Remaining Life Prediction of Cores Based on Data-driven and ...

Remaining Life Prediction of Cores Based on Data-driven and Physical Modeling Methods

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Xiang Li, Wen Feng Lu, Lianyin Zhai, Meng Joo Er, and Yongping Pan

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Model for Analysis of Time-to-Failure Data in Product Life Cycle Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Model for Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics of Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Analysis of Life Data: An Illustrative Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condition Prediction Using Enhanced Online Learning Sequential-Fuzzy Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Architecture of Fuzzy Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Online Sequential Learning Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multistep Prediction Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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X. Li (*) Singapore Institute of Manufacturing Technology (SIMTech), Singapore e-mail: [email protected] W.F. Lu • L. Zhai Department of Mechanical Engineering, Faculty of Engineering, National University of Singapore, Singapore e-mail: [email protected]; [email protected] M.J. Er • Y. Pan School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore e-mail: [email protected]; [email protected] # Springer-Verlag London 2015 A.Y.C. Nee (ed.), Handbook of Manufacturing Engineering and Technology, DOI 10.1007/978-1-4471-4670-4_57

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Abstract

This chapter presents development of enabling technologies that are able to assess the reliability of remanufactured products based on predictive modeling methods, to describe fast and accurate prediction algorithms that are able to predict condition of critical components or parts of manufactured products based on historical data. Machine health condition prediction of critical components under the situation of insufficient data, missing prior fault knowledge, and noisy measurement are studied using an enhanced online sequential learning-fuzzy neural network. Meanwhile, Weibull model-based reliability analysis is investigated in this chapter. Performance of various Weibull parameter estimation methods is compared using case studies. Results of this part of research have enabled the development of a product reliability analysis tool that is able to characterize the product failure modes, failure rate, and reliability profile.

Introduction As the primary goal of remanufacturing is part reuse, understanding of the quality/condition of the returned cores is very important for decision-making in remanufacturing processes. Hence, condition assessment and fault isolation of the returned cores becomes one of the most critical activities that determine the success of a remanufacturing process. Existing practices in remanufacturing typically carry out defect inspection and fault diagnosis only for isolated parts/components after the returned cores are disassembled. This may impose additional challenges and cost on the remanufacturing process such as fluctuation of schedule for remedial processes or treatments in the shop floor due to unexpected defects/faults identified after disassembly. In addition, it depletes the opportunity to assess the condition and performance of the products systematically based on their field operational data in each lifecycle before they are returned as cores, which is very important to establish reliability models of the products (Mazhar et al. 2010). On the other hand, for valuable machineries, such as mining trucks, a large amount of operational data is already being collected, typically on log sheets or by a control system. This process is usually not regarded as part of a condition monitoring or diagnostic program. However, there is a lot of valuable machinery performance and condition information buried in such operational data. In practice, the only challenge is how to extract useful information from such data. It is believed that the operational and inspection data collected on a machine, when properly interpreted, can produce an accurate picture of the machines health. In reality, the cores returned for remanufacturing may have experienced very different working conditions, and their components/parts may have diverse ages and different stress and strains arisen by the users, but remanufacturing companies usually do not involve the historical and field operational data of the cores when they make decisions for remanufacturing processes in the shop floor. Taking the engine of a mining truck as an example, operational data often is adequate to

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allow for calculations of engine efficiency and detection of its deviations, which can then be extended to evaluate the condition of the engines. Analysis such as the emerging operational trends of operational data not only can tell us that there is a performance problem with this engine, but it also can help to isolate fault sources well before the engine is sent for inspection in the remanufacturing workshop. Measures of this sort can greatly assist with prioritizing remanufacturing decisions and balance the time between reman cycles, and more importantly, it may simplify the reman workshop inspection procedure and minimize the inspection cost. It should be noted that most of the current practices in remanufacturing rely on rules of thumb or expert knowledge and lack rigorous reliability-based evaluation models to support remanufacturing shop-floor decision-making. Although recent years have seen some applications using visual and/or statistical analysis tools to assist the lifecycle assessment of remanufactured products, such tools remain inadequate in coping with large amount of field operational data with inherent variability, uncertainty, and nonlinearity. The proposed approach attempts to address critical issues for improving remanufacturing processes through effective analysis of field operational data. This research will fill up the gap in the current state of the art where existing remanufacturing practices lack rigorous and reliable analysis of operational data for support of remanufacturing shop-floor decisionmaking, despite the fact that an effective analysis of field operational data in various aspects of the products will provide invaluable information to facilitate sound decision-making and continuous improvement of remanufacturing processes. More specifically, the novelty of this research includes. A comprehensive approach to condition assessment and fault isolation through rigorous reliability-based evaluation models with progressive model learning capabilities; Fusion of statistical and machine learning techniques to form a fast real-time RUL prognosis tool that can scale well against large operational data with inherent variety and uncertainty in remanufacturing processes.

