Engine fault diagnosis based on sensor data fusion considering information quality and evidence theory

Abstract

Fault diagnosis is a problem processing variable information obtained from different sources in nature. Evidence theory, efficient to deal with information viewed as evidence, is widely used in fault diagnosis. However, a shortcoming of the existing fault diagnosis methods only gets probability distribution rather than the basic probability assignment. A novel method of generating basic probability assignment that takes information quality into account is proposed. The probability distribution is determined by the preliminary matrix and sampling matrix that are constructed by sensor data. And the quality of probability distribution is taken as the discount factor and the rest of belief is assigned to the universal set. Hence, the basic probability assignment is obtained. Then, basic probability assignment can be combined with Dempster and Shafer evidence theory to determine the status of the engine. An application of engine fault is shown to illustrate the practicability of the proposed method. Then by comparing the result of the method which takes information quality into account (the proposed method) and does not do it, the former is better than the latter. Finally, the reliability analysis shows that the proposed method has strong reliability because performance accuracy is 100% when the error rate is less than 10%.

Introduction

Fault diagnosis is important to system to correct timely and work smoothly. Up to now, fault diagnosis has been applied extensively to all kinds of profession, such as mechanics,^1–4 chemistry, ⁵ nucleus, ⁶ and electric. ⁷ A number of approaches to optimize the algorithm of fault diagnosis are proposed, such as the average current Park’s vector approach, ⁸ a fuzzy approach,^9,10 and optimized threshold de-noising method. ²

In real system, a larger quantity of information can be obtained from all kinds of sensors that detect the concrete values. Consequently, it is necessary for decision-makers to get rational result by considering all the complicated information^11–15 whose certainty maybe very high or low. And some methods of information fusion ¹⁶ used in fault diagnosis and other fields have been proposed, such as Kalman Filter,^17–19 neural network,^20–22 and fuzzy logic.^23–28

In addition, evidence theory^29,30 is efficient to deal with uncertainty^31–37 and Dempster’s rule can take the advantage of evidence combination from different sources without prior information. Up to now, on the basis of evidence theory, D-number,^38–40 Z-number,^41,42 and so on, have been studied by lots of scholars. As a result, evidence theory is not only used in medical diagnosis,^43–45 dependence assessment, ⁴⁶ correlation,^47,48 data stream, ⁴⁹ forensic crime investigations, ⁵⁰ traffic, ⁵¹ and target recognition^52,53 but also in fault diagnosis.^54–59 There are many researches about applying evidence theory into fault diagnosis. For example, the new combination rule ⁵⁷ is then built to allocate the conflicted information from multi-sensors based on the support degree of focal element. A novel-weighted evidence combination rule ⁶⁰ based on evidence distance and uncertainty measure is proposed. A novel method ⁶¹ that comprehensively analyzes vibration and temperature signals to diagnose bearing faults based on improved Dempster–Shafer (D-S) evidence theory is presented. A novel dissolved gas analysis (DGA) method ⁶² for power transformer incipient fault diagnosis based on integrated adaptive neuro fuzzy inference system (ANFIS) and Dempster–Shafer theory (DST) is presented. A weak thruster fault detection method ⁶³ is developed based on the combination of artificial immune system and single pre-processing. The architecture of an expert system ⁶⁴ that uses flame images grabbed during the combustion process in an experimental oil furnace as input parameters is presented. An effective method ⁶⁵ for precise fault diagnosis of planetary gearbox based on fusion of vibration and acoustic data using the DST is proposed. A new transformer fault diagnosis method ⁶⁶ based on a wavelet neural network optimized by adaptive genetic algorithm (AGA) and an improved D-S evidence theory fusion technique is proposed. In Basir and Yuan, ⁶⁷ to make rational decisions, a method is proposed with respect to engine quality and to evaluate the performance of the proposed information fusion system, a criterion is presented. But there are two issues needed to be improved, one is that the mass function obtained by calculating the distance between the measured features and fault prototypes is only probability distribution for singleton and cannot be called as basic probability assignment to some extent. And the other is that information quality is not taken into consideration.

To handle the above issues, in this article, a new method that Shannon entropy^68,69 of each probability obtained from each feature is used as discount factor^70–72 to acquire the basic probability assignment is proposed. All the measurements obtained from sensors can characterize two faults: (1) X₁: exhaust valve fault; (2) X₂: piston ring fault, but not all of the measurements can distinguish fault accurately in some situation, in other words, measurements have different information quality. Therefore, the mass function can be obtained by considering the information quality and on basis of probability distribution. The main advantages of this method are that Shannon entropy as the discount factor that is assigned to the universal set can obtain basic assignment instead of probability distribution and decrease conflicts to fuse efficiently.

