Online Support Vector Regression Approach for the ... - IEEE Xplore

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 37, NO. 5, SEPTEMBER 2007

Online Support Vector Regression Approach for the Monitoring of Motor Shaft Misalignment and Feedwater Flow Rate Olufemi A. Omitaomu, Member, IEEE, Myong K. Jeong, Member, IEEE, Adedeji B. Badiru, and J. Wesley Hines

Abstract—Timely and accurate information about incipient faults in online machines will greatly enhance the development of optimal maintenance procedures. The application of support vector regression to machine health monitoring was recently investigated; however, such implementation is based on batch processing of the available data. Therefore, the addition of new sample to the already existing dataset requires that the technique should retrain from scratch. This disadvantage makes the technique unsuitable for online systems that will give real-time information to field engineers so that corrective actions could be taken before there is any damage to the system. This paper presents an application of accurate online support vector regression (AOSVR) approach that efficiently updates a trained predictor whenever a new sample is added to the training set using shaft misalignment and nuclear power plant feedwater flow rate data. The results show that the approach is effective for online machine condition monitoring where it is usually difficult to obtain sufficient training data prior to the installation of the online systems. Index Terms—Data mining, machine health monitoring, nuclear power plant, online condition monitoring systems.

I. INTRODUCTION IMELY and accurate information about incipient faults in machines brings about a reduction in production costs by reducing machine downtime, avoiding overstocking of spare parts, enhancing product quality, and improving workers’ learning curves. It also increases customers’ reliability on prompt products delivery and improves productivity. There are basically two reasons for designing and scheduling machine maintenance in order to achieve these objectives: 1) to satisfy an imposed regulatory requirement; and 2) to enhance the reliability of deteriorating machine. The former reason is well established, and companies usually follow it dutifully. The second reason is the focus of several condition monitoring models in the literature.

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Manuscript received February 15, 2005; revised June 1, 2006. This work was supported in part by the National Science Foundation under Grant CMMI0644830. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the Air Force Institute of Technology, the Oak Ridge National Laboratory, UT-Battelle, U.S. Air Force, Department of Defense, Department of Energy, or the U.S. Government. This paper was recommended by Associate Editor D. McFarlane. O. A. Omitaomu is with the Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 3783-6017 USA (e-mail: [email protected]). M. K. Jeong is with the Department of Industrial and Information Engineering, University of Tennessee, Knoxville, TN 37996 (e-mail: [email protected]). A. B. Badiru is with the Department of Systems and Engineering Management, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH 45433-7765 (e-mail: [email protected]). J. W. Hines is with the Department of Nuclear Engineering, University of Tennessee, Knoxville, TN 37996 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCC.2007.900648

An effective maintenance system depends on a reliable, prompt, and accurate condition monitoring technique. Several quantitative and qualitative models have been developed for condition monitoring. The various quantitative models are based on linear and nonlinear techniques including principal components regression (PCR), partial least squares (PLS), kernel regression (KR), and artificial neural networks (ANN). Two of the common qualitative models are expert systems (ES) and qualitative trend analysis (QTA). A comprehensive review of these models and their applications in condition monitoring is presented in [1]. In addition, an overview of PCR and PLS in condition monitoring is given in [2]–[5]. However, these techniques vary in their accuracy, prediction efficiency, robustness, scalability, retraining ability, and transparency. As a result of the changing world of manufacturing, maintenance strategies are changing from periodic inspection strategies to condition-based predictive maintenance strategies [6]. Condition monitoring can enhance machine availability by providing field engineers with timely and accurate information about incipient failure while the machine is running. This process avoids offline inspections, which leads to a reduction in production costs. These benefits motivate the development of an online condition-monitoring methodology for machine diagnosis. Condition monitoring problems are based on continuously collected data where sample size is usually sparse. In addition, the condition-monitoring learning concepts evolve over time. One technique that has been outstanding for regression problems, in cases where sample data is sparse, is support vector regression (SVR). The SVR technique has been applied successfully to a wide range of pattern recognition and prediction problems [7], [8], and its application in rotating machinery using high-dimensional power spectrum data has been investigated [9]. However, the addition of a new data sample to the already existing dataset requires that support vector regression must retrain from scratch. To handle the problem of retraining the entire training set each time, a new sample is added to the training set or a training sample is modified or removed from the set, Ma et al. [10] present accurate online support vector regression (AOSVR) algorithm that combines the advantages of SVR with the capability of efficiently updating trained support vectors whenever a sample is added to the training set. These enhanced features motivate the implementation of the AOSVR technique for machine condition monitoring with applications to motor shaft misalignment and nuclear power plant feedwater flow rate data. From our understanding, this is the first attempt to use AOSVR to solve machine condition-monitoring-related

