# A Modified FNN Fault Diagnosis on PCVD Microwave System

Authors: Zhenyu Li; Hongsheng Li
Journal: Advances in Fuzzy Systems
 (2015)

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## Abstract

A modified FNN fault diagnosis algorithm is presented in this paper for microwave subsystem of Plasma Chemical Vapor Deposition (PCVD). The symptom variables are selected as the crisp inputs, and the corresponding membership functions are obtained from premeasured data as well as experts’ diagnostic experience/knowledge. The prior probability and the restriction coefficients are combined into the FNN algorithm via matrix operator. This modified FNN algorithm is verified for PCVD fault diagnosis application and realizes the MIMO for multifault mode diagnosis.


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## Body

## 1. Introduction

In PCVD equipment (Figure1) which is used for making optical fiber core rod [1, 2], the plasma is ignited in a substrate tube through a resonator by microwave power.Figure 1
PCVD illustration.The microwave power from the microwave generator is guided through the wave-guide into the resonator, generating microwave resonance in the cavity of the resonator to ignite the plasma inside the substrate quartz tube. And in the substrate tube, SiCl4/GeCl4 vapor and O2 are mixed and interreact in the plasma. SiO2 and GeO2, the resultant in the plasma reaction, are deposited on the inner surface of the tube.PCVD consists of some subsystems, which perform the complicated chemical and physical reaction, including microwave and plasma, vaporizer and gas supplier, furnace, transmission, reaction piping, electrical parts, and control. Within the production, the fault detection and diagnosis on these subsystems are very important and necessary. In this paper, a fault diagnosis algorithm is proposed for PCVD microwave and plasma system.


## 2. FNN for Fault Diagnosis

In fault diagnosis of complicated system, characteristics and reasons of faults are indefinite; consequently, fuzzy algorithm is better to be used for this kind of application. FNN (Fuzzy Neural Network) is an algorithm combining fuzzy and neural networks and implements fuzzy logical inference via neural network, in which sets of input and output are specified. Sugeno Model is usually used as inference model with 5 layers [3].Layer 1 is fuzzification layer, to produce membership grades of the crisp inputs(X,Y) which belong to each of the fuzzy sets by using the membership functions. Neurons in the layer output results to the neurons in the next layer.Layer 2 is the rule layer. Each of the neurons outputs the firing strengthWi according to the definition of Π, such as maximum operator or product operator.Layer 3 is the normalization layer. This layer receives inputs from all neurons in the rule layer and calculates the normalized firing strength of a given rule:(1)W-i=Wi∑j=1NWj.Layer 4 is node function layer. For the 2-input node shown in Figure2, the first-order function is(2)z=ax+by+c.Figure 2
FNN with Sugeno Model.Output of the layer is(3)O=W-·z=W-·ax+by+c,where a, b, and c are consequence parameters.Layer 5 sums all inputs as total output.Some applications of fault diagnosis use FNN algorithm to predict parameters (Figure3); then, the faults are revealed based on the difference between the measured concentration values and FNN predictions [4].Figure 3
FNN prediction for fault diagnosis.In this application, the FNN prediction is trained by a historical set of data collected during fault-free operation of the process; the process parameters are collected from online sensors. FNN is just used for parameter prediction, not for fault diagnosis directly.In another application, the relationship between fault phenomena and fault reasons is defined with membership. Fuzzy mathematical method is used to establish diagnostic matrix; then, according to the diagnosis matrix and by BP algorithm, fault diagnosis model is generated; finally, fault reasons are output. The expert experience and historical data are both combined with membership matrix, to draw the fault reasons [5].In a fault diagnosis application in nuclear power plant, a diagnostic solution is proposed based on the combination of FNN and data fusion (Figure4). The method is suitable for nuclear power plant monitoring and fault diagnosis to deal with the large quantities of online measurement data. FNN is applied in local diagnosis area and data fusion is applied in global diagnosis area; they complement independent fault diagnosis by using different inputs, thereby, referring each other and generating a more reliable result [6].Figure 4
The integration of FNN and data fusion.


## 3. Modified FNN for Fault Diagnosis

### 3.1. Modified FNN Algorithm

In the fault diagnosis application of the microwave system, a modified FNN is presented in this paper. By this algorithm, the probability of faults can be output directly based on a few of the real-time input signals. The FNN with Sugeno Model has obvious advantage for the nonlinear application, which involves complex computation for the identification of consequence parameters [7, 8]. Nevertheless, a new inference model in FNN Layer 4 is proposed and more suitable for this application, instead of Sugeno Model, which describes the relationship between characteristic signals and fault probability; that is, this approach is a multi-input and multioutput fault diagnosis algorithm.The modified FNN model is shown in Figure5.Figure 5
Modified FNN fault diagnosis where Π is maximum operator.The modification compared with FNN shown in Figure2 is presented on Layer 4 and Layer 5.According to Layer 4, based on the fault prior probability which is from experimental data as well as experts’ diagnostic experience/knowledge to distinguish the actual fault from various external static factors [9], the probability of every fault mode is output.According to Layer 5, outputs from Layer 4 are input into this layer, which makes the fault decision with the restriction of constraint matrix, by maximum rule.


