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Process Fault Detection Using Directional Kernel Partial Least Squares Yingwei Zhang, Wenyou du, yunpeng fan, and Lingjun Zhang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie501502t • Publication Date (Web): 12 Feb 2015 Downloaded from http://pubs.acs.org on February 18, 2015

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Industrial & Engineering Chemistry Research

Process Fault Detection Using Directional Kernel Partial Least Squares

Yingwei Zhang1 and Wenyou Du, Yunpeng Fan and Lingjun Zhang1

1

Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, Liaoning 110004, P. R. China

* Correspndence concerning this article should be addressed to Yingqwei Zhang at [email protected] or [email protected]

ACS Paragon Plus Environment


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Abstract—In this paper, a directional kernel partial least squares (DKPLS) monitoring method is proposed. The contributions are as follows: (1) by analyzing the relevance between the input residual and output variables, the kernel

partial

least

squares

(KPLS)

residual

subspace

still

contains

output-relevant variation. (2) a new KPLS algorithm which is called directional KPLS (DKPLS) algorithm is proposed to extract the output-relevant variation. Compared to the conventional algorithm, the DKPLS algorithm build a more direct relationship between the input and output variables. (3) based on the proposed DKPLS algorithm, a process monitoring method is proposed. In this monitoring method, kernel latent variables are used to explain the extracted output-relevant variation and calculate monitoring indices. Faults are detected accurately by the proposed DKPLS method. The DKPLS monitoring method is used to monitor the numerical example and the electro-fused magnesium process. The experiment results show the effectiveness of the proposed DKPLS method.

Keywords—Directional partial least squares; Output -relevant variation; Residual subspace; Quality-concerning process monitoring.


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1. Introduction Multivariate statistical process monitoring (SPM) techniques have been developed over the past two decades1-5. They are widely used in the monitoring of industrial processes6-11. In many kinds of industrial processes such as continuous process12, batch process13,14, dynamic process15, multivariate SPM techniques play important roles. Multivariate statistical methods such as principal component analysis (PCA)16-19, partial least squares or projection to latent structures (PLS)20-23 and more recently independent component analysis (ICA)24-27, have received great success in practice. Some extended methods based on PCA, PLS and ICA are also proposed to solve the practical problems in industrial processes28-33. As a data-driven method, PLS has been widely used in the modeling, monitoring and diagnosis of industrial processes and it has shown a good performance. The objective of PLS is to extract the covariation in both process and quality variables and to model the relationship between them34. With this relationship, the process can be monitored and new output variables can be predicted when the input variables are known. The PLS method is important for predicting and controlling the quality of products. Some extended PLS methods have been also proposed. For example, Helland et al. proposed a recursive PLS algorithm (RPLS) to update the PLS model with the latest process data35. For nonlinear input and output data, polynomial PLS36, neural PLS37,38 and kernel PLS39,40 are proposed. In these methods, kernel PLS (KPLS) is the most popular method. With kernel method, the nonlinear input data is mapped into a high-dimensional feature space41. In this feature space, the input data is nearly linear and then the PLS method is applied to the input data. MBPLS is one of such methods, which builds the variable correlation model within each block under the



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influence of other blocks. Zhang reported the super scores of MBPLS are identical to the scores of regular MPLS and thus achieved the same process monitoring performance42. Although PLS is used widely, there are still some problems in process monitoring using PLS. In this paper, two problems are explored. Problem 1 is that there is still variation which is related to the output variables in the PLS residual subspace. In PLS, variations in the PLS residual subspace are considered to be unrelated to output variables43. Indeed, the PLS residual subspace still contains variation which is related to the output variables. It is called output-relevant variation in this paper. This problem results in that PLS has the limitations in detecting output-relevant faults with the PLS latent variables. There are still large variations in the PLS residual subspace, making the residual subspace inappropriate to be monitored by the SPE statistic. PLS does not extract variances in the input space in a descending order unlike PCA. Some latent variables containing large variations are left in the residual subspace44. KPLS is the PLS in a high-dimensional feature space so that it also has the two problems mentioned above. Zhou et al. proposed a total projection to latent structures (T-PLS) monitoring method where PCA is applied to extract variation in the PLS residual subspace44. However, this method just decreases large variations in the PLS residual subspace. The output-relevant variation in the PLS residual subspace is not extracted and not used to monitor the process. In this paper, the KPLS algorithm is further analyzed in order to solve the problems mentioned above. The KPLS residual subspace still contains output-relevant variation. Then the output-relevant variation is deduced based on the derivation procedure and a new KPLS algorithm which is called directional KPLS algorithm (DKPLS) is proposed. Compared with the KPLS algorithm, the DKPLS algorithm build a more


