Incipient Sensor Fault Diagnosis Based on Average Residual

Article pubs.acs.org/IECR

Incipient Sensor Fault Diagnosis Based on Average Residual-Difference Reconstruction Contribution Plot Jiyang Xuan, Zhengguo Xu,* and Youxian Sun State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China ABSTRACT: This paper presents a novel method to diagnose incipient single sensor fault for data-driven process monitoring. As the traditional fault detection methods using statistical indices are not sensitive to incipient faults and the reconstruction-based contribution plot (RBCP), which reduces the fault smearing effect of the traditional contribution plot (TCP) also does not take incipient faults into consideration, a new contribution plot based on average residual-difference and reconstruction is defined to detect incipient faults with high detection sensitivity and to overcome two main disadvantages, lack of consideration of contribution plot in the normal condition and using the data at one sampling time, of the RBCP when incipient faults are treated. Then, fault magnitude can be estimated after the correct identification without the need of a fault direction set, which is different from the fault reconstruction method. The effectiveness of the proposed fault diagnosis method is verified by a Monte Carlo numerical simulation and the benchmark quadruple-tank process.

1. INTRODUCTION Multivariate statistical process monitoring (MSPM) techniques have been applied successfully to monitoring chemical processes.1−3 The key to MSPM is to detect faults early and to identify faults correctly, as faults can cause processes to deviate from their normal operating conditions. Principal component analysis (PCA) has been widely used to detect abnormal operating conditions utilizing process information acquired from historical process data, and it is capable of handling high dimensional, noisy, and correlated data by projecting the data onto a lower dimensional subspace, which explains the most pertaining features of the process. When a fault is detected, it is necessary to find the root cause of this fault, so MacGregor and Kourti4 proposed the traditional contribution plot (TCP) used for fault identification, and this method has been widely used as well. The TCP reveals which variable contributes most significantly to some multivariate statistical indices, such as Hotelling’s T2 and squared prediction error (SPE). The assumption behind the TCP is that faulty variables have large contributions to the fault detection indices. There are, however, reports that the TCP involves fault smearing effect which can lead to misdiagnosis. Alcala and Qin5 showed that the TCP approach failed to guarantee the correct identification of the faulty variable. Furthermore, they proposed the reconstruction-based contribution plot (RBCP) to locate the faulty variable without the fault smearing effect. The contribution plot method is an effective way, and another available method for fault identification is fault identification via reconstruction developed by Dunia and Qin,6,7 which can guarantee correct fault diagnosis provided that the fault direction is known and is in the candidate set of fault directions. After detecting and identifying a fault, it is necessary to estimate the fault magnitude finally, which can help operators to know how serious the fault is. However, most of the existing fault estimation methods are model-based, and data-driven methods are scarce. Fault estimation based on fault reconstruction theory is a feasible way that requires a set of known fault directions.6−8 In addition, without the assumption that the fault © 2014 American Chemical Society

magnitude was far greater than the normal measurement, which was an essential requirement for fault reconstruction method to extract process fault direction, Zhang et al.9 estimated the fault magnitude by solving a unary quadratic equation after getting the fault direction. There are different types of faults in chemical processes, such as sensor faults, actuator faults, and process faults. Incipient faults that happen in the initial phase of these mentioned faults are difficult to be detected, as the traditional PCA-based statistical indices are not sensitive enough to these faults with small magnitudes. A single sensor with an incipient bias fault is considered in our work. In this paper, a numerical example is first used to show that the detecting methods adopting common PCA-based statistical indices fail to detect incipient sensor faults, and the RBCP cannot identify incipient faults correctly. Two drawbacks can be found in the RBCP method if we take incipient sensor faults into consideration. First, the RBCP method does not consider the contribution plot of the normal condition, which is also vital as they can affect the contribution values of the observed variables. Second, the RBCP is derived at one sampling time, so the stochastic noises could decrease their effectiveness for identifying incipient sensor faults. To address these problems mentioned above, we propose a new contribution plot named average residual-difference reconstruction contribution plot (ARdR-CP). When dealing with sensor faults, the ARdR-CP has the capability of detecting and identifying faults simultaneously. After the faults are detected and identified, we directly utilize the fault estimation equation derived from the fault reconstruction method to estimate faults, but our fault diagnosis method does not include the procedure of fault identification via reconstruction that needs a known fault direction set. Received: Revised: Accepted: Published: 7706

