計測自動制御学会東北支部 40 周年記念学術講演会
2004 年 12 月 22 日，23 日 日本大学工学部

講演番号：B02

異常診断のための FDA 法の拡張
○裴旭東 山下善之 坂倉義康 松本繁 (東北大学)

A Simple Modification of Fisher Discriminant Analysis Method for Fault Diagnosis∗Pei Xudong, Y. Yamashita,Y. Sakakura and S. Matsumoto (Tohoku University)
Abstract– This paper concerns about the application of the Fisher discriminant analysis(FDA) method to diagnose root
causes of faults from process time series. A simple modification of the FDA method was developed so as to enhance theaccuracy of the diagnosis, which uses the absolute values of the residuals in the calculation of the scatter matrices. The
method was applied to a continuous stirred tank reactor(CSTR) and compared with the usual FDA method. The resultshows excellant improvement in the accuracy of diagnosis for several cases.
Key Words: Fisher Discriminant Analysis, Fault Diagnosis, Process Monitoring

Let X ∈ <n×m be a set of n samples x ∈ <m andinclude two subsets X1 and X2 , each of which contains
n1 and n2 rows of X corresponding to the samples fromclass 1 and class 2 respectively. Let x̄i be the mean ofsamples for class i(i = 1, 2)

1 Introduction
Fault detection and diagnosis plays an important part inprocess engineering. The methods of fault detection anddiagnosis can be categorised into three classes: quantitative model-based methods, qualitative model-based methods and process history based methods [1][2][3]. Quantitative model-based methods are the methods that develop apriori domain knowledge from a fundamental understanding of the process in terms of mathematical functional relationships between the inputs and outputs of the system.
Qualitative model-based method requires qualitative functional relationships from past experience with the process
to express the a priori knowledge. Process history based
methods create knowledge of a diagnostic system by feature extraction processes based on the process history data.
Fisher discriminant analysis (FDA) was originally developed for pattern classification, and soon was applied topattern recognition problems [4]. The FDA can project themulti-dimensional patterns to one-dimensional ones withthe maximum ratio of between-class scatter to within-classscatter, and was used as a data-driven method for fault diagnosis [5]. Chiang et al [6] applied the FDA to the fault
diagnosis of the Tennessee Eastman chemical plant simulator, and showed the FDA has superior diagnosis accuracy
than the well known PLS or PCA.
In this work, we propose a simple modification of FDA
method to improve the accuracy of fault diagnosis. The
proposed method uses the absolute values of the residuals
in the calculation of the scatter matrices. We named thismethod as the absolute-value-based FDA (AFDA). The
method is demonstrated on a continuous stirred tank reactor(CSTR). The fault diagnosis results are compared withthe result of usual FDA.

1 Xxni(1)

21 X 1(xi − x̄i )(xi − x̄i )T
2 x ∈X ni(2)x̄i =x∈Xi

then
Sw =iiXi=1

is the within-class scatter matrix, andSb = (x̄1 − x̄2 )(x̄1 − x̄2 )T(3)is the between-class scatter matrix.
In terms of Sb and Sw , the Fisher criterion function isdefined asw T Sb w
(4)
J(w) = Tw Sw w
where w is a projection direction, and the vector w thatmaximizes J(w) must satisfy
Sb w = λSw w(5)for some constant λ, which is a generalized eigenvalue
problem. If Sw is nonsingular, Eq.(5) can be rewritten as−1Sw
Sb w = λw(6)which is a conventional eigenvalue problem, and theFisher optimal discriminant direction is the eigenvector w
corresponding to the maximal eigenvalue.

2 Fisher Discriminant Analysis
The FDA method is one of the promising methods todiagnose a root cause of a fault. It shows an optimal
one-dimensional mapping to maximize the separability ofgiven data classes in one-dimensional subspace. It decides the optimal projection vector which maximizes thebetween-class scatter matrix while minimizing the withinclass scatter matrix.