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refined relationship between the input and output variables. After the output-relevant variation is separated from the KPLS residual subspace, it is proved that rest of the KPLS residual subspace doesn’t contain output-relevant variation. Based on the proposed DKPLS algorithm, a process monitoring method is proposed. In this monitoring method, kernel latent variables are used to explain the extracted output-relevant variation and calculate monitoring indices. Faults which affect output variables are detected by the proposed DKPLS monitoring method more accurately than KPLS monitoring method. In the simulation studies, the DKPLS monitoring method is used to monitor Monte Carlo simulation and Electro-fused magnesium furnace (EFMF) process, where the KPLS monitoring method also is applied. For the detection of two output-relevant faults, the proposed DKPLS monitoring method is more effective than the KPLS monitoring method in detecting the output-relevant faults, especially the faults which occur in the output-relevant variation. It is indicated that the proposed DKPLS monitoring method is more proper to monitor quality-concerning processes. The remaining sections of this paper are organized as follows. In Section 2, relevance between the input residual and the output variables is investigated. Then the output-relevant variation in the KPLS residual subspace is extracted to build the DKPLS algorithm. At last it is proved that there is no relevance between rest of the KPLS residual subspace and output variables. In Section 3, a monitoring scheme based on the proposed DKPLS algorithm is given. In Section 4 the Monte Carlo simulation and the EFMF process are employed to illustrate the feasibility of the proposed DKPLS monitoring method. At last the conclusions are drawn in Section 5.

2.

Directional KPLS algorithm

In this section, the directional KPLS algorithm (DKPLS) is proposed. Considering



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the output-relevant variation in the KPLS residual subspace, the DKPLS algorithm is deduced based on the KPLS algorithm.

2.1 KPLS algorithm In the KPLS algorithm, the input and output variables are decomposed to be the form as follows: i  T Φ ( x ) = ∑ t l p l + Φ i ( x )  l =1  i  Y = t qT + F ∑ l l i  l =1

(1)

where t i is the principal component of KPLS and i is the number of principal component which is determined by cross-validation45. Φi (x) and Fi are the residual of input and output variables respectively. Φi (x) spans the KPLS residual subspace. There are Φ0 (x) = Φ( x) and F0 = Y . Computing procedure of KPLS is described in Ref. [40]. A KPLS based on PLS is proposed in Tab. 1. In the second step of iteration, the normalized loadings vector w i is denoted by w i = Φi ( x )T ui Φi ( x )T ui

where w i can not be calculated since the mapping function Φi ( x ) is unknown. From Using K i = Φi (x )Φi (x)T , the scores vector is represented by t i = Φi ( x ) w i = Φi ( x )Φi ( x )T ui

(u

1

T i

Φi ( x )Φi ( x )T ui ) 2 = K i ui

ui T K i ui

In the sixth step of iteration, residual of Φi ( x ) is deflated by Φi +1 ( x ) = ( I − t i t i T t i T t i ) Φi ( x )

In the computation procedure, since K i = Φi (x )Φi (x)T , we can obtain: K i+1 = ( I − t i t iT t iT t i ) K i ( I − t i t iT t iT t i )

This residual is used for computing the next latent variable.