November 14, 2013 February 8, 2014 April 10, 2014 April 10, 2014 dx.doi.org/10.1021/ie403857f | Ind. Eng. Chem. Res. 2014, 53, 7706−7713

Industrial & Engineering Chemistry Research

Article

The contributions of our work are as follows: (1) We propose a new ARdR-CP that can be used for incipient sensor fault detection with high sensitivity and reduces the false identification compared with the RBCP when incipient sensor faults occur. (2) We replace the step of fault identification via reconstruction by the ARdR-CP to perform fault estimation, thus the predefined fault set is not necessary and fault estimation will be faster. The rest of the paper is organized as follows. Some basic theories are reviewed in section 2. Section 3 uses a numerical example to illustrate the limitations of some existing methods about fault detection and identification; then we propose the newly defined ARdR-CP and the procedure of incipient sensor fault diagnosis. In section 4, two simulations, including a Monte Carlo simulation and the benchmark quadruple-tank process, are illustrated to verify the effectiveness of the proposed method. In the end, conclusions are given in section 5.

2.1.2. Fault Detectability. A natural question about a multivariate statistical method for fault detection is whether all possible faults can be detected. Fault detectability for datadriven process monitoring was addressed for the SPE-based index by Dunia and Qin.6−8 In the presence of a single sensor fault, the sample vector xf can be represented using the fault magnitude f i and the fault direction vector ξi that is the ith column of the identity matrix

2. BASIC THEORIES 2.1. Fault Detection. 2.1.1. Principal Component Analysis. The use of PCA for monitoring process operation is now well established.10,11 Provided with the historical data collected from the normal process condition, most commoncause variations in a process can be captured. An essential requirement for the normal data is that the data should be so rich in normal variations that they can be representative to the normal variability of the process. Let Xn ∈ 9 n × m denote the raw data matrix with n samples (rows) and m variables (columns). Xn is first scaled to a matrix X with zero mean and unit variance. The scaled matrix X is decomposed as follows:

(6)

X = TPT + E

x f = x n + fi ξi

where xn represents the measurement vector under the normal operating condition, and it is unknown when a fault has occurred. To guarantee that the fault can be detected by the SPE, it is required that SPE ≤ δα2 for all possible normal values of xn, which leads to the following inequality6

|fi | >

(1)

m

SPE =

m

∑i = 1 λi

(2)

where λ1,...,λl are the l largest eigenvalues of the covariance matrix S = (1/(n − 1))XTX in the descending order. After establishing a principle component model, to detect possible faults indicated by a new sample vector x, Hotelling’s T2 and the SPE are often used. In the previous literature, the SPE was often applied instead of the T2 statistic to test the breakdown of the sensor correlations, avoiding the ambiguity of the T2 statistic in discriminating faults from normal changes. The SPE statistic indicates how well each sample conforms to the principle component model, measured by the projection of the sample vector on the residual subspace 2

⎡ c11 ̃ ··· c1̃ m ⎤ ⎢ ⎥ C̃ = ⎢ ⋮ ⋱ ⋮ ⎥ ⎢⎣ c ̃ ··· c ̃ ⎥⎦ m1 mm

(8)

If the ith sensor has a bias fault and the fault magnitude is sufficiently large, false identification will happen when c̃ii < c̃ji(i ≠ j). That is, a fault in the ith sensor is smeared into the jth contribution. When an incipient sensor fault is taken into consideration, the situation will be much more complicated because it can also cause incorrect diagnosis if c̃ji is less than but close enough to c̃ii. The reconstruction-based contribution of the variable xi to the SPE, 5 RBCSPE i , is expressed as

(3)

The process is considered normal if SPE ≤ δα2

(7)

where x̃2i is the contribution to the SPE from the ith variable. Although the TCP approach is popular, there are, however, reports that the TCP involves the fault smearing effect that can lead to misdiagnosis.1,17 Alcala and Qin5 proposed the reconstruction-based contribution plot (RBCP), which could reduce the fault smearing effect. They analyzed the influence of the projection matrix C̃ on the identification results, where

