A novel multi-dimensional feature pattern classification method and its

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A novel multi-dimensional feature pattern classification method and its application to fault diagnosis Qun-Xiong Zhu, Qianqian Meng, and Yan-Lin He Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b00027 • Publication Date (Web): 11 Jul 2017 Downloaded from http://pubs.acs.org on July 16, 2017

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Industrial & Engineering Chemistry Research

1

A novel multi-dimensional feature pattern classification method and its

2

application to fault diagnosis

3

Qun-Xiong Zhua, b, Qian-Qian, Menga,b, Yan-Lin Hea, b,*

4 5

a

College of Information Science & Technology, Beijing University of Chemical Technology, Beijing, 100029, China; b

Engineering Research Center of Intelligent PSE, Ministry of Education of China, Beijing 100029, China;

6

* Corresponding author: Tel.: +86-10-64413467; Fax: +86-10-64437805. Email addresses: [email protected]

7

Abstract: With the development of modern process industries, data-driven

8

fault diagnosis methods have attracted more and more attention. In this

9

paper, a novel nonlinear fault diagnosis method based on multi-dimensional

10

feature pattern classification (MDFPC) is proposed. The proposed MDFPC

11

method integrates multi-kernel independent component analysis (MKICA)

12

with adaptive rank-order morphological filter (AROMF). Firstly, some

13

dominant independent components capturing non-linearity are extracted

14

from the historical process data using the MKICA algorithm, getting the

15

template signal and the testing signal of each fault pattern. Then, a

16

multi-dimensional signal classification method based on AROMF is

17

developed to achieve the diagnosis of fault patterns. The effectiveness of

18

the proposed fault diagnosis method is demonstrated by carrying out a case

19

study using the Tennessee Eastman process.

20 21

1

ACS Paragon Plus Environment


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1

1.Introduction

2

Modern process industries including the chemical process industry, the

3

electricity industry and the metallurgical industry have been becoming

4

larger in scale and more integrated and complicated.1, 2 Thus, fault diagnosis

5

plays an increasingly important role in ensuring the safety of production

6

processes, the quality of products, and the profit of operations.1,3 Generally,

7

fault diagnosis methods can be divided into three groups: knowledge-driven

8

methods, model-driven methods and data-driven methods. 4,5 With the rapid

9

advances in computer technologies and data mining technologies, the

10

historical process data collected from complex industrial processes can be

11

fully utilized and easily analyzed. Therefore, data-driven fault diagnosis

12

methods have become a popular and promising research field in recent

13

years. 6

14

Among data-driven methods, multivariate statistical techniques have

15

been successfully and widely applied to fault detection and diagnosis for

16

complex industrial processes over the past decades. 7-9 Principal component

17

analysis (PCA) and partial least squares (PLS) are two popular multivariate

18

statistical methods. 10-12 Based on historical process data, normal operating

19

regions can be well characterized and process faults can be successfully

20

identified using the methods of PCA and PLS. In the methods of PCA and

21

PLS, the Hotelling’s T2 and squared prediction error are adopted as the

22

statistical confidence limits. These two statistical confidence limits perform 2


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1

well under the assumption that historical process data follow a Gaussian

2

distribution.