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2.2 Relevance between the KPLS residual subspace and output

variables Relevance between the KPLS residual subspace Φi (x) and output variables Y is proved as follows: i -1

Φi (x)T Y = Φi (x)T (Fi −1 + ∑ t l qTl )

(2)

l =1

Because Φ j (x)T t h = 0

j≥h

(3)

so that Eq. (2) is transform to be the form as follows: Φ i (x)T Y = Φi (x)T Fi −1

(4)

then Φi (x)T Y = Φi (x)T Fi −1 = (Φi -1 (x)T − t i pTi )T Fi −1

(5)

= Φi -1 ( x)T Fi −1 − pi tTi Fi −1

Because the loading vectors pi and qi in KPLS are calculated as follows: p i = Φ i -1 ( x)T t i tTi t i

(6)

q i = FiT−1t i tTi t i

(7)

then Eq. (5) is transformed to be Φi (x)T Y = Φi ( x)T Fi −1 = Φi -1 ( x)T Fi −1 − Φi -1 (x)T t i qTi

(8)

= Φi -1 ( x) (Fi −1 − t i q ) T

T i

Eq. (8) is described in Fig. 1. In Fig. 1, R ( X) and R (Y) denote space of input variables Φ(x) and output variables Y , respectively. Φi -1 (x) and Fi−1 denote the input and output residual after i − 1 iterations, respectively. Φi (x) and Fi denote the input and output residual after i iterations, respectively. θ is the angle between the direction of Φi (x) and the direction of Fi−1 . When the covariance of Φ(x) and



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Y is maximized, the possible positions of t i are shown in the Figure. Its direction

changes as the direction of dotted line. Meanwhile, the direction of Φi (x) is changed together with t i , and is always orthogonal to t i . From Eq. (8), it is seen that when Fi −1 = t i qTi

, there is Φi (x)T Y = Φi (x)T Fi−1 = 0 . It means when the direction of t i is the

same as that of Fi−1 , Φi (x) will be orthogonal to Fi −1 and Φi (x) will be orthogonal to Y , which is also shown in Fig. 1. However, t i is the principal component of Φ( x )

and in most cases, its direction is not the same as that of Fi −1 . Therefore, in

most cases, Φi (x)T Y ≠ 0 , meaning that there is still relevance between input residual Φ i ( x)

and output variables Y . In other words, there is output-relevant variation in

the KPLS residual subspace. 2.3 Expression of the output-relevant variation In PLS derivation procedure, to maximize the covariance between the component of input and output variables, the optimization problem is gotten as follows: s = w1T ET0 F0 c1 − λ1 ( w1T w1 − 1) − λ2 (c1T c1 − 1)

(9)

where w1 and c1 are the axis vectors of input variables E0 and output variables F0

respectively. E0 = X and F0 = Y .

With Lagrangian multiplier method, the equations are obtained: T E0 F0 c1 = θ w1  T F0 E0 w1 = θ c1

(10)

In KPLS structure, Eq. (10) is rewritten to be the form as follows: T Φi -1 (x) F0 ci = θ wi  T  F0 Φi -1 ( x) wi = θ ci

(11)

then, there is Φi -1 ( x)Φi -1 ( x)T F0 ci = θ Φi -1 (x) wi  F0 F0T Φi -1 (x) wi = θ F0 ci 


(12)

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so that F0 ci = θ Φi -1 ( x) wi Φi -1 (x)Φ i -1 (x)T

(13)

Insert Eq. (13) into the below equation in Eq. (12) and because Φi -1 (x) wi = t i , there is F0 F0T Φi -1 (x)Φi -1 (x)T t i = θ 2 t i

(14)

In the PLS structure, there is pi = Φi -1 ( x)T t i tTi t i , θ 2 = tTi t i and F0 = Y , so that Eq. (14) is rewritten: YYT Φi -1 (x)p i = t i

(15)

Because Φi -1 (x)pi = t i and Φˆ i -1 (x)pi = t i , so that YYT Φi -1 (x) can be seen as the output-relevant variation in input residual Φi -1 (x) . Then the output-relevant variation is gotten as follows: % ( x ) YT Y Er = YYT Φ

(16)

where Φ% (x) is the input residual in KPLS structure which is Φi (x) in Eq. (1). 1 YT Y

is a matrix to normalize the output-relevant variation.