This article is organized as follows: in section “Preliminary,” the concepts and rule of evidence theory and notations and formulation of the Shannon entropy are introduced. In section “Proposed method,” we present the frame of discernment and the new evidence combination. In section “Application in fault diagnosis,” an application is used to illustrate efficiency and reliability of the proposed method. In section “Conclusion,” the study is briefly concluded.

Preliminary

Preliminary notion of the D-S evidence theory

Evidence theory is the classical mathematic theory of evidence which is initially based on Dempster’s work concerning lower and upper probability distribution families and expanded by Shafer. In this section, some basic concepts and functions are introduced as follows:

• 1. The frame of discernment: Let $\mathrm{\Theta }$ be a finite set of elements; an element can be a hypothesis or a fault in our case. We refer $\mathrm{\Theta }$ is the frame of discernment. The $\mathrm{\Theta }$ is denoted as follows^29,30

$$\mathrm{\Theta }=\{H_1,H_2,H_3,\dots ,H_N\}$$

(1)

the set consisting of all the subsets of $\mathrm{\Theta }$ is called as power set of $\mathrm{\Theta }$ and denoted as follows

$$2^{\mathrm{\Theta }}=\{\mathrm{\varnothing },\{H_1\},\{H_2\},\dots ,\{H_1,H_2\},\dots ,\mathrm{\Theta }\}$$

(2)

where $\mathrm{\varnothing }$ is null set consisting of nothing.

• 2. Mass function: A key point of the frame of discernment is the basic probability assignment (BPA), which is a mapping of the power set $2^{\mathrm{\Theta }}$ to a number between 0 and 1, that is

$$m:2^{\mathrm{\Theta }}\to [0,1]$$

(3)

$$m(\mathrm{\varnothing })=0,\sum _{A\subseteq \mathrm{\Theta }}m(A)=1$$

(4)

where $m(A)$ represents how strongly the evidence supports A.

• 3. Rules of evidence combination: Assuming $m_1$ and $m_2$ are two mass functions, having the discrepancy, obtained from different information sources. The Dempster’s rules of combination, called orthogonal sum and noted by $m=m_1\oplus m_2$ , are defined as follows^29,30

$$m(\mathrm{\varnothing })=0,m(A)=\frac{\sum _{B\cap C\subseteq A}=m_1(B)m_2(C)}{1-K}$$

(5)

$$K=\sum _{B\cap C\subseteq \mathrm{\varnothing }}{m}_1(B)m_2(C)$$

(6)

The $m(\mathrm{\varnothing })$ represents that the evidence gives it little support in the dominance of “closed-world.”K is a coefficient to reflect the conflict between $m_1$ and $m_2$ , determined by summing the products of the mass function of all sets where the intersection is null.

Shannon entropy

The idea of entropy, of which Shannon entropy is accepted by most people, is an important concept to the probability distribution on the space $X=\{x_1,x_2,\dots ,x_n\}$ . Then the Shannon entropy ⁶⁸ is

$$H(P)=-\sum _{i=1}^n{p}_i\ln (p_i)$$

(7)

where $p_i$ represents the probability of ith element in vector. Obviously, the entropy will have maximum value $\ln (n)$ when it is uniform distribution, $p_i=1/n$ for $i=1,2,\dots ,n$ and minimum value 0 when one $p_i=1$ and all other $p_i=0$ . Hence, intuitively, the larger is $H(P)$ , the less is the information provided by the probability. Conversely, the smaller is $H(P)$ , the more is the information. For example, we cannot decide which elements are best when $p=\{0.5,0.5\}$ but we can do it when $p=\{1,0\}$ .

Proposed method

In this article, the proposed method is in Figure 1 shown and detailed as follows:

Figure 1.

The flowchart of the proposed method.

Faults in the frame of discernment

To solve fault diagnosis problem by using evidence theory, the frame of discernment is necessary. Each element in the frame of discernment represents relevant fault that machine has when working. For example, we take $X=\{X_0,X_1\}$ where $X_0$ intimates the absence of all kinds of faults and $X_1$ suggests the presence of a fault as the frame of discernment. Thus, N faults are taken account of and the frame of discernment is established as follows

$$X=\{X_0,X_1,X_2,\dots ,X_N\}$$

where $X_N$ represents the presence of ith fault.