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OMITAOMU et al.: ONLINE SUPPORT VECTOR REGRESSION APPROACH

problems. Previous applications are in the area of time series predictions [10]. In order to make the AOSVR algorithm suitable for online condition monitoring implementations, we compute SVR parameters using the training data as explained in Section III. Other potential applications of AOSVR for machine condition monitoring system are in the chemical process plants, aircraft industry, and oil and gas industry. The online parameter updating is especially important in areas where it is expensive to monitor the condition of some major upstream machines on a periodic basis but where data from other upstream or downstream machines or processes can be collected periodically, and used to predict the condition of these major upstream machines in real time. Therefore, an online condition monitoring system has the benefits of the traditional methods of detecting and correcting a malfunctioned system, and the additional benefit of obtaining such information in real time. Because the machine monitoring system will give timely and accurate information about incipient failure conditions, field engineers can develop and schedule appropriate preventive maintenance in a timely manner with due consideration for other factors that influence preventive maintenance development and schedule. These factors include r the level of the required maintenance tasks (easy, difficult, or extremely difficult); r the available resources (time, money, materials, and personnel); r the significance of the affected machines (downstream or upstream; top- or low-priority machine) and the production schedule, the anticipated downtime, and the allowable downtime. Therefore, some of the anticipated benefits of instituting an online SVR methodology, which are also some of the benefits of a functional predictive maintenance, include [11] r an increase in the availability and safety of the plant and workers; r improvements in the quality of products and maintenance processes; r an improvement in the quality of the available information about machine failures, maintenance activities, and support services in the design and improvement of future machines; r a reduction in maintenance costs. The objective of this paper is to assess the applicability of using AOSVR as an online machine condition monitoring tool, and compare its performance to conventional SVR for the given data. The remainder of this paper is organized as follows. Section II gives an overview of SVR and AOSVR. Two experimental applications of the approach are presented in Section III. Section IV gives the conclusion. II. REVIEW OF ONLINE SVR A detailed version of the SVR and AOSVR presentations can be found in [10] and [12], respectively. The SVR is a special implementation of the support vector machine (SVM) technique for predictive data analysis [12], [13]; therefore, it shares many of the advantages of SVM [10]. The SVM is an approximation technique that arose from statistical learning theory, and are used for learning classification and regression rules from data [14].

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Fig. 1.

Illustration of ε-tube for support vector regression (adopted from [13]).

They are based on the structural risk minimization principle, which incorporates capacity control that prevents overfitting of the input data [15]. The SVM has a sound orientation toward real-world applications [16]. The two key features in the implementation of SVM are quadratic programming and kernel functions. The parameters are obtained by solving a quadratic programming problem with linear equality and inequality constraints. The flexibility of kernel functions allows the technique to search a wide range of the solution space. The objective of a regression problem is to determine a function that approximates future values accurately and smoothly. Consider a set of training data T = {(x1 , y1 ), . . . , (xl , yl )} where xi ∈ RN is the input space of the sample, yi ∈ R is the corresponding target, and N is the size of the training data. For nonlinear SVR, f (x) on a feature space F is given by f (x) = WT Φ (x) + b.

(1)

The goal is to find W (the weight vector) and b (the bias) where W ∈ RN , b ∈ R, and Φ(x) maps the input x to a vector in F . The W and b in (1) can be obtained by solving the following optimization problem using ε-insensitive loss function [13]: 1 T W W+C (ξi + ξi∗ ) 2 i=1 l

Minimize P = subject to

yi − (WT Φ(x) + b) ≤ ε + ξi (WT Φ(x) + b) − yi ≤ ε + ξi∗

(2)