Weibull Model for Analysis of Time-to-Failure Data in Product Life Cycle Management Management of products and materials at the end-of-life (EOL) is being recognized as an integral part of the product life cycle engineering. Among various EOL management strategies, remanufacturing as a sustainable manufacturing process has received more attention in recent years. In remanufacturing practices, understanding and communicating the failure risk and reliability of a critical part, component, or subsystem plays a crucial role as it has a significant impact on the lifecycle management of the product and also determines the success of the remanufacturing process. In such a context, it is envisaged that many of the life cycle engineering techniques will have a significant impact on remanufacturing practices such as reliability and remanufacturability analysis of valuable parts and components, remaining useful life prediction and warranty cost of remanufactured products etc (Mazhar et al. 2007).

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In life cycle engineering as well as in remanufacturing industry, estimation of product mean life is an important task as it provides valuable information for effective life cycle management, in particular, the core management and inventories in remanufacturing. Usually a product’s mean life is determined by analyzing its time-to-failure data from a wide range of the same category of products operated under the same conditions of use (Li 2004). Another important issue in life cycle engineering and remanufacturing is the quantitative analysis of product or component reliabilities, based on which the product or the component’s expected useful life can be estimated. Understanding the probability of product or component failures at different stages of its life can be very useful to make optimized decisions in life cycle management and remanufacturing practices (Mazhar et al. 2007).

Weibull Model for Reliability Analysis In life cycle management, one of the simplest approaches to predicting failure is based on statistical reliability models of past failures (Gu and Li 2012). Reliability is defined as the probability that a product will continue to perform its intended function without failure for a specified period of time under stated conditions (Pham 2006; Calixto 2013). Usually, reliability predictions are used to estimate future failure based on past failure records by applying a probability distribution such as the exponential distribution. However, one of the principal shortcomings of using the exponential distribution is that it imposes a “Markov” assumption, meaning that the future prediction of a failure is independent of the history of the unit given the current measurement (Lourenco and Mello 2012). In this respect, Weibull distribution (Weibull 1951) for prediction provides an alternative reliability method as it relaxes the assumption of constant failure rates as well as the Markov assumption (Groer 2000). In fact, the most common distribution function in EOL management is Weibull distribution due to its ability to fit a greater variety of data and life characteristics by changing its shape parameter (Artana and Ishida 2002). Today, Weibull analysis is the leading method in the world for fitting and analyzing life data. In most cases of application, Weibull distribution is able to provide the best fit of life data. This is due in part to the broad range of distribution shapes that are included in the Weibull family. Many other distributions are included in the Weibull family either exactly or approximately, including the normal, the exponential, the Rayleigh, and sometimes the Poisson and the Binomial (Abernethy 2006). Compared with classic statistical methods, Weibull analysis uses failure reference and mean-time-to-failure (MTTF) to forecast failures, whereas statistical pattern analysis uses test data to identify a statistical pattern such as trend lines (Fitzgibbon et al. 2002). Another most salient feature to be noted for Weibull analysis is its ability to provide reasonably accurate failure analysis and failure forecasts with extremely small samples of life data, where most of other distributions fail to give meaningful result (usually when the sample size is smaller than 20) (Abernethy 2006). This feature of Weibull analysis makes it very valuable in remanufacturing decision-making practices because it is a common

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case that the life data of very big and especially very expensive parts/components collected in remanufacturing process are either incomplete or small in size.

Basics of Weibull Distribution In general, a typical Weibull probability distribution function (PDF) is defined by f ðt Þ ¼


 β t β1 ðηt Þβ e η η

(1)

where t  0 represents time, β > 0 is the shape or slope parameter, and η > 0 is the scale parameter of the distribution. Equation 1 is usually referred to as the 2-parameter Weibull distribution. Among the two parameters, the slope of the Weibull distribution, β, is very important as it determines which member of the family of Weibull failure distributions best fits or describes the data. It also indicates the class of failures in the “bathtub curve” failure modes as shown in Fig. 1. The Weibull shape parameter β indicates whether the failure rate is increasing, constant, or decreasing. If β < 1, it indicates that the product has a decreasing failure rate. This scenario is typical of “infant mortality” and indicates that the product is failing during its “burn-in” period. If β ¼ 1, it indicates a constant failure rate. Frequently, components that have survived burn-in will subsequently exhibit a constant failure rate. If β > 1, it indicates an increasing failure rate. This is typical for products that are wearing out. To summarize: • β < 1 indicates infant mortality. • β ¼ 1 means random failures (i.e., independent of time). • β > 1 indicates wear-out failures. The information about the β value is extremely useful for reliability-centered maintenance planning and product life cycle management. This is because it can provide a clue to the physics of the failures and tell the analyst whether or not scheduled inspections and overhauls are needed. For instance, if β is less than or equal to one, overhauls are not cost effective. With β greater than one, the overhaul period or scheduled inspection interval can be read directly from the plot at an

