# Sensor Fault Diagnosis Based on Fuzzy Neural Petri Net

Authors: Jiming Li; Xiaolin Zhu; Xuezhen Cheng
Journal: Complexity
 (2018)

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## Abstract

This study aims to improve the operating stability of the resistance strain weighing sensor and eliminate fuzzy factors in fault diagnosis. Based on fuzzy techniques for fault diagnosis, the proposed fuzzy Petri net model uses the fault logical relationship between a sensor and an improved Petri net model. A formula for confidence-based reasoning is proposed using an algorithm, which combines neural network regulation algorithm with a transition-enabled ignition judgment matrix. This formula can yield an accurate assessment of the operating state of the sensor. Backward inference and the minimum cut set theory are also combined to obtain the priority of faults, which helps avoid blind and ambiguous maintenance. The sensor model was analyzed, and its accuracy and validity were verified through statistical analysis and comparison with other methods of fault diagnosis.


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## Body

## 1. Introduction

The resistance strain weighing sensor (hereinafter referred to as “the sensor”) is a core component of electronic weighing instruments, and its quality directly influences the accuracy of measurement. In practical applications, due to the influence of raw materials, manufacturing processes, installation methods, service conditions, and the external environment, electronic weighing instruments are prone to various faults with uncertainty. Therefore, accurately predicting and diagnosing faults in these instruments are significant to ensure their accuracy and stability.As an effective method of parallel computing and behavioral analysis [1], the Petri net has a rigorous mathematical formulation as well as a straightforward graphic description. In [2–5], fuzzy technology (a new technology based on fuzzy mathematics) was combined with the Petri net to propose the fuzzy Petri net (FPN) method of modeling, which has exhibited powerful parallel processing capability. However, optimizing the model structure and developing the matrix implementation remain to be further researched. It is important to find a model that is representative of real environments.For calculations in FPN, although the problem in the FPN related to matrix reasoning was solved in [1, 6], its weight and other parameters remain undetermined, and accurate data are needed to ensure the correctness of the diagnosis.Based on the operator’s diagnostic experience, a method for fault diagnosis in expert systems (ES) can be used as the operating logic of the protection relay and circuit breakers and has been applied to power systems. Methods of fault analysis based on ES have been reported in the literature [7–9]. For example, an advanced logic-based ES was applied in [7]. The General Diagnosis Engine was used to analyze place information and evaluate security [9]. However, ES-based methods of analysis have shortcomings, such as requiring complex knowledge acquisition and maintenance and slow reasoning. Modeling based on directivity was proposed in [10] to reduce the dimensionality of the incidence matrix and simplify the calculation model, but it fails to provide a sufficient description of weight. The method proposed in [11] significantly improves the fault tolerance of the Petri net, but the Petri net model based on a time sequence does not apply to a static Petri net with adjustable weights.Owing to fuzzy behavior in the FPN, a number of methods for data determination have been proposed. The BP (back propagation) algorithm endows the Petri net with the capability of self-learning [12–14], resulting in clear weight values. However, the model does not improve accordingly. The BP neural network has been combined with the traditional fuzzy fault Petri net to develop the adaptive FPN [15, 16], which improves the capability of the traditional fuzzy fault Petri net to learn weights. However, it fails to explicitly show how to determine the transition confidence coefficient, leaving the system with a large number of uncertainties. The forward-backward algorithm was used to implement reasoning pertaining to unobservable place events in the model [17]. Many other fault diagnosis methods, such as data fusion and the support vector machine (SVM), were proposed as well. Reference [18] has been applied to effectively solve such problems as nonlinearity and high dimensionality. However, due to the characteristics of the SVM, multiple dichotomies are currently used to solve multiclassification problems; in this context, the excessive classification is associated with unnecessary complexity of calculation, and hence a faster method is needed to ensure system stability.This study proposes a method to diagnose sensor faults based on fuzzy neural Petri net. With the resistance strain weighing sensor as the research object, its FPN fault model is created. The neural network is applied to adjust the weight, with the abandonment of the transition confidence coefficient. The MYCIN confidence reasoning algorithm is optimized based on the sigmoid function and, consequently, fault diagnosis is accomplished based on the minimum cut set of fault rate.