3

non-Gaussian distribution because of the high nonlinearity of processes, the

4

various changes of production strategies, the shifts of operation conditions,

5

and so on. 13,14 In this respect, the results of fault detection and diagnosis

6

using the PCA method or the PLS method may be inaccurate. Fortunately,

7

another multivariate statistical technique named independent component

8

analysis (ICA) is proposed to deal with non-Gaussian processes.15-17

9

Historical process data can be projected into a subspace with some rather

10

lower dimensional independent components using the ICA method. ICA is

11

an extension method of PCA.18 Apart from extracting the independent

12

components, ICA can also reduce the correlation between variables.19 The

13

difference between ICA and PCA/PLS lies in that the latent variables of

14

mutual independence are extracted by ICA through maximizing the

15

non-Gaussianity based on higher-order statistics instead of the variance or

16

covariance based on the second-order statistics. That is why ICA can well

17

deal with non-Gaussian processes. To handle the high nonlinearity of

18

processes, kernel ICA (KICA) or some other modified KICA methods are

19

presented.20-22 Kernel based multivariate statistical methods have been

20

successfully used for data feature extraction and fault diagnosis of nonlinear

21

processes.23-25

However,

historical

process

data

3


usually

follow

a


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1

In our study, we proposed a novel nonlinear fault diagnosis method

2

based on multi-dimensional feature pattern classification (MDFPC). Pattern

3

classification techniques have been successfully applied to make fault

4

diagnosis of complex processes.26-28 Generally, there are two steps in

5

pattern classification techniques for fault diagnosis. Firstly, empirical rules

6

are needed to be established by extracting regular patterns from historical

7

process data. The rules contain the interrelationships of fault forms. Then,

8

the extracted patterns are used to judge the process state for telling that

9

which kind of faults that the process system is suffering from. Li and Xiao

10

proposed a pattern classification method with the aid of PCA and

11

one-dimensional adaptive rank-order morphological filter (AROMF).29

12

AROMF is used for pattern classification. AROMF is an improved model

13

of rank-order morphological filter (ROMF). The mathematical morphology

14

based ROMF is a kind of signal processing method with good filtering

15

function. The filtering characteristic is directly related to the structuring

16

elements and the percentile.30 As a kind of supervised transformation

17

technology, ROMF can reduce the difference between the supervised signal

18

and the test signal in a similar state. Although a PCA-based

19

one-dimensional feature pattern fault diagnosis method was proposed in the

20

reference 29, there is still much improvement room. PCA is used to extract

21

the fault signal feature. Based on our above analyses, most of processes

22

data follow a non-Gaussian distribution. Hence, PCA may not effectively 4


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1

extract fault features. What is more, only one-dimensional component is

2

used for pattern classification. Single principal component may miss the

3

key information because of the excessive dimension reduction of PCA.

4

Some of the components of different faults may be similar. If only one

5

component is used for pattern classification, the classification accuracy may

6

be biased. Multi-kernels were successfully used to enhance the model

7

performance.31 So, the basic KICA may not well deal with the highly

8

complex process data due to only one kernel used. In order to solve these

9

problems, our proposed MDFPC method integrates multi-kernel ICA

10

(MKICA) with AROMF (MKICA-AROMF). MKICA with multi-kernel

11

functions are built to effectively extract the multi-dimensional feature

12

patterns from non-Gaussian process fault data. Then multi-dimensional

13

feature patterns classification is carried out using the method of AROMF.

14

Finally, a novel effective fault diagnosis method integrating multi-kernel

15

independent component analysis with adaptive rank-order morphological

16

filter

17

MKICA-AROMF method, it is applied to Tennessee Eastman process for

18

fault diagnosis. The simulation results show that the proposed

19

MKICA-AROMF method can achieve higher accuracy, especially for

20

hard-to-diagnose faults.

is

presented.

To

test

the

performance

of

the

proposed

21

The organizations of the remaining parts of this paper are as follows:

22

Section 2 briefly reviews the kernel independent component analysis and 5



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1

the adaptive rank-order morphological filter; the proposed novel effective

2

fault diagnosis method integrating multi-kernel independent component

3

analysis with adaptive rank-order morphological filter is developed in

4

Section 3; in Section 4, a case study of applying the proposed fault

5

diagnosis method to the Tennessee Eastman process is given, including the

6

simulation results and the analyses. Finally, conclusions are presented in

7

section 5.

8

2.Preliminary methods

9

2.1 Kernel independent component analysis

10

KICA is derived to not only extract independent components but also

11

capture the nonlinearity of process data. The key principle of KICA can be

12

expressed as follows: for the sake of making better use of nonlinear data,

13

the original data space is mapped into high dimensional feature space F

14

via a nonlinear function ϕ (·) .

15

Set the number of samples as F, and then the original process data are

16

expressed as X = [ x1 , x2 ,..., xF ] . The map function ϕ (·) will be used to map

17

xi to ϕ ( xi ) . These mapping results are employed to comprise high

18

dimensional space, which expressed as φ = ϕ ( x1 ) ,ϕ ( x2 ) ,...,ϕ ( xF )  . Since

19

the map function ϕ (·) is usually unknown, the realization of nonlinear

20

transformation needs to draw support from a kernel function k ( xi , x j ) .

21

Define the kernel matrix as K , and each element value is calculated as

22

follows: 6


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K ij = k ( xi , x j ) = ϕ ( xi ) ϕ ( x j )， i, j = 1,L, F T

1

(1)

2

The widely used kernel functions include the polynomial kernel, the

3

sigmoid kernel and the radial basis kernel, etc. For instance, the radial basis

4

kernel function is shown in Eq. (2): k ( xi , x j ) = e

5 6 7

10

xi − x j c

K = K − I F K − KI F + I F KI F

1 . F

Besides, the normalization is shown as follows: Kx =

K

(4)

( )

trace K / F

The eigenvalue decomposition of K x is shown as Eq. (5). Then, the D

largest

eigenvalues

λ1 ≥ λ2 ≥ L ≥ λd

13

first

14

eigenvectors α1 ,α 2 ,L,α d can be obtained.