ξi , ξi∗ ≥ 0, i = 1, . . . , l where C > 0 is a prespecified value representing penalty weight. According to (2), all data points whose y-values differ from f (x) by more than ε are penalized. The slack variables ξi and ξi∗ correspond to the size of the difference for lower and upper deviations, respectively, as represented graphically in Fig. 1. The ε-tube is the largest deviation, and all the data points inside this tube do not contribute to the regression function. Other points are used for learning and they are called support vectors. The corresponding Lagrangian of (2) can be written as LP =

l l 1 T W W+C (ξi + ξi∗ ) − (ηi ξi + ηi∗ ξi∗ ) 2 i=1 i=1

−

l i=1

αi ε + ξi + yi − WT Φ (xi ) − b

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−

l

αi∗

ε+

ξi∗

− yi + W Φ (xi ) + b T

where θi determines both αi and αi∗ . In addition, they define a margin function h(xi ) for the i th sample xi as [10]

i=1

s.t. αi , αi∗ , ηi , ηi∗ ≥ 0

(3)

where α, α∗ , η, η ∗ are the Lagrange multipliers. Equation (3) in turn leads to the dual given by l l 1 ∗ ∗ i=1 j=1 Qij (αi − αi ) αj − αj Maximize D = 2 +ε li=1 (αi + αi∗ ) − li=1 yi (αi + αi∗ ) l ∗ i=1 (αi − αi ) = 0 subject to (4) ∗ 0 ≤ αi , αi ≤ C where Qij = Φ(xi )T Φ(xj ) = K(xi , xj ). The kernel function K(xi , xj ) must be positive definite in order to guarantee a unique optimal solution to the quadratic optimization problem. This kernel function allows nonlinear function approximations with SVR technique while maintaining the simplicity and computational efficiency of linear SVR approximations. Some of the common kernel functions are linear xTi , x, polynomial xi , xd , and radial basis function (RBF) exp(−xi − x2 /2 p2 ), where p is the width parameter. The Lagrange of (4) can be represented as [10]

+ε

l

(αi −αi∗ )−

i=1

+

l

l

yi (αi − αi∗ )−

i=1

l

(δi αi −δi∗ αi∗ )

i=1 l

(αi − αi∗ )

i=1

(5) (∗)

(∗)

where δi , ui , and ζ are the Lagrange multipliers. Optimizing (5) leads to the following Karush–Kuhn–Tucker (KKT) conditions [10] ∂LD = Qij αj − αj∗ + ε − yi + ζ − δi + ui = 0 ∂αi j=1 l

l

∂LD =− Qij αj − αj∗ + ε + yi − ζ − δi∗ + u∗i = 0 ∗ ∂αi j=1 (∗)

≥ 0,

(∗)

≥ 0,

δi

ui

(∗) (∗)

δi αi = 0 (∗) (∗) αi − C = 0 ui

(6)

where ζ in (6) is equal to b in (1) at optimality [18]. According to the KKT conditions in (6), at most one of αi and αi∗ will be nonzero, and both are nonnegative. Therefore, a coefficient difference θi was defined in [7] as θi = αi − αi∗

Qij θj − yi + b.

(8)

j=1

By combining (6), (7), and (8), the following five conditions were obtained [10]: h(xi ) ≥ε,

θi = −C

h(xi ) =ε,

−C < θi < 0

−ε ≤ h (xi ) ≤ε,

θi = 0

h(xi ) = −ε,

0 < θi < C

h(xi ) ≤ −ε,

θi = C.

(9)

These five conditions are used to identify the subset that each sample in the training set can be classified. To achieve this, the five conditions in (9) can be represented by three subsets [10]. 1) E Set: Error support vectors E = {iθi | = C}

(10)

2) S Set: Margin support vectors (11)

3) R Set: Remaining samples

[ui (αi − C) + u∗i (αi∗ − C)] + ζ

i=1

l

S = {i|0 < |θi | < C}

1 = (αi − αi∗ ) αj − αj∗ 2 i=1 l

LD

h(xi ) ≡ f (xi ) − yi =

(7)

R = {i|θi = 0}.

(12)

The algorithm starts the initialization of the predictor with some samples (in this case, two samples) to generate SVR coefficients, and use these coefficients to train the remaining part of the training segment using three samples at a time. When the new sample is not a support vector, i.e., the sample lies inside the ε-tube, then the sample can be safely added without influencing the regression tube, and no parameter update is necessary. However, if the added sample falls outside the ε-tube, the regression parameters must be incrementally increased or decreased using the incremental algorithm as proposed in [10]. Once the training is completed, the online testing can start. III. CONDITIONS AND PROCEDURES FOR ONLINE CONDITION MONITORING Online monitoring of large machines and complex processes continue to gain popularity in order to improve reliability and avoid sudden failures. Online monitoring is a continuous monitoring process in which regular information is collected for fault identification or process malfunctioning. Online condition monitoring is also called real-time condition monitoring. Therefore, an online condition monitoring process involves using newly acquired samples to update the predictor, without the need to retrain from scratch. However, it is usually difficult to obtain sufficient training data prior to the installation of the online condition monitoring system. The AOSVR technique facilitates gradual improvement of system performance in real