## 2. Improved Fuzzy Fault Petri Net

### 2.1. Structure of the Fuzzy Fault Petri Net

Based on the Petri net and fuzzy Petri net theory [19–22], a nine-parameter model is defined as SP=P,T,I,O,K,W,α,f,λ.The variables are as follows:(1)P=(p1,p2,…,pn)T is a set of place faults, including all faults relating to the sensor, such as “broken gate of the output-adjusting resistance,” “overloaded weighing,” and “excessively large sensitivity of diaphragm shunting.”(2)T=t1,t2,…,tnT is a set of transitions. If a transition is enabled, tj=1; otherwise, tj=0.(3)I is the input matrix of the Petri net.(4)O is the output matrix of the Petri net.(5)K=k1,k2,…,knT denotes the place label vector. When a fault occurs in place i, kj=1; otherwise, kj=0.(6)W=ωij is an n×m matrix of the weight of the place. When ∀t∈T,Pk∈It, ∑j=1nωk=1.(7)α=α1,α1,…,αnT is an n-dimensional vector of the confidence coefficients of the place, denoting the confidence of occurrence of a fault event.(8)f=f1,f1,…,fnT is a set of probabilities of the fuzzy occurrence of place events, where fj denotes the probability of occurrence of place event pi.(9)λ=λ1,λ1,…,λnT is the threshold vector of transition.


### 2.2. Structure of the Improved Fuzzy Fault Petri Net

The connection of sensor components is tight, multiple mappings between faults, with the complex and diverse fault propagation mode. Based on this, firstly, the structure of the sensor is analyzed, according to the fuzzy relation to obtain the fault logic relationship, and then the FPN mode is established following the basic rules of Petri net, where the confidence reasoning algorithm is optimized based on the sigmoid function. In other words, based on the original fuzzy Petri net, the sigmoid function replaces the initial transition confidenceμ to describe the rules to deduce the FPN model and the expression of fuzzy information. The confidence values of fault events occurring in different places can be obtained through reasoning, which provides the necessary conditions for the positive and negative instances of reasoning pertaining to faults. Figure 1 shows the basic elements of the improved Petri net.Figure 1
Improved Petri net model.


## 2.1. Structure of the Fuzzy Fault Petri Net

Based on the Petri net and fuzzy Petri net theory [19–22], a nine-parameter model is defined as SP=P,T,I,O,K,W,α,f,λ.The variables are as follows:(1)P=(p1,p2,…,pn)T is a set of place faults, including all faults relating to the sensor, such as “broken gate of the output-adjusting resistance,” “overloaded weighing,” and “excessively large sensitivity of diaphragm shunting.”(2)T=t1,t2,…,tnT is a set of transitions. If a transition is enabled, tj=1; otherwise, tj=0.(3)I is the input matrix of the Petri net.(4)O is the output matrix of the Petri net.(5)K=k1,k2,…,knT denotes the place label vector. When a fault occurs in place i, kj=1; otherwise, kj=0.(6)W=ωij is an n×m matrix of the weight of the place. When ∀t∈T,Pk∈It, ∑j=1nωk=1.(7)α=α1,α1,…,αnT is an n-dimensional vector of the confidence coefficients of the place, denoting the confidence of occurrence of a fault event.(8)f=f1,f1,…,fnT is a set of probabilities of the fuzzy occurrence of place events, where fj denotes the probability of occurrence of place event pi.(9)λ=λ1,λ1,…,λnT is the threshold vector of transition.


## 2.2. Structure of the Improved Fuzzy Fault Petri Net

The connection of sensor components is tight, multiple mappings between faults, with the complex and diverse fault propagation mode. Based on this, firstly, the structure of the sensor is analyzed, according to the fuzzy relation to obtain the fault logic relationship, and then the FPN mode is established following the basic rules of Petri net, where the confidence reasoning algorithm is optimized based on the sigmoid function. In other words, based on the original fuzzy Petri net, the sigmoid function replaces the initial transition confidenceμ to describe the rules to deduce the FPN model and the expression of fuzzy information. The confidence values of fault events occurring in different places can be obtained through reasoning, which provides the necessary conditions for the positive and negative instances of reasoning pertaining to faults. Figure 1 shows the basic elements of the improved Petri net.Figure 1
Improved Petri net model.


## 3. Algorithms for FPN Fault Reasoning

To clearly and concisely present the reasoning and calculation of each matrix during the reasoning for the FPN model, the Petri net is used to describe the capability of the concurrency system and the mathematical theory of the FPN to define five special operators [23]:(1) The comparison operator◇: C=A◇B, where A, B, and C are m×n matrices. When aij>bij, cij=1; when aij<bij, cij=0, i=1,2,…,m; j=1,2,…,n.(2) The minimum operator∧: C=A∧B, where A, B, and C are m×n matrices; cij=min⁡aij,bij, where i=1,2,…,m; j=1,2,…,n.(3) The maximum operator∨: C=A∨B, where A, B, and C are m×n matrices; cij=max⁡aij,bij, where i=1,2,…,m; j=1,2,…,n.(4) The direct product operator∗: C=A∗B, where A,B, and C are m×n matrices; b is ann-dimensional vector; cij=aij∗bi, where i=1,2,…,m; j=1,2,…,n.(5) The multiplication operator⊗: C=A⊗B, where A, B, and C are m×q, q×n, and m×n matrices, respectively; cij=maxi≤k≤q⁡aij,bij, where i=1,2,…,m;j=1,2,…,n.


