Journal of Control Science and Engineering

Volume 2018, Article ID 9182783, 11 pages

https://doi.org/10.1155/2018/9182783

## Estimating Remaining Useful Life for Degrading Systems with Large Fluctuations

High-Tech Institute of Xi’an, Xi’an, Shaanxi 710025, China

Correspondence should be addressed to Chang-Hua Hu; moc.anis@uer_hch

Received 16 March 2018; Accepted 3 April 2018; Published 14 May 2018

Academic Editor: Darong Huang

Copyright © 2018 Dang-Bo Du et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Remaining useful life (RUL) prediction method based on degradation trajectory has been one of the most important parts in prognostics and health management (PHM). In the conventional model, the degradation data are usually used for degradation modeling directly. In engineering practice, the degradation of many systems presents a volatile situation, that is, fluctuation. In fact, the volatility of degradation data shows the stability of system, so it could be used to reflect the performance of system. As such, this paper proposes a new degradation model for RUL estimation based on the volatility of degradation data. Firstly the degradation data are decomposed into trend items and random items, which are defined as a stochastic process. Then the standard deviation of the stochastic process is defined as another performance variable because standard deviation reflects the system performance. Finally the Wiener process and the normal stochastic process are used to model the trend items and random items separately, and then the probability density function (PDF) of the RUL is obtained via a redefined failure threshold function that combines the trend items and the standard deviation of the random items. Two practical case studies demonstrate that, compared with traditional approaches, the proposed model can deal with the degradation data with many fluctuations better and can get a more reasonable result which is convenient for maintenance decision.

#### 1. Introduction

With the development of industrial system, the technology of intelligent vehicle has become more and more mature, and its safety and reliability have become the key factor which restricts its development. An intelligent driving system of a vehicle is a complicated system, which usually needs to monitor the running state, position, and environment [1]. The navigation system is one of its monitoring systems. It can provide information about the position, direction, speed, and acceleration. With the development of intelligent driving technology, navigation system is more and more important. As a key part of the navigation system, the performance of the gyroscope will directly affect the performance of the whole system, and then influence the safety of the entire vehicle system. Therefore, it is necessary to forecast its remaining useful life for better maintenance or health management strategies, using the available condition monitoring information [2, 3].

Generally, the current methods for estimating RUL can usually be divided into two types: physics of failure based methods and data driven methods. Physics of failure based methods rely on the physics of the failure mechanisms. However, with advances in design and production technologies, engineering systems become more complex and large-scale, so it is difficult to obtain the physical failure mechanisms in advance. In contrast, data driven methods achieve RUL estimation via data fitting mainly including machine learning and statistics-based approaches. Therefore, data driven methods attempt to derive models directly from collected CM data and life data and have gained much attention in variety of industrial assets [4, 5]. Considering that the life data may not be available for cost-expensive or highly reliable products, we mainly focus on the statistical data driven approaches for RUL estimation in this paper.

Under a recognized definition of lifetime, that is, first hitting time (FHT) [6, 7], many RUL estimation approaches have been reported in literature [8–10]. References [6, 11] provide a detailed overview of the degradation modeling and RUL predicting method. It can be founded that these models mainly concern the system degrading with a soothing wave. However, some degradation processes have high dynamics and the observed degradation data exhibit many fluctuations. In this case, it is difficult to model those degradation data for estimating the RUL via traditional approaches [12, 13], while the fluctuation of degradation can reflect the stability of the system. With the deterioration of the system, the stability of the system may worsen gradually, which causes the fluctuation’s degree of the degradation data to usually increase over time. Such large fluctuating characteristics can be described by a stochastic degradation process with time-varying mean and variance. Hence, in order to simplify the expression, we use “large fluctuation” to describe the characteristics of these degradation data with time-varying mean and variance. Therefore, the issue of how to model those data with many fluctuations is of practical significance.

In this paper, it is assumed that the degradation process of the system is a stochastic process with time-dependent expectation and standard deviation at each time. Considering that the fluctuation can be represented by a stochastic degradation of standard deviation, which is a new key performance factor, then the whole system RUL can be estimated by considering the stochastic degradation data and the fluctuation. The model parameters can be estimated by maximum likelihood estimation method. Finally, a numbering validation method and INS gyro study are provided to show the superiority of our approach.

The remaining parts are organized as follows. In Section 2, the problem of failure prognosis is formulated and defined. In Section 3, a degradation model for prognosis is developed based on the degradation data and the standard deviation. Section 4 provides two case studies to illustrate the application and usefulness of the developed model. Section 5 draws the main conclusions.

#### 2. Problem Formulation

In this section, based on the fluctuation and the stochastic degradation data, a new model for RUL estimation is proposed. Firstly, we introduce the variables used in this paper.