λα = K xα

15

18

(3)

where I F is a constant matrix, each element value of which is

12

17

(2)

be obtained via Eq. (3):

11

16

2

The zero-mean processing of data is essential. The centralizer of K can

8

9

−

and

their

(5)

The eigenvector matrix V = [v1 , v2 ,L, vd ] can be calculated as

follows: V = φ HΛ −1/2

(6)

19

where H = [α1 ,α 2 ,L,α d ] , Λ =diag ( λ1 , λ2 ,L, λd ) . Then the whitening

20

matrix Qϕ will be obtained as follows:

7



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1  Q =V  Λ F 

2

−1/2

ϕ

1

Page 8 of 34

= F φ H Λ −1

(7)

Accordingly, data can be whitened as follows:

Z = Qϕϕ ( x ) = F Λ −1H T K x

3

(8)

4

After obtaining the whitening matrix Z in the feature space, the

5

independent component analysis (ICA) algorithm is used to get independent

6

components (ICs).

7

The basic principle of ICA is listed as follows: the separation matrix W

8

is found when the observed signal Z is known and the independent

9

component signal Y and mixing coefficient matrix A are unknown.

10

The relationship between Z and A is shown in Eq. (9):

11

Z = AS

12

where S is the independent component matrix. When the separation matrix

13

W is found from the observed signal Z, the independent component signal Y

14

can be got using Eq. (10)

15 16

Y = WZ = WAS = Sˆ

(9)

(10)

where Sˆ is the estimated value of the independent component S.

17

Obviously, the key problem of the ICA algorithm is to calculate the

18

separation matrix W to make the components in the reconstruction matrix

19

Sˆ as independent as possible. In this paper, the Fast-ICA algorithm is

20

selected to calculate the separation matrix W. The steps of the Fast-ICA

21

algorithm are described as follows:

22

(a) Select the number of independent components N, and set the number of 8


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1

iterations as P=1;

2

(b) Randomly assign an initial vector wP ;

3

(c) Let

4

{

} {

}

wP ← E zg ( wPT Z ) − E g ' ( wPT Z ) wP ,where

g ( •)

is

a

nonlinear function;

∑ ( w w )w ; P −1

T P

5

(d) Orthogonalization: wP =wP -

6

(e) Normalization: wP = wP / wP ;

7

(f) If wP does not converge, return (c); If wP converges, go to next step;

8

(g) Let P = P + 1 , if P ≤ N , return (b)

9

(h) End

j =1

j

j

10

2.2 Rank-order morphological filter

11

The rank-order morphological filtering (ROMF) can be regarded as a

12

data sorting methods it is an integration and development of the median

13

filter and the Minkowski structure.32 The main principle of ROMF is shown

14

as follows: after sorting a small segment of a signal, select an optimal

15

sample point to represent the entire segment of the selected signal. ROMF

16

can avoid the limitation of only choosing the maximum or minimum value

17

of the selected signal segment.

18

Rank-order morphological filtering is defined as follows: Set f (t) is

19

an one-dimensional discrete signal; B = j1 , j2 ,L jN B is a structural

20

element; µ ( B )

21

Sorting

22

order: f ( t1' ) ≤ f ( t2' ) ≤ L ≤ f t N' B .

{

values

(0 < µ ( B ) = N of

f ( B)

}

< +∞ ) is the number of elements in B.

B

defined

on

B

( )

9


in

an

ascending


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1 2

( )

Denote o-th ordered value of f (t) on B as f to'

5 6

ord {o, f B} = f ( to' ) Then one-dimensional ROMF

o = 1, 2,L, N B

( f ⓟB ) ( t ) can

by Eq.

(11)

be defined using Eq.

(12):

( f ⓟB ) ( t ) = ord {o, f Bt' } = ord {( N B − 1) p + 1, f Bt' }

7

Bt' = {t − j , j ∈ B}

8

morphological filter percent, o is the result of

9

to the nearest integer:

,

p = 0,1 / ( N B − 1) ,L,1

is

called

( N B − 1) p + 1

o = ( N B − 1) p + 1

10 11

, calculated

(11):

3 4

Page 10 of 34

(12) rank-order

being rounded

(13)

where [ a ] is used to find the integer nearest to a.