### 3.1. Confidence Algorithm

The confidence algorithm is modified to achieve higher computational efficiency. Following the reasoning calculation, the confidence values of all places are obtained and function as the basis of fault evaluation and diagnosis.Weight matrixW=ωij, where ωij∈0,1. When there is a directional arc pi to tj, ωij is the weight from pi to tj. When there is a directional arc tj to pi, ωij=0.The reasoning formula is(1)αk+1=αk∨1+exp⁡ρk⁡-1whereρk=O∙WT·αk, ai+1=αi; when and only when the reasoning is concluded; otherwise, it is continued.


### 3.2. Forward Reasoning

The forward reasoning based on the FPN model reflects the characteristics of fault propagation and predicts faults according to the work environment, the detection of components, or symptom-related information obtained by professionals. The faults that may occur are evaluated through the judgment matrix of transition firing and the flow of fault-state marking, and the corresponding response measures are then taken.


#### 3.2.1. Transition Judgment

Definition 1.
∀ P I j ∈ I t, ∑j=1nα(PIj)∙ωIj≥λt, where t is enabled by potential transition.Definition 2.
If transitionT can trigger ignition, there is a new confidence coefficient in the output place POj; if not, the output place is 0.(2)αPOj=exp⁡∑jαPIj∙ωj+1-1,∑jαPIj∙ωIj≥λt,0,∑jαPIj∙ωIj<λt,The transition-triggering ignition matrix isY, where Y=y1,y2,…,ynT.(3)Y=WT∙α◇λIf the ignition conditions are met,yi=1; otherwise, yi=0. According to the rules of ignition, the transition-enabled ignition matrix reasoning corresponding to the token containing the place is calculated out based on [1](4)Yi=y∧I∗Ki-1T∙lm,i=1,Yi=y∧I∗Ki-1-Ki-2T·lm,i=2,3,…,whereKi-1,Ki-2 denote the label vector of thei-1th ignition and lm=1,1,…,1T is an m-dimensional vector.


#### 3.2.2. Reasoning Matrix of Fault-State Label Vector

(5) K i = K i - 1 ⊕ A ⊗ Y iwhereA is the incidence matrix, A=aij∈Rn×m, n is the number of places, and m is the number of transitions [10].


### 3.3. Backward Reasoning

FPN backward reasoning deduces the cause of a fault if it occurs. To avoid blind maintenance and improve the efficiency of tracking the source of the fault, the minimum cut set is introduced as the basis of fault derivation and diagnosis.Definition 3.
If the minimum cut setG=p1,p2,…,pn, the rate of fault occurrence is(6)fG=α1+α2+…+αnn,n>0The input and output places of FPN backward reasoning are the output and input places of FPN forward reasoning, respectively; namely,I-=O,O-=I.The backward reasoning matrix is given by(7)Y-=I-∆Ki-1-⊗lm∧y,Ki-=Ki-1-⊕O-⊗Yi-,i=1,2,3,…,whereYk- is the backward-enabled transition sequence of thekth backward ignition.


## 3.1. Confidence Algorithm

The confidence algorithm is modified to achieve higher computational efficiency. Following the reasoning calculation, the confidence values of all places are obtained and function as the basis of fault evaluation and diagnosis.Weight matrixW=ωij, where ωij∈0,1. When there is a directional arc pi to tj, ωij is the weight from pi to tj. When there is a directional arc tj to pi, ωij=0.The reasoning formula is(1)αk+1=αk∨1+exp⁡ρk⁡-1whereρk=O∙WT·αk, ai+1=αi; when and only when the reasoning is concluded; otherwise, it is continued.


## 3.2. Forward Reasoning

The forward reasoning based on the FPN model reflects the characteristics of fault propagation and predicts faults according to the work environment, the detection of components, or symptom-related information obtained by professionals. The faults that may occur are evaluated through the judgment matrix of transition firing and the flow of fault-state marking, and the corresponding response measures are then taken.


### 3.2.1. Transition Judgment

Definition 1.
∀ P I j ∈ I t, ∑j=1nα(PIj)∙ωIj≥λt, where t is enabled by potential transition.Definition 2.
If transitionT can trigger ignition, there is a new confidence coefficient in the output place POj; if not, the output place is 0.(2)αPOj=exp⁡∑jαPIj∙ωj+1-1,∑jαPIj∙ωIj≥λt,0,∑jαPIj∙ωIj<λt,The transition-triggering ignition matrix isY, where Y=y1,y2,…,ynT.(3)Y=WT∙α◇λIf the ignition conditions are met,yi=1; otherwise, yi=0. According to the rules of ignition, the transition-enabled ignition matrix reasoning corresponding to the token containing the place is calculated out based on [1](4)Yi=y∧I∗Ki-1T∙lm,i=1,Yi=y∧I∗Ki-1-Ki-2T·lm,i=2,3,…,whereKi-1,Ki-2 denote the label vector of thei-1th ignition and lm=1,1,…,1T is an m-dimensional vector.


