Unified Analysis of Diagnosis Methods for Process Monitoring

Proceedings of the 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes Barcelona, Spain, June 30 - July 3, 2009

Unified Analysis of Diagnosis Methods for Process Monitoring Carlos F. Alcala ∗ , S. Joe Qin ∗∗ ∗

University of Southern California, Los Angeles, CA 90089 USA (e-mail: [email protected]) ∗∗ University of Southern California, Los Angeles, CA 90089 USA (e-mail: [email protected]) Abstract: Several diagnosis methods have been proposed for statistical process monitoring. They have been developed from different backgrounds and considerations. In this paper, five existing diagnosis methods are analyzed and compared. It is shown that these methods can be unified into three more general methods, making the original diagnosis methods special cases of the general methods. An analysis of the diagnosability of the general methods shows that some diagnosis methods guarantee correct diagnosis for simple sensor faults with large magnitudes, while others do not. For the case of sensor faults with modest fault magnitudes, a Monte Carlo simulation is used to compare the performance of the diagnosis methods. Keywords: process monitoring; fault diagnosis; contribution analysis; diagnosability. 1. INTRODUCTION Process monitoring is used in industry to detect and diagnose abnormal behavior of processes. Multivariate statistical methods and model-based methods are employed in process monitoring. Among the statistical methods, a popular method used in industry is principal component analysis (PCA) (Nomikos and MacGregor [1995]; Wise and Gallagher [1996]). PCA partitions the measurement space into a principal component subspace (PCS) and a residual subspace (RS). Fault detection makes use of fault detection indices. A fault is detected when one of the fault detection indices is beyond its control limit. After a fault has been detected, it is necessary to diagnose its cause. There exist several methods to perform fault diagnosis. Some methods look at a function of a variable and a fault detection index. The idea is that the functions of faulty variables have high values. Some of the methods for fault diagnosis that have been proposed are complete decomposition contributions (CDC), partial decomposition contributions (PDC), diagonal contribution (DC), reconstruction-based contributions (RBC), and anglebased contributions (ABC). Some of these methods have been used with just one or two fault detection indices. We can see in Table 1 which diagnosis method was developed for and used with each index. As it is seen, there are some diagnosis methods that have not been used with some fault detection indices. Because of that, one may wonder if a diagnosis method could perform better with an index different from the one originally used with. In this paper, we provide general expressions for the diagnosis methods so that they can be used with any fault detection index. In order to do that, we will first provide a general expression for the fault detection indices. Also, we show that the diagnosis methods can be unified into more general diagnosis methods. Furthermore, an analysis

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of the diagnosability of the unified methods is performed and the results are compared for the different diagnosis methods. Table 1. Diagnosis methods Method CDC PDC Diagonal RBC Angle-based methods

SP E

Index T2

Miller et al. (1993) None

Wise et al. (PLS-Toolbox) Nomikos (1996)

ϕ None

None Cherry and None Qin et al. (2000) Qin (2006) Alcala and Qin (2008, 2009) Raich and Cinar (1996) None Yoon and MacGregor (2001)

2. PCA FOR FAULT DETECTION 2.1 PCA Model In a process where n variables are measured, we build a PCA model using m measurements of the process under normal conditions. The data measurements, x ∈
X = [x(1) x(2) · · · x(m)] . (1) After scaling the data to zero mean and unit variance, we calculate its covariance matrix as 1 S= XT X. (2) m−1 Then, the covariance matrix is eigendecomposed to obtain the principal and residual loadings of the model as h iΛ 0 h iT ˜ ˜ S= P P P P (3) ˜ 0 Λ where P ∈
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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

ˆ and Projections to the principal and residual subspaces, x ˜ , are calculated as x ˆ = PPT x = Cx, x ˜P ˜ T x = Cx ˜ ˜=P x ˜ are the projection matrices to the PCS where C and C and RS. 2.2 Fault Detection Indices Fault detection indices are used to monitor the behavior of a process. There are several definitions of fault detection indices, among them, the most popular ones are the squared prediction error (SPE), the Hotteling’s T 2 statistic and a combination of both indices. A summary of these fault detection indices is presented here. More details are given in the paper of Qin [2003]. Squared prediction error, SPE. The SPE is defined as ˜P ˜ T x = xT Cx. ˜ SP E = xT P (4)
2 2 SP E 2 SP E χα h Its control limit δ is calculated as δ = g SP E with (1 − α) × 100% confidence Pn level, g Pn= θ2 /θ21 and SP E 2 h = θ1 /θ2 . Also, θ1 = i=l+1 λi , θ2 = i=l+1 λi , and λi is the ith eigenvalue of S. Hottelling’s T 2 statistic. The T 2 index is defined as T 2 = xT PΛ−1 PT x = xT Dx (5) with a control limit τ 2 = χ2α (l), and confidence level of (1 − α) × 100%. Combined index ϕ. The combined index, proposed by Yue and Qin (2001), is defined as ϕ = xT Φx