Then, the DKPLS structure is expressed as follows: i  T Φ(x) = ∑ t l p l + Er + Eir  l =1  i  Y = ∑ t l qTl + Fi  l =1

(17)

where Er is the output-relevant variation in the KPLS residual subspace and Eir is the rest of the KPLS residual subspace, which will be proved to be irrelevant to output variables. Prediction model of the DKPLS is inferred to the form as follows: ˆ = Φ ( x)B Y r

where B r = Φ(x)T UTT (TT T) −1[Φ(x) + YΦ% (x)(Y T Y)−1 ]B and B = Φ(x)T U (TT Φ(x)Φ(x)T U ) −1 TT Y .


(18)


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2.4 Irrelevance between rest of input residual and output variables Rest of the KPLS residual subspace is obtained as follows: % (x) − YYT Φ % ( x ) YT Y Eir = Φ

(19)

% ( x) − YYT Φ % ( x) Y T Y ) YT Eir = YT (Φ

(20)

Therefore,

Because YT Y is a square matrix with full rank, YT Y(YT Y)−1 = I where I is unit matrix. So that % (x) − YT YYT Φ % ( x) Y T Y YT Eir = YT Φ % (x) − YT Φ % ( x) = YT Φ

(21)

=0

Therefore, there is no relevance between rest of the KPLS residual subspace and output variables. It is indicated all the output-relevant variations in input residual are extracted.

3. Monitoring scheme based on directional KPLS algorithm In section 2, the DKPLS algorithm is developed as Eq. (17) and Er is the output-relevant variation in the KPLS residual subspace. To monitor the quality-concerning processes, effective variation in Er need to be extracted and used. KPCA can extract latent variables of a set of data and explain them46-48. In this paper, KPCA is used to decompose Er to extract the effective variation as follows. Considering Eq. (16), Er is rewritten to be the form as % ( x) Er = CΦ

(22)

where C = Y(YT Y)−1 YT If

there are

i

latent variables in

Φi (x)Φi (x)T = K i +1 . K i+1

KPLS,

% ( x ) = Φ ( x) Φ i

and

there is

is the gram matrix after i iteration and it is calculated in

KPLS. K1 is the initial gram matrix and Φ(x)Φ(x)T = K1 . Assume there are N


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samples in input data, then Φ(x) = [φ( x1 ), φ(x 2 ),..., φ(x N )]T , the covariance matrix is gotten as S = (1 N )CΦ i (x)Φ i (x)T CT

(23)

Latent variables of CΦi (x) can be obtained by finding eigenvectors of S with the equation as follows: SPr = λ Pr

(24)

where Pr is the eigenvectors and λ is the eigenvalues of S . For λ ≠ 0 , solution Pr

(eigenvector) can be regarded as a linear combination of CΦi ( x) to be Pr = Φi (x)T CT A

(25)

Combine Eq. (23), Eq. (24) and Eq. (25), there is (1 N )CK i +1CT A = λ A

(26)

Therefore, A is the eigenvalues of (1 N )CK i +1CT . The latent variables of CΦi (x) are calculated as Tr = CΦi (x)Pr = CΦi (x)Φi (x)T CT A

(27)

= CK i +1C A T

Define Td = [T, Tr ] and Pd = [P, Pr ] , therefore the directional KPLS structure is rewritten to be T Φ(x) = Td Pd + Eir  T  Y = Td Qd + Fir

(28)

where Td = [Td ,1 , Td ,2 ,..., Td ,i ] and Qd = [qd ,1 , qd ,2 ,..., qd ,i ] . qd ,l l = 1, 2,...i is calculated as follows: qd ,l = YlT-1t d ,l tTd ,l t d ,l


(29)


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To calculate the DKPLS algorithm, for the model data x ∈ R N ×M , normalize it and carry KPLS on them. Then, U = [u1 , u 2 ,..., ui ] is gotten. The initial gram matrix K1 is calculated as follows: K1raw ,ij = k ( x i , x j ) = exp( −

xi - x j

2

)

c

K1 = K1raw - K1raw E - EK1raw + EK1raw E

(30) (31)

where each element of E is equal to 1/ N and E ∈ R N ×N . c is specified a priori by the user. Then with KPLS, Eq. (27) and Eq. (28), Eq. (29) can be calculated. Then the monitoring statistics are calculated as follows: Td2 = Td Λ −1TdT

(32)

SPEd = Φi (x) − CΦi ( x)

2

= (I − C)Φi (x)

2

(33)

= θ K i +1 2

where Λ = TdT Td , θ is the maximum eigenvalue of (I − C) . The confidence limits of Td2

statistic and SPEd statistic are calculated from their characteristic distributions49.