### 3.2. Algorithm Realization

The following presents the modified FNN fault diagnosis algorithm.The crisp inputs are classified into different fuzzy sets with specified membership functions.Suppose(4)x1=PrefP;x2=Pref,where P is forward power of microwave; Pref is reflective power of microwave.Membership functions are defined by statistical probability of fault:(5a)Px1=0x1<a12a1-b13x1-a12x1-3b1-a12a1≤x1<b11b1≤x1.Actually,a1 = 5%, b1 = 15%, and(5b)
μ
A
i
x
1
=
0
x
1
<
a
i
x
1
-
a
i
b
i
-
a
i
a
i
≤
x
1
<
b
i
c
i
-
x
1
c
i
-
b
i
b
i
≤
x
1
<
c
i
0
c
i
≤
x
1
i
=
1,2
(5c)
μ
A
3
x
1
=
0
x
1
<
a
3
x
1
-
a
i
b
i
-
a
i
a
3
≤
x
1
<
b
3
1
b
3
≤
x
1
.Membership function of fault detected byPref/P is shown as in Figure 6(b), which is described by (5b) and (5c). The membership function is based on the statistical probability of fault (Figure 6(a)):(6a)
P
x
2
=
0
x
2
<
a
2
x
2
-
a
2
b
2
-
a
2
a
2
≤
x
2
<
b
2
1
b
2
≤
x
2
.Figure 6
(a) Fault probability byPref/P. (b) Membership function of fault by Pref/P.
(a)
(b)In production,a2=300, b2=1000 (unit: Watt), and(6b)
μ
B
1
x
2
=
1
x
2
<
300
1000
-
x
2
700
300
≤
x
2
<
1000
0
1000
≤
x
2
,
(6c)
μ
B
2
x
2
=
0
x
2
<
300
x
2
-
300
700
300
≤
x
2
<
1000
1
1000
≤
x
2
.Membership function of fault detected byPref is shown as in Figure 7(b), which is described by (6b) and (6c). The membership function is based on the statistical probability of fault (Figure 7(a)).Figure 7
(a) Fault probability byPref. (b) Membership function of fault by Pref.
(a)
(b)The realization of the fault diagnosis (Figure5) is illustrated as below.In production, microwave mismatch is a common fault in the microwave system; the phenomena appear as the measurement value ofPref or/and Pref/P higher than normal.The mismatch fault is divided into 4 fault modes (Figure8), wave-guide alignment problem, substrate-tube bend, autotuner fault, and magnetron fault. The fault weight w is calculated from statistic data.Figure 8
Weight of fault.w
i
,
1, wi,2, and wi,3 are corresponding to different fuzzy rule combination; for example, wi,1 is corresponding to Rule A1+B1 (Figure 5).Based on the above data of weight, the algorithm of Layer 4 is able to design as below:(7)zi=μF1,iμF2,iμF3,iμF4,i=wi¯·w1,iw2,iw3,iw4,ii=1,2,3.Decision algorithm of Layer 5 is designed as below:(8)z=μF1μF2μF3μF4=∑i=13Zi·r1,1r1,2r1,3r1,4r2,1r2,2r2,3r2,4r3,1r3,2r3,3r3,4r4,1r4,2r4,3r4,4,where constraint matrix r is determined by physical condition of fault mode, including intrinsic restriction and dependent restriction, and ri,j (i=j) is intrinsic restriction coefficient to judge the probability of fault mode Fi, determined by the factors related to the fault mode directly; and ri,j (i≠j) is dependent restriction coefficient to judge the probability of fault mode Fi, determined by the factors related to the fault mode indirectly.For example, the intrinsic restriction coefficient is a function to describe the working status of the specified part in the equipment. As for the wave guide, its alignment is not qualified absolutely after reinstallation; after confirmation by running, it keeps stable for a certain period; then, after the stable period, the alignment will turn to abnormal status caused by movement vibration and structure deformation. So we define the function of intrinsic restriction coefficient in Figure9.Figure 9
Intrinsic restriction coefficient function of wave-guide alignment.The intrinsic restriction coefficient of fault modeF1 (wave-guide alignment) is r1,1=1, if the wave guide was reinstalled before running, and r1,1=0.1, if the wave guide was not changed during 60 days (Figure 9).Fault modeF4 (magnetron fault) r4,4 is dependent on the reliability function of magnetron life time (Figure 10 and (9)). Normally, the life time of magnetron is about 5000 hours. According to the function, the fault probability is very low if the usage time of magnetron is less than 4000 hours; that is, r4,4=0.1:(9)Rt=0.1t<a2a-b3t-a2t-3b-a2a≤t<b1b≤t.Figure 10
Reliability function.Actually,a=4000 and b=6000 (unit: hour).Dependent restriction coefficient represents the associative influence factors from other faults. For example, the magnetron fault could be caused by the autotuner fault based on the theory of microwave application. In case the moding fault generated by autotuner fault appears in the microwave system, the filament in magnetron would be damaged. That means fault modesF3 (autotuner fault) and F4 (magnetron fault) are associative and F3 affects F4; then, r4,3>0, that is, to increase the weight of F4 by F3. Conversely, since there is no associative fault mode to affect F1, the dependent restriction coefficients of F1 are zero totally.


