1.Introduction

Recently,prognostics and health management(PHM)[1]has emerged as a promising technology with considerable benefits for various degrading systems. One of the essential objectives of PHMis the health assessment of systems subject to condition monitoring,thereby helping practitioners to optimize spare part requirements, develop optimal maintenance policies,and maximize the availability of degrading systems in critical applications[2].

To guarantee safety requirements,advanced sensors are applied to degrading systems to monitor their real time performance indicators.For an Engineering system subject to condition monitoring,a soft failure will be encountered if the implemented sensors indicate that the degradation signal exceeds the predefined critical level.It is also possible that the system functionality stops abruptly due to the physical failure,which is termed as the hard failure[3].In such cases,the degrading systems are considered to be subject to the competing risks,i.e.,a soft failure and a hard failure.Literature review indicates that competing risks[4]have received modest attention.Zhao et al.[5]investigated the competing risks problem where a Brownian motion process was applied to model the soft failure and a Weibull distribution was used to describe the occurrence of thehard failure.In[6],Bocchetti et al.put forward a method to characterize the competing risks for cylinder liners applied in Sulzer RTA 58 engines,where the wear degradation failure,i.e.,the soft failure was driven by a stochastic process,and the thermal cracking failure,i.e.,the hard failure was characterized by a Weibull distribution.For a long-storage-short-run system,Guo et al.[7]studied a competing risks model where the soft failure was driven by a Gamma process with a predefined critical level and the hard failure was described by a Weibull distribution.Considering both the soft and the hard failure modes in practical engineering systems,Wang and Gao[8]established a reliability evaluation method which was verified for an aircraft engine.The degradation process was characterized by a non-homogeneous Gamma process and the time to hard failure was described by a Weibull distribution.

Note that one of the common features of the aforementioned works is that there is no dependent relationship between the soft and the hard failures.However,in most practical applications,the occurrence of the hard failure depends not only on the operational time,but it is also more likely to occur when the degradation level is higher[9,10],which indicates that the soft failure and the hard failure are dependent.Thus,the independence assumption of the soft and hard failure modes may cause underestimation or overestimation of the system health condition.Recently,there has been a growing interest in modeling dependent competing risks of various failure modes for system health evaluation.Guo et al.[11]proposed a copula function method for a system subject to dependent competing risks driven by two degradation processes and random shocks.In[12],Huynh et al.applied a Gamma process to characterize the system degradation failure.To describe the dependent relationship,the degradation level was assumed to be a factor in the shock failure modeling.Other models considering the dependent failure modes can also be found in[13,14].For the competing risks of the dependent soft and hard failures, the model formulation and evaluation are quite challenging and have not been well addressed in the literature.Thus,much work should be carried out to facilitate the health assessment for degrading systems subject to the competing risks of dependent soft and hard failures.

Motivated by the lack of research in this area,a health evaluation method for degrading systems subject to competing risks where a dependent relationship exists between the soft and hard failures is proposed.The degrading system is considered to be in the failure state when either the degradation process exceeds the predefined critical level or the system’s functionality stops abruptly.The degradation process is characterized by a non-stationary Gamma process[15]considering the time-varying deterioration rate encountered in real systems.The dependent relationship between the soft and hard failures is modeled by incorporating the degradation process into a Cox’s proportional hazards(PHs)model as a covariate process to characterize the failure rate of time to hard failure.A transition probability-matrix method is developed which enables the analysis of the proposed model and derivation of the explicit formulas for the health characteristics.The calculation of the health characteristics can then be performed with simple operations on the transition probability matrix,which is quite computationally efficient and can support the online health evaluation.

This paper presents a comprehensive health assessment method for degrading systems subject to condition monitoring and dependent competing risks,which distinguishes it significantly from our preliminary work[16]presented at 2016 Prognostics and System Health Management Conference in two directions.First,the methodology development is more systematic,and theoretical details for the derivation and calculation of the health characteristics are presented.Second,the applications for the online health assessment and the implications for real applications,which were not included in[16],are illustrated and discussed.