### 3.2.2. Reasoning Matrix of Fault-State Label Vector

(5) K i = K i - 1 ⊕ A ⊗ Y iwhereA is the incidence matrix, A=aij∈Rn×m, n is the number of places, and m is the number of transitions [10].


## 3.2.1. Transition Judgment

Definition 1.
∀ P I j ∈ I t, ∑j=1nα(PIj)∙ωIj≥λt, where t is enabled by potential transition.Definition 2.
If transitionT can trigger ignition, there is a new confidence coefficient in the output place POj; if not, the output place is 0.(2)αPOj=exp⁡∑jαPIj∙ωj+1-1,∑jαPIj∙ωIj≥λt,0,∑jαPIj∙ωIj<λt,The transition-triggering ignition matrix isY, where Y=y1,y2,…,ynT.(3)Y=WT∙α◇λIf the ignition conditions are met,yi=1; otherwise, yi=0. According to the rules of ignition, the transition-enabled ignition matrix reasoning corresponding to the token containing the place is calculated out based on [1](4)Yi=y∧I∗Ki-1T∙lm,i=1,Yi=y∧I∗Ki-1-Ki-2T·lm,i=2,3,…,whereKi-1,Ki-2 denote the label vector of thei-1th ignition and lm=1,1,…,1T is an m-dimensional vector.


## 3.2.2. Reasoning Matrix of Fault-State Label Vector

(5) K i = K i - 1 ⊕ A ⊗ Y iwhereA is the incidence matrix, A=aij∈Rn×m, n is the number of places, and m is the number of transitions [10].


## 3.3. Backward Reasoning

FPN backward reasoning deduces the cause of a fault if it occurs. To avoid blind maintenance and improve the efficiency of tracking the source of the fault, the minimum cut set is introduced as the basis of fault derivation and diagnosis.Definition 3.
If the minimum cut setG=p1,p2,…,pn, the rate of fault occurrence is(6)fG=α1+α2+…+αnn,n>0The input and output places of FPN backward reasoning are the output and input places of FPN forward reasoning, respectively; namely,I-=O,O-=I.The backward reasoning matrix is given by(7)Y-=I-∆Ki-1-⊗lm∧y,Ki-=Ki-1-⊕O-⊗Yi-,i=1,2,3,…,whereYk- is the backward-enabled transition sequence of thekth backward ignition.


## 4. Fault Analysis of the Resistance Strain Weighing Sensor

### 4.1. Determination of FPN Data

In reasoning relating to the fuzzy Petri net, the confidence coefficient of the initial place (the bottom place of FPN model) needs to be entered externally, whereas those of the middle place and the concluding places are generally obtained by the reasoning. Thus, the determination of the confidence coefficient (fuzzy token) pertaining to place mainly refers to the initial place. In this study, the method proposed in [24] is used to combine historical data with expert opinion to set the confidence coefficient of the initial place.In the fuzzy Petri net, weightω represents the degree to which each condition influences the conclusion and is mainly determined based on past studies. This is significantly subjective and uncertain. As the improved fuzzy Petri net possesses certain characteristics of a neural network, the neural network algorithm can be used to train, learn, and adjust the network. The adjustment algorithm is as follows:d i is the due output (expected output) of the ith element and yi is its actual output. The element’s error signal is given by(8)ei=di-yi(9)yi=vi∙∑αi∙ωi(10)vix=1exp⁡x+1(11)xi=∑αi∙ωiThe adjustment of weight is mainly reflected in the backpropagation of the error, where the square errorE=(1/2)ei2 is propagated as a regulatory signal. The gradient of the modifier is(12)∂E∂ωi=∂E∂ei∂ei∂yi∂yi∂xi∂xi∂ωi=-eivi′∑αi∙ωiαiThe correction value of the weight is∆ωi=-η(∂E/∂ωi), where η is the learning rate. A new weight ωi(1)=∆ωi+ωi is obtained and substituted back into the above formula to conduct an iterative operation. Weight adjustment is complete when the square error is within the range of tolerance.


### 4.2. Determining the Model

The sensor is primarily composed of a strain gauge and a measuring circuit. The fault model is established by analyzing the structure of the weighing sensor and fault sampling and by considering the influence of the external environment on the sensor, as shown in Figure8 in Appendix. See Appendix for fault events corresponding to each place.As the scale of the model is large, writing the input and output matrices is cumbersome. To show the reasoning and calculation process, the “bridge circuit fault” is used as an example in this study to illustrate faults in the resistance strain weighing sensor, and its FPN fault model is shown in Figure2. The remaining part of the reasoning process is the same as the example.Figure 2
Part of the FPN model.


