1) The threshold T (τ ) is continuous in 0 ≤ τ ≤ ˜l and is left differentiable at ˜l. 2) The covariance function ρ(ν, τ ) is positive definite and has continuous first-order partial derivatives in {(ν, τ ) : 0 ≤ ν ≤ τ ≤ ˜l}, where appropriate left or right derivatives are taken at ν = 0, ν = τ, and τ = ˜l. 3) lim[∂ρ(τ, ˜l)/∂τ − ∂ρ(τ, ˜l)/∂ ˜l] = ξ, with 0 < ξ < ∞. τ →˜l
Then the PDF of the time when {B(˜l), ˜l ≥ 0} crosses the time-varying threshold T (˜l) is described using fB (˜l) = b(˜l)pB (˜l)
(9)
where E[I(τ, B)(T (τ ) − B(τ ))|B(˜l) = T (˜l)] b(˜l) = lim ˜l − τ τ →˜l 2 ˜ ( l) T 1 . pB (˜l) =
exp − 2˜l 2π˜l
(10) (11)
With the assumption [13] that if the degradation process crosses the threshold at a certain time t, then the probability that such a process passed the threshold before t would be assumed to be negligible, (9) is further changed to be fB (˜l) ∼ =
T (˜l) dT (˜l) − ˜l d˜l
1 T 2 (˜l)
. exp − 2˜l 2π˜l
(12)
C. Inverse Transformation Since {B(˜l), ˜l ≥ 0} is transformed from {s(t), t ≥ tk }, the probability distribution of the RUL F (l|sk ) has the following relationship with the probability distribution FB (˜l). ˜ F (l|sk ) = Pr(s(l + tk ) ≤ λ) = Pr (˜ s(l) ≤ λ) ˜ l)) = FB (˜l) = Pr (B(˜l) ≤ ϕ(λ,
(13)
˜ = λ − sk is the FT of the process {˜ s(l), l ≥ 0}. where λ By calculating the derivatives of F (l|sk ) and FB (˜l) with respect to l, the following equation is acquired. dψ(l) . f (l|sk ) = fB (˜l) dl
(14)
Based on Theorem 1, Lemmas 1 and 2, and (14), the RUL PDF of the degradation process in (3) is derived in the following theorem. Theorem 2: For the degradation process {s(t), t ≥ tk } in (3), which has a state value sk at tk and a FT λ with sk < λ, the
RUL PDF of the degradation process at tk is expressed as
l λ − sk − 0 μ(s(τ + tk ), τ + tk )dτ ∼ f (l|sk ) =
l 2 0 σ (τ + tk )dτ μ(s(l + tk ), l + tk ) − σ 2 (l + tk ) σ 2 (l + tk )
l 2π 0 σ 2 (τ + tk )dτ
l (λ − sk − 0 μ(s(τ + tk ), τ + tk )dτ )2 . exp −
l 2 0 σ 2 (τ + tk )dτ (15) ×
The proof of Theorem 2 is given in the Appendix. By now, the analytical solution of the RUL PDF has been derived from the general expression of stochastic process models. Based on the general expression and the analytical solution, a new RUL prediction method is developed in the following section. III. PROPOSED RUL PREDICTION METHOD The schematic of the proposed method is shown in Fig. 2(a). The rationale of this method is explained as follows. Equation (15) gives the analytical solution of the RUL PDF. It is noticed that the future degradation trajectories {s(τ + tk ), 0 ≤ τ ≤ l} are required in the calculating procedure of
l 0 μ(s(τ + tk ), τ + tk )dτ and μ(s(l + tk ), l + tk ). The future degradation trajectories, however, are unknown in real applications. And there are countless possible degradation trajectories because of the stochasticity of the degradation process. To acquire a series of possible degradation trajectories, the degradation process is simulated through step-by-step state transition. The RUL probability density of each degradation trajectory is calculated by inputting the state values of this trajectory into a discrete version of (15). The continuous RUL PDF is approximated by integrating the discrete RUL probability densities acquired from all of the simulated degradation trajectories. Since the proposed method is on the basis of degradation process simulation (DPS), it is named as the DPS-based method. The flowchart of the DPS-based method is shown in Fig. 2(b), and the steps are described as follows. Step 1: Set the initial time interval as Δl0 . Generate a series of initial particles {sn (tk ) = sk }n =1:N , where N is the number of particles. Step 2: The state of each particle is transmitted step-by-step through the following transition function: sn (li + tk ) = sn (li−1 + tk ) + μ(sn (li−1 + tk ), li−1 +tk )Δli−1 + Vi−1
(16)
where sn (li + tk ) is the state value of the nth degradation tra Δl and jectory at li + tk , i ∈ N+ = {1, 2, 3, ...}, li = i−1 j j =0 distribution Vi−1 is the transition noise following a uniform √ U (−vi−1 , vi−1 ) with vi−1 = 3σ(li−1 + tk ) Δli−1 .