Failure rate

Fig. 1 The “bathtub curve” failure modes

β1:wear-out failures

Time

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acceptable or allowable probability of failures. For wear-out failure modes, if the cost of an unplanned failure is much greater than the cost of a planned replacement, there will be an optimum replacement interval for minimum cost. On the other hand, the scale parameter, or spread, η, sometimes also called the characteristic life, represents the typical time-to-failure in Weibull analysis. It is related to the mean-time-to-failure (MTTF). In Weibull analysis, η is defined as the time at which 63.2 % of the products will have failed (Pasha et al. 2006). There are basically two fitting methods for parameter estimation in widespread use in reliability analysis, namely, the maximum likelihood estimation (MLE) and regression methods. MLE involves developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. Regression method generally works best with data sets with smaller sample sizes that contain only complete data (i.e., data in which all of the units under consideration have been run or tested to failure). This failureonly data is best analyzed with rank regression on time, as it is preferable to regress in the direction of uncertainty. In Weibull analysis, median-rank regression (MRR) method which uses median ranking for regression fitting is often deployed to find out the shape and scale parameters for complete life data (Abernethy 2006). The probability of failure at time t, also referred to as the Weibull distribution or the cumulative distribution function (CDF), can be derived from Eq. 1 and expressed as t β

FðtÞ ¼ 1  eðηÞ

(2)

Thus, the Weibull reliability at time t, which is 1  F(t) ¼ R(t), is defined as t β

RðtÞ ¼ 1  FðtÞ ¼ eðηÞ

(3)

This can be written as t β 1 ¼ eðηÞ 1  FðtÞ

(4)

Taking two times the natural logarithms of both sides gives an equation of a straight line:
ln ln

1 1  FðtÞ

 ¼ βlnt  βlnη

(5)

Equation 5 represents a straight line in the form of “y ¼ ax + b” on log/log(Y) versus log(X), where the slope of the straight line in the plot is β, namely, the shape parameter of Weibull distribution. Through the above transformation, the life data samples can be fitted in the Weibull model and the two Weibull parameters can be estimated.

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Table 1 Life data of a critical component in a diesel engine

No. 1 2 3 4 5 6 7 8 9 10

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Failure time (h) 38,456 48,334 50,806 51,521 61,544 66,667 72,605 75,521 80,785 84,894

The mean of the Weibull PDF, T , which is the MTTF in Weibull analysis, is given by


1 T ¼ηΓ þ1 β

 (6)

where Γ is the gamma function. It is noted that when β ¼ 1, MTTF is equal to η. In fact, as a rough approximation, in practices of Weibull analysis where β is equal to or slightly larger than 1, the characteristic life can be approximated as MTTF. However, for β that is much larger than 1, MTTF should be calculated using Eq. 6. This will be further discussed in the example elaborated in the next section.

Weibull Analysis of Life Data: An Illustrative Case Study In the life cycle management of a certain type of heavy-duty diesel engine, it is required to quantify the life characteristics of a critical component in order to understand its reliability and remanufacturability. The engine manufacturer has provided a past record of 10 failure cases of the said component under normal use conditions. The complete life data, i.e., the failure time of each sample, is shown in Table 1. Assume that our objectives in this case study include: 1. Determine the Weibull parameters and derive the Weibull distribution model for the data given. 2. Estimate the average life of the component (i.e., the MTTF or mean life). 3. Estimate the time by which 5 % of the components will fail or the time by which there is a 5 % probability that the component will fail. 4. Estimate the reliability of the components after a given number of hours of operation.

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Probability - Weibull

99.000

Point A

10.000

X = 35221.228 Y = 5.053

1.000

0.100 10000.000

Time (Hr)

100000

Fig. 2 Weibull probability plot

5. Estimate the warranty time for the component if the manufacturer does not want failures during the warranty period to exceed 5 %. In the following sections, Weibull analysis will be conducted to address the above objectives. In this case study, the 2-parameter Weibull analysis is deployed to analyze the life data characteristics of the diesel engine component. First of all, the parameters are estimated based on the 2-parameter Weibull analysis, in which the standard ranking method and median-rank regression are used to fit the given data in Table 1. As discussed earlier in section “Basics of Weibull Distribution,” regression method should be selected to fit the data when the data sample is small and contains complete life data. The fitting plot is shown in Fig. 2 and the two Weibull distribution parameters are calculated as follows: β ¼ 4.40 and η ¼ 69,079.89 (hours). After the two parameters of β and η are determined, the Weibull PDF expressed by Eq. 1 can be obtained as shown below: f ðtÞ ¼

3:4 4:4 t 4:4  t eð69079:89Þ 69079:89 69079:89

After simplification the Weibull PDF is plotted as shown in Fig. 3.

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Probability Density Function

3.000E-05

2.400E-05

1.800E-05

1.200E-05

6.000E-06

0

0

60000.000

120000.000

180000.000

Time (Hr) Fig. 3 Weibull probability density function

As discussed earlier, the MTTF or mean life is a very important indicator of the life data characteristics in life cycle engineering. MTTF can be either approximated by the value of η in cases where β is slightly larger but close to 1 or calculated using Eq. 6 if β is much larger than 1 or a more accurate value is required. In this case study, β ¼ 4.40 and therefore Eq. 6 is used to calculate the exact MTTF instead. The MTTF calculated is 62,952.73 h, and it is much smaller than the value of characteristic life η, which is 69,079.89 h. As shown in Fig. 2, in the Weibull probability fitting plot of the case study, the x-axis represents time using a logarithm scale, and the probability of failure is displayed on the y-axis using a double log reciprocal scale. Such a Weibull probability plot is able to tell very important information about the characteristics of the failures. From the plot, the probability of failure at a given time, or vice versa, can be obtained. For example, it may be of interest to determine the time at which 1 % of the population will have failed. For more serious or catastrophic failures, a lower risk may be required, for instance, six-sigma quality program goals often equate to 3.4 parts per million (PPM) allowable failure proportion. Such important information can be easily obtained from the Weibull probability plot. In this case study, for the red dot (point A) shown on the plot in Fig. 2, the x and y coordinates are x ¼ 35,221.228 h and y ¼ 5.053, respectively, which can be interpreted in the following way: the failure probability of the component at the time of 35,221.228 h