### 4.3. Original Data

According to the method mentioned in Section4, the vector form of the confidence coefficient of the underlying place was obtained as follows: α0= (0.89, 0.87, 0.84, 0.71, 0.88, 0.93, 0.89, 0.8, 0.87, 0, 0, 0.69, 0, 0.88, 0.9, 0, 0, 0, 0)T.Taking transitionT5, T7 as an example, the weights of P4, P5, P6, P7, and P8 are adjusted using the method described in Section 4. Hypothesis: ω45=0.3, ω55=0.4, ω65=0.3, ω77=0.33, ω87=0.65. From statistical calculations, the expectation αP30=0.6937, αP32=0.7003, η=0.1, the largest number of learning steps was set to 4,000, and the square error was 0.1×10-5. The training results are shown in Figures 3 and 4.Figure 3
Adjustment curve of square error E.Figure 4
Weighing adjustment curve.The square error after 3928 steps of iterative operation was within the allowable range. The weights ofP4, P5, P6, P7, and P8 obtained at this point were 0.3936, 0.5160, 0.0904, 0.5177, and 0.4823, respectively. Weight matrix is used to calculate the confidence of the entire Petri net:(13)ω=1000000000000000001000000000000000001000000000000000001000000000000000000.3900000000000000000.5200000000000000000.091000000000000000000.5200000000000000000.480.4200000000000000000.58000000000000000001000000000000000001000000000000000001000000000000000001000000000000000001000000000000000001000000000000000001000000000000000001000000000000000000According to the results of the calculation and an analysis of this model, the threshold value of all transitions was set to 0.5.α 0, W, O, and I were substituted into (1). The reasoning was not concluded until α4=α3,α3= (0.89, 0.87, 0.84, 0.71, 0.88, 0.93, 0.89, 0.8, 0.87, 0.7089, 0.72, 0.69, 0.7, 0.88, 0.9, 0.84, 0.67, 0.71, 0.71)T. Thereafter, the confidence coefficients of each place were obtained and used as the basis for the forward and backward reasoning.


### 4.4. Forward and Backward Reasoning

#### 4.4.1. Forward Reasoning

The sensor ran normally and no fault occurred, but symptoms of fault were detected, including “excessively high supply voltage,” “broken output lead,” “insufficient soldering of cable,” “humid environment,” and “broken gate of the output adjusting resistance.” The initial labeling vector obtained wasK0=(1,0,0,1,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0)T, and α3 was substituted into (2) and (3) to calculate the potential transition-enabled matrix y = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)T. K0 and y were substituted into (4) to conduct the reasoning calculation. The final results gained were K3=(1,0,0,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,1)T, and Y3=Y4. The reasoning was concluded, Y3=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)T, the final labeling vector was K3, and the fault transmission path is shown in Figure 5. The foregoing conclusions can be used as the basis for fault checking and maintenance to improve the operational stability of the sensor.Figure 5
Token distribution after forward reasoning.


#### 4.4.2. Backward Reasoning

Backward reasoning was carried out using the example of “no signal output or small signal output after loading.” The initial labeling vectorK0-=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)T, and the potential transition-enabled matrix obtained by the forward reasoning y-=(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)T.K 0 - , y - , I - , 0 - were substituted into (6). As indicated by the reasoning calculation, when y3-=y2-, the reasoning was concluded, and the labeling vector and backward transition matrix were obtained: K2-=(1,1,1,1,1,1,1,0,0,1,1,1,1,0,0,0,1,0,1)T and y2-=(1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0)T. The distribution of places is shown in Figure 6.Figure 6
Token distribution after backward reasoning.As shown in Figure6, the minimum cut set enabling P49 was G1=p1, G2=p2, G3=p3, G4=p4, G5=p4∙p5, G6=p5, G7=p5∙p6, G8=p4∙p5∙p6, and G9=p7. According to (6),f(G1) = 0.89, f(G2) = 0.87, f(G3) = 0.84, f(G4) = 0.88, f(G5) = 0.795, f(G6) = 0.71, f(G7) = 0.93, and f(G8) = 0.89. The fault occurrence probability is shown in Figure 7.Figure 7
Fault occurrence probability of the minimum cut set.Figure 8
FFPN model diagram of the sensor fault system.As indicated by Figure7, the order of diagnosis should be G7, G1, G9, G6, G2, G3, G5, and G4. Thus, the speed of diagnosis can be improved.