LI et al.: REMAINING USEFUL LIFE PREDICTION BASED ON A GENERAL EXPRESSION OF STOCHASTIC PROCESS MODELS
Fig. 2.
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(a) Schematic of the DPS-based method. (b) Flowchart of the DPS-based method.
Step 3: Count the number of failed trajectories that cross the FT at the ith transition, and denote the number as Mi . The time interval is self-adjusted using (17), as shown at the bottom of this page, where Mam is the allowed maximum number of the failed trajectories at one-step transition. The principle of the self-adjustment function is explained as follows. 1) Mi = 0 means that no trajectory fails during the time interval Δli−1 . In these cases, the time interval remains unchanged. 2) 0 < Mi ≤ Mam means that Δli−1 is so short that too few trajectories fail during Δli−1 . In these cases, the time interval should be extended, but be kept shorter than Δl0 . 3) Mi > Mam implies that Δli−1 is so long that too many trajectories fail during Δli−1 . In these cases, the time interval should be shortened. Since the number of the failed trajectories exceeds the allowed maximum number, this transition step is invalid and should be repeated with the shortened time interval. Step 4: Repeat steps 2 and 3 until all of the trajectories have exceeded the FT. Step 5: The nth degradation trajectory crosses the FT at the n −1 Δli . Based on the Ln th transition. Its RUL lL n equals Li=0 analytical solution of the RUL PDF in (15), the RUL probability
density of the nth trajectory is calculated by inputting the state values {sn (li + tk ),i = 0 : Ln − 1} into the discrete version of the RUL PDF, as shown by (18) at the bottom of this page. Step 6: Equation (18) gives the probability density of the nth degradation trajectory crossing the FT at lL n + tk . In reality, there may be some trajectories that have the same RUL but different probability densities. In this case, the mean value of the probability densities at the same RUL is taken as the estimated result. The RUL PDF is estimated by integrating the probability densities acquired from all of the trajectories using the following equation: N pn (l = lL n ) δ(l − lL n ) (19) fˆ (l|sk ) ∼ = n =1 N n =1 δ(l − lL n ) where δ(·) is the Dirac delta function. It should be mentioned that both the DPS-based method and the SMC-based method acquire the RUL PDF with the help of the simulated degradation trajectories. To explain the differences between them, the schematic and the flowchart of the SMC-based method are shown in Fig. 3(a) and (b), respectively. It is seen that the SMC-based method misses two steps compared with the DPS-based method, i.e., steps 3 and 5 in Fig. 2. Without step 3, the time interval is unable to be adjusted dur-
⎧ ⎪ ⎨ Δli = Δli−1 , i = i + 1, Δli = min(Mam Δli−1 / Mi , Δl0 ), i = i + 1, ⎪ ⎩ Δli−1 = Mam Δli−1 / Mi , i = i, pn (l = lL n ) ∼ =
λ − sk −
L n −1
i=0 lL n
μ(sn (li + tk ), li + tk )Δli
Mi = 0 0 < Mi ≤ Mam
(17)
Mi > Mam
μ(sn (lL n + tk ), lL n + tk ) − σ 2 (lL n + tk )
σ 2 (τ + tk )dτ n −1 2 σ 2 (lL n + tk ) μ(sn (li + tk ), li + tk )Δli ) (λ − sk − Li=0 × exp −
l l 2 0 L n σ 2 (τ + tk )dτ 2π 0 L n σ 2 (τ + tk )dτ 0
(18)
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Fig. 3.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 64, NO. 7, JULY 2017
(a) Schematic of the SMC-based method. (b) Flowchart of the SMC-based method.
