1.Introduction

Prognostics and health management(PHM)has been widely studied during the last several decades.Accurate remaining useful life(RUL)prediction results can provide effective basis for formulating maintenance schedules,and then increase systems’reliability and decrease their risk[1].As a consequence,RUL prediction has been considered as acrucial procedure in PHM[2,3],which is generally defined as the probability density function(PDF)of the first hitting time(FHT)[4,5].

From the existing literature,data driven RUL prediction approaches are usually divided into two classes:failure time data based approaches[6,7]and performance degradation data based approaches[8,9].Note that,for most practical equipment,the performance degradation data are easier to be achieved than the failure time data.In addition,the performance degradation data can reveal the detailed degradation mechanism of a specific system.Therefore,the performance degradation data based approaches are more desired and credible than failure time data based approaches.Many efforts have been made to achieve effective RUL prediction by degradation modeling.Some typical degradation models based on the Wiener process,the Gamma process or random coefficient regression(RCR)are proposed.However,most of the reported methods neglect the interdependency between degradations or only consider one degradation[10–12].In our previous work,an RUL prediction method for multi-degradation systems with public noise has been proposed[13].However,this method is based on an assumption that all the degradations are linear.In addition,the RUL of the whole system is not given.

However,in practical industry processes,equipment is generally composed by several parts or components,and there may exist correlations among the degradations of the system.Therefore,neglecting the correlations among degradations will result in inefficient prediction.Therefore,the correlations among degradations should be taken into account.As all the parts or components of a system work in the same circumstance and affect each other,there must exist public elements in their degradations.In this paper,the noise which impacts every degradation of the system is called the public noise.Here our main interest is to estimate the RUL of each single degradation and the whole system,respectively.

This paper is structured as follows.Section 2 introduces a nonlinear multi-degradation model.An estimation method is presented in Section 3 to estimate the degradation states as well as unknown parameters in the degradation model.In Section 4,we derive the PDF of FHT for each degradation and the whole system.In Section 5,a practical case is provided to prove the proposed method.Finally,conclusions of this paper are given in Section 6.

2.Multi-degradation model

To predict the RUL for a class of nonlinear multi-degradation systems, a method is presented. In the real industrial processes, systems are usually composed by several parts or components,and these parts or components are working in the same environment,thus the degradations of these parts or components will be influenced by common factors.To describe such a phenomenon in degradations,a multi-degradation model with public noise is proposed.To identify the degradation states and the unknown parameters,an iterative estimation method is proposed by using the Kalman filter and the expectation maximization(EM)algorithm.Next,with known thresholds,the RUL of each degradation can be predicted by using the FHT.In addition,the RUL of the whole system can be obtained by a Copula function.Finally,a practical case is used to demonstrate the proposed method.To characterize the degradation of a multi-degradation system influenced by public noise,we establish a nonlinear multidegradation model by adding a public Brownian motion to the traditional Wiener process with nonlinear drift. In order to obtain the RUL prediction, the multi-degradation system needs to satisfy the following assumptions:

临床药师主导的慢性病管理在提高老年高血压患者对药物了解程度方面的作用 ………………………… 武丹威等（23）：3251

在现有体制下，政府是资源的分配者和扶持政策的制定者，对中高职衔接贯通培养起到宏观性的引导作用。主要体现在以下几个方面：一是作为管理者，通过制度建设，吸引、鼓励企业更多地参与到校企合作；二是对中高职衔接贯通给予政策支持，构建优质的外部环境；三是对高职院校在开展中高职衔接贯通工作中，给予经费支持，保障中高职衔接的顺利进行。

宋代园林和明清时期园林在寻找象征与诗意方面呈现的是当时园林文化的自觉自为，园林艺术都呈现出一种独特的文化内涵。从园内的物质内容到精神功能，从园林的立意布局到园内景区的主题分配，从景物本身的表义内涵到景物之间的符号关系都有了深刻独到的见解和相应的表现原则。[2]这一切都是中国园林艺术的文化内涵之所在。