In fault diagnosis, possessing information is the prerequisite of all work. Accordingly, various sensors, such as the vibration sensor and the acoustic sensor, are used to detect the corresponding characteristic values that describe some features to determine the status of the engine and decide which fault takes place. Besides information obtained from sensors when the engine is working, the relevant parameter of the specific engine is also significant, or decision made cannot make a comparison. The specific feature value corresponding to the engine being in specific fault is necessary to be obtained. Consequently, a preliminary feature matrix of $(N+1)\times M$ is obtained as follows

$$H=\left[\begin{array}{cccc}{h}_{01}\hfill & h_{02}\hfill & \cdots \hfill & h_{0M}\hfill \\ \multicolumn{1}{c}{h_{11}}& h_{12}\hfill & \cdots \hfill & h_{1M}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{h_{N1}}& h_{N2}\hfill & \cdots \hfill & h_{\mathrm{NM}}\hfill \end{array}\right]$$

(8)

where N represents the number of faults taking place in the engine and M represents all feature values that the sensor can obtain.

After preliminary feature matrix is established, the values obtained from all sensors when the engine is working, it is used to construct sampling matrix compared by preliminary feature matrix. Let $S_k$ represent the measurement vector obtained from the kth sampling from all sensors

$$S_k=\{S_{k1},S_{k2},\dots ,S_{\mathrm{kM}}\}$$

where k represents the kth sampling and $j=1,2,3,\dots ,M$ represents jth feature value. Then the sampling matrix composed of all the measurement vectors is established.

A key problem is that which method we choose to be used to calculate the basic probability assignment. Obviously, if the more similar the element of measurement vector S _k we get from sensors is to the corresponding row vector {h_j1h_j2 . . . h_jM} of the preliminary matrix, the more probable the corresponding fault X_j occurs. Inversely, the less similar it is, the less probable it does. There are many measures to quantify the distance between the measured feature obtained from sensors and the relevant parameter value. The absolute distance measure is used in this article as follows

$$d_{\mathrm{kji}}=|S_{\mathrm{kj}}-h_{\mathrm{ij}}|$$

(9)

where k is the kth sample, i is the ith fault, and j is the jth feature that sensors detect.

The distances between all sensor measurements and the relevant parameter values can be captured in a matrix form ⁷³

$$D=\left[\begin{array}{cccc}(d_{110},d_{111},\dots ,d_{11N})\hfill & (d_{120},d_{121},\dots ,d_{12N})\hfill & \cdots \hfill & (d_{1M0},d_{1M1},\dots ,d_{1\mathrm{MN}})\hfill \\ \multicolumn{1}{c}{(d_{210},d_{211},\dots ,d_{21N})}& (d_{220},d_{221},\dots ,d_{22N})\hfill & \cdots \hfill & (d_{2M0},d_{2M1},\dots ,d_{2\mathrm{MN}})\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{(d_{L10},d_{L11},\dots ,d_{L1N})}& (d_{L20},d_{L21},\dots ,d_{L2N})\hfill & \cdots \hfill & (d_{\mathrm{LM}0},d_{\mathrm{LM}1},\dots ,d_{\mathrm{LMN}})\hfill \end{array}\right]$$

(10)

Each third dimension in the matrix represents the distance between measurements obtained from sensors and all fault including non-fault. The smaller the distance $d_{\mathrm{kji}}$ is, the more probable the ith fault is. Therefore, defining $p_{\mathrm{kji}}$ as

$$p_{\mathrm{kji}}=\frac{1}{d_{\mathrm{kji}}}$$

(11)

After normalization, a probability matrix of $L\times M\times (N+1)$ is obtained as follows

$$P=\left[\begin{array}{cccc}(p_{110},p_{111},\dots ,p_{11N})\hfill & (p_{120},p_{121},\dots ,p_{12N})\hfill & \cdots \hfill & (p_{1M0},p_{1M1},\dots ,p_{1\mathrm{MN}})\hfill \\ \multicolumn{1}{c}{(p_{210},p_{211},\dots ,p_{21N})}& (p_{220},p_{221},\dots ,p_{22N})\hfill & \cdots \hfill & (p_{2M0},p_{2M1},\dots ,p_{2\mathrm{MN}})\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{(p_{L10},p_{L11},\dots ,p_{L1N})}& (p_{L20},p_{L21},\dots ,p_{L2N})\hfill & \cdots \hfill & (p_{\mathrm{LM}0},p_{\mathrm{LM}1},\dots ,p_{\mathrm{LMN}})\hfill \end{array}\right]$$