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1.000

Reliability vs. Time

Poin t B 0.800

0.600

0.400

0.200

0

0

40000.000

80000.000

120000.000

Time (Hr) Fig. 4 Weibull reliability plot

is 5.053 %, or the average time by which 5.053 % of the components will fail is 35,221.228 h. It is known that reliability analysis is a very important issue in life cycle engineering. In this case study, the Weibull reliability function can be calculated based on Eq. 3 and its plot is shown in Fig. 4. Figure 4 can be easily used to answer the estimate of reliability of the component after a certain number of hours of operation. For example, for point B shown on the plot in Fig. 4, the x and y coordinates are x ¼ 32,090.306 h and y ¼ 0.966, respectively, which can be interpreted in the following way: the reliability of the component after 32,090.306 h of operation is 96.6 %. This is in fact the reverse interpretation of the coordinates of x and y in Fig. 4. For example, if the manufacturer does not want failures during the warranty period to exceed 3.4 % (i.e., the required reliability is 96.6 %), then the maximum warranty time promised to customers should not exceed 32,090.306 h, as shown by point B in Fig. 4. In life cycle engineering, failure rate is another important indicator of life data characteristics. Failure rate is usually defined as the frequency with which a product or component fails, and it is often expressed in failures per unit of time

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Failure Rate vs.Time

0.003

0.002

0.002

0.001

6.000E-04

0

0

60000.000

120000.000

180000.000

Time (Hr) Fig. 5 Failure rate versus time

(e.g., per hour in this case study). The failure rate of a product usually depends on time, with the rate varying over the life cycle of the product, as shown in the “bathtub curve” failure modes in Fig. 1. The failure rate of the case study is calculated and shown in Fig. 5. The increasing failure rate shown in the figure confirms that the life data in Table 1 follow a wear-out failure mode.

Condition Prediction Using Enhanced Online Learning Sequential-Fuzzy Neural Networks Machine health condition (MHC) prediction is useful for preventing unexpected failures and minimizing overall maintenance costs since it provides decisionmaking information for condition-based maintenance (CBM) (Vachtsevanos et al. 2006). Typically, MHC prediction methods can be divided into two categories, namely, model-based data-driven methods (Jardine et al. 2006). Due to the difficulty of deriving an accurate fault propagation model (Yu et al. 2012;, Yu et al. 2011), researches have focused more on the data-driven method in recent years (Si et al. 2011). The neural network (NN)-based approach, which falls under the category of the data-driven method, has been considered to be very promising for MHC prediction due to the adaptability, nonlinearity, and universal function

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approximation capability of NNs (Tian and Zuo 2010). Batch learning and sequential learning are two major training schemes of NNs. MHC prediction is essentially an online time-series forecasting problem which should perform realtime prediction while updating the NN. Thus, to save updating time and to maintain consistency of the NN, the sequential learning should be employed in such a problem. The most popular NNs applied to MHC prediction are recurrent NNs (RNNs) and fuzzy NNs (FNNs). In Gu and Li (2012), an extended RNN which contains both Elman and Jordan context layers was developed for gearbox health condition prediction. In Zhao et al. (2009), a FNN in Brown and Harris (1994) was applied to predict bearing health condition. In Wang et al. (2004), an enhanced FNN was developed to forecast MHC. Next, in Wang (2007) and Liu et al. (2009), a recurrent counterpart of the approach in Wang et al. (2004) and a multistep counterpart of the approach in Wang (2007) were presented to predict MHC, respectively. An interval type-2 FNN was also proposed to predict bearing health condition under noisy uncertainties in Chen and Vachtsevanos (2012). Note that the batch learning was employed in Tian and Zuo (2010), Zhao et al. (2009), and Chen and Vachtsevanos (2012). Common conclusions from Tian and Zuo (2010), Zhao et al. (2009), and Wang et al. (2004), Wang (2007), Liu et al. (2009), and Chen and Vachtsevanos (2012) are that the RNN usually outperforms the feedforward NN and the FNN usually outperforms the feedforward perceptron NN, feedforward radial basis function (RBF) NN, and RNN. Recently, to improve prediction performance under measurement noise, an integrated FNN and Bayesian estimation approach was proposed for predicting MHC in Chen et al. (2012), where a FNN is employed to model fault propagation dynamics offline and a first-order particle filter is utilized to update the confidence values of the MHC estimations online. In Chen et al. (2011), a high-order particle filter was applied to the same framework of Chen et al. (2012). A question in the approaches of Chen et al. (2011, 2012) is that the FNNs should be trained by the system state data (rather than the output data) which are assumed to be immeasurable. Extreme learning machine (ELM) is an emergent technique for training feedforward NNs with almost any type of nonlinear piecewise continuous hidden nodes (Huang et al. 2006). The salient features of ELM are as follows (Huang et al. 2006): (i) All hidden node parameters of NNs are randomly generated without the knowledge of the training data; (ii) it can be learned without iterative tuning, which implies that the hidden node parameters are fixed after generation and only output weight parameters need to be turned; (iii) both training errors and weight parameters need to be minimized so that the generalization ability of NNs can be improved; and (iv) its learning speed is extremely fast for all types of learning schemes. ELM demonstrates great potential for MHC prediction due to these salient features. Nonetheless, the original ELM proposed in Huang et al. (2006) is not appropriate for predicting MHC since it belongs to the batch learning scheme. To enhance the efficiency of ELM, online sequential ELM (OS-ELM) was developed in Liang et al. (2006) and was further applied to train the FNN in Rong et al. (2009).