≥ 80%

̃ ||2 SPE = || x̃ ||2 = ||(I − PPT )x || = || Cx

∑ xĩ 2 i=1

l

∑i = 1 λi

2δα || ξĩ ||

where ξ̃i = C̃ ξi, is the projection of the fault direction ξi in the residual subspace. This condition guarantees that the fault is sufficiently detectable. However, if the sufficient condition is not satisfied, the faults could possibly be undetected. 2.2. Fault Identification and Estimation. 2.2.1. Traditional and Reconstruction-based Contribution Plots. The SPE is able to detect when the process is out of control, but it cannot indicate which variables are responsible for the malfunction. The traditional contribution plot (TCP)15,16 is a well-known diagnostic tool for fault identification, which is based on the idea that the variables with the largest contributions to the fault detection indices are most likely the faulty variables. The contributions are actually the effects of the faults on the observed vector of measurements, and the TCP approach does not require any information about the faults to generate the plot. In the TCP method, the SPE is broken down into m elements

where T ∈ 9 n × l , P ∈ 9 m × l and E ∈ 9 n × m are the score matrix, the loading matrix and the residual matrix, respectively. The principal component projection reduces the original set of variables to l principal components. In our work, the cumulative percent variance (CPV) criterion12,13 is utilized to choose the number of principal components, which satisfies CPV =

(5)

(4)

δ2α

where is the upper control limit for SPE with a significance level α. Jackson and Mudholkar14 developed a confidence limit expression for the SPE assuming that x followed a normal distribution.

RBCi SPE = 7707

xĩ 2 ciĩ

(9)

dx.doi.org/10.1021/ie403857f | Ind. Eng. Chem. Res. 2014, 53, 7706−7713


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where c̃ii is the ith diagonal element of the projection matrix C̃ . As pointed out by Alcala and Qin, the reconstruction-based and traditional contributions to the SPE differ from each other only by a scaling coefficient c̃ii. However, since c̃ii varies with i, the reconstruction-based and traditional contributions to the SPE are fundamentally different. 2.2.2. Fault Reconstruction. When a sensor fault is detected, it is desirable to determine the necessary adjustments to let the process return to the normal condition, so the task of fault reconstruction is to estimate the normal vector xn. Fault identification via reconstruction is quite distinguishing from the procedures of contribution plot approaches. This method needs a predefined fault set {-1, ..., -m} which includes the faults happened before. To identify the actual fault among all possible faults, this method has to reconstruct xn from xf along all possible fault directions, {ξ1,...,ξm}. For each assumed fault -i , the sample xf is moved back in the direction ξi such that the distance between xf and the residual subspace is minimized. The reconstructed vector xir corresponding to fault -i is obtained by correcting xf along the fault direction ξi: x r i = x f − fi ξi

sensitive to incipient sensor faults and the RBCP can cause false identification when incipient sensor faults occur. The process model to be used is ⎡ x1 ⎤ ⎡−0.2310 ⎢x ⎥ ⎢ ⎢ 2 ⎥ ⎢−0.3241 ⎢ x3 ⎥ ⎢−0.217 ⎢x ⎥ = ⎢ ⎢ 4 ⎥ ⎢−0.4089 ⎢ x5 ⎥ ⎢−0.6408 ⎢ ⎥ ⎢ ⎣ x6 ⎦ ⎣−0.4655

− 0.0816 0.7055 − 0.3056 − 0.3442 0.3102 − 0.433

− 0.2662 ⎤ ⎥ − 0.2158 ⎥ ⎡ t ⎤ 1 − 0.5207 ⎥ ⎢ ⎥ ⎥ ⎢ t 2 ⎥ + noise − 0.4501 ⎥ ⎢ ⎥ ⎣t ⎦ 0.2372 ⎥ 3 ⎥ 0.5938 ⎦ (15)

where t1 ∼ 5 (10,0.1), t2 ∼ 5 (20,0.2), and t3 ∼ 5 (30,0.3) are normally distributed variables. The random noise included in this process is also normally distributed with the mean of zero and the standard deviation of 0.01. The simulation period is from t = 1 to t = 1000 and the sampling interval is 1s, so in order to build the principal component model, 1000 normal samples are generated. The data are scaled to zero-mean and unit variance. With the help of the CPV criterion, two principal components are selected. As an example of the sensor fault, a step change is set to x2 with the amplitude of 0.05 when t = 501. Figure 1 shows the