(12)

$$\sum _{i=0}^N{p}_{\mathrm{kji}}=1$$

Accordingly, the probability is obtained from the above matrix. Each vector obtained from the measurement and preliminary parameter in the matrix can be used as the probability because each feature can detect the fault. According to the preceding analysis, each fault is determined by many features. However, some features do not correctly distinguish the types of faults in some situation. For example, the feature value is equal to 10 when fault $X_1$ occurs, while it is also equal to 10 when fault $X_2$ takes place. Nevertheless, the other feature values are different once the faults are not same. It is important to consider the information quality of each probability varying a lot. Therefore, the Shannon entropy $H(P)=-\sum _{i=1}^n{p}_i\ln (p_i)$ introduced in section “Shannon entropy,” representing information quality, is taken as the discounting factor. Consequently, the basic probability assignment can be obtained by considering information quality and on the basis of the probability distribution. Define ${\omega }_{\mathrm{kj}}=H(P)$ as the discount factor

$$\mathrm{\Omega }=\left[\begin{array}{cccc}{\omega }_{11}\hfill & {\omega }_{12}\hfill & \cdots \hfill & {\omega }_{1M}\hfill \\ \multicolumn{1}{c}{{\omega }_{21}}& {\omega }_{22}\hfill & \cdots \hfill & {\omega }_{2M}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{\omega }_{L1}}& {\omega }_{L2}\hfill & \cdots \hfill & {\omega }_{\mathrm{LM}}\hfill \end{array}\right]$$

(13)

To make ${\omega }_{\mathrm{kj}}\in [0,1]$ , variance for all ${\omega }_{\mathrm{kj}}$ is used to calculate

$${\omega }_{\mathrm{kj}}=1-\frac{{\omega }_{\mathrm{kj}}}{\max \{{\omega }_{\mathrm{kj}}\}+\mathrm{variance}}$$

(14)

Then, for $p_{\mathrm{kj}}=\{p_{\mathrm{kj}0},p_{\mathrm{kj}1},\dots ,p_{\mathrm{kjN}}\}$ corresponding to non-fault, 1st fault, …, Nth fault, define $m_{\mathrm{kj}}=w_{\mathrm{kj}}\times p_{\mathrm{kj}}$ as the basic probability assignment. Let $m_{\mathrm{kj}}(X)=1-\sum _{i=0}^N{m}_{\mathrm{kj}}$ , hence, uncertainty is assigned to the universal set to decrease the conflict. Therefore, the discounting basic probability assignment is obtained as follows

$$m_{\mathrm{kj}}=\{m_{\mathrm{kj}1},m_{\mathrm{kj}2},\dots ,m_{\mathrm{kjN}},m_{\mathrm{kj}}(X)\}$$

(15)

Evidence combination in fault diagnosis

There are many rules proposed to fuse information which are based in D-S evidence theory. Although these rules^74,75 measure and decrease conflict or make result be rational intuitively, they also have some drawbacks, increasing computation complexity or lacking associative property. Due to the above basic probability assignments having fewer conflict when information quality is taken consideration, the classic Dempster’s rule, having fewer computation complexity, is the best choice for us to fuse information. Consequently, according to equations (5) and (6), a mass function can be obtained as follows

$$\begin{array}{c}\multicolumn{1}{c}{m(\{X_0\})=p_0,m(\{X_1\})=p_1,\dots ,m(\{X_N\})}\\ =p_N,m(X)=p_{N+1}\hfill \end{array}$$

(16)

The decision is made on basis of the mass function. The rule decision-makers use is maximum support rule, in which the hypothesis with maximum belief function is chosen to represent the state of the engine. Intuitively, if $m(\{X_0\})$ is greater than the others obviously, the engine is free of any fault and if $m(\{X_1\})$ has the largest value, the fault $X_1$ occurs.

In addition, a situation where an unknown fault may occur when the engine is working is qualitatively analyzed. If this fault occurs, because we have not added this fault in the frame of discernment, the quality of information obtained by the sensors is reduced. Then, based on $m_{\mathrm{kj}}(X)=1-\sum _{i=0}^N{m}_{\mathrm{kj}},m_{\mathrm{kj}}(X)$ , increases. Therefore, after fusion, $m(X)$ increases. Because an unknown fault leads to an increase in uncertainty which is assigned to the universal set, we could infer from the value of $m(X)$ whether there is an unknown fault.