time as new samples are acquired. That is, a model is updated using incoming data and at the same time makes predictions based on that model. Therefore, some of the conditions for implementing AOSVR for online condition monitoring are given. r There is small sample size, and it is not possible to acquire more training sample before the installation of the system. r The addition, updating, and removal of data points are frequent and real-time decision is needed. r When the systems operate automatically and/or critical safety issues are concern, sensor data can be processed in real time for decision making before major damages would occur to the systems. To implement the AOSVR algorithm for online condition monitoring, the following procedure is used. r The type of kernel to be used and the value of the kernel parameter(s) are selected. r The dataset is scaled with respect to the selected kernel. r The values of SVR hyperparameters (C and ε) are selected based on the users’ knowledge of the characteristics of the data. r A few data samples are collected for training purposes, and are used to initialize the AOSVR algorithm. The remaining data samples are used for the online testing. Some of the potential areas of application of AOSVR technique for machine condition monitoring are in the nuclear power plant, chemical process plants, oil and gas industry, and aircraft industry. An online condition monitoring system can also be used in situation where it is expensive to monitor some major machines on a periodic basis, but where data from upstream or downstream machines, sensors, or processes can be used to predict the performance of such major machines; this is possible because of the correlation between the variables of these machines or sensors. Another application of this methodology is in rotating machinery for motor shaft misalignment prediction. Therefore, Section IV describes the application of this methodology to shaft misalignment prediction and feedwater flow rate data in order to show the advantages of the AOSVR approach. IV. SHAFT MISALIGNMENT PROBLEM A typical mechanical system consists of a driver machine, a driven machine, and a coupling as depicted in Fig. 2. The main function of any coupling is to transfer rotating power from the driver machine to the driven machine; therefore, the objective of any coupling maintenance effort is to maximize this power transfer in order to increase the lifespan of rotating machinery [19]. A coupling could be a rotating shaft, rigid or elastic joints, or belt and gear trains. A shaft transmission system is one of the most fundamental and important parts of rotating machinery; the ability to detect and predict shaft misalignment accurately can significantly enhance the predictive maintenance task of a mechanical system. A proper shaft alignment is inevitable because it reduces excessive axial and radial forces on the most vulnerable parts of a machine system such as the bearings, seals, and couplings [19], [21]. It also minimizes the amount of shaft bending, thereby, permitting full transmission of power from the driver machine to the driven

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Fig. 2. [20]).

Schematic diagram of a driver-coupling-driven system (adapted from

Fig. 3. Illustration of parallel, angular, and combined misalignment conditions.

machine, and eliminates the possibility of shaft failure from cyclic fatigue. In addition, it minimizes the amount of wear in the coupling components, reduces mechanical seal failure, and lowers vibration levels in machine casings, bearing housings, and rotor [19]. The understanding of shaft misalignment problem and its effect on machine vibration have been presented in [6] and [22]–[24]. Therefore, monitoring and predicting shaft alignment condition is important in order to make intelligent decisions on when to perform coupling maintenance, which plays an essential role in increasing rotating machinery reliability and reducing maintenance costs. Shaft misalignment is one of the prevalent faults associated with rotating machines, and it occurs when the shaft of the driven machine and the shaft of its driver machine do not rotate on a common axis; that is, the shafts are not coaxial. Shaft misalignment is a measure of how far apart the two centerlines are from each other [19], [20], [25]. Such shift in centers can be in offset position (when the centerlines of the two shafts are parallel with each other, but at a constant distance apart), in angular position (when the centerlines are at an angle to each other), or a combination of these positions [6], [19] as shown in Fig. 3, where v is the vertical displacement in parallel alignment and a is the angular displacement in angular alignment. Misalignment of shafts causes noise and vibration, which leads to accelerated wear. The noise and vibration are the symptoms of a misalignment situation, while the wear is the effect that should be avoided [21]. Even though a perfect shaft alignment is unlikely during operating cycles, there is a limit to the maximum amount of misalignment that is allowable. Therefore, shaft alignment can be classified into four grades: unsafe, poor, acceptable, and excellent [21]. Two of these grades (poor and