The remainder of this paper is organized as follows.In Section 2,we present the modeling method of the dependent soft and hard failures as well as a transition probability-matrix method which enables the model analysis and further development.Section 3 presents the explicit formulas for the related health characteristics.To show the effectiveness of the proposed method,a numerical study is carried out in Section 4.Finally,the conclusions are given in Section 5.

2.Degradation and failure process modeling

Before presenting the details of the proposed method, several assumptions adopted in this paper can be summarized as follows:

通过投加一定比例的PAM和PAC（聚合氯化铝），进一步降低预曝气池中的固体悬浮物浓度。污水先流经PAC絮凝桶进行混合，产生细小絮状体，再流入PAM絮凝桶产生较大絮状体，从而能在初次絮凝沉淀池内较快地沉淀，降低固体悬浮物和部分COD。

(i)When the degradation signal reaches the predefined critical level,the soft failure occurs,which is observable.

对于小学较低年级的英语教学，在与学生通过唱歌、做游戏等环节学习英语时，需要特别明确教学目标，让学生明白游戏是为了练习哪一部分内容，在游戏过程中有所侧重。对于小学高年级学生，可在教学中适当引导学生，告知其课堂上的学习知识点与初中的哪一部分有关，以便其在学习到新知识时可以做出联想。

(ii)The degrading system may stop its functionality abruptly,i.e.,it is subject to hard failure.

目前，国内普遍采用预处理+生化处理+膜深度处理工艺处理垃圾渗沥液。其中，生化处理一般采用硝化-反硝化+MBR工艺，以保证脱氨效果。生物处理反硝化单元中主要为异养菌，异养菌脱氮过程主要是针对硝化过程中产生的硝氮进行反硝化脱氮。整个过程中典型的硝化反应过程如公式（1）所示，氨氮在自养菌作用下硝化形成硝氮，硝氮在异养菌作用下反硝化形成氮气，如公式（2）所示，从而完成脱氮。反硝化过程需要消耗有机物［1］，因此，需要保证废水中有足够的有机碳源。

关于第二语言习得中的母语迁移，前人还是做了很多的实证研究，但也存在有些学者只是简单地总结前人的研究，自己没有真正去做实证研究。另外学者们关于正迁移做的实证研究比较多，负迁移做的实证研究比较少，可能与正迁移在二语习得中逐渐受到重视有关，未来学者们可以在负迁移领域多做些实证研究。

Practically,these assumptions can be reasonably justified,and an example is a degrading system equipped with expendable devices such as fuses or canaries[17].When the degradation signal exceeds the predefined critical level,the corresponding protecting actions will be activated and the system is able to notify the user about its health state automatically.In this case,the system runs into the soft failure state and its residual life expires.On the other hand,the degrading system,especially the critical one,is not allowed to operate any longer when the operation time is larger than the maximum allowed time limit.

沉管法目前已成为我国修建跨江穿海隧道应用最广泛的方法之一。然而，沉管隧道处在复杂的海洋环境中，其稳定性问题比陆地结构复杂得多。据国土资源部历年发布的海洋灾害统计显示，因热带风暴在东部沿海登陆，我国每年约有40 km防波堤遭到毁坏。风暴后常有海底输油、气管线断裂事故发生[1]。作为重要的近海工程结构，沉管隧道在波浪作用下的稳定性是一个不容忽视的问题。

2.1 Model formulation

Assume that a degrading engineering system is subject to condition monitoring.For each system,there exists some observable degradation process related to its health condition that can be monitored along with the system operation.In most cases,the degradation process is monotonically increasing or decreasing and it is not self-recoverable without any interventions or maintenance,e.g.,the crack growth of bearings.Considering the monotonicity of the degradation process,we choose the Gamma process[18],which is considered as an accumulation of a sequence of small positive increments,to model the system degradation.Suppose that the degradation process is characterized by a Gamma process.Thus,the probability density function(PDF)of the degradation process can be expressed as