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Due to its extremely high learning speed, the OS-ELM-based FNN in Rong et al. (2009) seems to be suitable for MHC prediction. Yet, there are two drawbacks in Rong et al. (2009) as follows: (i) It is not good to yield generalization models since only tracking errors are minimized and (ii) it may encounter singular and ill-posed problems while the number of training data is smaller than the number of hidden notes. To further improve the efficiency of MHC prediction, a novel FNN with an enhanced sequential learning strategy is proposed in this paper. The design procedure of the proposed approach is as follows: Firstly, a ellipsoidal basic functions (EBFs) FNN is proposed; secondly, the FNN approximation problem is transformed into the bi-objective optimization problem; thirdly, an enhanced online sequential learning strategy based on the ELM is developed to train the FNN; and finally, a multistep direct prediction scheme based on the proposed learning strategy is presented for MHC prediction. The developed enhanced online sequential learning-FNN (EOSL-FNN) is applied to predict bearing health condition by the use of real-world data from accelerated bearing life. Comparisons with other NN-based methods are carried out to show the effectiveness and superiority of the proposed approach.

Architecture of Fuzzy Neural Network For MHC prediction, the n-input single-output system is considered. Yet, the following results can be directly extended to the multi-input multi-output (MIMO) system. The FNN is built based on an EBF NN. It is functionally equivalent to a Takagi-Sugeno-Kang (TSK) fuzzy model that is described by the following fuzzy rules (Wu et al. 2001): Rule Rj : IF x1 is A1j and  and xn is Anj THEN y^ is wj

(7)

where xi  ℝ and y^  ℝ are the input variable and output variable, respectively; Aij is the antecedent (linguistic variable) of the ith input variable in the jth fuzzy rule; wj is the consequent (numerical variable) of the jth fuzzy rule, i ¼ 1, 2, . . ., n, j ¼ 1, 2, . . ., L; and L is the number of fuzzy rules. As illustrated in Fig. 6, there are in total four layers in the FNN. In Layer 1, each node is an input variable xi and directly transmits its value to the next layer. In Layer 2, each node represents a Gaussian membership function (MF) of the corresponding Aij as follows: h  i   2 μAij xi j cij , σij ¼ exp  xi  cij =2σ2ij

(8)

where cij  ℝ and σ ij  ℝ+ are the center and width of the ith MF in the jth fuzzy rule, respectively. Note that the MF in Eq. 8 is an EBF since all its widths σij are

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Fig. 6 Architecture of fuzzy neural network

different (Wu et al. 2001). In Layer 3, each node is an EBF unit that denotes a possible IF-part of the fuzzy rule. The output of the jth node is as follows: h Xn  i   2 ϕj xjcj , σ j ¼ exp  i¼1 xi  cij =σ 2ij (9) where x ¼ ½x1 , x2 ,  , xn T  ℝn , cj ¼ [c1j, c2j,   , cnj]  ℝn, and σ j ¼ [σ 1j, σ 2j,   , σ nj]  ℝn. In the last layer, the output y^is obtained by the weighted summation of ϕj as follows: y^ ¼ f^ðxjW, c, σÞ ¼ Φðxjc, σÞW

(10)

where f^ðÞ : ℝnþLð1þ2nÞ 7!ℝ , Φ ¼ [ϕ1, ϕ2,   , ϕL]  ℝL, c ¼ [c1, c2,   , cL]T  ℝL  n, σ ¼ [σ 1, σ 2,   , σ L]T  ℝL  n, and W ¼ [w1, w2,   , wL]T  ℝL. For the TSK model, the THEN-part wj is a polynomial of xi which can be expressed as follows: wj ¼ α0j þ α1j x1 þ    þ αnj xn

(11)

where α0j, α1j,   , αnj  ℝ are weights of input variables in the jth fuzzy rule. The following lemma shows the universal function approximation property of the proposed FNN. Lemma 1 (Lin and Cunningham 1995) For any given continuous function f ðxÞ : D7!ℝ and arbitrary small constant e  ℝ+, there exists a FNN in Eq. 10 with proper parameters W, c, and σ such that supx  D jf ðxÞ  f^ðxjW, c, σÞj < ε where D  ℝn is an approximation region.