## 4.1. Determination of FPN Data

In reasoning relating to the fuzzy Petri net, the confidence coefficient of the initial place (the bottom place of FPN model) needs to be entered externally, whereas those of the middle place and the concluding places are generally obtained by the reasoning. Thus, the determination of the confidence coefficient (fuzzy token) pertaining to place mainly refers to the initial place. In this study, the method proposed in [24] is used to combine historical data with expert opinion to set the confidence coefficient of the initial place.In the fuzzy Petri net, weightω represents the degree to which each condition influences the conclusion and is mainly determined based on past studies. This is significantly subjective and uncertain. As the improved fuzzy Petri net possesses certain characteristics of a neural network, the neural network algorithm can be used to train, learn, and adjust the network. The adjustment algorithm is as follows:d i is the due output (expected output) of the ith element and yi is its actual output. The element’s error signal is given by(8)ei=di-yi(9)yi=vi∙∑αi∙ωi(10)vix=1exp⁡x+1(11)xi=∑αi∙ωiThe adjustment of weight is mainly reflected in the backpropagation of the error, where the square errorE=(1/2)ei2 is propagated as a regulatory signal. The gradient of the modifier is(12)∂E∂ωi=∂E∂ei∂ei∂yi∂yi∂xi∂xi∂ωi=-eivi′∑αi∙ωiαiThe correction value of the weight is∆ωi=-η(∂E/∂ωi), where η is the learning rate. A new weight ωi(1)=∆ωi+ωi is obtained and substituted back into the above formula to conduct an iterative operation. Weight adjustment is complete when the square error is within the range of tolerance.


## 4.2. Determining the Model

The sensor is primarily composed of a strain gauge and a measuring circuit. The fault model is established by analyzing the structure of the weighing sensor and fault sampling and by considering the influence of the external environment on the sensor, as shown in Figure8 in Appendix. See Appendix for fault events corresponding to each place.As the scale of the model is large, writing the input and output matrices is cumbersome. To show the reasoning and calculation process, the “bridge circuit fault” is used as an example in this study to illustrate faults in the resistance strain weighing sensor, and its FPN fault model is shown in Figure2. The remaining part of the reasoning process is the same as the example.Figure 2
Part of the FPN model.


## 4.3. Original Data

According to the method mentioned in Section4, the vector form of the confidence coefficient of the underlying place was obtained as follows: α0= (0.89, 0.87, 0.84, 0.71, 0.88, 0.93, 0.89, 0.8, 0.87, 0, 0, 0.69, 0, 0.88, 0.9, 0, 0, 0, 0)T.Taking transitionT5, T7 as an example, the weights of P4, P5, P6, P7, and P8 are adjusted using the method described in Section 4. Hypothesis: ω45=0.3, ω55=0.4, ω65=0.3, ω77=0.33, ω87=0.65. From statistical calculations, the expectation αP30=0.6937, αP32=0.7003, η=0.1, the largest number of learning steps was set to 4,000, and the square error was 0.1×10-5. The training results are shown in Figures 3 and 4.Figure 3
Adjustment curve of square error E.Figure 4
Weighing adjustment curve.The square error after 3928 steps of iterative operation was within the allowable range. The weights ofP4, P5, P6, P7, and P8 obtained at this point were 0.3936, 0.5160, 0.0904, 0.5177, and 0.4823, respectively. Weight matrix is used to calculate the confidence of the entire Petri net:(13)ω=1000000000000000001000000000000000001000000000000000001000000000000000000.3900000000000000000.5200000000000000000.091000000000000000000.5200000000000000000.480.4200000000000000000.58000000000000000001000000000000000001000000000000000001000000000000000001000000000000000001000000000000000001000000000000000001000000000000000001000000000000000000According to the results of the calculation and an analysis of this model, the threshold value of all transitions was set to 0.5.α 0, W, O, and I were substituted into (1). The reasoning was not concluded until α4=α3,α3= (0.89, 0.87, 0.84, 0.71, 0.88, 0.93, 0.89, 0.8, 0.87, 0.7089, 0.72, 0.69, 0.7, 0.88, 0.9, 0.84, 0.67, 0.71, 0.71)T. Thereafter, the confidence coefficients of each place were obtained and used as the basis for the forward and backward reasoning.


## 4.4. Forward and Backward Reasoning

### 4.4.1. Forward Reasoning

The sensor ran normally and no fault occurred, but symptoms of fault were detected, including “excessively high supply voltage,” “broken output lead,” “insufficient soldering of cable,” “humid environment,” and “broken gate of the output adjusting resistance.” The initial labeling vector obtained wasK0=(1,0,0,1,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0)T, and α3 was substituted into (2) and (3) to calculate the potential transition-enabled matrix y = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)T. K0 and y were substituted into (4) to conduct the reasoning calculation. The final results gained were K3=(1,0,0,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,1)T, and Y3=Y4. The reasoning was concluded, Y3=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)T, the final labeling vector was K3, and the fault transmission path is shown in Figure 5. The foregoing conclusions can be used as the basis for fault checking and maintenance to improve the operational stability of the sensor.Figure 5
Token distribution after forward reasoning.