Application in fault diagnosis

A practical application of the proposed method

In this section, an application for detecting the state of the engine^76,77 is given to illustrate the preceding proposed fault diagnosis method. Three sensors (two acceleration sensors and one acoustic sensor) are used to detect the state of the engine. One acceleration sensor is mounted on the cylinder cover near the outlet valve. Another is located on the cylinder cover near the inlet valve. Their peak-to-peak value (P-to-P) in the time domain and the frequency of the maximum spectrum $(F_{\max })$ are calculated as four features represented as ${\alpha }_1,{\alpha }_2,{\alpha }_3,\mathrm{and}{\alpha }_4$ . In concern of the acoustic signal, its mean pressure level (MPL) and the centrobaric frequency of the spectrum $(F_c)$ are calculated as another two features, represented as ${\alpha }_5$ and ${\alpha }_6$ . Because only one sensor may be unreliable in some situation, Three sensors are chosen together to get a rational result. Two faults $(X_1:\mathrm{exhaust}\mathrm{valve}\mathrm{fault};X_2:\mathrm{piston}\mathrm{ring}\mathrm{fault})$ are defined when engine is working and the free of fault is also taken into consideration when the engine works smoothly. Therefore, the frame of discernment is defined as follows

$$X=\{X_0,X_1,X_2\}$$

(17)

Then, the power set of X is established as follows

$$\begin{array}{c}\multicolumn{1}{c}{2^X=\{\mathrm{\varnothing },\{X_0\},\{X_1\},\{X_2\},\{X_0,X_1\},}\\ \{X_0,X_2\},\{X_1,X_2\},X\}\hfill \end{array}$$

(18)

Six features are taken into consideration for each state of the engine and their characteristic values are obtained as follows

$$H=\begin{array}{cc}\hfill & \begin{array}{cccccc}{\alpha }_1\hfill & {\alpha }_2\hfill & {\alpha }_3\hfill & {\alpha }_4\hfill & {\alpha }_5\hfill & {\alpha }_6\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}{X}_0\hfill \\ \multicolumn{1}{c}{X_1}\\ \multicolumn{1}{c}{X_2}\end{array}}& \left(\begin{array}{cccccc}{h}_{01}\hfill & h_{02}\hfill & h_{03}\hfill & h_{04}\hfill & h_{05}\hfill & h_{06}\hfill \\ \multicolumn{1}{c}{h_{11}}& h_{12}\hfill & h_{13}\hfill & h_{14}\hfill & h_{15}\hfill & h_{16}\hfill \\ \multicolumn{1}{c}{h_{21}}& h_{22}\hfill & h_{23}\hfill & h_{24}\hfill & h_{25}\hfill & h_{26}\hfill \end{array}\right)\hfill \end{array}$$

(19)

The values of H that are from Basir and Yuan’s ⁶⁷ research are shown in Table 1.

Table 1.

Features according to previous knowledge.

	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$	${\alpha }_4$	${\alpha }_5$	${\alpha }_6$
$\{X_0\}$	313.5	559.6	378.6	557.4	152.9	762.7
$\{X_1\}$	1850.7	550.8	1734.5	597.2	152.3	808.2
$\{X_2\}$	2669.3	546.6	2567.4	534.8	152.7	724.1

Then, the characteristic values of the features are collected and a feature matrix is constructed by data obtained from sensors after four samplings as follows

$$S=\begin{array}{cc}\hfill & \begin{array}{cccccc}{\alpha }_1\hfill & {\alpha }_2\hfill & {\alpha }_3\hfill & {\alpha }_4\hfill & {\alpha }_5\hfill & {\alpha }_6\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}{S}_0\hfill \\ \multicolumn{1}{c}{S_1}\\ \multicolumn{1}{c}{S_2}\\ \multicolumn{1}{c}{S_3}\end{array}}& \left(\begin{array}{cccccc}{S}_{01}\hfill & S_{02}\hfill & S_{03}\hfill & S_{04}\hfill & S_{05}\hfill & S_{06}\hfill \\ \multicolumn{1}{c}{S_{11}}& S_{12}\hfill & S_{13}\hfill & S_{14}\hfill & S_{15}\hfill & S_{16}\hfill \\ \multicolumn{1}{c}{S_{21}}& S_{22}\hfill & S_{23}\hfill & S_{24}\hfill & S_{25}\hfill & S_{26}\hfill \\ \multicolumn{1}{c}{S_{31}}& S_{32}\hfill & S_{33}\hfill & S_{34}\hfill & S_{35}\hfill & S_{36}\hfill \end{array}\right)\hfill \end{array}$$

(20)

The values of S that are from Song and Jiang’s ⁷⁸ research are shown in Table 2.