(12)

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Online Sequential Learning Strategy For training FNNs, consider a data set with N arbitrary distinct training samples: N N ¼ fðxl , yl ÞgNl¼1 , where xl ¼ [xl1, xl2,   , xln]T  ℝn, yl  ℝ, and l is the number of the sampling point. If a FNN with L hidden nodes can approximate these N samples with zero error, then there exist proper parameters W,c, and σ such that Φðxl jc, σÞW ¼ yl

(13)

for all l ¼ 1, 2, . . ., N. Since wj in Eq. 11 can be rewritten into wj ¼ xTle αj with  T xle ¼ 1, xTl  ℝnþ1 and αj ¼ [α0j, α1j,   , αnj]T  ℝn + 1, one gets  T W ¼ xTle α1 , xTle α2 ,   , xTle αL :

(14)

Substituting Eq. 14 into Eq. 13 for all l ¼ 1, 2, . . ., N, applying the definition of Φ, and making some manipulations, one gets 3 2 3 y1 xT1e ðϕ1 α1 þ ϕ2 α2 þ    þ ϕL αL Þ 6 xT ðϕ1 α1 þ ϕ2 α2 þ    þ ϕL αL Þ 7 6 y2 7 7 ¼ 6 7: 6 2e 5 4⋮5 4 ⋮ T yN xNe ðϕ1 α1 þ ϕ2 α2 þ    þ ϕL αL Þ 2

From the above expression, it is easy to show that 32 3 2 3 y1 α1 xT1e ϕ1 , xT1e ϕ2 ,   , xT1e ϕL 6 xT ϕ1 , xT ϕ2 ,   , xT ϕL 76 α2 7 6 y2 7 2e 2e 76 7 ¼ 6 7 6 2e 54 ⋮ 5 4 ⋮ 5 4 ⋮ αL yN xTNe ϕ1 , xTNe ϕ2 ,   , xTNe ϕL 2

which can be written into the following compact form: H ðX,c,σÞQ ¼ Y

(15)

where X ¼ ½x1 , x2 ,  , xN T  ℝNn , Y ¼ [y1, y2,   , yN]T  ℝN  1, and Q ¼ [αT1 , αT2 ,   , αTL]T  ℝ(n + 1)L  1 is the consequent parameter matrix and H  ℝN  (n + 1)L is the hidden matrix weighted by the fired strength of fuzzy rules given by   3 xT1e f 1 ðx1 , c1 , σ1 Þ,   , xT1e f L x1 , cL , σL  6 xT f 1 ðx2 , c1 , σ1 Þ,   , xT f L x2 , cL , σL 7 2e 2e 7: H ðX,c,σÞ ¼ 6 5 4 ⋮   T T xNe f 1 ðxN , c1 , σ1 Þ,   , xNe f L xN , cL , σL 2

(16)

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From ELM theory, the parameters c and σ in Eq. 16 can be randomly generated and fixed after generation, i.e., the updating of antecedent parameters is not necessary. Usually, the equality in Eq. 15 cannot be obtained due to the limitation of FNN scale. Consider the following minimizing problem: min ðjjHQ  Yjj2 þ λjjQjj2 Þ

(17)

Q

where ||  || denotes the Euclidean norm and λ is a real positive constant. The leastsquares solution of Q in Eq. 17 is as follows:   ^ ¼ H T H þ λI 1 H T Y: Q

(18)

0 Now, give an initial data set N 0 ¼ fðxl , yl ÞgNl¼1 . From Eq. 18, one immediately gets

^0 ¼ K 1 H T Y 0 Q 0 0

(19)

K 0 ¼ HT0 H 0 þ λI

(20)

 T where Y 0 ¼ y1 , y2 ,   , yN 0 , H0 ¼ H ðX0 , c, σÞ, and X0 ¼ ½x1 , x2 ,  , xN0 T . Let y^l be the estimation of yl with l ¼ 1, 2,   . The FNN output at the initial phase is as follows: ^0 Y^0 ¼ H 0 Q

(21)

 T where Y^0 ¼ y^1 , y^2 ,   , y^N0 . Then, present the (k + 1)th chuck of new observations: N kþ1 ¼ fðxl , yl Þg with l ¼ ∑ jk ¼ 0Nj + 1, ∑ jk ¼ 0Nj + 2,   , ∑ jk ¼+ 10Nj, where Nj denotes the number of observations in the (k + 1)th chunk. From Liang et al. (2006), one obtains the RLS solution for Q in Eq. 17 as follows:

^kþ1 Q

K kþ1 ¼ K k þ H Tkþ1 H kþ1   ^k þ K 1 H T Y kþ1  Hkþ1 Q ^k ¼Q kþ1 kþ1

xX k

,  , xXkþ1

(22) (23) T

where H kþ1 ¼ H ðXkþ1 ,c,σÞ , Xkþ1 ¼ N þ1 N j , and j¼0 j j¼0

T yXk , yXk ,   , yXkþ1 Y kþ1 ¼ þ1 þ2 N N N j . The FNN output at the j j j¼0 j¼0 j¼0 learning phase is as follows:

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^kþ1 Y^kþ1 ¼ H kþ1 Q



y^Xk , y^Xk ,  , y^Xkþ1 where Y^kþ1 ¼ þ1 þ2 N N Nj j j j¼0 j¼0 j¼0

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(24)

T .