### 3.3. Production Instance

The fault diagnosis algorithm is illustrated as below.Crisp inputs are(10)x1=9%,x2=500.According to (5b) and (5c), (11)μA1x1=0.20,μA2x1=0.80,μA3x1=0.According to (6b) and (6c), (12)μB1x2=0.60,μB2x2=0.30.The prior probability is(13)w1,1w1,2w1,3w2,1w2,2w2,3w3,1w3,2w3,3w4,1w4,2w4,3=0.400.250.150.300.300.250.150.150.250.150.300.35,(14a)z1=0.160.120.060.06i,(14b)z2=0.150.180.090.18i,(14c)z3=0000.And the other conditions are as follows:The resonator is replaced recently; the magnetron has been used for 4500 hours.Dependent restriction coefficient (F3 to F4) r4,3 is set to 0.4:(15)r=100001000010000.40.5.So,(16)z=0.310.300.150.24100001000010000.40.5=0.310.300.150.18.The results of FNN fault diagnosis of 60 sets of historical data are listed in Table1.Table 1
The results of FNN fault diagnosis.
Time (hr)
Fault symptom
Fault mode probability
Actual fault
I.R.C.
P
ref
/
P
P
ref
F
1
F
2
F
3
F
4
F
1
F
2
F
3
F
4
0~4000
0.1
0.16
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.13
800
0.7268
0.1279
0.0569
0.0884
T
0.1
0.10
600
0.7495
0.1224
0.0503
0.0778
T
0.1
0.07
400
0.7566
0.1174
0.0533
0.0726
T
0.1
0.20
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.16
800
0.7077
0.1317
0.0638
0.0968
T
0.1
0.12
600
0.7526
0.1202
0.0516
0.0755
X
0.1
0.08
400
0.7548
0.1187
0.0526
0.0739
T
0.1
0.25
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.20
800
0.7077
0.1317
0.0638
0.0968
T
0.1
0.15
600
0.7307
0.1242
0.0602
0.0850
X
0.1
0.10
400
0.7521
0.1206
0.0514
0.0759
T
0.1
0.33
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.27
800
0.7077
0.1317
0.0638
0.0968
T
0.1
0.20
600
0.7307
0.1242
0.0602
0.0850
X
0.1
0.13
400
0.7566
0.1174
0.0533
0.0726
T
0.1
0.50
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.40
800
0.7077
0.1317
0.0638
0.0968
T
0.1
0.30
600
0.7307
0.1242
0.0602
0.0850
T
0.1
0.20
400
0.7518
0.1173
0.0569
0.0740
T
5000
0.5
0.16
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.13
800
0.4289
0.0755
0.0335
0.4621
T
0.5
0.10
600
0.4647
0.0759
0.0312
0.4282
T
0.5
0.07
400
0.4698
0.0729
0.0331
0.4242
T
0.5
0.20
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.16
800
0.3998
0.0744
0.0361
0.4897
T
0.5
0.12
600
0.4669
0.0746
0.0320
0.4264
T
0.5
0.08
400
0.4685
0.0737
0.0326
0.4252
T
0.5
0.25
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.20
800
0.3998
0.0744
0.0361
0.4897
T
0.5
0.15
600
0.4296
0.0730
0.0354
0.4621
T
0.5
0.10
400
0.4665
0.0748
0.0319
0.4268
T
0.5
0.33
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.27
800
0.3998
0.0744
0.0361
0.4897
T
0.5
0.20
600
0.4296
0.0730
0.0354
0.4621
T
0.5
0.13
400
0.4698
0.0729
0.0331
0.4242
T
0.5
0.50
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.40
800
0.3998
0.0744
0.0361
0.4897
T
0.5
0.30
600
0.4296
0.0730
0.0354
0.4621
T
0.5
0.20
400
0.3854
0.0601
0.0291
0.5253
T
6000
1
0.16
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.13
800
0.3494
0.0615
0.0273
0.5617
T
1
0.10
600
0.3840
0.0627
0.0258
0.5275
T
1
0.07
400
0.3940
0.0612
0.0278
0.5171
T
1
0.20
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.16
800
0.3231
0.0601
0.0291
0.5876
T
1
0.12
600
0.3884
0.0620
0.0266
0.5229
T
1
0.08
400
0.3915
0.0616
0.0273
0.5197
T
1
0.25
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.20
800
0.3231
0.0601
0.0291
0.5876
X
1
0.15
600
0.3537
0.0601
0.0291
0.5570
T
1
0.10
400
0.3876
0.0622
0.0265
0.5238
T
1
0.33
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.27
800
0.3231
0.0601
0.0291
0.5876
T
1
0.20
600
0.3537
0.0601
0.0291
0.5570
T
1
0.13
400
0.3940
0.0612
0.0278
0.5171
T
1
0.50
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.40
800
0.3231
0.0601
0.0291
0.5876
T
1
0.30
600
0.3537
0.0601
0.0291
0.5570
T
1
0.20
400
0.3854
0.0601
0.0291
0.5253
T
I.R.C.: intrinsic restriction coefficient.
T: diagnosis result identical to actual fault.
X: actual fault.Where the matrix of prior probability is shown as (13), the matrix of restriction coefficient is shown as below.When usage time of magnetron is in the duration of 0 to 4000 hours,(17)r=100000.1500000.100000.1.When usage time of magnetron is about 5000 hours,(18)r=100000.1500000.10000.60.5.When usage time of magnetron is about 6000 hours,(19)r=100000.1500000.10000.51.In particular, in some circumstances, the fault diagnosis algorithm is still effective. For example, under this kind of circumstance: the wave guide has kept no change for 10 to 20 days, the substrate tube is qualified, the autotuner is of normal status, the usage time of magnetron is in the duration of 0 to 4000 hours, and it seems that every part is staying in normal status, but actually the faults still appear as the alignment turning worse by vibration and moving, or the substrate-tube deforming by heating or stress. Table2 shows the diagnosis results by the algorithm.Table 2
The results of FNN fault diagnosis.
Time (hr)
Fault symptom
Fault mode probability
Actual fault %
P
ref
/
P
P
ref
F
1
F
2
F
3
F
4
F
1
F
2
F
3
F
4
0~4000
0.16
1000
0.1758
0.3626
0.1758
0.2857
14
73
13
0
0.13
800
0.2101
0.3699
0.1644
0.2556
15
70
15
0
0.10
600
0.2303
0.3761
0.1545
0.2391
19
66
15
0
0.07
400
0.2371
0.3681
0.1672
0.2276
25
60
15
0
0.20
1000
0.1758
0.3626
0.1758
0.2857
15
70
15
0
0.16
800
0.1949
0.3626
0.1758
0.2666
16
68
16
0
0.12
600
0.2333
0.3726
0.1601
0.2341
20
66
14
0
0.08
400
0.2354
0.3701
0.1640
0.2305
20
66
14
0
0.25
1000
0.1758
0.3626
0.1758
0.2857
20
65
15
0
0.20
800
0.1949
0.3626
0.1758
0.2666
21
61
18
0
0.15
600
0.2134
0.3626
0.1758
0.2481
19
60
21
0
0.10
400
0.2327
0.3732
0.1590
0.2350
20
60
20
0
0.33
1000
0.1758
0.3626
0.1758
0.2857
23
56
21
0
0.27
800
0.1949
0.3626
0.1758
0.2666
25
55
20
0
0.20
600
0.2134
0.3626
0.1758
0.2481
25
56
19
0
0.13
400
0.2371
0.3681
0.1672
0.2276
25
54
21
0
0.50
1000
0.1758
0.3626
0.1758
0.2857
25
55
20
0
0.40
800
0.1949
0.3626
0.1758
0.2666
26
56
18
0
0.30
600
0.2134
0.3626
0.1758
0.2481
24
56
20
0
0.20
400
0.2325
0.3626
0.1758
0.2290
22
57
21
0
0.16
1000
0.1758
0.3626
0.1758
0.2857
24
55
21
0
0.13
800
0.2101
0.3699
0.1644
0.2556
25
53
22
0
0.10
600
0.2303
0.3761
0.1545
0.2391
25
55
20
0As the fault decision is by maximum rule, the results of the diagnosis match actual faults very well, with high diagnostic accuracy (Table2). Of course, if the calculation results of faults probability are approximate, the diagnosis decision is hard to make correctly. So the new input should be introduced into the diagnosis algorithm, in order to improve the accuracy.