For a critical system applied in military and space areas,there exists a maximum allowed age limit TAthat the operation time of the system is not expected to exceed due to safety matters.For the second part of the proposed discretization technique,the monitoring time axis is discretized using an equidistant time interval of length δ,δ＞ 0 and thereby the maximum allowed age limit,i.e.,the whole life cycle of the system can be represented as TA=Nδ(N ≥1).In this way,all the monitoring activities and health assessment will be conducted in the confined time interval[0,Nδ].The monitoring activities can only be performed at integer multiples of tw=wδ,0≤w＜N,w∈N,and the monitoring mechanism will become more flexible when δ goes to zero.Thus,the discrete counterpart of the transition probability(6)can be developed as

Considering the evolution of the monitoring time,the transitions can only occur progressively with the increment of δ time and thus other elemental blocks,for example,Pw1,w2(w2/=w1+1),will be 0.It is also worth noting that the system will not continue any longer when the operation time equals its whole life cycle time TA=Nδ,thus,Φ((N-1)δ)=0.In this way,the system failure probability due to either the soft failure or the hard failure can be included in the last column of P.Also,note that the whole life cycle transition probability matrix is a sparse matrix with only a series of Φ(wδ)above its main diagonal.This appealing feature will essentially accelerate the multiplication and inversion of the matrix.In the following development,it can be seen that these are the main operations involved in the calculation of health characteristics,which are computationally efficient and can be implemented in practical applications.

where Γ(a,x)is the incomplete Gamma function.

To combine the operating age and the degradation information in the modeling of the hard failure,a Cox’s PH model is selected to describe the failure rate of the time to hard failure.Generally,the Cox’s PH model is composed of two multiplicative factors,i.e.,a function h0(t)which describes the baseline hazard at time t,and a positive function ψ(X(t);β)which is applied to incorporate the effect of covariates.Thus,the hazard function can be expressed as

where β denotes the regression coefficient and X(t)is the covariate process which can be either an external environmental variable or an internal diagnostic variable.In this paper,to show the effect of the degradation on the system failure,the degradation process X(t)is used in the Cox’s PH model as a time-varying covariate process.Using this model,the operating age and the degradation information can be integrated to describe the hard failure mechanism.In this way,the dependent relationship between the soft and hard failures is modeled by incorporating the degradation process in the Cox’s PH model.Thus,the PDF of the hard failure time This given by the following formula:

It can be seen that the probability distribution of the competing failure time for the soft and hard failures can be explicitly expressed by(2)and(4),respectively.The reliability function for a system is then given by

Nevertheless,considering the non-stationary characteristic of the degradation process and dependent relationship between failure modes,the straightforward derivation and evaluation of(5)are quite challenging,or even impossible.Moreover,the conditional health characteristics given the monitoring degradation state and the operation age, such as the conditional reliability function(RF),conditional mean residual life (MRL) [19], and the corresponding confidence interval which are essential for the online health assessment,are also quite challenging to develop and evaluate.To overcome these challenges,a method is proposed to enable the derivation and calculation of the health characteristics.

2.2 Transition-probability-matrix method

In[20],Brook and Evans proposed a method based on the Markov chain to calculate the average run length(ARL)in quality control.Considering the similarities between ARL and MRL,a method inspired by[20–22]is proposed to enable the above model evaluation and thereby calculate health characteristics for degrading systems subject to dependent soft and hard failures.

To consider both the soft and hard failures simultaneously,we propose a joint process V(t)=(X(t),Tc＞t)(t∈R+)to integrate the degradation process X(t)and the failure time Tc.For any given time instants 0=t0＜t1＜ t2＜ ···＜ tn-1＜ tn＜ ∞,the transition probability of the joint process is able to be expressed as

It can be seen that the joint process V(t)possesses the Markov property.With the joint degradation process,both the degradation behavior and failure information including the soft failure time Tsand the hard failure time Th can be integrated into(6)and their evaluations can be performed together.Note that both the degradation process and the monitoring time axis in(6)are continuous and thus the state space of the joint process is in finite.To make the model evaluation feasible,the state space of the joint process needs to be finite and a twofold discretization technique is proposed.