To avoid the singular problem for the matrix inversion of Kk + 1in Eq. 23 while 1 N0 < L, one makes P0 ¼ K and applies the Woodbury identity to calculate P0 as 0 follows (Huynh and Won 2011):  1 P0 ¼ I=λ  H T0 λI þ H 0 H T0 H0 =λ:

(25)

Similarly, to avoid the ill-posed problem so that the computational cost for the matrix inversion of Kk + 1 in Eq. 23 while Ni  L can be reduced, one makes 1 1 ^ and Pk + 1 ¼ K Pk ¼ K  k k + 1 and applies the updating law of Qkþ1 as follows:  1 Pkþ1 ¼ Pk  Pk H Tkþ1 I þ H kþ1 Pk H Tkþ1 H kþ1 Pk ,   ^k þ Pkþ1 H T Y kþ1  Hkþ1 Q ^kþ1 ¼ Q ^k : Q kþ1

(26) (27)

Multistep Prediction Scheme MHC prediction is essentially an online time-series prediction problem which should carry out updating and prediction concurrently. To carry out multistep direct prediction, consider the nonlinear autoregressive with exogenous input (NARX) model as follows: y s ðk þ r Þ

       nr , ¼ f ys ðkÞ, ys k  r , ys k  2r ,   , ys k   xs ðkÞ, xt k  r , ys k  2r ,   , xs k  nr

(28)

step, n + 1 is the maximum lag, i.e., the order of the system. Then, give a timen0 series data set T ¼ fðxs ðiÞ, ys ðiÞÞg1 i¼1 , its initial set T 0 ¼ fðxs ðiÞ, ys ðiÞÞgi¼1 with n0 > (n + 1)r, and choose the root-mean-square error (RMSE) as the performance index. Based on the proposed learning strategy, the multistep direct prediction scheme of time-series is presented as follows: Step 1) Offline initialization: Obtain the initial 0 N 0 ¼ fðxl , yl ÞgNl¼1 , where N0 ¼ n0  (n + 1)r and      xl ¼ xs ðlÞ, xs l þ r ,   , xs l þ nr ,

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yl ¼ ys ðl þ ð1 þ nÞr Þ:

(30)

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(a) Randomly generate parameters c and σ. (b) Calculate H 0 ¼ HðX0 ,c,σÞ by Eq. 16, where X0 ¼ ½x1 , x2 ,  , xN0 T . ^0 using Eq. 19 with Eq. 20 (if N0  L ) or with Eq. 25 (c) Calculate Q (if N0 < L ).   (d) Calculate the initial training performance: RMSE train Y^0 , Y 0 with Y^0 ¼ H 0  ^0 and Y0 ¼ y , y ,   , y T . Q 1

N0

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(e) Predict the next r step’s time-series:   ^0 : y^N0 þr ¼ H xTN0 þr ,c,σ Q   ^0 . (f) Let Y 10 ¼ yN0 þ1 and Y^10 ¼ y^N0 þ1 ¼ H xTN0 þ1 ,c,σ Q (g) Set the training step: k ¼ 0. Step 2) Online sequential prediction: Present the (k + 1)th training data set   N kþ1 ¼ xN0 þkþ1 , yN0 þkþ1 , where xN0 þkþ1 and yN0 þkþ1 are given by Eqs. 29 and 30, respectively.   (a) Calculate H kþ1 ¼ H xTN 0 þkþ1 ,c,σ by Eq. 16.

 (b) Update the prediction performance by Eq. 31: RMSEPred Y^ðkþ1Þk , Y ðkþ1Þk h iT h T iT ¼ Y Tkðk1Þ , yN 0 þkþ1 , Y^ðkþ1Þk ¼ Y^kðk1Þ , y^N0 þkþ1 , and ^k . y^N0 þkþ1 ¼ Hkþ1 Q ^kþ1 using Eq. 23 with Eq. 22 (if Nk + 1  L ) or by Eq. 27 with (c) Update Q Eq. 26 (if Nk + 1 < L ).   (d) Update the training performance by Eq. 31: RMSEtrain Y^kþ1 , Y kþ1 , Y kþ1  T  ^kþ1 , and Xkþ1 ¼ XT , xN þkþ1 T . ¼ Y T , yN þkþ1 , Y^kþ1 ¼ H ðXkþ1 ,c,σÞQ 0 k

k

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Simulation Studies The applied MHC monitoring data were collected from PRONOSTIA, an experimental platform dedicated to test and validate bearings fault detection, diagnostic, and prognostic approaches (Nectoux et al. 2012). As shown in Fig. 7, the PRONOSTIA is composed of three main parts: a rotating part, a degradation generation part, and a measurement part. The main objective of PRONOSTIA is

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to provide real experimental data that characterize the degradation of ball bearings along their whole operational life. This platform allows accelerating bearing degradation in only few hours. An example of the vibration raw signal gathered during a whole experiment is shown in Fig. 8. The non-trendable and nonperiodical