Table 2.

Features according to samplings.

	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$	${\alpha }_4$	${\alpha }_5$	${\alpha }_6$
$S_0$	1830.6	553.9	1780.5	600.2	152.5	780.3
$S_1$	1883.5	549.9	1702.4	590.0	151.9	813.6
$S_2$	1854.0	551.7	1738.1	595.4	152.1	797.5
$S_3$	1882.2	555.2	1757.3	575.5	152.5	802.4

According to equations (9)–(12), the probability matrix can be established as follows

$$P=\left[\begin{array}{cccc}(p_{110},p_{111},p_{112})\hfill & (p_{120},p_{121},p_{122})\hfill & \cdots \hfill & (p_{160},p_{161},p_{162})\hfill \\ \multicolumn{1}{c}{(p_{210},p_{211},p_{212})}& (p_{220},p_{221},p_{222})\hfill & \cdots \hfill & (p_{260},p_{261},p_{262})\hfill \\ \multicolumn{1}{c}{(p_{310},p_{311},p_{312})}& (p_{320},p_{321},p_{322})\hfill & \cdots \hfill & (p_{360},p_{361},p_{362})\hfill \\ \multicolumn{1}{c}{(p_{410},p_{411},p_{412})}& (p_{420},p_{421},p_{422})\hfill & \cdots \hfill & (p_{460},p_{461},p_{462})\hfill \end{array}\right]$$

(21)

The result of calculation is shown in Table 3.

Table 3.

Probability distribution.

	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$
$S_0$	(0.0128, 0.9641, 0.0231)	(0.2763, 0.5080, 0.2157)	(0.0301, 0.9164, 0.0536)
$S_1$	(0.0197, 0.9411, 0.0393)	(0.0679, 0.7323, 0.1997)	(0.0228, 0.9422, 0.0350)
$S_2$	(0.0021, 0.9938, 0.0040)	(0.0883, 0.7750, 0.1368)	(0.0026, 0.9931, 0.0043)
$S_3$	(0.0189, 0.9433, 0.0378)	(0.3981, 0.3981, 0.2037)	(0.0158, 0.9572, 0.0269)
	${\alpha }_4$	${\alpha }_5$	${\alpha }_6$
$S_0$	(0.0628, 0.8961, 0.0411)	(0.2000, 0.4000, 0.4000)	(0.5144, 0.3245, 0.1611)
$S_1$	(0.1634, 0.7400, 0.0965)	(0.2105, 0.5263, 0.2632)	(0.0910, 0.8573, 0.0517)
$S_2$	(0.0440, 0.9284, 0.0276)	(0.1579, 0.6316, 0.2105)	(0.2116, 0.6881, 0.1003)
$S_3$	(0.4388, 0.3660, 0.1952)	(0.2000, 0.4000, 0.4000)	(0.1197, 0.8196, 0.0607)

All the measurements obtained from sensors can characterize two faults, but not all of the measurements can distinguish fault accurately in some situation, in other words, measurements have different information quality. The discount factor obtained from the Shannon entropy, representing information quality, is used to construct basic probability assignments. Therefore, $\mathrm{\Omega }$ , namely the discount factor, can be obtained by equations (7) and (14) as follows