To avoid casualties caused by burning through,blast furnaces are usually stopped in a conservative manner.Thus the total life cycle data of blast furnace walls are hard to acquire.Here we only consider the FHT of the degradation passing a set threshold.The thresholds are supposed to be 175° for degradation 1,130° for degradation 2,and 190°for degradation 3.Then,the failure time of the three degradations is the 2 848th hour,the 2 784th hour and the 2 776th hour,respectively.According to Corollary 1,we obtain the RULs of the degradations(as shown in Fig.2).To verify the superiority of the presented method,we also obtain the RUL predictions by a traditional method[10]which neglects the public noise (as shown in Fig. 2).Moreover,to quantify the performance of the methods,the mean square error(MSE)is employed(as shown in Table 2).Note that,it means the failure of the whole blast furnace,if any point of the blast furnace wall burns through,i.e.the failure mode of the blast furnace is competing failure.We simply assume that the RUL of the blast furnace wall is determined by the three degradations.Therefore,the failure time is the 2 776th hour.

(iii)The failure mode of the system is competing failure,i.e.,it is regarded as the failure of the system,once any degradation of the system exceeds the threshold.

(iv)The threshold of the each degradation is known.

From Corollary 1,the RUL of each single degradation can be obtained.To predict the RUL of the whole system,we use a Copula function to combine the marginal distribution of each degradation.Firstly,we calculate the cumulative distribution function(CDF)of RUL of each degradation by

It can be easily proved that the sum of two independent Brownian motions is also a Brownian motion.Therefore,for a single degradation,we use a Brownian motion B′(i)(t)to replace B(i)(t)and B(t).Set δk=t-tk,then the ith degradation can be expressed by(27)at the sampling time tkgiven t≥tk.

whereis the initial value of theith degradationY(i)(t),i=1,2,...,m,ψ(i)(t;Θ)denotes the drift coefficient of the ith degradation,Θ is a set of the unknown parameters in ψ(i)(t;Θ),λ(i)and γ(i)are diffusion coefficients.B(t)and B(i)(t)are standard Brownian motions,where B(t)characterizes the public noise which influences all the degradations,andB(i)(t)is distinct from each other in each degradation because it characterizes the private noises.Without loss of generality,here we assume

As the m degradations are observed by m sensors,we can formulate the measurement equation as follows:

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To describe different degradation paths of various equipment,the drift coefficient ψ(i)(t;Θ)can be any form.To obtain a closed-form expression of RUL,we only require that ψ(i)(t;Θ)is continuous with respect to t.To embody the degradation,we use three different forms ofψ(i)(t;Θ).

The corresponding degradation processes can be denoted as

3.Identification of the degradation model

In order to estimate the RUL of the multi-degradation system,the degradation states and the parameters in the mdegradation model should be estimated firstly.For M1,we have While,for M2andWe propose the joint estimation of yk and Θ by combining the Kalman filter with the EM algorithm in this section.

3.1 Estimation of degradation states

Firstly,we use a Kalman filter[14]to estimate ykaccording to the measurement vector zk.

(i)Time-update:According to the last estimation,update the one-step predictions and variances by

whereare the estimates of degradation states,=andare variances of the estimated degradation states,Thus,the degradation states can be estimated by Kalman filter(10)–(14)after obtaining

3.2 Estimation of parameters

Note that the EM algorithm[15]can identify parameters by using incomplete or missing data.In order to identify the unknown parameters online with estimated degradation states,we propose a recursive identify method in this section.For simplicity,we defineand,for M1,M2and M3,respectively.Based on the estimated degradation states obtained by(10)–(14),the EM estimation in the lth iterative step can be given by

(ii)M-step:Calculate the estimates by maximizing

Owing to the nonlinear drift coefficient,the analytical solutions of the estimates are intractable to deduce by making the partial derivatives equal to zero.Here we use their numerical solutions obtained by the Matlab function“fminsearch”.