不典型的脑型脂肪栓塞临床诊断较困难。下肢长骨骨折的患者，尤其是多发骨折，早期没有出现明显症状，出现不明原因的血红蛋白持续降低，要怀疑脂肪栓塞综合征发生的可能。出现难以解释的脑症状时，即使没有典型的肺和皮肤表现，也应怀疑脂肪栓塞。如有条件，及时行颅脑MRI检查，排除其他疾病，明确诊断。

First,we discretize the continuous degradation process X(t)into several discrete states.Particularly,it is supposed that such a continuous degradation process is approximated and represented as a Markovchain whose state space Ω consists of M+1 states.Denote the state spacewhere S0represents the“good as new”state and SM-1represents the worst operation state.When the system deteriorates into the absorbing state SM,it is considered as encountering a soft failure and the system must be stopped for maintenance or replacement.Because the degradation process is continuous,the probability that the degradation process will be in the same discrete degradation state in a certain time period is corresponding to the probability that the degradation path increment does not exceed a small value.For the discretization of the degradation process,the width of the discretization interval is determined as a=2Y/(2M-1),and Xi=i·a is denoted as the degradation level of stateThereby X(t)is supposed to be in state i when the degradation level stays in the interval[lbi,ubi],wherelbiandubiare the lower and upper bounds for the degradation state i.They can be expressed as

With such a discretization,the continuous degradation process is transferred to its discrete counterpart and the number of the degradation states is finite.As an illustration,Fig.1 shows the failure mechanisms of the degrading system with the discretization of the degradation process.In Fig.1,it can be seen that the system is able to run into either the soft or the hard failure from any discrete degradation states.

Fig.1 Failure mechanisms of the degrading system with degradation process discretization

其实现功能的流程是：由电源模块输出供电给设计的整个电路，单片机最小系统作为控制模块，利用DDS产生三种波形信号，由控制部分实现波形的输出以及调节频率的大小和电压幅度的大小。控制部分的外围电路有按键电路，用来辅助控制DDS产生的波形，产生的信号经过放大电路放大输出。显示电路由3个LED组成，用来表示不同类型的波形输出。

作者贡献声明 杨帆：参与选题，酝酿和设计实验，撰写论文，统计分析，对编辑部的意见进行核修。张红：参与选题，酝酿和设计实验，起草论文关键性理论。黄磊、康杨、吴琼：参与选题，酝酿和设计实验。杜春宇：酝酿和设计实验，对编辑部的意见进行核修。胡琦：参与选题，酝酿和设计实验，起草论文关键性理论，对学术问题进行解答

where Xi,Xjdenote the degradation levels for degradation states i,j,respectively,0≤w1＜w2＜N.By discretizing the degradation path and the monitoring time axis,the number of the states of the joint process V(t)becomes finite,which is the key feature to enable the model evaluation.

As the fundamental element of the probability behavior of the joint process,we will focus on the transition probability behavior in a time interval of length δ,i.e.,the one step transition probability matrixwhose element can be obtained and derived as

In Fig.4,it clearly shows the conditional RF for various monitoring time,where the monitoring time t=0.1,1.6,3.1,respectively,given the monitoring state 0.As can be seen,the conditional RF gets steeper along the monitoring time axis.These trends can be attributed to the decrease of the distance between the monitoring time and the real failure time.