12

Based on the analysis of the above definition, the main effects of the

13

performance of filter are µ ( B ) the number of element in B and the

14

percentile value of rank-order morphological filter p. An adaptive algorithm

15

named adaptive rank-order morphological filter (AROMF) is presented 29.

16

Assume xi = f ( ti ) = si + ni

( i = 1, 2,3L)

is a discrete signal with

17

noise; si = d i is the desired signal without noise; ni is the noise. The

18

calculation principle of AROMF and the adaptive update procedure of

19

parameters are shown in Eq. (14) and Eq. (15)

20

yi (

itN )

{(

= ord N t(i

itN )

)

− 1 pi(

itN )

+ 1; f Bt(i

10


itN )

}

(14)

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(

)

(

)

m(itN +1), j = m(itN ), j + α d − y ( itN ) sgn f ( t − j ) − y (itN ) + 2 p ( itN ) − 1 , ∀j ∈ B (itN ) i i i i i i ti    i  (itN +1) itN itN +1 , j (15) = j ∀j ∈ Bi( ) , mi( ) > thmB  Bi   pi( itN +1) = Bi(itN ) + 2 β di − yi( itN ) N t(i itN ) − 1 

{

1

}

(

)(

)

In Eq. (14) and Eq. (15), ord is the rank-order processing operator

2

( itN )

3

given in Eq. (12). itN represents the current iteration, B

4

itN structural element for all samples in the itN-th iteration, and pi( ) is the

5

rank-order morphological filter percent of the i-th sample ti in the itN-th

6

iteration; for ∀j ∈ Bti

7

preset threshold; α

8

N t(i

9

can be updated adaptively in AROMF.

10

( itN )

itN )

(

= µ Bt(i

itN )

),

Bt(i

( itN ), j

assigned with the value of mi

and β itN )

is the

, thmB is the

are the updating step of parameters.

{

= ti − j , ∀j ∈ Bt(i

itN )

} . The parameters of B and p

3.MKICA-AROMF based fault diagnosis method

11

Due to complexity of the process systems, only one independent

12

component (IC) cannot reconstruct the complete information of system

13

states. And the collected process data usually follows non-Gaussian

14

distribution. Considering these reasons, a novel fault diagnosis method

15

based on multi-dimensional feature pattern classification integrating

16

multi-kernel independent component analysis with adaptive rank-order

17

morphological filter (MKICA-AROMF) is proposed.

18

3.1 MKICA

19

In our study, three kernel functions of the linear kernel function, the

20

sigmoid kernel function, and the Radial basis kernel function are selected to 11



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Page 12 of 34

1

establish multi-kernel ICA (MKICA) for better extracting the features of

2

each fault. Multi-dimensional feature signals of fault data are extracted so

3

that the information of each fault pattern can be completely reconstructed.

4

There are two main steps for building the MKICA model: first, the three

5

selected kernel functions of the linear kernel function, the sigmoid kernel

6

function, and the Radial basis kernel function are used to expand the data

7

space to multi-kernel space; then the Fast-ICA algorithm is used to obtain

8

the independent components (feature space) of the data in the multi-kernel

9

space. Suppose that the original process data represented as X ∈ RT ×F ,

10

where T is the number of process variables for the original data, F is the

11

number of samples for original data. Then the process data X is expanded to

12

three kernel matrices. We use K L ∈ R F ×F , K S ∈ R F ×F and K R ∈ R F ×F to

13

represent the linear kernel matrix, the sigmoid kernel matrix and the Radial

14

basis kernel matrix, respectively. The whitening process is carried out for

15

each kernel matrix. Z L ∈ R d ×F

16

represent the whitened matrices of the linear kernel matrix, the sigmoid

17

kernel matrix and the Radial basis kernel matrix, respectively. Finally, the

18

three

19

of Z =  Z LT

20

combined matrix Z can be obtained by the Fast-ICA algorithm. Based on

21

the analyses above, the flowchart of the proposed MKICA is shown in

22

Figure 1.

whitened

ZST

matrices

,

Z S ∈ R d ×F and Z R ∈ R d ×F are used to

are

put

together

in

a

matrix

T

Z R T  . Next, the independent components(ICs) of the

12


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Original data space

Original data for all fault patterns

nonlinear mapping by three kernel functions respectively Multi-kernel space linear kernel space