ing the state transition process. More and more trajectories will cross the FT at the same transition step with the loss of the RUL. As a result, the discrete probability densities included in the estimation result will become too sparse to describe the continuous RUL PDF. Without step 5, the probability densities of the RULs can only be approximately estimated by counting the numbers of the trajectories failing at the same transition steps. The accuracy of the estimation result highly relies on the quality and quantity of the simulated trajectories. In the DPS-based method, the time interval is self-adjusted during the state transition process to keep the diversity of the discrete probability densities. Furthermore, the probability densities of the RULs are calculated by inputting the future state values of the simulated degradation trajectories into the analytical solution of the RUL PDF. The accuracy of the result is determined by the analytical solution instead of the simulated trajectories. In conclusion, the DPS-based method inherits the basic idea of the SMC and takes advantage of the exactness of the analytical solution. Therefore, it is expected to be more effective in dealing with the issue of RUL prediction than the SMC-based method. IV. EXPERIMENTAL DEMONSTRATIONS A. Experimental Data Introduction In this section, a classical fatigue-crack-growth dataset [24], [25] is utilized to demonstrate the effectiveness of the proposed method. As shown in Fig. 4, this dataset includes crack size observations of 21 alloy specimens. A 0.9-inch notch was seeded in each specimen before cyclic loading. The crack sizes were recorded every 0.01 million cycles. The FT of the test specimens was 1.6 inch. The fatigue experiment for each specimen was terminated after the crack size reached the FT or after 0.12 million cycles, whichever came first. After the experiment, 12 specimens reached the FT in total. B. RUL Prediction To describe the degradation processes of the fatigue cracks, three degradation models following the expression of (3) are
Fig. 4.
Fatigue-crack-growth data.
employed in this paper t t ⎧ √ b−1 ⎪ ⎪ s(t) = s + abτ dτ + c + dτ dB(τ ) M1 k ⎪ ⎪ ⎪ tk tk ⎪ ⎪ t t ⎨ √ ab exp(bτ )dτ + c + dτ dB(τ ) M2 s(t) = sk + ⎪ tk tk ⎪ ⎪ ⎪ t t√ ⎪ ⎪ ⎪ s(t) = s + ⎩ as(τ )b dτ + c + dτ dB(τ ) M3. k tk
tk
(20) The drift coefficient functions of Model 1 (M1) and Model 2 (M2) are μ1 (s(τ ), τ ) = abτ b−1 and μ2 (s(τ ), τ ) = ab exp(bτ ), respectively, including a polynomial function and an exponential function depending on the age. They are two commonly used age-dependent drift coefficient functions in the stochastic process models [5], [8], [13], [15]. In Model 3 (M3), a new state-dependent drift coefficient function is employed, i.e., functions of the μ3 (s(τ ), τ ) = as(τ )b . The diffusion coefficient √ three degradation models are σ(τ ) = c + dτ . To describe the unit-to-unit variability among different specimens [11], a random variance is introduced into the parameter a. Consequently, a is assumed to follow a normal distribution N (μa , σa2 ). There are five parameters in each model, i.e., μa , σa2 ,
LI et al.: REMAINING USEFUL LIFE PREDICTION BASED ON A GENERAL EXPRESSION OF STOCHASTIC PROCESS MODELS
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Fig. 5. RUL PDF of Case 12 using (a) M1 at 0.07 million, (b) M2 at 0.07 million, (c) M3 at 0.07 million, (d) M1 at 0.11 million, (e) M2 at 0.11 million, and (f) M3 at 0.11 million.