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unsafe) are wake-up calls for maintenance actions. A shaft is said to be in a poor grade when the alignment is within the manufacturer’s allowances but outside the machine’s recommended limits; so it is a warning grade. An unsafe grade means the alignment is outside the manufacturer’s design specification allowance and must be attended to. Industry invests significant amount of time and money on shaft misalignment on a yearly basis because it causes a decrease in motor efficiency and makes the machine more prone to failure due to increased loads on the shaft support devices such as the bearings, seals, and couplings. Frequent failures of the shaft support devices are indicators for alignment work; however, such failures can be very expensive in terms of economic and personnel obligations. It has been generally agreed that proper alignment is critical to the life of a machine since coupling wear or failure, bearing failures, bent rotors, and bearing housing damage are direct evidence of poor shaft alignment [26]. One way to overcome this problem and reduce its associated cost is to monitor the condition of the shaft position during machine operations, and collect data to predict the state of the shaft.

Fig. 4.

Plot of the input data for one of the misalignment conditions. TABLE I MANUFACTURERS’ TOLERANCES FOR THE COUPLINGS USED [30]

A. Experimental Setup The data used was collected using a three-phase 50-hp ac motor attached through two different types of flexible couplings (a Zurn gear coupling and a Rexnord elastomer coupling) to a 150-hp dynamometer at controlled parallel and angular misalignment conditions. Data for the input currents and voltages at these misalignment conditions were stored on digital tape using a data recorder. The motor was attached to a 20 in × 20 in × 8 in block of aluminum. There were two threaded rods fitting into the aluminum block that were used to adjust the motor’s position in the horizontal plane. The misalignment measurements were made using a laser alignment system with the motor in off line position. Two minutes of analog data were recorded for each of the coupling type for each alignment condition. The digital tape recorder was then connected to a computer to convert the analog data to digital computer media. The data was acquired by digitizing two of the three phase currents and voltages off of the digital tape at a sampling rate of 6000 Hz. The third of the three phase currents and voltages were calculated using a variation of the Ohm’s law (Power = Voltage × Current). Advanced signal processing techniques were then used to transform the raw voltage and current data into the power time waveform. Therefore, for this analysis, the input is the motor power time waveform and the output is the misalignment condition. The size of the dataset is 18 × 72 000 for the elastomer coupling and 10 × 72 000 for the gear coupling. The output data for the gear coupling ranges from 0 to 50 mils for parallel misalignment and from 0 to 15 mils/in for angular misalignment. For the elastomer coupling, the output data ranges from −45 to 90 mils for parallel misalignment and from 0 to 40 mils/in for angular misalignment. Fig. 4 is a time waveform plot of the input data for one of the misalignment conditions. The outputs in this case are 0 mils for parallel misalignment (perfect parallel condition) and 30 mils/in for angular misalignment (imperfect angular condition) for the elastomer coupling.

B. Data Modeling A shaft misalignment monitoring index (SMMI) is calculated to represent the misalignment condition of each output data [9]. The SMMI is defined as the relative alignment condition SMMI =

˙ X(t) ˜ X(t)

(13)

˙ where X(t) is the actual instantaneous (recorded or predicted) ˜ is the maximum allowable alignment. If the alignment and X(t) SMMI is much greater than 1.0, the shaft grade can be said to be within an unsafe grade; and when it is approximately 1.0, the shaft grade is in a poor grade, which is a warning index. The manufacturers’ specifications for maximum parallel and angular misalignment information are given in Table I [30]. The dataset is preprocessed (scaled) in relation to the chosen kernel. Scaling the data also helps avoid numerical difficulties during the computations. The datasets were divided into training and test sets using approximately two-thirds of the data for training and approximately one-third for testing. The idea for dividing the dataset was adapted from [31], which suggests splitting datasets for bootstrapping method according to the ratio −0.632 for training and 0.368 for testing. The AOSVR software used was adapted from [10]. All computations were carried out in MATLAB environment. The developed models are evaluated using the mean square error (MSE) defined as MSE =

(yi − yî )2 n

(14)