3Absolute Value Based FDA

The AFDA method is different from the FDA in the following two aspects:• Calculation of Sw• Calculation of Sb29

F : leakage of 0.01m3 /min in reactant inlet;
Fj : leakage of 0.01m3 /min in coolant inlet;

We use the absolute values of the right hand side of∗Eq.(2) and Eq.(3) to calculate Swand Sb∗ .
2X
1|xi − x̄i ||xi − x̄i |T
2 x ∈X ni1∗
Sw=For both faults, ten data sets including 100 normal samples and 50 abnormal samples were generated by ten simulation runs with different random seeds of white noise.
Typical examples of the generated data are shown in Fig.
2.
Eight combinations of sample variables were investigated to evaluate the effect of selection of the variables
Tj , CA and cv for each method. To evaluate the effect of
the number of fault samples, five data sets with differentnumbers of abnormal samples were evaluated.
The result of the fault diagnosis is summarized in Table
2 based on the accuracy of the correct recognition(%). Serial numbers in the first column of the table indicate theeight combinations of sample variables as shown in thesecond column. Numbers in the third column correspondto the fault of leakage in reactant inlet (I) and the fault ofleakage in coolant inlet (II). The elements in the columnof root cause are the variables diagnosed as the root causeof the fault I or II. The numbers 30, 35, 40, 45 and 50 in
the top of the table are the numbers of abnormal samples
in each data set. The numbers in FDA and AFDA columns
indicate the numbers of identified fault in ten simulations.
The last two columns summarize the accuracy of the fault
diagnosis for each method based on the result of all thesimulations.

(7)

ii
Xi=1Sb∗ = |x̄1 − x̄2 ||x̄1 − x̄2 |T(8)

According to the Fisher criterion function shown inEq.(4), we can get the same formula with Eq.(6) denoting∗the relation among the new within-class scatter matrix Sw,∗
between-class scatter matrix Sb , eigenvalue λ and eigen∗is nonsingular.
vector w if Sw
From Eqs.(7), (8) and (6), we can find the optimal discriminant direction w which is the eigenvector corresponding to the maximal eigenvalue, and the variable correlated
to the maximum of absolute value of each element in vector w is estimated to be the root cause of the fault.

4 Case Study
We applied the AFDA method to fault diagnosis on aCSTR with temperature control.
4.1 Data preparation
The schematic diagram of the CSTR [7] is shown in Fig.
1. The outlet temperature T of the reaction mixture wascontrolled by regulating the flowrate of the cooling water
Fj . The initial condition for the CSTR simulation is shownin Table 1, and the setpoint of the temperature T ∗ is 360K.
Proportional-integeral(PI) algorithm was used to control
the outlet temperature T .
The following six variables were selected as sample
variables :4.2

The controller changes the value of cv based on the measured value of Fj . In case of leakage of inlet coolant flow,
the controller will give larger cv than in the normal operation to compensate T to its set point. As the result,relationship between cv and Fj will be different between
leakage of inlet coolant flow and normal operation.
For this fault, Table 2 shows that the accuracy of thedetection of the AFDA is completely identical to that of
the usual FDA for all the combination of sample variables.
Moreover, estimated root causes of these two methods arealso the same in the ten simulations no matter which variable combination or which abnormal dataset is used.
As the result, we conclude that the two methods havethe same performance for the diagnosis of the fault of theleakage of inlet coolant flow.

T : outlet temperature of the reaction mixture
Tj : outlet temperature of the cooling water
CA : outlet concentration of the reaction mixture
cv : setpoint of the flowrate of cooling water
Fj : flowrate of the cooling water
F : inlet flowrate of the reaction mixture
Zero-mean white noises having 0.25% standard deviation were added to F and Fj .
The following two faults were considered:

T*TC

Leakage of inlet coolant

F T0CA0

Table 1: Initial condition for the CSTR simulation

Tj

Variable
FT0CA0Fj
Tj0TTjCAT
Fj Tj0TCA
Fig. 1: Schematic diagram of the CSTR

30

Value
0.0524
294.4
8.24
0.099
294.4
340
340
3.0

Unit
m3 /minK
kmol/m3
m3 /min
K
K
K
kmol/m3

361

361

359

360

T

T

360
0

20

40

60

80

100

120

140

160

180

359

200

335
330

0

20

40

60

80

100

120

140

160

180

330

200

CA

CA
0

20

40

60

80

100

120

140

160

180

200

cv

cv
020

40

60

80

100

120

140

160

180

Fj

Fj

2040

60

80

100

120

140

160

1800200

0.05

0.04

F

F

0

0.06

020

40

60

80

100

120

140

160

180

0.02

200

361

20

40

60

80

100

120

140

160

180

359

200

T
Tj

Tj
0

20

40

60

80

100

120

140

160

180

330

200

CA

CA
0

20

40

60

80

100

120

140

160

180

200

cv

cv

60

80

100

120

140

160

180

200

020

40

60

80

100

120

140

160

180

200

020

40

60

80

100

120

140

160

180

200

020

40

60

80

100

120

140

160

180

200

020

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

020

40

60

80

100

120

140

160

180

200

020

40

60

80

100

120

140

160

180

200

020

40

60

80

100

120140160

180200

020

40

60

80

100

120

140

160

180200

020

40

60

80

100

120

140

160

180

200

0.2

0.1
020

40

60

80

100

120

140

160

180

0.1
0200

0.2
Fj

0.2
Fj

40

10.5(d)