Sigmoid kernel space

Radial basis kernel space

Preprocessing by Eq. (3-8)



Fast-ICA Determine objective function bases on negative entropy maximization theory

Obtain the separating matrix W via Newton iterative method Convergent

NO

YES Gain independent components via W*Z

Feature space

IC1

IC2

……

ICN

1 2 3 4

Figure 1. Flowchart of the proposed MKICA

3.2 MDFPC based on MKICA-AROMF After the MKICA model is built, the multi-dimensional feature pattern 13



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Page 14 of 34

1

classification can be carried out based on MKICA-AROMF. AROMF is a

2

filtering method. In AROMF, the desired signals are regarded as

3

supervisory signals to filter the test signals. Different output signals y can

4

be obtained via the same noised test signal x under different supervisory

5

signals d. If the supervisory signal is an undesired signal, the recovery

6

performance of the noisy signal will be poor, and then the distance between

7

the output signal y and the desired signal d will not achieve the smallest

8

value.

9

To carry out the multi-dimensional feature pattern classification method,

{

} from

10

the first step is to obtain the supervisory signals d i = d i1 , d i2 ,L d iN

11

the training data, where i = 1, 2,L, M is the i-th fault pattern. M is the

12

number of supervisory signals.

13

The training data of each fault pattern will be mapped into multi-kernel

KL

14

space

15

data Z i =  Z iLT

16

fault pattern.

17

,

KS Z iS T

and

KR

,

then

the

T

Z iRT  can be get, where i = 1, 2,L, M is the i-th

After getting the whitened data Z i of the i-th fault pattern, and the

18

separation matrixW = [ w1 , w2 ,L, wN ] of the MKICA model

19

supervisory signals di = di1 , d i2 ,L, d iN

20

di = WZ i

21 22

whitened

T

{

}

,

then the

is obtained as follows: (16)

For a certain testing data, carry out the same steps of dealing with the

{

training data. Then the test signal x = x1 , x 2 ,L, x N 14


}

can be obtained.

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{

1

The output signal yi = yi1 , yi2 ,L, yiN

2

calculated via Eq. (17):

}

of the i-th fault pattern is

yin = F ( x, din )

3

(17)

4

where F ( a, b ) stands for the adaptive rank-order morphological filter

5

between a and b.

6 7

Define the distance between the output signal yin and the supervisory signal d in as Ein calculated using Eq. (18).

(y

− diln )

8

Ein = ∑ l =1

9

where L is the number of signal sampling points; di

L

n il

2

(18) are the

10

multi-dimensional supervisory signals for the i-th fault pattern; d in is the

11

component signal on the n-th IC of di ; yin is the output signal obtained

12

under the supervision of d in .

13 14

Define the mean distance between yi and di as Ei calculated using Eq. (19).

Ei =

15

1 N

∑

n

Ein

(19)

is the number of ICs extracted from the MKICA model; yi are

16

where N

17

the multi-dimensional output signals obtained under the supervision of di .

18 19

The flowchart for calculating the mean distance of multi-dimensional signals is shown in Figure 2.

15



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Page 16 of 34

Unknown multi-dimensional feature signal x obtained by MKICA

i-th template signal obtained by MKICA from i-th fault-pattern

test signal multi-dimensional AROMF

supervisory signal

output signal yi obtained under supervision of di

Calculate the distance Ein between corresponding ICs of di and yi

Calculate the mean distance between output signal yi and supervisory signal di 1 2

Figure 2. Calculate the mean distance between the multi-dimensional output

3

signal yi and the i-th multi-dimensional supervisory signal d i

4

Therefore, the main procedures of the proposed multi-dimensional

5

fault signal (feature) pattern classification method are listed as follows:

6

1) According to the Eq. (17), recover the different output signals

7

yi = { yi1 , yi2 ,L, yiN } from the unknown test signal x by regarding each

{

8

template (supervisory) signal d i = d i1 , d i2 ,L, d iN

9

as different supervisory signals;

}

of the i-th fault patterns

( i = 1, 2,L, M )

10

2) Calculate the mean distance Ei

11

signal yi and the template (supervisory) signal di using Eq. (18) and Eq. 16


between the output

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1

(19).

2

3) Find the minimum Ei

3

fault.

4

( i = 1, 2,L, M ) , then the fault pattern is the i-th

Based on the above analyses, the steps of the MKICA-AROMF based

5

fault diagnosis method are summarized as follows:

6

1) Data preprocessing: denoising and normalization of the original training

7

data and the testing data.

