# A Neutrosophic Set Based Fault Diagnosis Method Based on Multi-Stage Fault Template Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. The Proposed Method

- Step 1
- Collect the multi-stage data of fault types under each feature. Suppose that there are m fault types ($F=\{{F}_{1},{F}_{2},\dots ,{F}_{m}\}$) with n features ($C=\{{C}_{1},{C}_{2},\dots ,{C}_{n}\}$). Firstly, collect the multi-stage data of each fault type under each feature. Each stage’s data for each fault type under each feature are obtained by continuously collecting within the time interval (T). Suppose that data from k stages of every fault type under every feature are obtained. The multi-stage data of each fault type under each feature are shown as follows:$$\begin{array}{cc}& \begin{array}{ccccccccccccc}\hfill & & \cdots & & & {C}_{j}& & & \cdots & & & & \end{array}\\ \begin{array}{c}\\ \vdots \\ \\ {F}_{i}\\ \\ \vdots \\ \end{array}& \left[\begin{array}{ccc}\ddots & \vdots & \ddots \\ \ddots & k\phantom{\rule{4pt}{0ex}}stages\phantom{\rule{4pt}{0ex}}data\phantom{\rule{4pt}{0ex}}of\phantom{\rule{4pt}{0ex}}{F}_{i}\phantom{\rule{4pt}{0ex}}under\phantom{\rule{4pt}{0ex}}{C}_{j}& \ddots \\ \ddots & \vdots & \ddots \end{array}\right]\end{array}$$
- Step 2
- Generate the SNS for an unknown fault sample (S) based on the multi-stage data of each fault type under each feature. For each stage’s data for each fault type under every feature, and for the data of every feature of the unknown fault sample (S), a normal distribution model is established which is obtained by using the arithmetic average (m) and variance (${\sigma}^{2}$) of a stage’s data as the arithmetic average and standard deviation of the normal distribution model, denoted as $N(m,\phantom{\rule{4pt}{0ex}}{\sigma}^{2})$. Then, k normal distribution models and k normal distribution figures are generated according to k stages of data of each fault type under each feature. In addition, a normal distribution model is generated based on the data of the unknown fault sample under each feature. The normal distribution figures generated from the data of ${C}_{j}$ of unknown fault sample S and k stages of data for ${C}_{j}$ of ${F}_{i}$ are shown in Figure 2. As the figure shows, each stage’s data collected drift to a certain extent in a certain range. In particular, there are distinct differences between the fault type’s data collected in the fourth stage and the data of the unknown fault sample.The normal distribution function indicates the distribution probability density of the data. The membership degree of SNS is defined as the ratio of the maximum value of the vertical coordinate of the intersection point between the unknown fault sample and the fault type and the peak value of the unknown fault sample. The two normal distribution curves (Figure 3) and the definition of the membership degree ($\mu $) are as follows:$$\mu =\frac{{y}_{h}}{{y}_{m}},$$As the figure shown, the intersection points of distribution between the unknown fault sample and ${F}_{i}$ are marked with X, and the peak point of S’ distribution is marked with X in the same way. Then, from the Equation (6), the membership degree is generated.In this paper, it is assumed that the non-membership degree and the membership degree are interdependent. The indeterminacy-membership degree indicates the uncertainty degree of neutrosophic information. Entropy represents the uncertainty of the information and has been widely used in many fields. Shannon introduced the quantitative and qualitative model of communication as a statistical process that underlies information theory [55], which is a formalism that was originally applied to digital communication. The indeterminacy-membership degree and non-membership degree are defined as follows:$$\begin{array}{c}\left(1\right)\phantom{\rule{1.em}{0ex}}\nu =1-\mu ,\hfill \\ \left(2\right)\phantom{\rule{1.em}{0ex}}\pi =\mu lo{g}_{2}(\frac{1}{\mu})+\nu lo{g}_{2}(\frac{1}{\nu}),\phantom{\rule{1.em}{0ex}}\mu \ne 0,\nu \ne 0.\hfill \end{array}$$The indeterminacy-membership degree ($\pi $) represents the Shannon entropy of the membership degree ($\mu $) and the non-membership degree ($\nu $), and $\pi $ equals 0 if $\mu $ or $\nu $ equal 0. Hence, the SNS can be obtained. The generated SNS is shown in Table 1:
- Step 3
- Aggregate the generated SNS based on each fault type under each feature. In this paper, it is assumed that the weights of data from k stages collected under the same working conditions are equal. The k SNNs of each fault type under each feature are fused via the SNWA operator, as shown in Equation (5). For instance,$${\alpha}_{11}=SNWA({\alpha}_{11}^{1},{\alpha}_{11}^{2},\dots ,{\alpha}_{11}^{k}).$$Then, the fused SNS matrix (A) is as follows:$$A=\begin{array}{cc}& \begin{array}{ccccccccc}\hfill & \cdots & & & {C}_{j}& & & \cdots & \end{array}\\ \begin{array}{c}\\ \vdots \\ \\ {F}_{i}\\ \\ \vdots \\ \end{array}& \left[\begin{array}{ccc}\ddots & \vdots & \ddots \\ \ddots & ({\mu}_{ij},\phantom{\rule{4pt}{0ex}}{\pi}_{ij},\phantom{\rule{4pt}{0ex}}{\nu}_{ij})& \ddots \\ \ddots & \vdots & \ddots \end{array}\right]\end{array}$$
- Step 4
- Aggregate the fused SNS based on all features of each fault type. If the weights of n features are equal, n SNNs of each fault type are fused via the SNWA operator, as shown in Equation (5). For instance,$${\alpha}_{1}=SNWA({\alpha}_{11},{\alpha}_{12},\dots ,{\alpha}_{1n}).$$Then, the fused SNS matrix (F) is as follows:$$F=\begin{array}{cc}\begin{array}{c}{F}_{1}\\ {F}_{2}\\ \vdots \\ {F}_{m}\end{array}& \left[\begin{array}{c}({\mu}_{1},\phantom{\rule{4pt}{0ex}}{\pi}_{1},\phantom{\rule{4pt}{0ex}}{\nu}_{1})\\ ({\mu}_{2},\phantom{\rule{4pt}{0ex}}{\pi}_{2},\phantom{\rule{4pt}{0ex}}{\nu}_{2})\\ \vdots \\ ({\mu}_{m},\phantom{\rule{4pt}{0ex}}{\pi}_{m},\phantom{\rule{4pt}{0ex}}{\nu}_{m})\end{array}\right].\end{array}$$
- Step 5
- Determine the fault type of the unknown fault sample. Considering the fuzziness of the unknown fault sample and the fault types, direct application of the defuzzification method can intuitively reflect the results of the fault diagnosis and reduce the amount of calculation in the process of fault diagnosis. The crisp number of each SNN is defuzzied and calculated as follows [56]:$${C}_{i}={\mu}_{i}+\left({\pi}_{i}\right)\left(\frac{{\mu}_{i}}{{\mu}_{i}+{\nu}_{i}}\right).$$${C}_{i}$ is the degree to which the information extracted from the data of untested fault supports each fault type. As a result, the ranking order of all the fault types can be determined according to the descending order of their crisp numbers (${C}_{i}$).