(10)

where f i is an estimate of the fault magnitude which measures the displacement in the direction ξi. If the real fault is exactly the assumed fault -i , the corresponding reconstructed vector xir will be closest to xn. The distance between xir and the residual subspace is given by the magnitude of the residual vector ∥xir̃ ∥, or the SPE for the reconstructed vector xir 2

SPEir = || x̃ r i ||

2 = || x̃ f − fi ξĩ || o 2

= || x̃ f − fi ̃ ξĩ ||

(11)

where ξoĩ = (ξĩ )/∥ξĩ ∥ and fĩ = f i∥ξĩ ∥. If the assumed fault is the true fault, the greatest reduction from faulty condition to reconstructed condition in the SPE is expected. This feature will be used for the identification of sensor faults. The minimization of SPEir leads to8 6

dSPEir oT o = 2ξĩ (x̃ f − fi ̃ ξĩ ) = 0 ̃ dfi

(12)

oT oT fi ̃ = ξĩ x̃ f = ξĩ x f

(13)

Figure 1. SPE statistic monitoring plot of Monte Carlo for sensor fault 2.

or

monitoring performance of this sensor fault using the SPE statistic. The SPE statistic fails to detect this incipient sensor fault, which indicates that the SPE is not sensitive to this incipient sensor fault. Generally, incipient sensor faults do not conform to the sufficient condition of fault detectability described in section 2, so they have the possibility of being undetected by the SPE. Figure 2 is the result of fault identification using the RBCP method, and this method identified the fifth sensor as the faulty one, which causes false identification. Thus, when treating incipient sensor faults, the RBCP can lead to incorrect identification. The RBCP is easy to calculate, yet it does not always provide correct identification results, especially for incipient sensor faults. There are two major drawbacks in the RBCP method. First, the RBCP does not take the contribution plot of the normal condition into consideration, and it will give rise to false

and oT

ξĩ x f fi = = || ξĩ || || ξĩ || fi ̃

(14)

It should be noticed that only one fault direction selected from the fault direction set satisfies eq 12, which leads to the correct fault identification after the fault direction is obtained. Thus, an estimate of the fault magnitude can be obtained using eq 14, which is a crucial step in sensor fault diagnosis.

3. PROPOSED METHOD 3.1. Motivational Example. A simulated simple process model called Monte Carlo simulation model5,9,18 is employed to illustrate that the PCA-based detecting index, SPE, is not 7708



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unluckily. Therefore, a new type of contribution plot need to be designed to overcome these limitations mentioned above. 3.2. Definition of a New Contribution Plot. To avoid the possible false identification by the RBCP method, we propose a new type of contribution plot, namely average residualdifference reconstruction contribution plot (ARdR-CP), which can improve the performance of detecting and identifying single incipient sensor fault. Let xñ and xf̃ denote the residuals under the normal and possible faulty conditions, respectively, and the normal condition is named the historical normal condition here. According to PCA, we can get ̃ n x̃ n = Cx

(16)

̃ f x̃ f = Cx

(17)

Then, we will use the difference between xf̃ and xñ to generate the new contribution plot: m

|| x̃ f − x̃ n ||2 =

Figure 2. Reconstruction-based contribution plot.