$$\mathrm{\Omega }=\begin{array}{cc}\hfill & \begin{array}{cccccc}{\alpha }_1\hfill & {\alpha }_2\hfill & {\alpha }_3\hfill & {\alpha }_4\hfill & {\alpha }_5\hfill & {\alpha }_6\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}{S}_1\hfill \\ \multicolumn{1}{c}{S_2}\\ \multicolumn{1}{c}{S_3}\\ \multicolumn{1}{c}{S_4}\end{array}}& \left(\begin{array}{cccccc}{\omega }_{11}\hfill & {\omega }_{12}\hfill & {\omega }_{13}\hfill & {\omega }_{14}\hfill & {\omega }_{15}\hfill & {\omega }_{16}\hfill \\ \multicolumn{1}{c}{{\omega }_{21}}& {\omega }_{22}\hfill & {\omega }_{13}\hfill & {\omega }_{14}\hfill & {\omega }_{15}\hfill & {\omega }_{26}\hfill \\ \multicolumn{1}{c}{{\omega }_{31}}& {\omega }_{32}\hfill & {\omega }_{13}\hfill & {\omega }_{14}\hfill & {\omega }_{15}\hfill & {\omega }_{36}\hfill \\ \multicolumn{1}{c}{{\omega }_{41}}& {\omega }_{42}\hfill & {\omega }_{13}\hfill & {\omega }_{14}\hfill & {\omega }_{15}\hfill & {\omega }_{46}\hfill \end{array}\right)\hfill \end{array}$$

(22)

The value of $\mathrm{\Omega }$ is shown in Table 4.

Table 4.

The discount factor.

	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$	${\alpha }_4$	${\alpha }_5$	${\alpha }_6$
$S_0$	0.4259	0.1136	0.2784	0.2466	0.1112	0.1165
$S_1$	0.3354	0.1527	0.3370	0.1506	0.1149	0.2078
$S_2$	0.7612	0.1618	0.7415	0.3019	0.1267	0.1392
$S_3$	0.3421	0.1110	0.3919	0.1119	0.1112	0.1835

After the basic probability assignment matrix constructed by considering information quality is obtained, the Dempster’s rule of evidence combination is used to get mass function in this article. According to equations (5) and (6), we can get the final result as follows

$$\begin{array}{c}m\{X_0\}=0.000566\hfill \\ m\{X_1\}=0.998134\hfill \\ m\{X_2\}=0.000544\hfill \\ m\{X\}=0.000756\hfill \end{array}$$

To show the difference of mass function intuitively, the histogram is shown as follows:

The rule decision-makers use is maximum support rule, in which the hypothesis with maximum belief function is chosen to represent the state of the engine. Consequently, from Figure 2, the basic probability assignment (0.9981) of $X_1$ of the proposed method is obviously larger than the others. It can prove that the fault is $X_1$ , namely, exhaust valve fault.

Figure 2.

The values of mass function for the methods which take information quality into account and do not do it.

In Figure 2, both methods get effective results that the fault is $X_1$ . By comparing the result of the method which takes information quality into account (the proposed method) and does not do it, the basic probability assignment of $X_1$ of the former is significantly larger than that of the latter. In this article, the proposed method not only includes multiple subsets but also considers information quality. Accordingly, the proposed method is illustrated as practicability.

Reliability analysis and comparison with the typical method

In this section, the reliability of the proposed method is analyzed by the procedure of simulating random errors in actual measurement. ⁶⁷ And maximum support rule is used as a decisive standard in this analysis.

When the sensors perform measurement, there is a certain error for data obtained from sensors due to its own measurement error. Therefore, on this basis, we would give a random error rate no more than a certain percent (5%, 10%, 15%, and 20%) to the data obtained from Table 2 for reliability analysis. Use the proposed method to process the new data obtained, then according to maximum support rule, performance accuracy is obtained and it is shown in Table 5. We can find that when the error rate is less than 10%, performance accuracy is 100%, so the proposed method has strong reliability.

Table 5.

Performance accuracy.

The proposed method					The typical method
Random error rate (%)	5	10	15	20	0
Performance accuracy (%)	100	100	95	83	94

Basir and Yuan ⁶⁷ had used three sensors (two acceleration sensors and one acoustic sensors) to test the efficiency of the typical method. The performance accuracy is displayed on the left side of the Table 5. By comparing performance of the new method and the typical method, the former is slightly better than the latter.

Conclusion

In this article, a new method that entropy of each probability obtained from each feature is used as the discount factor to get the basic probability assignment is proposed. First, it is important to construct the frame of discernment. Second, the preliminary matrix is established by expert knowledge. Third, the basic probability assignment matrix can be obtained from calculating the distance between the values from samplings and preliminary matrix. Last, the Dempster’ rule is used to combine evidence to get final result. The proposed method could be demonstrated efficient from the preceding example. And the proposed method also has strong reliability. In addition, in this article, there is a limitation of this article which is that multiple subsets are not taken into account because the engine may have multiple faults at the same time. Therefore, the future research direction is to extend the single subset to multiple subsets to more effectively identify one or more possible faults of the engine at the same time.