### 4.4.2. Backward Reasoning

Backward reasoning was carried out using the example of “no signal output or small signal output after loading.” The initial labeling vectorK0-=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)T, and the potential transition-enabled matrix obtained by the forward reasoning y-=(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)T.K 0 - , y - , I - , 0 - were substituted into (6). As indicated by the reasoning calculation, when y3-=y2-, the reasoning was concluded, and the labeling vector and backward transition matrix were obtained: K2-=(1,1,1,1,1,1,1,0,0,1,1,1,1,0,0,0,1,0,1)T and y2-=(1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0)T. The distribution of places is shown in Figure 6.Figure 6
Token distribution after backward reasoning.As shown in Figure6, the minimum cut set enabling P49 was G1=p1, G2=p2, G3=p3, G4=p4, G5=p4∙p5, G6=p5, G7=p5∙p6, G8=p4∙p5∙p6, and G9=p7. According to (6),f(G1) = 0.89, f(G2) = 0.87, f(G3) = 0.84, f(G4) = 0.88, f(G5) = 0.795, f(G6) = 0.71, f(G7) = 0.93, and f(G8) = 0.89. The fault occurrence probability is shown in Figure 7.Figure 7
Fault occurrence probability of the minimum cut set.Figure 8
FFPN model diagram of the sensor fault system.As indicated by Figure7, the order of diagnosis should be G7, G1, G9, G6, G2, G3, G5, and G4. Thus, the speed of diagnosis can be improved.


## 4.4.1. Forward Reasoning

The sensor ran normally and no fault occurred, but symptoms of fault were detected, including “excessively high supply voltage,” “broken output lead,” “insufficient soldering of cable,” “humid environment,” and “broken gate of the output adjusting resistance.” The initial labeling vector obtained wasK0=(1,0,0,1,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0)T, and α3 was substituted into (2) and (3) to calculate the potential transition-enabled matrix y = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)T. K0 and y were substituted into (4) to conduct the reasoning calculation. The final results gained were K3=(1,0,0,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,1)T, and Y3=Y4. The reasoning was concluded, Y3=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)T, the final labeling vector was K3, and the fault transmission path is shown in Figure 5. The foregoing conclusions can be used as the basis for fault checking and maintenance to improve the operational stability of the sensor.Figure 5
Token distribution after forward reasoning.


## 4.4.2. Backward Reasoning

Backward reasoning was carried out using the example of “no signal output or small signal output after loading.” The initial labeling vectorK0-=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)T, and the potential transition-enabled matrix obtained by the forward reasoning y-=(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)T.K 0 - , y - , I - , 0 - were substituted into (6). As indicated by the reasoning calculation, when y3-=y2-, the reasoning was concluded, and the labeling vector and backward transition matrix were obtained: K2-=(1,1,1,1,1,1,1,0,0,1,1,1,1,0,0,0,1,0,1)T and y2-=(1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0)T. The distribution of places is shown in Figure 6.Figure 6
Token distribution after backward reasoning.As shown in Figure6, the minimum cut set enabling P49 was G1=p1, G2=p2, G3=p3, G4=p4, G5=p4∙p5, G6=p5, G7=p5∙p6, G8=p4∙p5∙p6, and G9=p7. According to (6),f(G1) = 0.89, f(G2) = 0.87, f(G3) = 0.84, f(G4) = 0.88, f(G5) = 0.795, f(G6) = 0.71, f(G7) = 0.93, and f(G8) = 0.89. The fault occurrence probability is shown in Figure 7.Figure 7
Fault occurrence probability of the minimum cut set.Figure 8
FFPN model diagram of the sensor fault system.As indicated by Figure7, the order of diagnosis should be G7, G1, G9, G6, G2, G3, G5, and G4. Thus, the speed of diagnosis can be improved.


## 5. Statistics and Verification of Fault Reasoning

### 5.1. Fault Statistics

In this study, the maintenance record of CSY-3000 (an instrument manufactured by Zhejiang Golink Technology Development Co., Ltd., China) for the last two years (2016–2017) and data from the manufacturer’s reliability manual (“other” fault causes were introduced due to loss of data records; we render the data true and reliable, including the statistics) were statistically analyzed and compared with the results of reasoning. The data on “no signal output or small signal output after loading” were sorted out, as shown in Table1.Table 1
Fault-related data.
Causes of fault Number of faults/frequency (xi) Confidence of place (yi) P 1 60 0.89 P 2 46 0.87 P 3 40 0.84 P 4 47 0.88 P 5 70 0.93 P 6 58 0.89 P 29 34 0.7089 P 30 32 0.72 Others 4 0The correlation coefficient can be obtained based on the data mentioned in Table1, which can then be used to verify the correctness of the results of reasoning. The correlation coefficient is calculated as follows:(14)r=∑i=1nxi-x-yi-y-∑i=1nxi-x-2∙∑n=1iyi-y-2The average values were calculated first:x-=43.4 and y-= 0.74766. xi and yiwere substituted into (14) to obtain the correlation coefficientr= 0.8865. As indicated by the results of the calculation, those of the diagnosis were strongly correlated with actual statistics.