(6)

where ˜ C D + 2. (7) δ2 τ The control limit of this index is ζ 2 = g ϕ χ2α (hϕ ), where Φ=

   l θ1 g = / + τ2 δ2 2    θ1 l θ2 l ϕ h = + 2 / + 4 τ2 δ τ4 δ ϕ



θ2 l + 4 4 τ δ

The control limit of Index (x) can be calculated using the results of Box [1954] as η = gIndex χ2α (hIndex ) (11) where tr{SM}2 gIndex = (12) tr{SM} and 2 [tr{SM}] . (13) hIndex = tr{SM}2 The expression tr{A} denotes the trace of matrix A. 3. FAULT DIAGNOSIS METHODS 3.1 Complete Decomposition Contributions The Complete Decomposition Contribution (CDC) decomposes the fault detection index as the summation of the variable contributions. This method is widely used in industry. The CDC for the SPE is called contribution plots and was originally proposed by Miller et al. [1993]. Wise et al. [2006] proposed the CDC for the T 2 index. The CDC is defined as 1

Index(x) = xT Mx = kM 2 xk2 n  n 2 X X 1 = ξiT M 2 x = CDCiIndex (14) i=1

i=1 1

1

CDCiIndex = xT M 2 ξi ξiT M 2 x. Here, ξi is the ith column of the identity matrix, ξi = [0

T

0 · · · 1 · · · 0] .

(15) (16)

3.2 Partial Decomposition Contributions As it name suggests, the Partial Decomposition Contribution (PDC) partially decomposes a fault detection index as the summation of variable contributions. It was first proposed by Nomikos [1997] for the T 2 index. The PDC for T 2 index was defined as T 2 (x) = xT Dx = xT DIx ! n X T T =x D ξi ξi x

(8)

i=1

(9) =

with (1 − α) × 100% confidence level.

n X

xT Dξi ξiT x

i=1 2 P DCiT

General Index. We can see that the fault detection indices are quadratic forms. Thus, we can simplify our notation by considering just one general index, Index(x), as Index(x) = xT Mx (10) where M is shown in Table 2 for each detection index. Table 2. Values of M Index M

SP E ˜ C

T2 D

ϕ Φ

= xT Dξi ξiT x. (17) The P previous result is obtained using the relationship n I = i=1 ξi ξiT . In the general case, we just substitute D by M to obtain P DCiIndex = xT Mξi ξiT x. (18) 3.3 Diagonal Contributions The Diagonal Contribution (DC) removes the cross-talk among variables. It was proposed by Qin et al. [2001] for the T 2 index. Cherry and Qin [2006] used it with the combined index ϕ. It was defined for the T 2 index as

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2

1

DCiT = kPΛ− 2 PT ξi ξiT xk2 = xT ξi ξiT PΛ−1 PT ξi ξiT x = xT ξi ξiT Dξi ξiT x. The definition of this contribution for any index is DCiIndex = xT ξi ξiT Mξi ξiT x.

(19)

4.2 Reconstruction-Based Contributions (20) Since ABC is a scaled version of RBC, we will use RBC as a general case for both diagnosis methods. The definition of RBC has been given in Equations 21 and 22.