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TABLE II SHAFT MISALIGNMENT PREDICTION RESULTS BASED ON MSE

where yî is the predicted value of the actual output value yi . C. Experimental Results The conventional SVR trains with the entire training set at a time and test, with the entire test set at a time. The AOSVR approach, on the other hand, starts training with two training samples in order to generate the SVR coefficients that are then used to train the remaining samples of the training set using three samples at a time. The elastomer coupling model uses 12 samples for training and six samples for testing; whereas the gear coupling model uses seven samples for training and three samples for testing. The testing is also done online using one test sample at a time. The SVR models were developed using RBF kernel of width 2.0, ε-insensitive loss function C = 0.6573, and ε = 0.3857. The values of the SVR metaparameters (C and ε) used are selected using the approach proposed in [32]. The results generated for elastomer and gear couplings using online SVR and conventional SVR are presented in Table II. From Table II, the MSE for both parallel and angular misalignment is about 0.09. Therefore, the predictions in both the cases are of about the same value. This is a good average error range for online systems. Using the conventional SVR, the average error range is from 0.11 for parallel alignment to 0.13 for angular alignment. The results further show that the range of mean error using AOSVR technique is about 0.20 for angular alignment and 0.27 for parallel alignment. Using conventional SVR, the mean error range is about 0.26 for angular alignment and 0.34 for parallel alignment. It can also be seen from the results in Table II that AOSVR performs better than conventional SVR in all cases. In addition, elastomer coupling has smaller average error; this performance may be attributed to the fact that elastomer coupling has greater number of training samples than gear coupling, since SVM performance depends on the number of samples. However, AOSVR approach gives better prediction errors than SVR. V. FEEDWATER FLOW RATE PROBLEM In nuclear power plant, the correct prediction of an important variable, such as feedwater flow rate, can reduce periodic monitoring. Such prediction can be used to access sensor performance, thereby, reducing maintenance costs and increasing reliability of the instrument. Feedwater flow rate directly estimates the thermal power of a reactor. Nuclear power plants use venturi meters to measure feedwater flow rate. These meters are sensitive to measurement degradation due to corrosion products in the feedwater [33]. Therefore, measurement error due to feedwater fouling results in feedwater flow rate overestimation. As

Fig. 5.

Plot of the original feedwater flow rate signals.

a result, the thermal power of the reactor is also overestimated, and the reactor must be adjusted to stay within regulatory limits, which is an unnecessary action and involves unnecessary costs. To overcome this problem, several online inferential sensing systems have been developed to infer the “true” feedwater flow rate. They include using neural nets [34], multivariate state estimation techniques [35], and regularization techniques [36], [37]. Inferential sensing is the use of correlated variables for prediction. Inferential measurement is different from conventional prediction where a parameter value is estimated at time tn+1 , based on the information about other parameters at time tn . In inferential measurements, a parameter is estimated at time tn based on the information about other parameter also at time tn . Inferential sensing is an inverse problem, and it can produce inconsistent results due to minor changes in the data. A detailed description of this problem is available in [33]. To further illustrate the advantages of AOSVR approach for machine condition monitoring related problems, a set of realworld operating data from a nuclear power plant feedwater-flow venturi meter is used. This data contains 24 selected variables based on engineering judgment and on their high correlation with feedwater flow variable. Therefore, the problem may be ill posed [38], as there may not be enough information to estimate a true value of the feedwater flow. In addition, different empirical model metaparameters can provide different information about the true value of the feedwater flow rate. Furthermore, the estimations can be unstable due to small perturbations in the input data or due to different preprocessing techniques. These are some sources of uncertainty in the estimation of the feedwater flow rate. A plot of the original values of the response variable (feedwater flow rate) is shown in Fig. 5. The dataset is preprocessed (scaled) in relation to the property of the chosen kernel (radial basis function kernel). This helps eliminate the effect of any influential observations and helps avoid numerical difficulties during computations. The dataset is trained with different number of data points and tested with 700 data points. The values of the SVR metaparameters (C and ε)

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TABLE III FEEDWATER FLOW RATE PREDICTION RESULTS BASED ON DRIFT VALUES

Fig. 6.