### 3.4. Comparison with Information Fusion

Next, we will compare the diagnosis result of Information Fusion Diagnosis with the modified FNN diagnosis.In the Information Fusion Diagnosis, Dempster-Shafer evidence combination rules with strong classification capability for faults of evidences data overlapping are applied to obtain Information Fusion results [10]. First, the data (evidences) from sensing sources are pretreated, and then the basic confidence degree is combined by D-S rules; finally, the fusion results are selected according to specified decision rule (Figure 11).Figure 11
Fault diagnosis of D-S Information Fusion.As the limitation of the fusion algorithm, some conditions, such as magnetron usage time and wave-guide reinstallation time, cannot be introduced into the D-S Information Fusion model.According to D-S rules, set the recognition framework(20)Θ=Wave-guidealignment,Substrate-tubebend,Auto-tunerfault,Magnetronfaultdenoted by(21)Θ=F1,F2,F3,F4.Corresponding evidences are Pref/P and Pref.Conventionally, evidences (Pref/P, Pref) are analog variable, and that involves probability density [10, 11]. This will lead to enormous calculation and lower real-time property of fault diagnosis. So, the detection signal grading in PCVD control system is used to simplify variables value, such as Normal (N), Over High Limitation (H), Over Higher Limitation (HH), Over Low Limitation (L), and Over Lower Limitation (LL).Based on simplified value, the distribution of belief functions is determined via historical statistic data:(22)
B
L
F
1
P
r
e
f
P
=
0,0
,
0,0
;
P
r
e
f
P
=
L
L
0.25,0.60,0.15,0
;
P
r
e
f
P
=
L
0.20,0.66,0.14,0
;
P
r
e
f
P
=
N
0.15,0.70,0.15,0
;
P
r
e
f
P
=
H
0.23,0.56,0.21,0
;
P
r
e
f
P
=
H
H
,
B
L
F
2
P
r
e
f
=
0,0
,
0,0
;
P
r
e
f
=
L
0,0
,
0,0
;
P
r
e
f
=
L
0.20,0.60,0.20,0
;
P
r
e
f
=
N
0.24,0.56,0.20,0
;
P
r
e
f
=
H
0.20,0.65,0.15,0
;
P
r
e
f
=
H
H
.Suppose Θ is a recognition framework; m1 and m2 are called basic belief distribution function of frame Θ. Ω is the power set of Θ; B and C are elements of Ω. D-S evidence combination result is (23)mA=∑Bi∩Cj=Am1Bim2Cj1-k,where(24)k=∑Bi∩Cj=Φm1Bim2Cj.Equation (25) is D-S evidence combination formula for 2 evidences, denoted by (25)mA=m1B⊕m2C.So, multievidence combination formula is(26)mA=m1B⊕m2C⊕m3D⊕m4E⋯,where B, C, D, E,… are elements of power set Ω.The below example shows D-S Information Fusion application in fault diagnosis. Since the number of evidences is 2, the evidences are set into several permutations with 2 evidences [12]. First, the belief functions are calculated according to (23) and (25), and then faults are determined by maximal value (Table 3).Table 3
Calculation of D-S Information Fusion.
m
F
1
F
2
F
3
F
4
m
1: Pref/P=L
0.25
0.60
0.15
0
m
2: Pref=N
0.20
0.60
0.20
0
m
1
⊕
m
2
0.1136364
0.818182
0.068182
0
m
1: Pref/P=L
0.25
0.60
0.15
0
m
2: Pref=H
0.24
0.56
0.20
0
m
1
⊕
m
2
0.1408451
0.788732
0.070423
0
m
1: Pref/P=HH
0.23
0.56
0.21
0
m
2: Pref=HH
0.20
0.65
0.15
0
m
1
⊕
m
2
0.1041903
0.824462
0.071348
0From the result of calculation in Table3, the distribution value of belief function for fault F2 is obviously maximal; that is, the probability of F2 is maximal; the distribution value of belief function for fault F4 is zero, that is, probability zero. In this application, maximum principle is the criterion of fault determination. Obviously, the result is not identical to the actual situation. For example, on the magnetron usage time over 5000 hrs, fault F1 and fault F4 are possible (shown in Table 1). As the data (evidences) from sensing sources are not sufficient and the additional restriction conditions are not able to be introduced, the diagnosis results deviate from the actual ones. Of course, some correction method could be applied for the Information Fusion; that will be presented in another paper.To demonstrate the superiority of the proposed algorithm, the simulated diagnosis results (shown as in Tables1, 2, and 3) are listed in Table 4 for comparison. The diagnosis results are described and evaluated by diagnosis error rate under specified fault condition.Table 4
Diagnosis result comparison of different algorithm.
Samples restriction
Algorithm
Time (hr)
Resonator replaced recently
Modified FNN
Information Fusion
0~6000
Yes
5.0%
58.3%
0~6000
No
9.6%
44.1%
0~4000
No
39.4%
39.4%In most circumstances, the proposed algorithm shows the superiority with much lower diagnosis error rate. Just in the specified circumstance, the two algorithms have identical results.