3.4 Reconstruction-Based Contributions The Reconstruction-Based Contribution (RBC) uses the amount of reconstruction of a fault detection index along a variable direction as the contribution of that variable to the index. It was proposed by Alcala and Qin [2008, 2009]. The RBC is defined as RBCiIndex = xT Mξi ξiT Mξi
2 ξiT Mx . = T ξi Mξi


−1

ξiT Mx

(22)

The ABC of Variable i is the squared cosine of the angle ¯ and ξ¯i between x =

!2


2 ξiT Mx = T ξi Mξi xT Mx

The diagonal contribution has been defined in Equation 20, and, since is a unique case, it will be treated individually. 5. ANALYSIS OF DIAGNOSABILITY

The Angle-Based Contribution (ABC) measures the cosine of the angle between a measurement and a variable direction. Angle-based information has been used for diagnosis by Raich and Cinar [1996] and by Yoon and MacGregor [2001]. For a fault sample x, the angled-based contribution of the ith variable is measured by the angle betweeen x and ξi after they are projected or rotated by M associated with a particular index. The projected vectors are 1 1 ¯ = M 2 x. ξ¯i = M 2 ξi , x

T ¯ ξ¯i x ¯ kξi kk¯ xk

4.3 Diagonal Contributions

(21)

3.5 Angle-Based Contributions

ABCiIndex

and CDC is a special case of GDC when β = 21 . This is obtained from 1 1 GDCiIndex = xT M 2 ξi ξiT M 2 x = CDCiIndex .

A fault in Sensor j is represented as x = x∗ + ξj f ; here, x∗ is the fault-free part of the measurement, ξj is the direction of the fault and f its magnitude. If the sensor fault is very large compared to the fault-free measurement, we can express the faulty measurement as x = ξj f. (26) This sensor fault is the simplest kind of fault that can happen in a process, and, if a diagnosis method is not able to diagnose it correctly, then there is no guarantee that such method will diagnose more complex faults correctly. 5.1 Diagnosability of General Decompositive Contributions Substituting the fault in Equation 26 into Equation 25 we get

(23)

RBCiIndex . (24) = Index(x) As we can see, ABC and RBC differ by a factor that is independent of i. Therefore, the diagnosis results will be the same for ABC and RBC. In the rest of the paper, RBC will be used for fault diagnosis. 4. UNIFIED DIAGNOSIS METHODS The five diagnosis methods presented previously can be classified into three general diagnosis methods, general decompositive contributions, reconstruction-based contributions and diagonal contributions. 4.1 General Decompositive Contributions The complete and partial decomposition contributions can be unified into a more general type of contribution, the General Decompositive Contribution (GDC). The GDC is defined as GDCiIndex = xT M1−β ξi ξiT Mβ x , 0 ≤ β ≤ 1. (25) PDC is a special case of GDC when β = 0 or β = 1. We can see that from GDCiIndex = xT Mξi ξiT x = P DCiIndex

GDCiIndex =

  1−β   β  2 M ij f for i 6= j  M ij

  1−β   β  M M jj f 2 for i = j jj  1−β    In this equation M = ξiT M1−β ξj and Mβ ij = ij ξiT Mβ ξj . Correct diagnosis is guaranteed if the GDC of the nonfaulty variable is less than or equal to the GDC of the faulty variable, this is, if  1−β   β      M M ij ≤ M1−β jj Mβ jj . (27) ij Here we have two cases. One is for β = 0 where Mβ is the identity matrix. In that case Mβ ij = 0 for i 6= j. Therefore, from Equation 27, correct diagnosis is guaranteed if 0 ≤ [M]jj . (28) Since this inequality always holds for positive semidefinite matrices, correct diagnosis is guaranteed for β = 0, which corresponds to the case of PDC. The other case is for β = 21 , which is the case of CDC, we guarantee correct diagnosis if

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h

1

M2

i2 ij

h 1 i2 ≤ M2 . jj

(29)

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However, in general, the relationship in Equation 29 does not hold; therefore, correct diagnosis is not guaranteed for CDC. 5.2 Diagnosis of Reconstruction-Based Contributions Substitution of the fault in Equation 26 into Equation 22 leads to  2  (mij ) 2   f for i 6= j mii RBCiIndex =    mjj f 2 for i = j In this case, mij = ξiT Mξj and mjj = ξjT Mξj . Correct diagnosis is guaranteed if the RBC value of the non-faulty variable is less than or equal to the RBC value of the faulty one. Thus, the following inequality has to hold 2