0.2

0.1
020

40

60

80

100

120

140

160

180

0.1
0200

0.06

0.06

0.04

0.04

FF

20

1.5

1

0.02

0335

1.5

0200

340

335

0

180

360

T
0

340

0.5

160

361

360

(c)

140

0.1

0.1

330

120

0.2

0.1

359

100

0.1
0200

0.2

080

0.2

0.1

060

1
0.5(b)

0.2

040

1.5

1
0.5

20

335

1.5(a)

0340
Tj

Tj

340

020

40

60

80

1001201401601800.02

200Fig. 2: Examples of generated data sets. (Data between the sample 1 and 100 are normal samples; Data between 101 and150 are samples with a leakage of inlet reactant; Data between 151 and 200 are samples with a leakage of inlet coolant.)
Table 2: Summary of the diagnosis

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4.3 Leakage of inlet reactant

5The value of Fj changes with cv in the normal operation. In case of leakage of inlet reactant flow, the controller
will give smaller cv than in the normal operation to keepT to its set point, and Fj will become smaller with thechange of cv. As the result, relationship between cv andFj will be the same between leakage of inlet reactant flow
and normal operation. At the same time, Tj will becomelarger with the deduction of Fj , and CA smaller with thededuction of F .
For this fault, Table 2 shows that the accuracy of thedetection of the AFDA is better than that of the usual FDA
in most cases, although the accuracy of these two methods
is identical for the case 2.
For the comparison of these two methods, we divide thevariable combinations into the following groups:

A simple modification of the FDA method was proposed
for improving the accuracy of fault diagnosis of process
plant. The method uses the absolute values of residuals
rather than the residuals of sample variables in the calculation of the between-class scatter matrix and within-classscatter matrix. The case study of a CSTR showed that theAFDA method is identical or superior than the usual FDA
method for the isolation of faults.

Conclusions

参考文献[1] Venkat Venkatasubramanian, Raghunathan Rengaswamy, Kewen Yin and Surya N.Kavuri (2003a),
A review of process fault detection and diagnosis.
Part I: Quantitative model-based methods. Computrers & Chemical Engineering, 27(3), 293–311.
[2] Venkat Venkatasubramanian, Raghunathan Rengaswamy and Surya N.Kavuri (2003b), A review ofprocess fault detection and diagnosis. Part II: Qualitative models and search strategies. Computrers &Chemical Engineering, 27(3), 313–326.

Related to Tj
case 1 & case 2case 3 & case 4case 5 & case 6[3] Venkat Venkatasubramanian, Raghunathan Rengaswamy, Kewen Yin and Surya N.Kavuri (2003c),
A review of process fault detection and diagnosis.
Part III: Process history based methods. Computrers& Chemical Engineering, 27(3), 327–346.

case 7 & case 8
Related to CA
case 1 & case 3[4] Richard O.Duda, Peter E.Hart, David G.Stork(2001), Pattern Classification. New York: Wiley.

case 2 & case 4case 5 & case 7[5] L. H. Chiang, E. L. Russell, and R. D. Braatz (2001),
Fault detection and diagnosis in industrial systems.
New York: Springer-Verlag.

case 6 & case 8
Related to cv
case 1 & case 5[6] Leo H. Chiang, Evan L. Russell, Richard D. Braatz(2000), Fault diagnosis in chemical process using
Fisher discriminant analysis, discrimininant partial
least squares, and principal component analysis.
Chemometrics and Intelligent Laboratory Systems,
50(2), 243–252.

case 2 & case 6case 3 & case 7
case 4 & case 8
From the comparison of the diagnosis result, we conclude the following two differences between the two methods:[7] K. Yamuna Rani, K. Gangiah (1991), Nonlinear dynamic matrix control of an open-loop unstable process with least-squares minimization for constraints.
Chemical Engineering Science, 46(5/6), 1520–1525

1. The AFDA method always have high accuracy in thediagnosis of root cause, and the accuracy does notchange by the difference of variable combinations.
However, the accuracy of the usual FDA method is
obviously decreased when both cv and CA are included in the variable combination. These results
show that the selection of sample variables for fault
diagnosis requires much attention in the usual FDA
than the AFDA.
2. The estimated root causes of these two methods are
not the same. The AFDA method diagnoses F and Fj
as the root causes in all the cases. But the usual FDA
method diagnoses F , Fj and cv as the root causes
when cv is used for fault diagnosis.

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