##### 2.1. Notations

The notations that will be used in this paper are listed as follows: : time of the th CM point (can be irregularly spaced) : failure threshold : random variable representing the degradation at time instant : degradation observation at : history of degradation observations for the system up to : trend item : fluctuation item : expectation of : standard deviation of : failure function : standard deviation function at the time instant : lifetime : parameters column vector : PDF of : function of and : function of and : function of and : RUL of the system at time instant : log-likelihood function : time instant of the change-point : number of the observation data at time instant : initial degradation : drift and diffusion parameters : standard Brownian motion : expectation operator : individual difference random variable.

##### 2.2. Problem Formulation of the New Approach for RUL Estimation

In this paper, we model the degradation process of the system as a stochastic process as , where is the degradation quantity a time . Therefore, the lifetime of the system can be defined as follows:where is the failure threshold.

From (1), it is important to establish the model of stochastic degradation process. In general, a stochastic degradation process can be represented aswhere is the trend term, is the stochastic fluctuating term, and both of them change over time.

It is assumed that is the complex stochastic process with the expectation and standard deviation . Note that determines the fluctuation of the stochastic degradation process. Thus, jointly modeling the standard deviation and trend items is necessary for RUL estimation. Generally, there are two methods to model the standard deviation: the first method is that the standard deviation is regarded as an independent failure factor, but the difficulty lies in how to set the failure threshold; the second method is to formulate a new failure function which establishes the linkage between the standard deviation and the trend . Accordingly, the lifetime of the system can be defined as follows:where is the failure function, is the standard deviation, which is the function of the time instant , and is the parameters column vector.

Based on the estimated lifetime , the RUL at time is obtained as where is the time at the CM points. In this way, the fluctuation in the degradation process could be incorporated into the RUL estimation.

From (3), we can observe that the fluctuation of the degradation data is taken into account in estimating the lifetime. Then the next section is to model the trend term and the stochastic fluctuating term .

#### 3. Degradation Modeling and RUL Estimation

In this section we describe how to model the degradation process and estimate the parameter based on the observed degradation data in detail and then how to obtain the RUL of the system via the proposed approach.

##### 3.1. Degradation Modeling

Based on the problem formulation of our approach, the main algorithm formulation for degradation modeling and RUL estimation is outlined as follows.

First, we define the stochastic disturbance term as a function of and :where both the expectation and the standard deviation are time-dependent and formulated as

As a result, the PDF of can be represented as

Suppose the trend is also a function of as follows:

Then the degradation process defined in (2) can be further described as follows:

In practice, the degradation process is often discretely monitored at time and let denote the degradation observation at time . Then, the set of the degradation observations up to is represented by . In this paper, we can utilize the history of the degradation observations to evaluate the parameters in (4). The details of algorithm for parameter estimation are summarized in Appendix A.

##### 3.2. Decomposing Trend Items and Fluctuation Items

As discussed in Section 3.1, the degradation process is decomposed into the trend item and fluctuation item at first. Nowadays the common methods of trend extraction, such as the average slope method, finite difference method, LPF (Low-Pass Filter) method, and least square fit method, need the form of the trend to be defined in advance, which leads to difficulty in applying those methods to the degradation signals with the unknown trend. Because the EMD (Empirical Mode Decomposition) method is an adaptive trend extraction method [14–16], which has been widely used for trend extraction, then, in our paper, the EMD method is adopted for trend extraction.

###### 3.2.1. Estimation Parameters of Fluctuation Items

Suppose are the trend items obtained by EMD method; let , so, as discussed in Section 3.1, denotes the observed data of the stochastic fluctuating process .

Since the stochastic disturbance is defined as a normal random process, so it could be concluded that the mean of stochastic fluctuating process is based on the good property of the normal random process. And it can be obtained as follows:where is defined as a function of .

In engineering practice, many systems have two stages of degradation. In the first stage, the trend of the degradation is not obvious, and the fluctuation is stable. In the second stage, the trend of the degradation has an obvious increase, and the fluctuation becomes higher over time. So, it is reasonable to adopt a two-stage model to describe the degrading systems with large fluctuations. The two-stage model can be expressed by [17]where is the time of the change-point. It is noted that the change-point assumed can be easily found and here we regard it as known information [17]. When , is the fixed value; when , which is the function of time .

When , , is a constant. could be evaluated by the maximum likelihood estimation based on the property of normal distribution [18] as follows:

In (12), denotes the number of the observed data at time and is the arithmetic average of the observed data before .