4.RUL prediction

4.1 RUL of a single component

The RUL of each part or component is very informational for scheduling maintenance actions.The RUL is usually defined as the time interval from the present time tkto the failure time Tf,namely,

The RUL at tkof the ith degradation with a known threshold η(i)can be formulated as

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Then the multi-degradation can be described by an mdimension vectorY(t)=[Y(1)(t),Y(2)(t),···,Y(m)(t)]T.Each degradation can be denoted by a Wiener process with a nonlinear drift and two independent Brownian motions,which is formulated by

随着科学技术的不断创新，“互联网+”理念的革新，使产业和产业之间的融合有了便捷交流的平台；而大数据时代的到来则为以体育特色小镇为核心的产业链构建提供了可能，通过对数据的分析预测规避了小镇的运营风险，提高了小镇的服务水平[7]。一系列以体育为借力点的跨界融合模式的推出，作为新的经济增长点，在满足大众消费需求的同时也成为拉动经济发展的内在动力。

To achieve the analytical form of RUL of each degradation,we use Lemma 1 to calculate the FHT of the stochastic process given by(27).

Lemma 1[17] For degradation(27),if ψ(i)(t;Θ)is continuous with respect to t,the PDF of RUL with the threshold η(i)can be approximated as According to Lemma 1,we can deduce the RULs for the specific degradations expressed by M1,M2and M3.The main results are provided in Corollary 1.

(i)The system includesmdegradations,and each dgradation can be observed by a sensor.

Here we suppose that the failure mode of the system is the competing failure.Therefore,we can define the RUL of the whole system as

4.2 RUL of the whole system

可见,带借代意义的词,之所以获得了借代意义,从源头上看,是由于人们在认知事物的过程中,不同的词涉及多个不同角度的关联性所使然。这种不同的角度在语言系统中又是相互依存、相互关联的,没有主次分别。不过,就其关联的事物的类别而言,同类关联和异类关联还是有着性质上的不同。也正因此,我们将这些不同的语义内在联系方式区分为同类理据和异类理据。从探求理据的角度看,同类理据要比异类理据相对容易一些。

The Copula function is a method of correlation analysis and multiple statistical analysis,which can obtain the joint distribution by combining several one-dimensional marginal distributions[18].The existence of the Copula function can be illustrated by the Sklar theorem.

Lemma 2(Sklar theorem)[19]Any multivariate joint distribution can be written by univariate marginal distribution functions with a Copula function,which describes the dependence between the variables.

Corollary1 For degradation processes(7),(8)and(9),the PDF of RUL with a threshold η(i)can be given as

Then by the Copula function,we can obtain the distribution of RUL of the whole system FLk(lk)as follows:

where C(·)is a Copula function,and θ is a matrix of parameters.Here,the maximum likelihood estimation(MLE)method is adopted to identify θ.The likelihood function is given by

where c(·)is the PDF of C(·).Then the estimation of θ can be obtained by maximizing

克劳斯反应器中的有机硫水解率主要取决于两个因素:一是反应温度。反应温度越低，则有机硫水解率越低。二是所使用的催化剂类型。在320℃的典型操作条件下，铝基催化剂活性稳定后，有机硫水解率为20%～40%，高含量钛基催化剂的有机硫水解率＞90%。因此，一级反应器要获得良好的水解效果，须组合使用钛基催化剂。此外，由于硫磺回收装置克劳斯单元水解反应受动力学控制，同时还需要控制一级反应器较高的床层温度，最好控制在300℃以上[16-17]。

To obtain the RUL of the whole system,the Gaussian Copula function is adopted here.With the CDF of RUL of each degradation obtained by (32) and the estimated matrix θ,the CDF of RUL of the whole system can be formulated by

where Φθ is an m-dimensional normal distribution with correlation matrix θ.The PDF of RUL of the whole system can be deduced by taking the derivative of FLk(lk).