The detailed calculation equations of confidence limit are given by Qin et al.50. For a new set of sample x new ∈ R1× M , normalize it first. Then K new is calculated as follows: = k (x new,i , x j ) = exp(−

x new,i - x j

(34)

raw raw raw K new = K raw new - K new E - E1 K new + E1 K new E

(35)

1 [1,...,1] ∈ R1× N N

c

. Then t new is calculated as follows:

t new = K new U K i +1, new

2

)

K

where K new ∈ R1× N and E1 =

raw new

is calculated as follows:


(36)

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K i +1, new = (I −

t i tTi t tT )K new,i (I − Ti i ) T ti ti ti ti

(37)

where t i is the i th component of T . K new,1 = K new . Then t r ,new is calculated as follows: t r , new = CK i +1, newCT A

(38)

t d , new = [t new , t r , new ]

(39)

Therefore, t d ,new is obtained as

The monitoring statistics are calculated as follows: Td2, new = t d , new Λ −1tTd , new SPEd , new = Φi (x new ) − CΦi (x new )

= (I − C)Φi (x new )

(40)

2

2

(41)

= θ ki (x new , x new ) 2

where ki (x new , x new ) = 1 − K1,mn

2 N

N

∑K

new, m

+

m =1

1 N2

N

N

∑∑ K

1, mn

. K new,m is the m th element of K new ,

m =1 n =1

is the mn th element of K1 .

To sum up, monitoring procedure of the DKPLS monitoring method is shown as follows: (1) Obtain the normal data X and Y , then normalize them; (2) Calculate the initial gram matrix K1 with Eq. (30) and Eq. (31), then calculate their confidence limits; (3) Carry KPLS on X and Y to obtain U , T and K i+1 ; (4) Calculate the monitoring statistics with Eq. (32) and Eq. (33); (5) Obtain a new set of data x new ∈ R1×M and normalize it; (6) Calculate K new as Eq. (34) and Eq. (35). (7) Calculate t new as Eq. (36);



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(8) Calculate K i +1,new as Eq. (37); (9) Calculate t r ,new as Eq. (38); (10) Obtain t d ,new as Eq. (39); (11) Calculate the monitoring statistics of the new set of date with Eq. (40) and Eq. (41). If any statistic of goes beyond their corresponding confidence limits, it means that a fault is detected. The calculating and monitoring procedure of directional KPLS is shown in Fig. 2.

4. Simulation study 4.1. Numerical example The purpose of this example is to compare the calculation result of the ∆x value using the method in this paper and reference [52] by Monte Carlo simulation. The process model to be used is  x1   −0.2310  x   −0.3241  2   x3   −0.217  =  x4   −0.4089 x    5   −0.6408  x6   −0.4655 

− 0.0816 − 0.2662   0.7055 − 0.2158 t  − 0.3056 − 0.5207   1  t2 + noise − 0.3442 − 0.4501     t3  0.3102 0.2372   − 0.433 0.5938 

(42)

where t1 , t2 and t3 are zero-mean random variables with standard deviations of 1, 0.8 and 0.6, respectively. The noise included in the process is zero-mean with standard deviation of 0.2 and is normally distributed. In order to build the model, 1000 samples are generated. 300 samples are used to test the KPLS and DKPLS monitoring methods. The data is scaled to zero mean and unit variance. After generating and scaling the data, DKPLS is applied to build the model. Four variables are selected as input variables and the other two variables are selected as output variables. The


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simulated faults are of the form