m

∑ Δxi 2 = ∑ zi i=1

identification, especially for those faults with small magnitudes. When the CPV criterion aforementioned is used to acquire the principal component model, just about 80% of the normal condition information is included. Hence, there exist residuals measured by the projection of the normal measurement vector x to the residual subspace, as well as by the contribution plot of the normal condition. Therefore, the residuals reflected in the contribution plot of the normal condition are not negligible for incipient sensor faults. Second, the RBCP is derived at one sampling time, and the stochastic noises could affect its validity for identifying faults. An illustration of the contribution plot of the normal condition for this example is provided in Figure 3. As is shown

i=1

(18)

where zi is the contribution from the ith variable at the sampling time when xf̃ is obtained. Next, we will average the contribution values so as to decrease the influence of random noises:

zi̅ =

1 k

k

∑ zi , j (19)

j=1

where zi,j is the contribution value of the ith variable at the jth sampling time and k is the sampling window size, and we name this type of contribution plot as average residual-difference contribution plot (ARd-CP). It is reasonable for us to assume that the random noise is included in zi with the mean of μ and the standard deviation of δ. However, the random noise in zi̅ is with the mean of μ and the standard deviation of δ/k. We can find that the standard deviation of random noise decreases from δ to δ/k; thus, its influence on the performance of fault diagnosis weakens. In addition, if we want to improve the performance of fault diagnosis, the sampling window size k should be enlarged. But, k cannot be too large for the consideration of timeliness and computation, and it should be selected as appropriate as possible based on these two aspects mentioned above. If the ARd-CP is used for fault identification, the fault smearing effect is inevitable. Thus, based on the idea in the RBCP method, we modify the proposed method following eq 18: z ci = i ciĩ (20) ci̅ =

Figure 3. Reconstruction-based contribution plot of the normal condition.

1 k

k

∑ ci ,j = j=1

1 k

k

∑ j=1

zi , j ciĩ

(21)

where ci,j is the contribution value of the ith variable at the jth sampling time and ci̅ is the ARdR contribution value of the ith sensor. Now, the ARd-CP and ARdR-CP methods are applied to the same example stated above to show that the ARdR-CP is superior to the ARd-CP. The sampling window size k is set to be 100. From Figure 4a and b, the superiority, without the fault smearing effect, of the ARdR-CP method over the ARd-CP

in the figure, the contribution value from the second variable is fairly small compared with that from the fifth variable which occupies the most part of the total contributions from all variables. When the second variable has such a tiny fault magnitude that its contribution value is still below that of the fifth one, as described in Figure 2, we will choose the fifth sensor as the faulty one, which causes false identification 7709



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Figure 5. Fault-free ARdR-CP contribution plot.

Figure 6. SPE statistic monitoring plot of Monte Carlo for sensor fault 1.

Figure 4. Contribution plots: (a) ARd-CP, (b) ARdR-CP.

method is clearly illustrated. The ARdR-CP method identifies the fault in the second sensor correctly, while the ARd-CP method mistreats the sixth sensor as the faulty one. 3.3. Procedure of Incipient Sensor Fault Diagnosis. In the motivational example, if all the sensor measurements are within the normal variation, the ARdR-CP is shown in Figure 5. It should be noted that the contribution values of the fault-free condition are not the same order of magnitude as that of the faulty condition in Figure 4b, and this difference can help us to judge whether a sensor fault has happened or not. For this reason, we can resort to the ARdR-CP for the purpose of detecting incipient sensor faults that the SPE statistic is not sensitive. Accordingly, the ARdR-CP method has the capability of detecting and identifying incipient single sensor fault simultaneously. Fault identification via reconstruction is an effective but not efficient enough approach for fault identification. In this method, a fault direction set {ξ1,...,ξm} should be determined first, and then each of these directions in the set is applied one by one to check which sensor is in the faulty condition. This will be time-consuming for a large number of sensors. The ARdR-CP method can identify a single sensor fault just by one

step with no need for a priori knowledge. Thus, if we can identify the single sensor fault correctly, the fault diagnosis approach, which combines the ARdR-CP with the fault estimation equation, eq 14, derived from fault reconstruction method, can be applied to the incipient single sensor fault diagnosis. For the same reason for reducing the effect of random noises, the mean fault magnitude within a sampling window, whose size is equal to that of the ARdR-CP, is computed at last. In conclusion, the ARdR-CP is used to detect and identify an incipient sensor fault simultaneously, and then the fault estimation equation is directly utilized to estimate the fault magnitude within a sampling window after correct identification.