is distributed as ,where the shape parameter is

Then,the first part of(7)can be written as

where 0≤i≤j≤M-1.The lower and the upper bounds are expressed as

For the second part in(7),a Cox’s PH model is applied to characterize the hazard rate of the time to hard failure.Because δ is considerably smaller than the maximum allowed time limit TA,it is reasonable to assume that the covariate process value is constant and equal to(Xi+Xj)/2 when a transition from degradation state i to j occurs.Thus,the element of the one-step transition probability matrix can be further obtained as(8).Using(8),the one-step transition probability matrix Φ(wδ)can be readily obtained.It should be noted,however,that due to the non-stationary behavior of the degradation process and the aging factor in the Cox’s PH model,the one step transition probability matrix is not constant and the transition elements will depend on time.To eliminate this non-stationary behavior and provide a unified framework for further model evaluation,a whole life cycle transition probability matrix P is proposed.It consists of a series of Φ(wδ)and is organized as shown in(11),

where eMis a unit vector of M elements,and 0 is the zero matrix of M×M.Denote the elemental block of the whole life cycle transition probability matrix as Pw,w+1=Φ(wδ).It can be seen that Pw,w+1represents the probabilities that the joint process successfully transits from the states at time instant wδ to the states at(w+1)δ.

where k(t)＞ 0 is the shape parameter,θ＞ 0 is the scale parameter,and Γ(k(t))is the Gamma function.It should be noted that the shape parameterk(t)depends on t,which suggests that the temporal variability of the degradation rates is able to be easily incorporated into the degradation process modeling.Denote Tsas the soft failure time to represent the time instant when X(t)exceeds the critical levelY.Based on the properties of the Gamma process,the probability distribution of the soft failure time[18]can be written as

Also,note that the failure probability of the system subject to dependent competing risks,i.e.,the probability that the joint process transits to either the soft failure or hard failure,is in the last column vector of the whole life cycle transition probability matrix.Thus,the probabilistic behavior of the joint process V(t)can be described by the whole life cycle transition probability matrix P.

3.Health characteristics calculation

Before proceeding to the derivation of the formulas for the health characteristics,it is necessary to analyze the structure and the properties of the matrix P in(11).Note that the last column of P describes the probability that the joint process transits to either soft or hard failures from state i at tw=wδ(w=0,1,...,N-1).Thus,the whole life cycle transition probability matrix P is divided into two parts,i.e.,the degradation part B and the failure part PF,

where B is an MN × MN matrix and is a zero vector of MN elements.PF=(I-B)eMNis a vector of MN elements,I is a unit matrix and eMNis a unit vector of MN elements.Based on this partitioned form,the related health characteristics are able to be calculated using the basic operations on the whole life cycle transition probability matrix.It should be noted that the evolution of the joint process V(t)can be presented with the multiplication of the whole life cycle matrix P.Considering the properties of the partitioned matrix,it follows immediately from(12)

Next,a new evaluation method is proposed to calculate health characteristics given the operation age and the degradation state information.The variable RL is defined as the residual life of a degrading system.From(13),it can be seen that(I-Br)eMNrepresents the probability that the degradation system will fail in the next rδ time units,which is equivalent to the cumulative distribution of RL given the degradation state X(wδ)=Xiat the monitoring time tw=wδ.Thus,we get

wherer = 1,2...,N -w,and πi(wδ) =(0,...,0,1,0,...,0)is the initial probability which is a 1×MN matrix with 1 for the(wM+i)th element and zeros elsewhere,and it can be obtained given the monitoring time constant twand the degradation state i.

Letbe the conditional probability mass function(PMF)of RL which is conditioned on both the monitoring time tw=wδ and the corresponding degradation state i.Based on(14),the conditional PMF of RL can be developed as

The residual life is a random variable and its expected value given the operation time and the corresponding degradation state,i.e.,the conditional MRL,is often applied as a point estimate for such a random variable.This health characteristic is essential to system optimization and maintenance policy making.Thus,it is appealing to derive the formula for the conditional MRL.Based on(15),the conditional MRL at monitoring time tw=wδ given that the corresponding degradation state is i can be obtained as