8

2) Establish MKICA model: three kernel functions of the linear kernel

9

function, the sigmoid kernel function, and the Radial basis kernel function

10

are selected to establish a multi-kernel ICA (MKICA) for better extracting

11

the ICs of fault patterns, expressed as: IC1、IC2……ICN.

12

3) Obtain the template signal: the template signals d = {d1 , d 2 ,L d M } are

13

obtained by projecting the single fault training data onto the MKICA model.

14

Where d i = d i1 , d i2 ,L d iN

15

number of ICs.

16

4) Obtain the testing signal: the test signal x = x1 , x 2 ,L, x N to be

17

classified is obtained via projecting the unknown original testing data onto

18

the MKICA model.

19

5) Determine fault patterns: the fault patterns of the test signals can be

20

diagnosed by using the multi-dimensional fault signal (feature) pattern

21

classification method mentioned above.

{

} , M is the number of fault patterns, N is the {

22 17


}


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2

The flowchart of MKICA-AROMF based fault diagnosis method is shown in Figure 3: Training stage The whole training data

Data preprocessing

Single fault training data

Building MKICA model

Data preprocessing

Template signals of M fault-patterns d1,d2 ,… … ,dM

Supervisory signal Testing stage Testing data

Data preprocessing

Test signal to be classified

The mean distance between test signal and each template signal di is calculated separately

E1

……

E2

EM

Min{ E1,..., EM}

Determine pattern

3 4 5

Figure 3. Flowchart of MKICA-AROMF based fault diagnosis method

4. Case study

6

This section provides the validation of our proposed fault diagnosis

7

method. Tennessee Eastman (TE) process is a famous benchmark

8

problem.33-36 The TE process shown in Figure 4 has been widely used as a

9

complicated and highly nonlinear process for fault diagnosis and academic

10

researches. In our study, the TE process is selected to verify the superiority

11

and efficiency of the proposed fault diagnosis method. In the TE process, 18


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1

there are 12 manipulated variables and 41 measuring variables. The process

2

variables

3

XOV(1)-XOV(41), respectively.

of

TE

can

be

expressed

as

XMV(1)-XMV(12)

and

4 5 6

Figure 4. Tennessee Eastman process

The simulation code for the TE process can be downloaded from

7

http://depts.washington.edu/control/LARRY/TE/download.html,

8

control strategy of TE is described in the reference 37. In our case study, the

9

first 7 of the 21 fault patterns in the TE process, i.e. IDV(1)–IDV(7), are

10

considered for fault diagnosis using the proposed MKICA-AROMF method.

11

The information of the selected 7 faults and the normal state defined as

12

IDV(0) are listed in Table 1. The variables of XMV(5), XMV(9) and

13

XMV(12) for each fault are constants. Thus, the remaining 50 process

14

variables of each single fault are used for getting template signals. The 19


and

the


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

sampling interval is selected as 1.5 minutes and each simulation run is set

2

as 24hrs. Thus, 960 samples can be generated for each simulation run. The

3

number of samples is cut down to 90 via interval sampling to reduce the

4

computational load. Single fault is introduced at the 8th hr. Table 1. The first 7 fault patterns in the TE process

5

Fault pattern

Description

IDV(0)

normal state

IDV(1)

A/C feed ratio, B composition constant (stream 4)

IDV(2)

B composition, A/C ration constant (stream 4)

IDV(3)

D feed temperature (stream 2)

IDV(4)

reactor cooling water inlet temperature

IDV(5)

condenser cooling water inlet temperature

IDV(6)

A feed loss (stream 1)

IDV(7)

C header pressure loss-reduced availability (stream 4)

6

According to the section 3, the simulation process of fault diagnosis is

7

carried out. To obtain the complete features of the TE process, the data of

8

the first 7 fault patterns and the normal state i.e. IDV(0)-IDV(7) are used to

9

establish the MKICA model. Then projecting the original training data of

10

each known fault pattern onto the MKICA model, many corresponding

11

template signals (i.e. ICs) are obtained. Four of these template signals (i.e.

12

IDV(0)~IDV(3)) in a simulation are shown in Figure 5. From Figure 5, we

13

can see that comparing template signals of each fault pattern with those of

14

the normal state, the trends and changes of template signals on the same IC

15

are totally different. However, the trends and changes of template signals on

16

the same IC for each fault pattern are similar. Hence, it is necessary to 20


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1

obtain multi-dimensional characteristic signals for improving the accuracy

2

of fault diagnosis. In our study, the first 4 ICs are selected in the established

3

MKICA model for fault diagnosis.