X failure = X* + ∆X = X* + ξi f i

(43)

where X* is generated according to the model given above and the fault magnitude f i is a constant number between 0 and 0.5. Here, we assume ξi = [1, 0, 0, 0, 0, 0]T and f i = 0.4 . That is to say, we assume the first variable x1 is faulty and the magnitude

of the fault is 0.4. Monitoring results of KPLS monitoring method are shown in Fig. 3 and those of DKPLS monitoring method are shown in Fig. 4. In Fig. 3 (a), KPLS T 2 statistic is blow its confidence limit before the 152th sample and goes beyond its confidence limit from the 151th samples sharply, indicating that the fault is detected by the KPLS T 2 statistic. In Fig. 3 (b), shape of the KPLS SPE statistic is almost the same as the KPLS T 2 statistic. Seen from the monitoring results of KPLS, magnitude of the fault is large enough to be detected by KPLS. In Fig. 4 (a), DKPLS Td2

statistic also detected the fault. Compared with KPLS T 2 statistic, value of

DKPLS Td2 statistic is larger than that of KPLS. In Fig. 4 (b), DKPLS SPEd statistic also detects the fault and is almost the same as that of KPLS. For this simple fault, performances of KPLS and DKPLS are almost the same, but the DKPLS monitoring method is still better. Relevance between the KPLS residual subspace and output variables is shown in Fig. 5. Relevance between the DKPLS residual subspace and output variables is shown in Fig. 6. Values of relevance between every input variables residual and every output variable are shown in these figures. It can be seen that DKPLS relevance values are much less than those of KPLS, indicating that DKPLS residual subspace has little relevance with output variables. 4.2. Electro-fused magnesium furnace process

EFMF is a sort of equipment for refining magnesia. Study of EFMF process is the



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important development direction of the Electro-fused magnesium metallurgical industry and plays an important role in energy-saving and environmental protection. The round shell of EFMF can facilitate melting process. The main parts of EFMF are shown in Fig. 7. There is trolley under the furnace to make the product cool when melting process has completed. The control object is to ensure the temperature of EFMF can meet the melting point of magnesium. The 'healthy' data are used for modeling. The 'faulty' data are used for monitoring. The average time of the whole EFMF process is 10 h. The current voltage value of three phases and the temperature of furnace all can be measured on-line, which providing abundant process information. In this section, EFMF process is employed to test the effectiveness of the proposed directional KPLS method. As a comparison the conventional KPLS method is also applied in the monitoring of this system. The temperature of EFMF is an important variable, which is determined by the voltage of three phases, the current of three phases and other parameters. In this simulation, the internal temperature of EFMF and the temperature of EFMF’s shell are selected as output variables and the voltage, environment temperature and the current are chosen as input variables. 300 normal samples are used to develop normal operating condition model. 300 samples are used for testing. Fault 1 is the case of leakage furnace. Fault 2 is the case of airway obstruction. Fault 1 is added in the input variables by increasing the current of phase A sharply from the 51th sample to the 150th sample. The fault data is obtained with the equation x1f = xnormal + ξ1 f1 where xnormal is the normal data. ξ1 is the fault variable matrix and f1

is the magnitude of

the fault and it is a constant here. Fault 2 is introduced in the input variables by increasing the voltage of phase C gradually from the 150th sample to the 300th sample. The fault data is obtained with the equation x 2f = xnormal + ξ 2 f 2


and f 2

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becomes larger over time. Both these two faults will affect the output variables. For fault 1, the monitoring results are shown in Fig. 8 and Fig. 9. In Fig. 8 (a), from the 51th to the 150th sample, most points of the KPLS T 2 statistic are under the confidence limit. It means the KPLS T 2 statistic isn’t able to detect the fault. Although the fault is relevant to output variables, it only affects the output-relevant variation in the KPLS residual subspace, so that the fault is not detected by the KPLS T 2 statistic. In Fig. 8 (b), the KPLS SPE statistic detects the fault and it goes beyond

the confidence limit from the 51th sample and below the confidence limit after the 151th sample. It means the variation in the KPLS residual subspace is affected indeed. The monitoring statistics of DKPLS monitoring method are shown in Fig. 9. It is seen that DKPLS Td2 and SPEd statistics monitor the fault. Both Td2 and SPEd statistics go beyond the confidence limits from the 51th sample and below the confidence limits after the 151th sample. As Td contains more output-relevant variations, DKPLS Td2 statistic which is calculated by Td can detect more output-relevant faults than the KPKS T 2 statistic. Because large variations in KPLS residual subspace are extracted, performance of DKPLS SPEd statistic is not as good as that of the KPLS SPE statistic. Besides the output-relevant variation, some other variations are also affected by the fault so that the DKPLS SPEd statistic goes beyond confidence limit. Seen from the monitoring results, the DKPLS monitoring method is more effective than the KPLS monitoring method for this fault. The Relevance between KPLS input variables residual and output variables is shown in Fig. 10. Relevance between DKPLS input variables residual and output variables is shown in Fig. 11. Values of DKPLS are much less than those of KPLS. For fault 2, the monitoring results are shown in Fig. 12 and Fig. 13. In Fig. 12, the