⑤要增加对违反水土保持相关法律法规行为的惩罚力度，增加建设和施工单位的违法违规成本。目前存在着违反水土保持法律法规成本过低的情况，很多项目只是在验收前付少部分费用补做水土保持监测报告等材料，对其挪用、占用了较多的水土保持经费却没有给予适当的处罚。所以，要进一步加强对违反水土保持相关法律法规行为的打击力度，加大生产建设项目违规成本。

With the proposed transition-probability-matrix method,the health characteristics can be calculated in a feasible and unified framework.Apparently,all the formulas for the health characteristics can be obtained with the simple manipulation of the whole life cycle transition probability matrix P.While P is a relatively large matrix especially when the time interval of length δ is small,however,this matrix is sparse and has non-zero block elements only along the main diagonal.

新的严重的ADR/ADE有83例（23.18%）。在严重类别上，导致其他重要医学事件211例，占57.81%，比例最高（表6）。严重ADR/ADE结果中，好转的279例（77.93%），痊愈42例（11.73%），不详21例（5.87%），未好转13例（3.63%），有后遗症2例（0.56%），死亡1例（0.28%）。

This feature essentially reduces the computational efforts such as the matrix multiplication and inversion.Thus,all the health characteristics evaluations can be obtained using a regular computational platform in an acceptable time.On the other hand,the practitioner is able to arbitrarily reduce the time interval of length δ so as to improve the accuracy of the evaluations with a powerful computational platform.Next,a numerical study will be conducted to verify and illustrate the proposed health evaluation method.

4.Numerical study

With the degradation histories and failure information of practical engineering systems,the likelihood function can be derived and the parameters in the proposed model can be calculated using the method of maximum likelihood(MML).Considering a practical degrading system,a numerical study using assumed parameters is presented to illustrate and verify the proposed method.

For the degradation process,the shape parameter of the non-stationary Gamma process is assumed to be k(t)=0.82·t1.34,and the scale parameter is assumed to be 1.92.Without loss of generality,such a degrading system is assumed to be new without any degradation at the beginning,i.e.,X(t0)=0,t0=0.If the degradation signal surpasses the critical level Y =4,the system enters the soft failure state and thus the expendable devices connected to the system will notify users.For the discretization technique,the degradation process is discretized into M=10 states,where state 0 is the initial state of the new or renewed system,and state 9 is the worst operation state.For the monitoring time axis discretization,the time interval of length δ is selected as 0.1.For the baseline function of the Cox’s PH model,it is assumed to be Weibull as

where the shape parameter α and the scale parameter β are assumed to be 4.31,2.45,respectively.For illustrative purposes,the degradation process X(t)is assumed to be perfectly observable and the regression coefficient β=0.15,and the link function is assumed to have an exponential form ψ(X(t);β)=exp(0.15 ·X(t)).Under this assumption,the conditional MRL given the corresponding degradation state 0,i.e.,is calculated as 3.82,thus the whole life cycle of the system is reasonably selected to be more than two times of Tmean,i.e.,TA=8.00.

总体来讲，培训后各项素质能力均较培训前有所提高，差异均有统计学意义，随着培训时间的延长呈现2015级研究生总体评分≥2016级研究生总体评分＞2017级研究生总体评分的正相关趋势。其中医患沟通、采集病史、体格检查、诊断鉴别诊断的临床思维、辅助报告阅读、门诊急诊中处理常见病、抢救危重病患者、常见病多发病的治疗原则等与实际操作技能相关的能力提升较多，差异均有统计学差异。但是医疗法律知识、处理医患纠纷和维护医院医生权益较其他项评分较低。在病例书写、查房、医德医风、团队合作、责任感、保存医疗档案这六个方面，各年级间相互比较差异无统计学意义。

Considering the excellent performance of the matrix calculations,Matlab is used to implement the proposed method.To begin with,the conditional PMF of RL,given the monitoring time t5=0.5 and the corresponding degradation state 0,can be calculated using(15)and the results are depicted in Fig.2.