5.Case study

To demonstrate the proposed method,an actual example of a blast furnace wall is adopted.The degradation d at a are collected from a blast furnace with a capacity of 1 000 m2 in Jiayuguan,China.The degradations of three positions of the blast furnace wall are collected.Since the temperature of the blast furnace wall can reflect the residual thickness,we use the temperature as the degradation to predict the RUL.The three degradations are shown in Fig.1.The degradation 1 is observed by a temperature sensor located with a height of 7.395 m and a depth of 0.15 m northeast,the degradation 2 is observed by a temperature sensor located with a height of 8.936 m and a depth of 0.4 m southwest,and the degradation 3 is observed by a temperature sensor located with a height of 10.139 m and a depth of 0.15 m northwest.

由上式可见，如果步进电机的齿数和拍数为定值时，电机转动速度和输入脉冲频率成正比，所以，可以通过改变出入脉冲频率来控制步进电机的转速。

Fig.1 Degradations of the three positions of a blast furnace wall

As the operation and the fluctuation of the furnace condition affect the whole blast,there may exist public noise in the degradations of different parts of the blast furnace wall.From Fig.1,we find that the three degradations have a common trend to some extent,even though they are located in different directions and deepness.To demonstrate the common trend among the three degradations,the co-integration test is employed[20,21].From the pairwise tests of the degradations,the common trend can be found between degradation 1 and degradation 2 with 96.38%confidence,between degradation 1 and degradation 3 with 86.26%confidence,and between degradation2 and degradation 3 with 99.80%confidence.Thus the proposed degradation model is more suitable for the degradations.Next,we compare our method with a traditional method[10]neglecting the public noise.

According to the presented method,a degradation model with public noise for the three degradations can be established.Here we use M2to establish the degradation model.Then the unknown parameters can be identified by the approach presented in Section 3.Main estimations are listed in Table 1.

Table 1 Estimations of main parameters

tion α(i) β(i) θ ion 1 2.103 9×10-9 2.980 1 ion 2 8.369 4×10-308.787 59 ion 3 4.986 0×10-195.795 0■ ■■1 0.956 0.938 0.956 1 0.979 0.938 0.979 1■ ■■

(ii)The degradations of the system are observed by the same kind of sensors.

首先，农民是一个庞大的集体，占据了中国总人口的50%以上，即使是在城市生活的农民，绝大多数也没有机会接触到优质的教育资源，而远程教育的智慧应用可以解决这个问题。

Table 2 MSEs of the RUL predictions

Degradation Our method Traditional method Degradation 1 31 265 37 853 Degradation 2 20 035 22 499 Degradation 3 6 459 8 287 The whole system 10 829 19 663

Fig.2 RUL prediction of each single degradation

Fig.3 shows the RULs of the blast furnace wall obtained by the Copula function with results achieved by the proposed method and the traditional method,respectively.And the MSEs of the whole system’s RUL predictions obtained by the two methods are also listed in Table 2.

Fig.3 Combined RUL prediction with the three degradations

The results given by Fig.2,Fig.3 and Table 2 demonstrate that the proposed RUL prediction method for single degradation and the whole system works well and efficiently for the degradations of the blast furnace wall,compared with the traditional method which neglects the public noise.

纵观古今，不少人用其一生来诠释“意义”。鲁迅高呼“无穷的远方，无数的人们，都和我有关”，将自己之生，系于他人之生；诺尔曼·白求恩断然告别了在加拿大的优渥生活，于救死扶伤中体现自己的价值；阿尔贝特·施韦泽在非洲的土地上坚韧前行，以一个个脚印向世间询问来此的意义，成就生命之惊鸿。

事实上，对于蔬菜类这一整体的营养评价很难用非好既坏的“二元论”方式评价，生蔬菜和熟蔬菜的营养价值应从不同角度来衡量，不同的蔬菜也应区别对待。

6.Conclusions

In order to predict the RULs of a kind of multi-degradation systems affected by the public noise,a nonlinear multidegradation model is proposed.To characterize the public noise,a public Brownian motion is formulated in each degradation process.In addition,the measurement noise is also taken into account.To identify the degradation states and unknown parameters,a joint online identification algorithm is presented.Then the RUL of each degradation is obtained in the sense of FHT.The RUL of the whole system is also obtained by a Copula function.A case study of a blast furnace wall is presented to demonstrate the validity of the presented method.

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