### 5.2. Case Analysis

To further verify the accuracy of this method, the techniques proposed in [13, 18] were used to analyze two cases: “no signal output or small signal output after loading” and “unstable indicating instrument.” The results are shown in Table 2.Table 2
Comparison and validation of different methods.
Reference [18] [13] This study Source of weight None BP algorithm BP algorithm Confidence of transition None Experts’ experience Sigmoid function Reasoning method Data integration Calculation and reasoning The modified reasoning method Case # Case 1 Case 2 Case 1 Case 2 Case 1, Case 2 Field fault p 1,p4,p5 p 9 p 1 , p 4 , p 5 p 9 p 1 , p 4 , p 5 p 9 Diagnosis results p 1,p4 p 9 p 1 , p 4 , p 5 p 9 p 1 , p 4 , p 5 p 9 Undetected p 5 None None Correlation  0.7516 0.8865It can be seen from the table that, in terms of effectiveness, compared with the results of [13], the results were verified as valid. From the aspect of fault tolerance, the authors of [13] and this paper observed no leakage detection, whereas the work in [18] reported leakage in “insufficient soldering of cable.” In terms of data selection, the other methods were excessively dependent on expert experience, whereas this paper used a neural network and the sigmoid function as trigger modes, thus increasing the value of the correlation coefficient in the final diagnosis and bringing it closer to the actual fault state.


## 5.1. Fault Statistics

In this study, the maintenance record of CSY-3000 (an instrument manufactured by Zhejiang Golink Technology Development Co., Ltd., China) for the last two years (2016–2017) and data from the manufacturer’s reliability manual (“other” fault causes were introduced due to loss of data records; we render the data true and reliable, including the statistics) were statistically analyzed and compared with the results of reasoning. The data on “no signal output or small signal output after loading” were sorted out, as shown in Table1.Table 1
Fault-related data.
Causes of fault Number of faults/frequency (xi) Confidence of place (yi) P 1 60 0.89 P 2 46 0.87 P 3 40 0.84 P 4 47 0.88 P 5 70 0.93 P 6 58 0.89 P 29 34 0.7089 P 30 32 0.72 Others 4 0The correlation coefficient can be obtained based on the data mentioned in Table1, which can then be used to verify the correctness of the results of reasoning. The correlation coefficient is calculated as follows:(14)r=∑i=1nxi-x-yi-y-∑i=1nxi-x-2∙∑n=1iyi-y-2The average values were calculated first:x-=43.4 and y-= 0.74766. xi and yiwere substituted into (14) to obtain the correlation coefficientr= 0.8865. As indicated by the results of the calculation, those of the diagnosis were strongly correlated with actual statistics.


## 5.2. Case Analysis

To further verify the accuracy of this method, the techniques proposed in [13, 18] were used to analyze two cases: “no signal output or small signal output after loading” and “unstable indicating instrument.” The results are shown in Table 2.Table 2
Comparison and validation of different methods.
Reference [18] [13] This study Source of weight None BP algorithm BP algorithm Confidence of transition None Experts’ experience Sigmoid function Reasoning method Data integration Calculation and reasoning The modified reasoning method Case # Case 1 Case 2 Case 1 Case 2 Case 1, Case 2 Field fault p 1,p4,p5 p 9 p 1 , p 4 , p 5 p 9 p 1 , p 4 , p 5 p 9 Diagnosis results p 1,p4 p 9 p 1 , p 4 , p 5 p 9 p 1 , p 4 , p 5 p 9 Undetected p 5 None None Correlation  0.7516 0.8865It can be seen from the table that, in terms of effectiveness, compared with the results of [13], the results were verified as valid. From the aspect of fault tolerance, the authors of [13] and this paper observed no leakage detection, whereas the work in [18] reported leakage in “insufficient soldering of cable.” In terms of data selection, the other methods were excessively dependent on expert experience, whereas this paper used a neural network and the sigmoid function as trigger modes, thus increasing the value of the correlation coefficient in the final diagnosis and bringing it closer to the actual fault state.


## 6. Conclusion

A method of fault diagnosis in Petri net sensors was proposed in this study based on a new confidence reasoning method and was applied to the fault prediction and diagnosis of a resistance strain weighing sensor.( 1 ) A fault diagnosis model of the resistance strain weighing sensor was established based on the structure, operating characteristics, and fault occurrence of the sensor.( 2 ) A neural network algorithm was applied to determine the parameters of the model, and a confidence reasoning formula proposed to deduce the pathway and mode of fault propagation, which improved speed and diagnosis efficiency.( 3 ) Forward and backward reasoning were combined to obtain the order of occurrence of faults for each component, which helps avoid blind detection and maintenance. The relationship between events was clearly presented by the Petri net diagrams.Despite the contributions of this study, the proposed method has some limitations. The logical relationship, the optimization of threshold setting in the sensor model, and the numerical simulation of the model will be studied in future work.



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*Source: 8261549-2018-10-23.xml*