Fig.2 Conditional PMF at monitoring time t=0.5 with degradation state 0

Next,the conditional RF is calculated using(16).Table 1 shows the results for the conditional RF given monitoring time t5=0.5 for several degradation states.In Fig.3,it depicts the conditional RF given the degradation states i=0,3,6,9.As can be seen,the shape of the conditional RF is highly affected by the corresponding degradation state.This result is consistent with the observations in real applications that the competing failures are more likely to occur when the degradation level is close to the pre-defined critical level.

Table 1 Conditional RF at monitoring time t=0.5 given various degradation states

Monitoring Degradation state time 0 3 6 9 0.6 0.992 6 0.985 3 0.964 3 0.846 5 1.1 0.915 5 0.850 3 0.706 7 0.315 5 1.6 0.759 6 0.627 0 0.410 9 0.094 9 2.1 0.546 6 0.383 6 0.189 3 0.023 7 2.6 0.333 0 0.192 4 0.069 9 0.005 0 3.1 0.169 7 0.079 0 0.020 9 0.000 9

Fig.3 Conditional RF at monitoring time t=0.5 given various degradation states

It can be seen that the first part in(7)represents the probability that the joint process successfully transits from degradation state i to j during the time interval of length δ without any failure events.Considering the property of the independent increments[18,23]of the Gamma process,the increment of the degradation process in the time interval[wδ,(w+1)δ],i.e.,

(iii)The degrading system will stop working when the operating time reaches a maximum allowable age limit,i.e.,the end of its whole life cycle.

两幅图画，两个屏幕，顺势而接，隔着两千五百年，依然天籁般洽和。可是现在田园将芜！田园已芜！采莲的男人走了，其他的男人也走了，男耕女织就这样颠覆错位了：

Fig.4 Conditional RF for several monitoring time instants with initial degradation state 0

To provide an illustration of the online assessment,the degradation history(t,X(t))of a practical degrading system is supposed to be(0,0),(1.5,2),(3,4),(4.5,7),(6,9).The conditional PMF of RL can be calculated by(15)and conditional MRL can be calculated by(17).Fig.5 shows the online assessment of the health characteristics when the monitoring time increases.As can be seen,the PDF gets narrower and the estimated conditional MRL is decreasing with the monitoring time increasing,which is consistent with the operations of real engineering systems.

Fig.5 Online assessment of the degrading system subject to condition monitoring

In practical applications,the practitioners are interested not only in the point estimate but also in the confidence interval.Based on(15),the corresponding 95%confidence interval can be calculated.Fig.6 shows the conditional MRL and the corresponding 95%confidence interval of RL with the monitored information.As can be seen,the confidence interval gets shorter when the monitoring time increases.For instance,given that the degradation state 4 is monitored at time 3.00,the conditional MRL is calculated as 0.769 5,and the corresponding95%confidence interval is calculated as(0,1.9).

Fig.6 Conditional MRL with the corresponding 95%confidence interval

It can be seen from this numerical analysis that the health characteristics of the system subject to dependent competing risks can be feasibly evaluated with the proposed method,and the obtained explicit formulas are computationally effective for real applications.

5.Conclusions

In this paper,we propose a new health evaluation method for degrading systems subject to competing risks of dependent soft and hard failures.A non-stationary Gamma process is applied to model the system degradation.As a time-varying covariate process,this degradation process is then incorporated into the Cox’s PH model to characterize the hazard rate of the time to hard failure.Discretization techniques are applied to both the degradation path and the monitoring time axis in order to facilitate the evaluation of the characteristics of the proposed model.The explicit formulas for the health characteristics have been derived based on the proposed transition-probability-matrix method.The calculations of these health characteristics can be performed with small computational efforts and the method is applicable to obtain the online health evaluation.The procedure and the effectiveness of the proposed method have been illustrated and verified by performing a numerical study.These illustrative results have shown that our method is able to be implemented effectively for a degrading system subject to condition monitoring so as to provide a basis for the optimal maintenance policy making and to prevent the possible catastrophic failures and their consequences.

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