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KPLS T 2 and SPE statistics are shown. In Fig. 12 (a), the KPLS T 2 statistic fails to detect the fault because the fault doesn’t affect the output-relevant variations in KPLS principal subspace. Only the output-relevant variation in the KPLS residual subspace is affected. Therefore the fault is detected by the KPLS SPE statistic, which goes beyond the confidence limit from the 200th sample. Because the fault is added gradually, the process variables can stay normal for a short time. In fault 2, the process is considered to be normal before the 200th sample. In Fig. 13, monitoring results of the DKPLS Td2 and SPEd statistics are shown. Both the DKPLS Td2 and SPEd statistics detect the output-relevant fault. These two statistics go beyond the

confidence limits from the 200th sample. Relevance between KPLS input variables residual and output variables is shown in Fig. 14. Relevance between DKPLS input variables residual and output variables is shown in Fig. 15. It is indicated that the DKPLS method is more effective in monitoring fault 2. From the monitoring results of two faults, it is indicated that an output-relevant fault may be not detected by the KPLS monitoring method, especially with the KPLS T 2 statistic. When the output-relevant variation is affected by a fault, the KPLS T 2

statistic can’t detect the fault. These two faults are detected by the KPLS SPE statistic. However, variation that is monitored by the KPLS SPE statistic is usually considered to be unrelated to output variables. Therefore, the fault which is monitored only by the KPLS SPE statistic may be considered to be irrelevant to output variables. Indeed, if the fault affects the output-relevant variation which exists in the KPLS residual subspace, this fault will affect output variables. The proposed DKPLS monitoring method is more sensitive to the changes in output-relevant variations so that the DKPLS Td2 statistic can detect the fault which occurs in output-relevant variation. The DKPLS SPEd statistic can detect the fault because the output-relevant


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fault affects other process variables too. Seen from the monitoring results, the DKPLS monitoring method is better to detect output-relevant faults. Therefore, the proposed DKPLS monitoring method is more appropriate to monitor the quality-concerning processes than the KPLS monitoring method.

5. Conclusions

In this paper, a directional KPLS algorithm (DKPLS) is proposed. By analyzing the relevance between the KPLS residual subspace and output variables, the output-relevant variation in the KPLS residual subspace is extracted and explained. By making full use of the variation in residual, a monitoring scheme based on the proposed method is developed. The case studies on Monte Carlo simulation and EFMF process are performed to test the performance of DKPLS monitoring method for monitoring output-relevant faults, where the KPLS monitoring method is also applied. Results of case studies show that the DKPLS monitoring method has a better monitoring performance than KPLS monitoring method. It is indicated that the proposed DKPLS monitoring method is more effective in the monitoring of quality-concerning processes.



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List of figures

Fig. 1.

Relevance between the KPLS residual subspace and the output variables.

Fig. 2.

Calculating and monitoring procedure of directional KPLS.

Fig. 3.

Monitoring results of Monte Carlo simulation with KPLS.

Fig. 4.

Monitoring results of Monte Carlo simulation with DKPLS.

Fig. 5.

Relevance between KPLS input variables residual and output variables in Monte Carlo simulation.

Fig. 6.

Relevance between DKPLS input variables residual and output variables in Monte Carlo simulation.

Fig. 7.

Electro-fused magnesium furnace.

Fig. 8.

Monitoring results of fault 1 with KPLS.

Fig. 9.

Monitoring results of fault 1 with DKPLS.

Fig. 10. Relevance between KPLS input variables residual and output variables of fault 1 in EFMF process. Fig. 11. Relevance between DKPLS input variables residual and output variables of fault 1 in EFMF process Fig. 12. Monitoring results of fault 2 with KPLS. Fig. 13. Monitoring results of fault 2 with DKPLS. Fig. 14. Relevance between KPLS input variables residual and output variables of fault 2 in EFMF process. Fig. 15. Relevance between DKPLS input variables residual and output variables of fault 2 in EFMF process.