When , is the function of . In order to simplify the modeling, it is assumed that is a linear function of time , expressed as

Based on the maximum likelihood estimation method, we can easily obtain the log-likelihood function . Taking the partial derivatives of the log-likelihood function to all parameters, we have

It is observed from (14) that the analytic solution of those parameters via maximum likelihood estimation could not be obtained. To be solvable, the numerical method is adopted to estimate , which is summarized in Appendix C.

###### 3.2.2. Modeling the Trend Items

Because the systems in the same batch may have different degradation paths, we adopt the random variable to describe such individual difference. As such, the real trend item can be represented aswhere is the trend item from initial time 0 to time instant and is the random variable reflecting individual difference.

From the above formulation, it could be concluded that is also random due to individual difference. So we could model the trend items by statistics-based data driven methods. Considering that the trend items may not be a monotonic process, we utilize Wiener process to describe this random trend process [18].

In order to simplify modeling, only the linear degradation model or the model that could be converted into linear form will be discussed in this paper. In general, a linear Wiener-process-based degradation model can be represented as follows [19]:where is the initial degradation, and are the drift and diffusion parameters, respectively, and denotes the standard Brownian Movement (BM), which represents the stochastic dynamics. We assume that is the stochastic coefficient while and are deterministic. And we further assume that when , ; thus .

Define . Then the joint sampling distribution can be calculated as

In Bayesian framework, it is assumed that the prior distribution of follows . Thus, we can obtain the following [20]:

Due to the property of the normal distribution, we can obtain the posterior estimate of as follows:with

It is obvious that when a new observation is available, the posterior estimate of can be easily updated. is used to denote the unknown parameters. Actually, we can also estimate those unknown parameters in via EM algorithm, which provides a possible framework for estimating the parameters involving hidden variables [19, 21, 22]. For example, let denote the estimated based on the observed data . Then, can be obtained as follows:The details of the derivation are summarized in Appendix D.

##### 3.3. Joint Model for RUL Estimation

If is a normal distribution, then

It is well-known that is frequently used to be the confidence interval of normal distributed random variable. Therefore, we assume that if does not reach the failure threshold , it is reasonable that the degradation data does not reach the failure threshold . For those highly critical systems, the maintenance after failure is too expensive or the consequence of failure is disastrous. Thus, for those systems, it is essential that the conservative method for PHM be adopted to avoid unexpected failure. Therefore, the failure function is defined as

Then, as we discussed before, the failure function can be further represented as follows:

If is used to denote the failure time, the following result can be obtained directly:

If is used to denote the current time, we havewhere .

Solving the equations in (25) and (26), we can get

Based on the definition of RUL in (1), the RUL estimated at time could be represented as

Further, utilizing the property of Wiener process, the PDF of the RUL estimation at time can be obtained:

Recall that follows ; Theorem 1 in [20] can be used to estimate the PDF of the RUL as

Clearly, there are five steps of RUL estimation in our approach, which are summarized below.

##### 3.4. A Procedure of Proposed Approach for RUL Estimation

As discussed in the previous subsection, the procedure of the approach proposed in this paper for RUL estimation is summarized as follows.

*Step 1. *Through EMD method, the degradation data are divided into the trend items and the fluctuation items when the monitored observations are available.

*Step 2. *The trend items and the fluctuation items are modeled by Wiener process and a normal stochastic process separately; the details of algorithm and parameter estimation are summarized in Section 3.2.1 and Appendices B, C, and D.

*Step 3. *According to the failure function defined in Section 3.2.2 and the model proposed in Step 2, the PDF of the estimated RUL can be obtained.

*Step 4. *When the monitored observation is available, let and go to Step 1. Otherwise, go to Step 5.

*Step 5. *Using the PDF of the estimated RUL, the system’s performance can be evaluated.

#### 4. Case Studies

In this section, two practical cases for gyros in the inertial navigation system (INS) are provided to illustrate the application of our model and compare the performance of our model with traditional models in [12, 23].

##### 4.1. Case Description

As discussed before, gyro plays a key role in intelligent vehicle. With the working hours increase, gyro may cause a kind of degradation that makes gyro drift. Usually, this gyro can be compensated using a specific compensating model. However, when gyro drift degrades to a predefined threshold, the gyro’s performance can not satisfy the vehicle INS system requirement. The gyro’s drift can not be monitored directly, and a performance indicator is used instead.

In the following, two practical cases are provided to illustrate the adaptability and rationality of our approach for RUL estimation.

##### 4.2. Two Practical Case Studies

###### 4.2.1. Case 1: The Fluctuations of Degradation Data Are Stable

In this section, we utilize the degradation data from INS to illustrate, where its operating time is 374 hours and the interval is 2 hours. The collected data are shown in Figure 1.