(mij ) ≤ mjj . mii

magnitudes, but the GDC with β = 12 does not. For sensor faults with medium size magnitudes, noise has an impact. It is desirable to know how the general diagnosis methods perform for this kind of fault. In order to assess the diagnosability for this kind of fault, a Monte Carlo simulation will be carried out in the next section. 6. SIMULATION STUDY The purpose of this example is to compare the rates of correct diagnosis for modest size faults given by the diagnosis methods. The rates of diagnosis of the GDC will be determined for different values of β and compared to the rates given by RBC. Also, the rates of correct diagnosis for CDC, PDC, DC and RBC will be compared. All the comparisons will be made for each of the cases when faults are detected by one of the SPE, T 2 and ϕ indices. The process model is

Using the Cauchy-Schwarz inequality, 2

(mij ) ≤ mii mjj the RBC value of the non-faulty variable is



  x1 −0.3879  x2   −0.4979     x3   −0.3023  x  =  −0.5441  4   x5   −0.3338 −0.3206 x6

(30)

2

(mij ) mii mjj ≤ = mjj . mii mii

(31)

The resulting expression is just the inequality needed to guarantee correct diagnosis. Therefore, correct diagnosis is guaranteed with RBC. It should be noted that, if M is positive definite, this is if M > 0, Equation 31 becomes 2

(mij ) < mjj mii

(32)

which implies a stronger diagnosability power of the RBC in this case. This is the case when the ϕ index is used for fault diagnosis because M = Φ > 0. 5.3 Diagnosis of Diagonal Contributions The substitution of the fault from Equation 26 into Equation 20 leads to DCiIndex

 =

0 for i 6= j mjj f 2 for i = j

These results are obtained knowing that ξiT ξj = 0 for i 6= j, ξiT ξi = 1 and mjj = ξjT Mξj . Again, correct diagnosis is guaranteed if the contribution of the non-faulty variable is less than or equal to the contribution of the faulty variable, this is, if the following inequality holds, 0 ≤ mjj . (33) Since M ≥ 0, the inequality always holds; therefore, correct diagnosis is guaranteed with diagonal contributions. However, a drawback of this approach is that it does not consider the correlation between variables, so it is similar to univariate monitoring. We have seen in this section that, for single sensor faults with large magnitudes, the GDC with β = 0, RBC and DC guarantee correct diagnosis for sensor faults with large

−0.7338 0.3126 0.2761 −0.0212 −0.2722 0.4616

 −0.0269 −0.0247  " #  t −0.3406  1 t2 + noise 0.6961   t 3  −0.6125 −0.1515

Here, t1 , t2 and t3 are zero-mean random variables with standard deviations of 1, 0.8 and 0.6, respectively. The noise in the process is normally distributed, has zero-mean and a standard deviation of 0.2. We build the PCA model with 3000 samples. First, the data are scaled to zero-mean and unit variance. The simulated faults are of the form xf aulty = x∗ + ξi f. (34) The fault-free measurements, x∗ , are generated with the process model; the fault magnitude f is uniformly distributed between 0 and 5; the direction ξi is uniformly random out of the six variable directions. In this case, we will generate 2000 faults. The simulation results can be seen from Figures 1 to 6. Figures 1 and 2 show the rates of correct diagnosis when the faults are detected with the SPE index. Figure 1 compares the rates of correct diagnosis for GDC with respect to β. The best diagnosis rates are obtained when β is near 0 or 1, and the worst rates are obtained when β is near 0.5. These results agree with the diagnosis results for sensor faults with very large magnitudes, where correct diagnosis was guaranteed with GDC for β = 0, 1 (PDC), but not for β = 0.5 (CDC). Overall, the highest rates are obtained when the GDC is used with the ϕ index. In Figure 2 we can see that when a fault is detected with the SPE, fault diagnosis results vary from one method to another; however, RBC and PDC give the best diagnosis rates, while CDC has the worst rates. If other indices are used for fault diagnosis, we can see that all the methods perform well when the ϕ index is used for diagnosis. Figures 3 and 4 show the diagnosis results when faults are detected with the T 2 index. In Figure 3 the qualitative diagnosis results are the same as those from Figure 1. In Figure 4 the diagnosis results are qualitatively similar to the ones of Figure 2 when the SPE or ϕ indices are used for

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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

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70

GDC - SPE 2

GDC - T GDC - Phi

60

50

40 0

GDC - SPE GDC - T2 GDC - Phi

80

70

60

50

0.2

0.4

0.6

0.8

40 0

1

0.2

β

90

90

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80 Rate of Correct Diagnosis (%)

Rate of Correct Diagnosis (%)