4. SIMULATION STUDIES In this section, the Monte Carlo simulation and the benchmark quadruple-tank process are used to demonstrate the advantages of our proposed method. 4.1. Monte Carlo Simulation. The introduction of this Monte Carlo simulation is described in the previous section, 7710



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Figure 7. Contribution plots: (a) RBCP, (b) ARdR-CP.

Table 1. Simulation Parameters param. A1;A3 A2;A4 a1;a3 a2;a4 k1;k2 v1;v2 γ1;γ2 g

unit 2

cm cm2 cm2 cm2 cm3 /(V s) V cm/s2

value 28 32 0.071 0.057 3.33, 3.35 3.00, 3.00 0.7, 0.6 981

Figure 8. SPE statistic monitoring plot of the quadruple tank for sensor fault 1.

and the simulation parameters are not changed. The projection matrix C̃ is ⎡ 0.7602 ⎢ ⎢−0.0782 ⎢−0.2395 C̃ = ⎢ ⎢−0.2267 ⎢ 0.1629 ⎢ ⎣ 0.2022

0.2022 ⎤ ⎥ −0.2663 0.3826 ⎥ 0.0315 0.7413 −0.2531 0.2196 0.1456 ⎥ ⎥ 0.0755 −0.2531 0.7493 0.2316 0.1143 ⎥ −0.2663 0.2196 0.2316 0.7236 0.0270 ⎥⎥ 0.3826 0.1456 0.1143 0.0270 0.6665 ⎦ −0.0782 −0.2395 −0.2267 0.1629 0.3591

0.0315

0.0755

Figure 9. Reconstruction-based contribution plot of the normal condition.

(22)

When we analyze the projection matrix C̃ , we can find that c̃22 < c̃62, which indicates that the fault smearing effect may exist in the second sensor. Therefore, the fault in the second sensor is used to show that the ARdR-CP is better than the ARd-CP as described in Figure 4. Then, we will observe another incipient sensor fault from the first sensor, and the fault magnitude is set to be 0.03. Figure 6 shows that the SPE cannot detect this incipient sensor fault, as we cannot judge whether there is a

fault or not. Moreover, this incipient sensor fault results in false identification in the RBCP, as shown in Figure 7a, which identifies the fifth sensor as the faulty one as well, yet the actual faulty sensor is the first one. From Figure 7b, we can observe that the ARdR-CP identifies the fault in the first sensor correctly. Until now, we have detected this incipient sensor fault by comparing Figures 7b and 5, and 7711



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identified the faulty sensor correctly, so the fault direction ξ1 = [1 0 0 0 0 0 ]T is obtained prepared for fault estimation. Then, we utilize eq 14 to calculate the estimated fault magnitude that is f i = 0.0315 and is close to the true fault magnitude. 4.2. Quadruple Tank Simulation. The quadruple-tank process was originally developed by Johansson19 and has been widely used as a novel multivariate laboratory process. There are four interconnected water tanks, two pumps, and associated valves in this process. A schematic diagram of the process can be found in the literature.20 v1 and v2 are the voltages supplied to the pumps, and h1−h4 are the water levels. The flow to each tank is adjusted using the associated valves γ1 and γ2. A nonlinear model is derived based on mass balances and Bernoulli’s law:20 (23)

γ k2 a dh 2 a = − 2 2gh2 + 4 2gh4 + 2 ν2 dt A2 A2 A2

(24)

(25)

(1 − γ1)k1 dh4 a = − 4 2gh4 + ν1 dt A4 A4

(26)