## 3.1. Modified FNN Algorithm

In the fault diagnosis application of the microwave system, a modified FNN is presented in this paper. By this algorithm, the probability of faults can be output directly based on a few of the real-time input signals. The FNN with Sugeno Model has obvious advantage for the nonlinear application, which involves complex computation for the identification of consequence parameters [7, 8]. Nevertheless, a new inference model in FNN Layer 4 is proposed and more suitable for this application, instead of Sugeno Model, which describes the relationship between characteristic signals and fault probability; that is, this approach is a multi-input and multioutput fault diagnosis algorithm.The modified FNN model is shown in Figure5.Figure 5
Modified FNN fault diagnosis where Π is maximum operator.The modification compared with FNN shown in Figure2 is presented on Layer 4 and Layer 5.According to Layer 4, based on the fault prior probability which is from experimental data as well as experts’ diagnostic experience/knowledge to distinguish the actual fault from various external static factors [9], the probability of every fault mode is output.According to Layer 5, outputs from Layer 4 are input into this layer, which makes the fault decision with the restriction of constraint matrix, by maximum rule.


## 3.2. Algorithm Realization

The following presents the modified FNN fault diagnosis algorithm.The crisp inputs are classified into different fuzzy sets with specified membership functions.Suppose(4)x1=PrefP;x2=Pref,where P is forward power of microwave; Pref is reflective power of microwave.Membership functions are defined by statistical probability of fault:(5a)Px1=0x1<a12a1-b13x1-a12x1-3b1-a12a1≤x1<b11b1≤x1.Actually,a1 = 5%, b1 = 15%, and(5b)
μ
A
i
x
1
=
0
x
1
<
a
i
x
1
-
a
i
b
i
-
a
i
a
i
≤
x
1
<
b
i
c
i
-
x
1
c
i
-
b
i
b
i
≤
x
1
<
c
i
0
c
i
≤
x
1
i
=
1,2
(5c)
μ
A
3
x
1
=
0
x
1
<
a
3
x
1
-
a
i
b
i
-
a
i
a
3
≤
x
1
<
b
3
1
b
3
≤
x
1
.Membership function of fault detected byPref/P is shown as in Figure 6(b), which is described by (5b) and (5c). The membership function is based on the statistical probability of fault (Figure 6(a)):(6a)
P
x
2
=
0
x
2
<
a
2
x
2
-
a
2
b
2
-
a
2
a
2
≤
x
2
<
b
2
1
b
2
≤
x
2
.Figure 6
(a) Fault probability byPref/P. (b) Membership function of fault by Pref/P.
(a)
(b)In production,a2=300, b2=1000 (unit: Watt), and(6b)
μ
B
1
x
2
=
1
x
2
<
300
1000
-
x
2
700
300
≤
x
2
<
1000
0
1000
≤
x
2
,
(6c)
μ
B
2
x
2
=
0
x
2
<
300
x
2
-
300
700
300
≤
x
2
<
1000
1
1000
≤
x
2
.Membership function of fault detected byPref is shown as in Figure 7(b), which is described by (6b) and (6c). The membership function is based on the statistical probability of fault (Figure 7(a)).Figure 7
(a) Fault probability byPref. (b) Membership function of fault by Pref.
(a)
(b)The realization of the fault diagnosis (Figure5) is illustrated as below.In production, microwave mismatch is a common fault in the microwave system; the phenomena appear as the measurement value ofPref or/and Pref/P higher than normal.The mismatch fault is divided into 4 fault modes (Figure8), wave-guide alignment problem, substrate-tube bend, autotuner fault, and magnetron fault. The fault weight w is calculated from statistic data.Figure 8
Weight of fault.w
i
,
1, wi,2, and wi,3 are corresponding to different fuzzy rule combination; for example, wi,1 is corresponding to Rule A1+B1 (Figure 5).Based on the above data of weight, the algorithm of Layer 4 is able to design as below:(7)zi=μF1,iμF2,iμF3,iμF4,i=wi¯·w1,iw2,iw3,iw4,ii=1,2,3.Decision algorithm of Layer 5 is designed as below:(8)z=μF1μF2μF3μF4=∑i=13Zi·r1,1r1,2r1,3r1,4r2,1r2,2r2,3r2,4r3,1r3,2r3,3r3,4r4,1r4,2r4,3r4,4,where constraint matrix r is determined by physical condition of fault mode, including intrinsic restriction and dependent restriction, and ri,j (i=j) is intrinsic restriction coefficient to judge the probability of fault mode Fi, determined by the factors related to the fault mode directly; and ri,j (i≠j) is dependent restriction coefficient to judge the probability of fault mode Fi, determined by the factors related to the fault mode indirectly.For example, the intrinsic restriction coefficient is a function to describe the working status of the specified part in the equipment. As for the wave guide, its alignment is not qualified absolutely after reinstallation; after confirmation by running, it keeps stable for a certain period; then, after the stable period, the alignment will turn to abnormal status caused by movement vibration and structure deformation. So we define the function of intrinsic restriction coefficient in Figure9.Figure 9
Intrinsic restriction coefficient function of wave-guide alignment.The intrinsic restriction coefficient of fault modeF1 (wave-guide alignment) is r1,1=1, if the wave guide was reinstalled before running, and r1,1=0.1, if the wave guide was not changed during 60 days (Figure 9).Fault modeF4 (magnetron fault) r4,4 is dependent on the reliability function of magnetron life time (Figure 10 and (9)). Normally, the life time of magnetron is about 5000 hours. According to the function, the fault probability is very low if the usage time of magnetron is less than 4000 hours; that is, r4,4=0.1:(9)Rt=0.1t<a2a-b3t-a2t-3b-a2a≤t<b1b≤t.Figure 10
Reliability function.Actually,a=4000 and b=6000 (unit: hour).Dependent restriction coefficient represents the associative influence factors from other faults. For example, the magnetron fault could be caused by the autotuner fault based on the theory of microwave application. In case the moding fault generated by autotuner fault appears in the microwave system, the filament in magnetron would be damaged. That means fault modesF3 (autotuner fault) and F4 (magnetron fault) are associative and F3 affects F4; then, r4,3>0, that is, to increase the weight of F4 by F3. Conversely, since there is no associative fault mode to affect F1, the dependent restriction coefficients of F1 are zero totally.