4 5

(a) Four-dimensional feature signals(ICs) of IDV(0)

6 7

(b) Four-dimensional feature signals(ICs) of IDV(1)

8 9

(c) Four-dimensional feature signals(ICs) of IDV(2)

10 11

(d) Four-dimensional feature signals(ICs) of IDV(3)

12

Figure 5. Template signals of IDV(0)~IDV(3)

13

After obtaining the template signals of the selected seven faults, the 21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

MKICA model is used to detect the characteristic signals to be classified as

2

the testing signals. Then determine the fault pattern using the proposed

3

MKICA-AROMF fault diagnosis method. Among the first 7 faults of TE

4

process, the third fault IDV(3) is difficult to diagnosis. Here, we take IDV(3)

5

as an example to show the diagnosis process. The simulation results of

6

IDV(3) based on the AROMF algorithm with different supervision signals

7

of IDV(0)~IDV(7) are shown in Figure 6.

8 9

(a) The multi-dimensional AROMF for IDV(3) under the supervision of IDV(0)

10 11

(b) The multi-dimensional AROMF for IDV(3) under the supervision of IDV(1)

12 13

(c) The multi-dimensional AROMF for IDV(3) under the supervision of IDV (2)

22


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1 2

(d) The multi-dimensional AROMF for IDV(3) under the supervision of IDV (3)

3 4

(e) The multi-dimensional AROMF for IDV(3) under the supervision of IDV (4)

5 6

(f) The multi-dimensional AROMF for IDV(3) under the supervision of IDV (5)

7 8

(g) The multi-dimensional AROMF for IDV(3) under the supervision of IDV (6)

23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 34

1 2

(h) The multi-dimensional AROMF for IDV(3) under the supervision of IDV (7)

3

Figure 6. The results of AROMF for test signal of IDV (3) with different template

4

(supervision) signals

5

Table 2. Distance Ei (i=0,1,2,3,4,5,6,7) simulation results of the first 7 fault patterns in

6

the TE process Supervisory

d0

d1

d2

d3

d4

d5

d6

d7

x0

8.96

27.52

31.04

28.75

25.81

25.51

36.80

27.23

x1

44.62

2.58

11.53

8.85

10.55

8.90

39.18

10.15

x2

44.35

8.74

3.39

9.61

11.48

8.92

38.84

10.06

x3

41.62

7.46

9.13

2.90

6.00

3.30

40.27

6.41

x4

41.12

7.09

8.29

4.46

3.86

4.57

38.23

4.41

x5

38.98

12.44

14.63

9.33

11.99

9.29

41.85

12.88

x6

58.21

53.61

57.24

54.65

52.44

54.63

3.76

53.38

x7

41.27

8.57

9.14

5.80

4.29

6.62

37.53

2.86

signal Test signal

7

Distance Ei (i=0,1,2,3,4,5,6,7) simulation results of the first 7 faults in

8

TE process are shown in Table 2. Data in each row of the Table 2 represent

9

the average distance between the test signal xi of the i-th testing fault

10

pattern (i=0,1,2,3,4,5,6,7) and the supervisory signal di (i=0,1,2,3,4,5,6,7).

11

We take the test signal x3 of the 3-rd testing fault pattern as an example to

12

analyze the results. We can see from the Table 2 that the distance E0 24


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1

between x3 and d 0 is 41.62. The values of E1 , E2 , E3 , E4 , E5 , E6 ,

2

and E7 are 7.46, 9.13, 2.90, 6.00, 3.30, 40.27, and 6.41, respectively. We

3

can find that the E3 achieved the least value for the 3-rd testing fault

4

pattern, which means that the average distance between the output signal

5

x3 of the 3-rd testing fault pattern and the supervisory signal d3 is smallest.

6

Thus, the 3-rd fault pattern is correctly diagnosed. The simulation results

7

confirm that the gap between the output signal and the supervisory signal is

8

the smallest when the supervisory signal of the same fault pattern of the

9

template signal is used. For the fault patterns of IDV (1) ~ IDV (7), 50

10

times of simulations are carried out in our study. In addition, other two

11

models of the AROMF based on one dimensional PCA (ODPCA) and

12

KICA-AROMF (ODKICA) methods are also developed for comparisons.

