APrognosticApproachforSystemsSubjecttoWiener
DegradationProcessZithCumulative-WypeRandomShocks
ZhengxinZhang,ChanghuaHu,XiaoshengSi,Jian[unZhang,QuanShi
DepartmentofAutomationEngineering,Xi’anResearchInstituteofHigh-Tech,Xi’an,710025
E-mail:zhangzhengxin13@gmail.com
Abstract:Conditionmonitoringdatahavebeenwidelyusedtoevaluatethehealthstateandreliability,aswellasestimatethe
remainingusefullife(RUL)fordegradingsystems.AmongvariousdegradationmodelingandRULestimatingmethods,Wiener
processbasedmodelsL
Vrecognizedbybothscholarsandengineersastheoneofthemosteffecttools,andthusbecomesvery
popular nowadays. In this paper, a prognostic approach is developed for degrading systems whose performance evolution is
captureGbyanintegrationofWienerprocessandcumulative-typeofrandomshocksdepictedbyacompoundPoissonpro-cess.
Under the concept of first hitting time, an approximate lifetime distribution in analytical form, which greatly reduces the
computingtimewhileprovinganadequatelyaccurateresult,hasbeenformulated. Theparameterestimationframeworkbased
expectation maximization (EM) algorithm has been derived. Simulations and caseV study of the degradation of Li-ion
batteriesareexecutedtoillustrateandvalidatetheproposedmethod.Theresultsdemonstratethattheapproachinthispapercan
notonlyhandlethepositiveshocksbutalsoprocessthenegativeshocks,whichoutperformsmostoftheexistingmodels.
Key Words: Reliability, Wiener Process, Degradation-shock Models, Compound Poisson Process, RUL Estimation
1 Introduction
The performance of an industrial system usually decreas-
es with its service time, due to the interaction of its inner
deterioration in materials performance with its working en-
vironment. The system fails once it cannot provide an an-
ticipant performance, and the unexpected failure of system
will cause economical losses and environmental damage. If
the lifetime or remaining useful lifetime (RUL) can be esti-
mated before failure, sequential managerial activities will be
arranged, to avoid disastrous failure, cut down costs and pro-
tect the environment. However, it is costly in both time and
money to obtain adequate lifetime data to evaluate the relia-
bility and estimate the lifetime and RUL for these degrading
systems, especially the ones with extremely long life or high
price. Fortunately, the development of condition monitor-
ing (CM) techniques makes it possible to collect sufficient
data revealing system’s health state, which further leads to
the boom stage of degradation modeling based approaches
[1, 2].
Indeed, there are various type of models for degradation
data analysis and RUL estimation, such as the failure-of-
physics based models, the data-driven models, the hybrid
methods, etc [3]. Since the degradation mechanism of a
system is often quite complicated, data-driven models draw
much for favor. The stochastic dynamics involved in the
degradation path, which usually caused by the uncertainty
during the degradation process, expedites the widest utiliza-
tion of stochastic process based models among other data-
driven methods. Particularly, Markovian chain [4], Gamma
process [5], Inverse Gaussian process[6], and Wiener pro-
cess [8] are all applied to degradation modeling. We con-
centrates on the Wiener process based degradation models,
because of its capacity to capture non-monotonous degrada-
tion processes which are frequently encountered in industry.
Besides, the degradation process may exhibit some ran-
dom shocks which suddenly increase or decrease the degra-
This work is supported by National Natural Science Foundation (NNS-
F) of China under Grants 61603398,61374126, 61104223, and 61573365.
dation level of the system. The underlying causes of the
random shocks are various, i.e., shocks in the environmen-
t, repair activities with ignorable time, recovery of material
nature, can all introduce shocks into the degradation pro-
cess. According to the work of in [7], there are general-
ly fine types of random shocks mentioned in the literature,
including the extreme shock, cumulative shock, 𝑚-shock,
run shock, and 𝛿 shock. Extreme shocks cause system fail-
ure when the size of any shock is beyond a specific critical
threshold. Systems experience cumulative shocks will fail
when the cumulated damage due to shocks exceeds a fail-
ure threshold. 𝑚-shocks are defined as the 𝑚 shocks whose
amplitude is larger than a threshold and will cause the fail-
ure of system, while run shock means a run of 𝑛 consec-
utive shock that a greater than a threshold. 𝛿 shocks are
two sequential shocks whose inter-arrival time is less than
a threshold 𝛿. Among all these kinds of random shocks,
cumulative-type kind of random shocks are most frequently-
encountered in industrial practice. Therefore, we carried out
our study based on cumulative-type of random shocks. Al-
though, some researchers have reported their work regard-
ing the degradation-shock prognostic model [10, 11], there
are still some issues deserving further address. At the very
first, compared to most degradation-shock models construct-
ed in the framework of general path models without consid-
ering the temporal dynamics of the degradation paths, ran-
dom shocks have seldom been incorporated into stochas-
tic process model, especially Wiener process based model.
Then, the computation of lifetime and RUL probability den-
sity functions (PDFs) in the existing method requires trun-
cation and is time costly. Additionally, the parameter es-
timation procedure for degradation-shock models is greatly
desired to make the degradation-shock model more practical.
Motivated by the aforementioned discussions, we pro-
posed a prognostic approach for degrading systems whose
performance evolution is described by a Wiener process with
cumulative-type of random-shocks. The lifetime and RUL
distributions are derived under the concept of the first hitting
time (FHT). Also, an approximate to the derived lifetime and
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