## 3.3. Production Instance

The fault diagnosis algorithm is illustrated as below.Crisp inputs are(10)x1=9%,x2=500.According to (5b) and (5c), (11)μA1x1=0.20,μA2x1=0.80,μA3x1=0.According to (6b) and (6c), (12)μB1x2=0.60,μB2x2=0.30.The prior probability is(13)w1,1w1,2w1,3w2,1w2,2w2,3w3,1w3,2w3,3w4,1w4,2w4,3=0.400.250.150.300.300.250.150.150.250.150.300.35,(14a)z1=0.160.120.060.06i,(14b)z2=0.150.180.090.18i,(14c)z3=0000.And the other conditions are as follows:The resonator is replaced recently; the magnetron has been used for 4500 hours.Dependent restriction coefficient (F3 to F4) r4,3 is set to 0.4:(15)r=100001000010000.40.5.So,(16)z=0.310.300.150.24100001000010000.40.5=0.310.300.150.18.The results of FNN fault diagnosis of 60 sets of historical data are listed in Table1.Table 1
The results of FNN fault diagnosis.
Time (hr)
Fault symptom
Fault mode probability
Actual fault
I.R.C.
P
ref
/
P
P
ref
F
1
F
2
F
3
F
4
F
1
F
2
F
3
F
4
0~4000
0.1
0.16
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.13
800
0.7268
0.1279
0.0569
0.0884
T
0.1
0.10
600
0.7495
0.1224
0.0503
0.0778
T
0.1
0.07
400
0.7566
0.1174
0.0533
0.0726
T
0.1
0.20
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.16
800
0.7077
0.1317
0.0638
0.0968
T
0.1
0.12
600
0.7526
0.1202
0.0516
0.0755
X
0.1
0.08
400
0.7548
0.1187
0.0526
0.0739
T
0.1
0.25
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.20
800
0.7077
0.1317
0.0638
0.0968
T
0.1
0.15
600
0.7307
0.1242
0.0602
0.0850
X
0.1
0.10
400
0.7521
0.1206
0.0514
0.0759
T
0.1
0.33
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.27
800
0.7077
0.1317
0.0638
0.0968
T
0.1
0.20
600
0.7307
0.1242
0.0602
0.0850
X
0.1
0.13
400
0.7566
0.1174
0.0533
0.0726
T
0.1
0.50
1000
0.6809
0.1404
0.0681
0.1106
T
0.1
0.40
800
0.7077
0.1317
0.0638
0.0968
T
0.1
0.30
600
0.7307
0.1242
0.0602
0.0850
T
0.1
0.20
400
0.7518
0.1173
0.0569
0.0740
T
5000
0.5
0.16
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.13
800
0.4289
0.0755
0.0335
0.4621
T
0.5
0.10
600
0.4647
0.0759
0.0312
0.4282
T
0.5
0.07
400
0.4698
0.0729
0.0331
0.4242
T
0.5
0.20
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.16
800
0.3998
0.0744
0.0361
0.4897
T
0.5
0.12
600
0.4669
0.0746
0.0320
0.4264
T
0.5
0.08
400
0.4685
0.0737
0.0326
0.4252
T
0.5
0.25
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.20
800
0.3998
0.0744
0.0361
0.4897
T
0.5
0.15
600
0.4296
0.0730
0.0354
0.4621
T
0.5
0.10
400
0.4665
0.0748
0.0319
0.4268
T
0.5
0.33
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.27
800
0.3998
0.0744
0.0361
0.4897
T
0.5
0.20
600
0.4296
0.0730
0.0354
0.4621
T
0.5
0.13
400
0.4698
0.0729
0.0331
0.4242
T
0.5
0.50
1000
0.3678
0.0759
0.0368
0.5195
T
0.5
0.40
800
0.3998
0.0744
0.0361
0.4897
T
0.5
0.30
600
0.4296
0.0730
0.0354
0.4621
T
0.5
0.20
400
0.3854
0.0601
0.0291
0.5253
T
6000
1
0.16
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.13
800
0.3494
0.0615
0.0273
0.5617
T
1
0.10
600
0.3840
0.0627
0.0258
0.5275
T
1
0.07
400
0.3940
0.0612
0.0278
0.5171
T
1
0.20
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.16
800
0.3231
0.0601
0.0291
0.5876
T
1
0.12
600
0.3884
0.0620
0.0266
0.5229
T
1
0.08
400
0.3915
0.0616
0.0273
0.5197
T
1
0.25