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List of Tables Tab. 1. KPLS algorithm


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R(X)

ti

Φi (x)

Φi-1(x)

θ

Fi Fi−1

R(Y)

Fig. 1. Relevance between the KPLS residual subspace and the output variables



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Fig. 2. Calculating and monitoring procedure of directional KPLS


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7

2 x 10

2

1.5 T

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1 0.5 0 0

50

100

150 Samples

200

250

(a) KPLS T 2 statistic of Monte Carlo simulation


300


7

2.5 x 10 2 SPE

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1.5 1 0.5 0 0

50

100

150 Samples

200

250

(b) KPLS SPE statistic of Monte Carlo simulation Fig. 3. Monitoring results of Monte Carlo simulation with KPLS


300

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7

2 x 10

2

1.5 Td

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1 0.5 0 0

50

100

150 Samples

200

250

(a) DKPLS Td2 statistic of Monte Carlo simulation


300


7

3 x 10 2.5

d

2 SPE

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1.5 1 0.5 0 0

50

100

150 Samples

200

250

300

(b) DKPLS SPEd statistic of Monte Carlo simulation Fig. 4. Monitoring results of Monte Carlo simulation with DKPLS


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-11

2.5 x 10

Output Variable 1 Output Variable 2

2 Relevance Value

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1.5 1 0.5 0

1

2

3

4

Input Variable

Fig. 5.

Relevance between KPLS input variables residual and output variables in Monte Carlo simulation.



-26

3 x 10

Output Variable 1 Output Variable 2

2.5 Relevance Value

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2 1.5 1 0.5 0

1

2

3

4

Input Variable

Fig. 6.

Relevance between DKPLS input variables residual and output variables in Monte Carlo simulation.


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Fig. 7. Electro-fused magnesium furnace (1-transformer, 2-short circuit network, 3-electrode holder, 4-electrode, 5-furnace shell, 6-trolley, 7-electric arc, 8-burden)



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(a) KPLS T 2 statistic of fault 1


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(b) KPLS SPE statistic of fault 1 Fig. 8. Monitoring results of fault 1 with KPLS



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(a) DKPLS Td2 statistic of fault 1


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(b) DKPLS SPEd statistic of fault 1 Fig. 9. Monitoring results of fault 1 with DKPLS



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Fig. 10. Relevance between KPLS input variables residual and output variables of fault 1 in EFMF process.


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Fig. 11. Relevance between DKPLS input variables residual and output variables of fault 1 in EFMF process.



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(a) KPLS T 2 statistic of fault 2


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(b) KPLS SPE statistic of fault 2 Fig. 12. Monitoring results of fault 2 with KPLS



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(a) DKPLS Td2 statistic of fault 2


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(b) DKPLS SPEd statistic of fault 2 Fig. 13. Monitoring results of fault 2 with DKPLS



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Fig. 14. Relevance between KPLS input variables residual and output variables of fault 2 in EFMF process.


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Fig. 15. Relevance between DKPLS input variables residual and output variables of fault 2 in EFMF process.



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Tab. 1. KPLS algorithm No.

For explanation

For calculation

1

Initializing u k

Initializing u k

2

Calculating the score matrix of Φ ( x )

t k = Φ(x) ⋅ Φ(x)T uk = Kuk

3

Normalizing the score vector

tk ← tk / tk

4

Calculating the load of the output variables

p k = YT t k

5

Calculating the

6

Normalizing the latent variable u k

u k ← uk / u k

7

The step (2) to (6) is repeated to obtain the

The step (2) to (6) is repeated to obtain the

uk

which has been convergent

8

9

uk

which has been iterated

which has been convergent

u k = Yp k

Calculating the residual information of matrix

K=(I − t k t k T )K(I − t k t k T )

of Φ ( x ) and Y

Y=Y − t k t k T Y

Go to step 2

Go to step 2


uk

Process Fault Detection Using Directional Kernel ... - ACS Publications

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