The parameter values of this process are given in Table 1.19 The data are generated by eqs 23−26, where γi and νi are corrupted by the Gaussian white noises with zero mean and the standard deviation of 0.005. The measured h1−h4 and f1−f420 are corrupted by the Gaussian white noise with zero mean and the standard deviation of 0.001. The simulation period is from t = 1 to t = 1000 and the sampling interval is 1s, so 1000 normal samples are generated to train the principal componet model when the quadruple-tank process is in its steady condition. Four principal components are kept in the model, which capture about 84% of the total variance. The sampling window size is 100 for calculating the ARdR-CP. A bias sensor fault is introduced to the first sensor which measures the level of tank 1 when t = 501, and the magnitude of the sensor fault is 0.015. First, the SPE statistic is used to detect this incipient sensor fault, but it does not succeed as shown in Figure 8, because we still cannot determine whether a fault has happened or not. Then, we will observe the RBCP of this quadruple-tank process in the normal condition, as shown in Figure 9 where the contribution value of the first sensor is lower than that of the third one, so a fault in the first sensor may cause the RBCP to misdiagnose the true fault. The ARdR-CP of the fault-free condition is provided in Figure 10, which is not the same order of magnitude as that of the faulty condition in Figure 11b, so we can judge that a fault has occurred. Next, the two types of mentioned contribution plots are used to identify the faulty sensor, as shown in Figure 11. Only our proposed method can locate the faulty sensor correctly. After the faulty sensor is located, the fault direction is also known as ξ1 = [1 0 0 0 0 0 ]T, and then the fault magnitude estimate, that is 0.0148 and is close to the true value, can be acquired.

Figure 10. Fault-free ARdR-CP.

γ k1 a dh1 a = − 1 2gh1 + 3 2gh3 + 1 ν1 dt A1 A1 A1

(1 − γ2)k 2 dh 3 a = − 3 2gh3 + ν2 dt A3 A3

Figure 11. Contribution plots: (a) RBCP, (b) ARdR-CP. 7712



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(13) Valle, S.; Li, W.; Qin, S. J. Selection of the number of principal components: the variance of the reconstruction error criterion with a comparison to other methods. Ind. Eng. Chem. Res. 1999, 38, 4389− 4401. (14) Jackson, J. E.; Mudholkar, G. S. Control procedures for residuals associated with principal component analysis. Technometrics 1979, 21, 341−349. (15) Miller, P.; Swanson, R.; Heckler, C. E. Contribution plots: A missing link in multivariate quality control. Appl. Math. Comput. Sci. 1998, 8, 775−792. (16) Yoon, S.; MacGregor, J. F. Fault diagnosis with multivariate statistical models part I: Using steady state fault signatures. J. Process Control 2001, 11, 387−400. (17) Westerhuis, J. A.; Gurden, S. P.; Smilde, A. K. Generalized contribution plots in multivariate statistical process monitoring. Chemom. Intell. Lab. Syst. 2000, 51, 95−114. (18) Alcala, C. F.; Joe Qin, S. Analysis and generalization of fault diagnosis methods for process monitoring. J. Process Control 2011, 21, 322−330. (19) Johansson, K. H. The quadruple-tank process: A multivariable laboratory process with an adjustable zero. IEEE Trans. Control Syst. Technol. 2000, 8, 456−465. (20) He, Q. P.; Qin, S. J.; Wang, J. A new fault diagnosis method using fault directions in Fisher discriminant analysis. AIChE J. 2005, 51, 555−571. (21) Zhang, Y.; An, J.; Ma, C. Fault detection of non-Gaussian processes based on model migration. IEEE Trans. Control Syst. Technol. 2013, 21, 1517−1526.

5. CONCLUSIONS In this paper we focus on a single sensor with an incipient bias fault, which has been rarely studied in the previous literature based on data-driven methods. A new contribution plot called ARdR-CP is proposed to sensitively detect incipient sensor faults to which the traditional fault detection indices are not sensitive and to identify incipient sensor faults more precisely compared with the RBCP method. It is noteworthy that our proposed ARdR-CP has the ability of detecting and identifying incipient sensor faults simultaneously. When the fault has been correctly identified by the ARdR-CP, fault estimation can be done without the predefined fault direction set which is indispensable for fault reconstruction method. In addition, exhaustive comparisons among different types of contribution plots have been implemented in the Monte Carlo simulation and the benchmark quadruple-tank process, and it is shown that our proposed ARdR-CP method is superior to the RBCP when incipient sensor faults are treated. In the future work, we can do some researches on incipient faults in more complicated processes, such as the multimode non-Gaussian process,21 and we will try to diagnose multiple sensor faults with tiny fault magnitudes.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (NSFC) under grants 61004074 and 61134001 and the 973 Program of China under grant 2012CB720500.

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REFERENCES

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