b, c, and d. These parameters are estimated using a commonly used algorithm, i.e., maximum likelihood estimation [11]–[13], [17]. There are 12 failed cases among the 21 test specimens. The RULs of the 12 failed cases are predicted one after another using the proposed method. When one specimen is selected as the prediction case, the remaining 20 specimens are treated as the training cases. The model parameters are estimated by inputting the observations of the training cases into the degradation models. To validate the advantage of the method based on DPS compared with the method based on SMC, a commonly used SMC-based method in [15] is also employed to predict the RULs of the 12 failed cases. The RULs of each case are predicted using three models, i.e., M1–M3. For the two age-dependent models M1 and M2, the RULs are calculated using the analytical solution, the DPSbased method and the SMC-based method, respectively. For the state-dependent model M3, since its RUL analytical solution is unable to be acquired, the RULs are just calculated using the DPS- and SMC-based methods. To visualize the superiority of the DPS-based method in describing the RUL PDF, the prediction results of Case 12 at two different ages are shown in Fig. 5. It is seen that the prediction results of the analytical solution and the DPS-based method for M1 and M2 almost coincide with each other. This phenomenon verifies that the DPS-based method has the same accuracy with the analytical solution for age-dependent models. This is because the DPS-based method calculates the RUL probability densities by inputting the simulated degradation trajectories into the analytical solution. Thus, it has the same results with the analytical
solution. Additionally, in the results of the SMC-based method, the discrete probability densities become sparse with the loss of the ground truth RUL. At 0.07 million cycles, the results of the SMC-based methods have enough discrete probability densities to describe the RUL PDF. At 0.11 million cycles, however, the discrete probability densities are too sparse to describe the RUL PDF. This is because the time interval of the SMC-base method keeps unchanged during the state transition process. With the loss of the ground truth RUL, more and more trajectories will fail at the same time. Compared with the SMC-based method, the DPS-based method provides more discrete values for the RUL PDF, and the number of the discrete values keeps stable with the loss of the RUL. The advantage of the DPS-based method is due to the self-adjustment of the time interval during the state transition process. When the RUL becomes shorter, the time interval will be shortened correspondingly. In conclusion, the DPS-based method provides the same prediction accuracy with the analytical solution and more detailed description for the RUL PDF than the SMC-based method. Taking Case 1 and Case 12 for example, the real-time RULs predicted by the DPS- and SMC-based methods are shown in Fig. 6. Since the analytical solution has the same prediction results with the DPS-based method, it is not presented here. It is seen that at the beginning of the lifetime, the results of Case 1 have large discrepancies. The results of Case 12 perform better than those of Case 1. As time goes on, all of them converge to the ground truth RUL gradually. And the 95% confidence intervals become narrow. It implies that the accuracy of the prediction results increases gradually as more measurements
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Fig. 6. RUL prediction results of (a) Case 1 using M1, (b) Case 1 using M2, (c) Case 1 using M3, (d) Case 12 using M1, (e) Case 12 using M2, and (f) Case 12 using M3.
Fig. 7.
Comparison of the CRA values among three models and two methods for 12 cases.