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statistical properties of this type of signals increase the difficulty of MHC prediction (Porotsky 2012). In this study, two bearing data sets are chosen under the operating conditions: 1,800 rpm speed and 4,000 N load to carry out simulation. For the NARX model in Eq. 28, set n ¼ 1 and r ¼ 1, 2, 5, or 10 and select xs as the standard deviation (STD) of each vibration data set which consists of 2,560 vibration signals and ys as the 5 % trimmed mean of the vibration signal. The prediction procedure is as follows: First, the offline initialization is carried out based on one data set to obtain an intimal FNN model; second, the online prediction is carried out based on another data set to forecast time-series of r steps ahead. To demonstrate the superiority of the proposed EOSL-FNN, the OS-ELM in Liang et al. (2006) and the NARX-NN are selected as the compared methods, where 10 notes are applied to the NARX-NN and 100 notes with λ ¼ 0.001 are applied to the EOSL-FNN and OS-ELM. Two performance indexes, namely, the RMSE and the mean absolute percentage error (MAPE), are defined as follows:   h   i1=2 ^ Y ¼ E Y^  Y 2 RMSE Y, ,

(31)

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MAPE ¼

 1 Xn ^ ð y  y Þ=y t t t  100%: t¼1 n

(32)

The accuracy index is defined as (100 %  MAPE). The initial training and online prediction performance of the proposed EOSLFNN are depicted in Figs. 9, 10, 11, and 12. One observes that high training and predicting accuracy is obtained under small ahead step, and satisfied training and predicting accuracy is still obtained under large ahead step. The performance comparisons of all prediction methods in terms of the time, RMSE, STD, and accuracy are shown in Table 2. Note that the results are obtained from averaging 10 times’ simulation results. One observes that both the EOSL-FNN and the OS-ELM are extremely faster (with small training and predicting time) and more stable (with small STD) than the NARX-NN, the EOSL-FNN performs similar or better (with small RMSE and accuracy) than the NARX-NN and OS-ELM, and the EOSL-FNN performs a little slower (with larger training and predicting time) than the OS-ELM since it contains more adjusting parameters.

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Summary In this chapter, both enhanced online sequential learning-fuzzy neural network (EOSL-FNN) and Weibull modeling are presented for predicting machine health condition and life cycle reliability analysis. The Weibull distribution is among the most popular in the field of life cycle engineering and reliability analysis because it is able to accommodate various types of failure data by manipulation of its parameters. The case study presented in this chapter has successfully shown that Weibull analysis can provide a simple and informative graphical representation of characteristics of life data, especially when the life data sample is small and other statistical tools fail to given useful information. Weibull analysis is able to answer many life cycle engineering problems such as mean life estimation, reliability of products at any operational time and warranty cost estimation, etc. The advantages of Weibull model in life data analysis can be extended to facilitate decision-making processes in many remanufacturing practices such as prediction of the number of cores returned for remanufacturing, estimation of spare parts or remedy resources needed for each failure mode, and so on. Future work

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of this study will include (1) the analysis of incomplete life data for the support of decision-making in product EOL management and remanufacturing processes and (2) comparison and optimization of parameter estimation methods in Weibull model. It is envisaged that the extension of this research will see more robust capabilities of Weibull analysis in life cycle engineering applications. The novel EOSL-FNN models have been developed and successfully applied to predict machine health condition. An online sequential learning strategy based on the ELM is developed to train the FNN. A multistep time-series direct prediction scheme is presented to forecast bearing health condition online. The proposed approach not only keeps all salient features of the ELM, including extremely fast learning speed, good generalization ability, and elimination of tedious parameter design, but also solves the singular and ill-posed problems caused by the situation that the number of training data is smaller than the number of hidden nodes. Simulation studies using real-world data from the accelerated bearing life have demonstrated the effectiveness and superiority of the proposed approach. Further work would focus on bearing long-term condition and remaining useful life prediction using online dynamic FNNs.

r ¼ 10

r¼5

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Step r¼1

NN type ESL-FNN OS-ELM NARX-NN ESL-FNN OS-ELM NARX-NN ESL-FNN OS-ELM NARX-NN ESL-FNN OS-ELM NARX-NN

Training Time (s) 0.0352 0.0312 1.5506 0.0334 0.0250 1.535 0.0388 0.0324 1.6427 0.0295 0.0264 1.5085 RMSE 0.0832 0.0865 0.1153 0.0987 0.1056 0.1220 0.1054 0.1181 0.1644 0.1250 0.1441 0.1255

Table 2 Performance comparisons of all methods STD 54.010e-4 34.100e-4 25.200e-4 5.6765e-4 6.9462e-4 197.00e-4 4.7654e-4 3.7799e-4 1474.0e-4 9.3490e-4 5.6543e-4 101.00e-4

Accuracy (%) 97.197 95.195 94.631 97.120 94.585 94.363 95.078 94.044 94.326 94.418 93.317 94.285

Prediction Time (s) 2.1145 2.0159 4.1824 2.2387 2.1141 4.2151 2.2416 2.1541 4.1434 2.3015 2.2784 4.0014 RMSE 0.2343 0.2641 0.3345 0.2645 0.2837 0.4744 0.3879 0.4562 0.4683 0.4561 0.5441 0.6344

STD 0.0354 0.0254 0.0191 0.0083 0.0232 0.2707 0.0141 0.0342 0.1815 0.0355 0.0341 0.1684

Accuracy (%) 98.565 97.548 96.744 98.018 97.453 95.970 97.365 95.343 95.832 95.096 93.992 94.630

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