100

70 60 50 40 30

0

CompDecomp Cont PartDecomp Cont Diag Cont RBC SPE

0.8

1

Fig. 3. Correct diagnosis when faults are detected with the T 2 index

100

10

0.6 β

Fig. 1. Correct diagnosis when faults are detected with the SP E index

20

0.4

T2

70 60 50 40 30 CompDecomp Cont PartDecomp Cont Diag Cont RBC

20 10 0

Phi

Fig. 2. Correct diagnosis when faults are detected with the SP E index diagnosis; however, in this case, if the T 2 index is used for diagnosis, the DC has improved rates of correct diagnosis. In Figures 5 and 6 the faults are detected with the ϕ index. We can see that, in this case, the diagnosis results are very similar to the results when the faults are detected with the SPE index. 7. CONCLUSION In this paper we have shown that the fault diagnosis methods of complete decomposition, partial decomposition, diagonal, reconstruction-based and angle-based contributions can be unifed into the three general diagnosis methods. Also, it has been shown that, for single sensor faults with large magnitudes, diagonal and reconstructionbased contributions guarantee correct diagnosis; however, the general decompositive contributions do not guarantee correct diagnosis, except for the special case of partial

SPE

T2

Phi

Fig. 4. Correct diagnosis when faults are detected with the T 2 index decomposition contributions. It is surprising that the CDC method, which is widely used in industry, does not guarantee correct diagnosis for the simplest kind of faults. For single sensor faults with modest sizes, different diagnosis methods perform better for different indices; however, in general, CDC underperforms and RBC, using the ϕ index, has the best performance diagnosing sensor faults correctly. 8. ACKNOWLEDGEMENTS We gratefully acknowledge the financial support given to this work by the Roberto Rocca Education Program and the Texas-Wisconsin-California Control Consortium. REFERENCES C. Alcala and S. J. Qin. Reconstruction-based contribution for process monitoring. In Proceedings of 17th IFAC

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GDC - SPE

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GDC - T2 GDC - Phi 60

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40 0

0.2

0.4

0.6

0.8

1

β

Fig. 5. Correct diagnosis when faults are detected with the ϕ index

August 1997. P. Nomikos and J.F. MacGregor. Multivariate SPC charts for monitoring batch processes. Technometrics, 37(1): 41–59, 1995. S. J. Qin. Statistical process monitoring: Basics and beyond. J. Chemometrics, 17:480–502, 2003. S. J. Qin, S. Valle-Cervantes, and M. Piovoso. On unifying multi-block analysis with applications to decentralized process monitoring. J. Chemometrics, 15:715–742, 2001. A. Raich and A. Cinar. Statistical process monitoring and disturbance diagnosis in multivariate continuous processes. AIChE J., 42:995–1009, 1996. B. M. Wise, N. B Gallagher, R. Bro, J. M. Shaver, W. Winding, and R. S. Koch. PLS Toolbox User Manual. Eigenvector Research Inc., 2006. B.M. Wise and N.B. Gallagher. The process chemometrics approach to process monitoring and fault detection. J. Proc. Cont., 6:329–348, 1996. S. Yoon and J.F. MacGregor. Fault diagnosis with multivariate statistical models, part I: using steady state fault signatures. J. Proc. Cont., 11:387–400, 2001.

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80 70 60 50 40 30 CompDecomp Cont PartDecomp Cont Diag Cont RBC

20 10 0

SPE

T2

Phi

Fig. 6. Correct diagnosis when faults are detected with the ϕ index World Congress, Seoul, Korea, July 2008. Carlos F. Alcala and S. J. Qin. Reconstruction-based contribution for process monitoring. Accepted for publication on Automatica, 2009. G.E.P. Box. Some theorems on quadratic forms applied in the study of analysis of variance problems, I. effect of inequality of variance in the one-way classification. Ann. Math. Statistics, 25:290–302, 1954. G. Cherry and S. J. Qin. Multiblock principal component analysis based on a combined index for semiconductor fault detection and diagnosis. IEEE Trans. on Semiconductor Manufacturing, 19(2):159–172, 2006. P. Miller, R.E. Swanson, and C.F. Heckler. Contribution plots: the missing link in multivariate quality control. In Fall Conf. of the ASQC and ASA, 1993. Milwaukee, WI. P. Nomikos. Statistical monitoring of batch processes. In Preprints of Joint Statistical Meeting, Anaheim, CA,

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