become available. It is also observed that the results of the DPSand SMC-based methods are close to each other. It is difficult to say which one is better. For quantitative comparison, the cumulative relative accuracy (CRA) [26] is utilized to measure the accuracy of the prediction results. A higher CRA value indicates a more accurate prediction result and the optimal value is 1. The CRA values of the 12 cases are shown in Fig. 7. From the CRA values it is also observed that the results of Case 12 are a little accurate than those of Case 1. The CRA values of DPS are a little higher than those of SMC for all of the 12 cases and the three models. It means that the DPS-based method acquires higher prediction accuracies than the SMCbased method, whichever model is used. To compare the performances of the three models for different cases, the winner among the six results of each case is highlighted with a pentagram. It is seen that the performances of the three models are distinct in different cases. In Cases 4 and 9,
M1 performs better than the other two models. M2 acquires the highest CRA values in Cases 1–3, 7, and 10. M3 performs best in the remaining five cases. From the comparison it is concluded that none of the three models is absolutely superior for all of the 12 cases. Models present various performances for different cases. This may be caused by the diversity of the degradation trends among different cases. V. CONCLUSION This paper derived the analytical solution of the RUL PDF from a general expression of stochastic process models, which is dependent on both the system age and the system state. A new RUL prediction method was proposed based on the general expression and the analytical solution. In this method, a series of degradation trajectories were first generated through DPS. Then, the RUL was estimated by inputting the simulated degradation
LI et al.: REMAINING USEFUL LIFE PREDICTION BASED ON A GENERAL EXPRESSION OF STOCHASTIC PROCESS MODELS
trajectories into the analytical solution. This method inherits the basic idea of the SMC and takes an advantage of the exactness of the analytical solution, thus performing better than the SMC-based method. In the experimental demonstrations, three models following the general expression were used to predict the RUL of the failed cases. It was concluded that different models present various superiorities for different cases. In the future research, more stochastic process models following the general expression will be constructed, and the optimal model selection strategies will be studied. It should be mentioned that the proposed approach just presents a general RUL prediction strategy from the perspective of theoretical analysis. There are still lots of work to do before applying it into industrial fields. For example, the interactions among multiple components and different fault patterns should be considered in the RUL prediction of a complex system. Additionally, the influence of time-varying operational conditions needs to be thought about in real applications.
APPENDIX Proof of Theorem 1 Let l = t − tk . Equation (1) is transformed as follows: s(t) − sk =
l+t k
μ(s(τ ), τ )dτ +
tk
=
l
0
l+t k
tk
σ(τ )dB(τ ) l 0
σ(τ + tk )dB(τ ).
Let s˜(l) = s(t) − sk , μ ˜(˜ s(τ ), τ ) = μ(s(τ + tk ), τ + tk ), and σ ˜ (τ ) = σ(τ + tk ). The former equation is further transformed into s˜(l) =
0
l
μ ˜(x(τ ), τ )dτ +
l
0
σ ˜ (τ )dB(τ ).
This completes the proof of Theorem 1.
B. Proof of Theorem 2 Through the translation transformation of Theorem 1, the stochastic process {s(t), t ≥ tk } has been changed into {˜ s(l), l ≥ 0}. In Lemma 1, let k1 = 1, k2 = k3 = 0, l0 = l1 =
l l2 = 0 and h(l) = exp(− 0 c2 (τ )dτ ) then
φ(˜ s, l) = h(l)
s(l + tk ) − sk − σ(l + lk )
l 0
According to the formula of the integration by parts, l 1 d (s(τ + tk ) − sk )h(τ )dτ σ(τ + tk ) 0 dτ l 1 = (s(τ + tk ) − sk )h(τ )d σ(τ + tk ) 0 l s (τ + tk )h(τ ) (s(l + tk ) − sk )h(τ ) − dτ = σ(τ + tk ) σ(τ + tk ) 0 l (s(τ + tk ) − sk )h(τ )c2 (τ ) + dτ . σ(τ + tk ) 0 Let c2 (τ ) = −σ (τ + tk )/σ(τ + tk ) − log(σ(tk ))/l. It is acs, l) is simplified as follows: quired that h(τ ) = σ(τ + tk ). ϕ(˜ l ϕ(˜ s, l) = s(l + tk ) − sk − μ(s(τ + tk ), τ + tk )dτ. 0
The standard transformation in (8) is changed to be ⎧ l ⎪ ⎪ ˜ ⎨B(l) = ϕ(˜ s, l) = s(l + tk )−sk − μ(s(τ + tk ), τ + tk )dτ 0 l . ⎪ 2 ⎪ ˜ ⎩l = ψ(l) = σ (τ + tk )dτ 0
Correspondingly, the FT is transformed into l ˜ l) = λ − sk − ϕ(λ, μ(s(τ + tk ), τ + tk )dτ . 0
μ(s(τ + tk ), τ + tk )dτ +
5717
According to Lemma 2, the PDF of the first time when the ˜ l) standard BM process {B(˜l), ˜l ≥ 0} crosses the threshold ϕ(λ, is denoted as ˜ l) dϕ(λ, ˜ l) ˜ l) ϕ(λ, 1 ϕ2 (λ,
− exp − fB (˜l) ∼ . = ψ(l) dψ(l) 2ψ(l) 2πψ(l) By inputting fB (˜l) into (14), the RUL PDF is derived
l λ − sk − 0 μ(s(τ + tk ), τ + tk )dτ ∼ f (l|sk ) =
l 2 0 σ (τ + tk )dτ μ(s(l + tk ), l + tk ) − σ 2 (l + tk ) σ 2 (l + tk ) × l 2π 0 σ 2 (τ + tk )dτ
l (λ − sk − 0 μ(s(τ + tk ), τ + tk )dτ )2 exp − .