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.20
800
0.3231
0.0601
0.0291
0.5876
X
1
0.15
600
0.3537
0.0601
0.0291
0.5570
T
1
0.10
400
0.3876
0.0622
0.0265
0.5238
T
1
0.33
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.27
800
0.3231
0.0601
0.0291
0.5876
T
1
0.20
600
0.3537
0.0601
0.0291
0.5570
T
1
0.13
400
0.3940
0.0612
0.0278
0.5171
T
1
0.50
1000
0.2914
0.0601
0.0291
0.6193
T
1
0.40
800
0.3231
0.0601
0.0291
0.5876
T
1
0.30
600
0.3537
0.0601
0.0291
0.5570
T
1
0.20
400
0.3854
0.0601
0.0291
0.5253
T
I.R.C.: intrinsic restriction coefficient.
T: diagnosis result identical to actual fault.
X: actual fault.Where the matrix of prior probability is shown as (13), the matrix of restriction coefficient is shown as below.When usage time of magnetron is in the duration of 0 to 4000 hours,(17)r=100000.1500000.100000.1.When usage time of magnetron is about 5000 hours,(18)r=100000.1500000.10000.60.5.When usage time of magnetron is about 6000 hours,(19)r=100000.1500000.10000.51.In particular, in some circumstances, the fault diagnosis algorithm is still effective. For example, under this kind of circumstance: the wave guide has kept no change for 10 to 20 days, the substrate tube is qualified, the autotuner is of normal status, the usage time of magnetron is in the duration of 0 to 4000 hours, and it seems that every part is staying in normal status, but actually the faults still appear as the alignment turning worse by vibration and moving, or the substrate-tube deforming by heating or stress. Table2 shows the diagnosis results by the algorithm.Table 2
The results of FNN fault diagnosis.
Time (hr)
Fault symptom
Fault mode probability
Actual fault %
P
ref
/
P
P
ref
F
1
F
2
F
3
F
4
F
1
F
2
F
3
F
4
0~4000
0.16
1000
0.1758
0.3626
0.1758
0.2857
14
73
13
0
0.13
800
0.2101
0.3699
0.1644
0.2556
15
70
15
0
0.10
600
0.2303
0.3761
0.1545
0.2391
19
66
15
0
0.07
400
0.2371
0.3681
0.1672
0.2276
25
60
15
0
0.20
1000
0.1758
0.3626
0.1758
0.2857
15
70
15
0
0.16
800
0.1949
0.3626
0.1758
0.2666
16
68
16
0
0.12
600
0.2333
0.3726
0.1601
0.2341
20
66
14
0
0.08
400
0.2354
0.3701
0.1640
0.2305
20
66
14
0
0.25
1000
0.1758
0.3626
0.1758
0.2857
20
65
15
0
0.20
800
0.1949
0.3626
0.1758
0.2666
21
61
18
0
0.15
600
0.2134
0.3626
0.1758
0.2481
19
60
21
0
0.10
400
0.2327
0.3732
0.1590
0.2350
20
60
20
0
0.33
1000
0.1758
0.3626
0.1758
0.2857
23
56
21
0
0.27
800
0.1949
0.3626
0.1758
0.2666
25
55
20
0
0.20
600
0.2134
0.3626
0.1758
0.2481
25
56
19
0
0.13
400
0.2371
0.3681
0.1672
0.2276
25
54
21
0
0.50
1000
0.1758
0.3626
0.1758
0.2857
25
55
20
0
0.40
800
0.1949
0.3626
0.1758
0.2666
26
56
18
0
0.30
600
0.2134
0.3626
0.1758
0.2481
24
56
20
0
0.20
400
0.2325
0.3626
0.1758
0.2290
22
57
21
0
0.16
1000
0.1758
0.3626
0.1758
0.2857
24
55
21
0
0.13
800
0.2101
0.3699
0.1644
0.2556
25
53
22
0
0.10
600
0.2303
0.3761
0.1545
0.2391
25
55
20
0As the fault decision is by maximum rule, the results of the diagnosis match actual faults very well, with high diagnostic accuracy (Table2). Of course, if the calculation results of faults probability are approximate, the diagnosis decision is hard to make correctly. So the new input should be introduced into the diagnosis algorithm, in order to improve the accuracy.