Plot of the feedwater flow rate prediction for test data.

are 10 and 0.1, respectively. The AOSVR procedure was implemented using radial basis function kernel of width 2.0. The predictions from both SVR and AOSVR procedures were evaluated using the mean of the sum of differences between the predicted values and the measured data, which is called drift. The measured values are as recorded by the Venturi meter. The results of the implementation are shown in Table III. The experimental results for feedwater flow rate in Table III show that AOSVR is more effective for predicting feedwater flow rate data than SVR. The reduction in drift for using AOSVR is more than 100% for each training set. It can also be seen that the value of drift is proportional to the number of training data points; even though, the number of test data remains constant (700 data points) in all cases. This shows that the models are not stable. The instability in the models may be attributed to the use of the same value for the metaparameters for all steps of training in case of AOSVR. However, the AOSVR approach predicts better than the conventional SVR using the same value of metaparameters; and the instability in the results is approximately constant, which further supports the conclusion from the shaft misalignment problem. Furthermore, the feedwater flow rate problem is an online system rather than a batch processing system. The ability to predict the feedwater flow rate using downstream sensor data can greatly enhance the maintenance design and implementation of the feedwater venturi meter in the nuclear power plant. A plot of the predicted and the actual feedwater flow rate prediction for data points 8001–8700 using AOSVR and 300 training samples is shown in Fig. 6.

VI. CONCLUSION The online SVR approach was implemented for machine condition monitoring with application to motor shaft misalignment and feedwater flow rate data. Our results show that online SVR performs better than conventional SVR technique, which indicate that it learns over time with the addition of new samples to the training dataset. In addition, online SVR, like conventional SVR, is a universal function approximator since it can handle both linear and nonlinear functions effectively. The online SVR methodology has no limit on the dimensionality of the input space, which indicates that it can be used in several practical cases with smaller number of samples and larger number of variables (small n, large p problems). The capability of the online SVR methodology to update trained support vector allows the addition of new samples without retraining the predictors from scratch. These benefits are the outstanding features of this approach, which make it suitable for online condition monitoring system that will give timely and accurate information about an incipient machine failure. One major benefit of these results is that real-time monitoring information can be obtained much more quickly during operation rather than during shut down, and field engineers can be alert of any impending damage before it actually occurs. An implementation of the approach was presented for motor shaft misalignment predictions using data collected under laboratory conditions; this kind of data may be difficult to obtain in the real world because there may be no suitable instruments for measuring shaft alignment positions when the machine is in operation, but it may be useful in a research and development setup. Therefore, the approach was also implemented for real-world data collected in a nuclear power plant. The approach is also suitable for other online condition monitoring situations. For example, real-time tool condition monitoring for automatic cutting processes [39], machine axial fluxes [40], and direct measurement of rotor variables [41]. Such data can be used in an online condition monitoring system to generate real-time information using AOSVR approach. One downside of the AOSVR approach is that it uses fixed values for the SVR metaparameters, even though the characteristics of the incoming data may not be fixed. Therefore, future research effort in this area is to develop techniques for computing adaptive SVR parameters in online systems so that the online SVR approach can use adaptive metaparameters rather than fixed metaparameters. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their comments in improving the quality of this paper. REFERENCES [1] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, “A review of process fault detection and diagnosis. Part III: Process history based methods,” Comput. Chem. Eng., vol. 27, pp. 327–346, 2003. [2] J. F. MacGregor, C. Jacckle, C. Kiparissides, and M. Koutondi, “Process monitoring and diagnosis by multiblock PLS methods,” Am. Inst. Chem. Eng. J., vol. 40, no. 5, pp. 826–838, 1994.


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Adedeji B. Badiru received the B.S. degree in industrial engineering, the M.S. degree in mathematics, and the M.S. degree in industrial engineering from the Tennessee Technological University, Cookeville, and the Ph.D. degree in industrial engineering from the University of Central Florida, Orlando, in 1979, 1981, 1982, and 1984, respectively. Currently, he is the head of systems and engineering management at the Air Force Institute of Technology, Wright Patterson Air Force Base, OH. Earlier, he was the department head of industrial and information engineering at the University of Tennessee, Knoxville, and previously a Professor of industrial engineering and Dean of University College at the University of Oklahoma, Norman. His research interests include artificial intelligence, expert systems, and project management. Prof. Badiru is a Fellow of the Institute of Industrial Engineers and the Nigerian Academy of Engineering. He is a Registered Professional Engineer.

J. Wesley Hines received the B.S. degree in electrical engineering from Ohio University, Athens, in 1985, and the M.B.A., M.S., and Ph.D. degrees in nuclear engineering from The Ohio State University, Columbus, in 1992 and 1994, respectively. He is currently a Professor in the Department of Nuclear Engineering at the University of Tennessee, Knoxville. He was a nuclear qualified submarine officer in the U.S. Navy. His research interests include advanced statistical and artificial intelligence applications in process monitoring and diagnostics. Prof. Hines is a member of the American Society of Mechanical Engineers, the American Nuclear Society, and the American Society for Engineering Education.