l 2 0 σ 2 (τ + tk )dτ This completes the proof of Theorem 2.
μ(s(τ + tk ), τ + tk ) σ(τ + tk )
c2 (τ )(s(τ + tk ) − sk ) − σ(τ + tk ) 1 d (s(τ + tk ) − sk ) h(τ )dτ. + dτ σ(τ + tk )
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Naipeng Li received the B.S. degree in mechanical engineering from Shandong Agricultural University, Tai’an, China, in 2012. He is currently working toward the Ph.D. degree in mechanical engineering in the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, China. His research interests include machinery condition monitoring, intelligent fault diagnostics, and remaining useful life prediction of rotating machinery. Yaguo Lei (M’15) received the B.S. and Ph.D. degrees in mechanical engineering from Xi’an Jiaotong University, Xi’an, China, in 2002 and 2007, respectively. He is currently a Full Professor of mechanical engineering at Xi’an Jiaotong University. Prior to joining Xi’an Jiaotong University in 2010, he was a Postdoctoral Research Fellow with the University of Alberta, Edmonton, AB, Canada. He was also an Alexander von Humboldt Fellow with the University of Duisburg-Essen, Duisburg, Germany. He has pioneered many signal processing techniques, intelligent diagnosis methods, and remaining useful life prediction models for machinery. His research interests include machinery condition monitoring and fault diagnosis, mechanical signal processing, intelligent fault diagnostics, and remaining useful life prediction. Prof. Lei is a member of the Editorial Boards of more than ten journals, including Mechanical System and Signal Processing and Neural Computing and Applications. He is also a member of the American Society of Mechanical Engineers (ASME). Liang Guo received the B.S. and Ph.D. degrees in mechanical engineering from Southwest Jiaotong University, Chengdu, China, in 2011 and 2016, respectively. He is currently a Postdoctoral Researcher in the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, China. His current research interests include machinery condition monitoring, intelligent fault diagnostics, and remaining useful life prediction. Tao Yan received the B.S. degree in mechanical engineering from Central South University, Changsha, China, in 2016. He is currently working toward the Ph.D. degree in mechanical engineering in the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, China. His research interests include machinery condition monitoring, intelligent fault diagnostics, and remaining useful life prediction of rotating machinery. Jing Lin received the B.S., M.S., and Ph.D. degrees from Xi’an Jiaotong University, Xi’an, China, in 1993, 1996, and 1999, respectively, all in mechanical engineering. He is currently a Professor in the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University. From July 2001 to August 2003, he was a Postdoctoral Fellow with the University of Alberta, Edmonton, AB, Canada, and a Research Associate with the University of Wisconsin–Milwaukee, Milwaukee, WI, USA. From September 2003 to December 2008, he was a Research Scientist in the Institute of Acoustics, Chinese Academy of Sciences, Beijing, China, under the sponsorship of the Hundred Talents Program. His current research directions are in mechanical system reliability, fault diagnosis, and wavelet analysis. Prof. Lin received the National Science Fund for Distinguished Young Scholars Award in 2011.