13

The simulation results of different fault diagnosis methods are shown in

14

Table 3 and Figure 7.

15 16 17

Table 3. Accuracy comparisons of fault diagnosis for the first 7 faults of TE process using MKICA-AROMF, ODPCA-AROMF, and ODKICA-AROMF methods Fault pattern

MKICA-AROMF

ODKICA-AROMF

ODPCA-AROMF

IDV(1)

100%

100%

98%

IDV(2)

100%

90%

96%

IDV(3)

66%

40%

28%

IDV(4)

76%

42%

26%

IDV(5)

100%

100%

98%

IDV(6)

100%

96%

96%

IDV(7)

100%

100%

98%

25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2

Figure 7. Accuracy of fault diagnosis for the first 7 faults of TE process using three

3

different methods

4

From Table 3, we can see that the diagnosis accuracy of the first 7 faults

5

IDV(1)~ IDV(7) using our proposed MKICA-AROMF method are 100%,

6

100%, 66%, 76%, 100%, 100%, and 100%, respectively; the diagnosis

7

accuracy of the first 7 faults IDV(1)~ IDV(7) using the ODPCA-AROMF

8

method are 98%, 96%, 28%, 26%, 98%, 96%, and 98%, respectively; the

9

diagnosis accuracy of the first 7 faults IDV(1)~ IDV(7) using the

10

ODKICA-AROMF method are 100%, 90%, 40%, 42%, 100%, 96%, and

11

100%, respectively. Overall, the average diagnosis accuracy of the proposed

12

MKICA-AROMF method, the ODKICA-AROMF method, and the

13

ODPCA-AROMF method is 91.7%, 81.1%, 77.1%, respectively. The

14

results in Table 3 indicate that the proposed MKICA-AROMF method can

15

achieve the highest accuracy. Especially, for the hard-to-diagnose fault of

16

IDV(3) and IDV(4) , the improvement in the diagnosis accuracy are much 26


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1

higher. The same conclusion can be made from Figure 7. In the method of

2

ODPCA-AROMF, PCA is used to extract the principal components (PCs)

3

of the fault patterns. Complex process data usually follow a non-Gaussian

4

distribution. PCA is not suitable. What is more, PCA is a linear model that

5

cannot well exact the PCs of fault patterns. In addition, only

6

one-dimensional feature (i.e. the first IC) is used for diagnosis. From Figure

7

5, we have concluded that the trends and changes of template signals on the

8

same IC for each fault pattern are similar. Thus, only one-dimensional

9

characteristic signals may reduce the accuracy, especially for the

10

hard-to-diagnose fault patterns. In our proposed MKICA-AROMF method,

11

an effective nonlinear MKICA model is used to extract ICs for fault

12

patterns. And multi-dimensional features are adopted for diagnosis. That is

13

why our proposed MKICA-AROMF method can achieve higher accuracy.

14

Through the case study using the TE process, the superiority of the

15

proposed MKICA-AROMF method is validated.

16

5. Conclusions

17

In this article, a novel multi-dimensional feature pattern classification

18

fault diagnosis method integrating multi-kernel independent component

19

analysis with adaptive rank-order morphological filter is proposed. A

20

multi-kernel independent component analysis is firstly developed for

21

effectively extracting the features of different fault patterns. Then,

22

multi-dimensional adaptive rank-order morphological filter is established 27



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Page 28 of 34

1

for fault diagnosis. Finally, the first 7 faults of the TE process are selected

2

as a case study to validate the performance of our proposed method.

3

Simulation

4

ODPCA-AROMF and ODKICA-AROMF, our proposed MKICA-AROMF

5

method could achieve highest diagnosis accuracy, especially for the

6

hard-to-diagnose fault.

7

Acknowledgements

8

This research is supported by the National Natural Science Foundation of

9

China under Grant Nos. 61533003 and 61473026.

results

show

that

compared

with

the

methods

of

10

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(37) Ricker, N. L. Decentralized control of the Tennessee Eastman Challenge Process. J. Proc. Cont, 1996, 6 (4), 205-221.

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TOC

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Original data space

Original testing data for all fault patterns

nonlinear mapping by three kernel functions respectively Multi-kernel space linear kernel space

Sigmoid kernel space

Radial basis kernel space

Whitening

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ICA Gain independent components via W*Z

Feature signal

Fault diagnosis

Test signal

Template signals

The proposed multi-dimensional feature pattern classification (MDFPC)

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A novel multi-dimensional feature pattern classification method and its

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