## 3.4. Comparison with Information Fusion

Next, we will compare the diagnosis result of Information Fusion Diagnosis with the modified FNN diagnosis.In the Information Fusion Diagnosis, Dempster-Shafer evidence combination rules with strong classification capability for faults of evidences data overlapping are applied to obtain Information Fusion results [10]. First, the data (evidences) from sensing sources are pretreated, and then the basic confidence degree is combined by D-S rules; finally, the fusion results are selected according to specified decision rule (Figure 11).Figure 11
Fault diagnosis of D-S Information Fusion.As the limitation of the fusion algorithm, some conditions, such as magnetron usage time and wave-guide reinstallation time, cannot be introduced into the D-S Information Fusion model.According to D-S rules, set the recognition framework(20)Θ=Wave-guidealignment,Substrate-tubebend,Auto-tunerfault,Magnetronfaultdenoted by(21)Θ=F1,F2,F3,F4.Corresponding evidences are Pref/P and Pref.Conventionally, evidences (Pref/P, Pref) are analog variable, and that involves probability density [10, 11]. This will lead to enormous calculation and lower real-time property of fault diagnosis. So, the detection signal grading in PCVD control system is used to simplify variables value, such as Normal (N), Over High Limitation (H), Over Higher Limitation (HH), Over Low Limitation (L), and Over Lower Limitation (LL).Based on simplified value, the distribution of belief functions is determined via historical statistic data:(22)
B
L
F
1
P
r
e
f
P
=
0,0
,
0,0
;
P
r
e
f
P
=
L
L
0.25,0.60,0.15,0
;
P
r
e
f
P
=
L
0.20,0.66,0.14,0
;
P
r
e
f
P
=
N
0.15,0.70,0.15,0
;
P
r
e
f
P
=
H
0.23,0.56,0.21,0
;
P
r
e
f
P
=
H
H
,
B
L
F
2
P
r
e
f
=
0,0
,
0,0
;
P
r
e
f
=
L
0,0
,
0,0
;
P
r
e
f
=
L
0.20,0.60,0.20,0
;
P
r
e
f
=
N
0.24,0.56,0.20,0
;
P
r
e
f
=
H
0.20,0.65,0.15,0
;
P
r
e
f
=
H
H
.Suppose Θ is a recognition framework; m1 and m2 are called basic belief distribution function of frame Θ. Ω is the power set of Θ; B and C are elements of Ω. D-S evidence combination result is (23)mA=∑Bi∩Cj=Am1Bim2Cj1-k,where(24)k=∑Bi∩Cj=Φm1Bim2Cj.Equation (25) is D-S evidence combination formula for 2 evidences, denoted by (25)mA=m1B⊕m2C.So, multievidence combination formula is(26)mA=m1B⊕m2C⊕m3D⊕m4E⋯,where B, C, D, E,… are elements of power set Ω.The below example shows D-S Information Fusion application in fault diagnosis. Since the number of evidences is 2, the evidences are set into several permutations with 2 evidences [12]. First, the belief functions are calculated according to (23) and (25), and then faults are determined by maximal value (Table 3).Table 3
Calculation of D-S Information Fusion.
m
F
1
F
2
F
3
F
4
m
1: Pref/P=L
0.25
0.60
0.15
0
m
2: Pref=N
0.20
0.60
0.20
0
m
1
⊕
m
2
0.1136364
0.818182
0.068182
0
m
1: Pref/P=L
0.25
0.60
0.15
0
m
2: Pref=H
0.24
0.56
0.20
0
m
1
⊕
m
2
0.1408451
0.788732
0.070423
0
m
1: Pref/P=HH
0.23
0.56
0.21
0
m
2: Pref=HH
0.20
0.65
0.15
0
m
1
⊕
m
2
0.1041903
0.824462
0.071348
0From the result of calculation in Table3, the distribution value of belief function for fault F2 is obviously maximal; that is, the probability of F2 is maximal; the distribution value of belief function for fault F4 is zero, that is, probability zero. In this application, maximum principle is the criterion of fault determination. Obviously, the result is not identical to the actual situation. For example, on the magnetron usage time over 5000 hrs, fault F1 and fault F4 are possible (shown in Table 1). As the data (evidences) from sensing sources are not sufficient and the additional restriction conditions are not able to be introduced, the diagnosis results deviate from the actual ones. Of course, some correction method could be applied for the Information Fusion; that will be presented in another paper.To demonstrate the superiority of the proposed algorithm, the simulated diagnosis results (shown as in Tables1, 2, and 3) are listed in Table 4 for comparison. The diagnosis results are described and evaluated by diagnosis error rate under specified fault condition.Table 4
Diagnosis result comparison of different algorithm.
Samples restriction
Algorithm
Time (hr)
Resonator replaced recently
Modified FNN
Information Fusion
0~6000
Yes
5.0%
58.3%
0~6000
No
9.6%
44.1%
0~4000
No
39.4%
39.4%In most circumstances, the proposed algorithm shows the superiority with much lower diagnosis error rate. Just in the specified circumstance, the two algorithms have identical results.


## 4. Conclusion

The proposed FNN fault diagnosis algorithm combined with prior probability and restriction coefficient is approved by PCVD fault diagnosis simulation. It is a significant exploratory method where the matrix of prior probability introduced to the algorithm establishes the connection between the diagnosis result and historical diagnosis data. Another improvement for FNN algorithm is application of restriction coefficient matrix for diagnosis decision, which contains dependent restriction coefficient and intrinsic restriction coefficient, corresponding, respectively, to interaction and intrinsic property of diagnosed parts. Through the mathematic method, the characteristic and interrelationship of specified objects and historical experience are integrated into the diagnosis algorithm directly, instead of linguistic description. During the application of the diagnosis, the intrinsic restriction coefficients should be set according to actual equipment situation; and the dependent restriction coefficients and prior probability should be finely adjusted and corrected based on accumulated statistic data after a certain period.



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*Source: 632456-2015-10-28.xml*