## 4. Illustrative Example and Discussion

- ${C}_{1}$: The vibration amplitude when the acceleration frequency of the rotor is the basic frequency, $1X$.
- ${C}_{2}$: The vibration amplitude when the acceleration frequency of the rotor is the frequency $2X$.
- ${C}_{3}$: The vibration amplitude when the acceleration frequency of the rotor is the frequency $3X$.
- ${C}_{4}$: The average amplitude of vibration displacement in the time-domain.

- Step 1
- Collect the multi-stage data of each fault type under each feature. There are three fault types set up on the test-bed:
- ${F}_{1}$: Rotor imbalance.
- ${F}_{2}$: Rotor misalignment.
- ${F}_{3}$: Support base loosening.

For each feature of each fault type, data from five stages were collected, and for each stage’s data, forty consecutive observation values were collected continuously within a time interval of 16 s. The data in this paper originated from Reference [57]. For instance, the first stage’s data of ${F}_{1}$ under ${C}_{1}$ was as follows:$$\begin{array}{c}{{F}_{1}}_{{C}_{1}}\phantom{\rule{4pt}{0ex}}First\phantom{\rule{4pt}{0ex}}Stag{e}^{\prime}s\phantom{\rule{4pt}{0ex}}Data=[0.1663\phantom{\rule{1.em}{0ex}}0.1590\phantom{\rule{1.em}{0ex}}0.1568\phantom{\rule{1.em}{0ex}}0.1485\phantom{\rule{1.em}{0ex}}0.1723\phantom{\rule{1.em}{0ex}}0.2006\phantom{\rule{1.em}{0ex}}0.1903\hfill \\ \phantom{\rule{1.em}{0ex}}0.1908\phantom{\rule{1.em}{0ex}}0.1986\phantom{\rule{1.em}{0ex}}0.1843\phantom{\rule{1.em}{0ex}}0.1785\phantom{\rule{1.em}{0ex}}0.1610\phantom{\rule{1.em}{0ex}}0.1579\phantom{\rule{1.em}{0ex}}0.1511\phantom{\rule{1.em}{0ex}}0.1532\phantom{\rule{1.em}{0ex}}0.1647\phantom{\rule{1.em}{0ex}}0.1628\phantom{\rule{1.em}{0ex}}0.1646\hfill \\ \phantom{\rule{1.em}{0ex}}0.1634\phantom{\rule{1.em}{0ex}}0.1642\phantom{\rule{1.em}{0ex}}0.1648\phantom{\rule{1.em}{0ex}}0.1640\phantom{\rule{1.em}{0ex}}0.1674\phantom{\rule{1.em}{0ex}}0.0661\phantom{\rule{1.em}{0ex}}0.1659\phantom{\rule{1.em}{0ex}}0.1650\phantom{\rule{1.em}{0ex}}0.1633\phantom{\rule{1.em}{0ex}}0.1632\phantom{\rule{1.em}{0ex}}0.1604\hfill \\ \phantom{\rule{1.em}{0ex}}0.1542\phantom{\rule{1.em}{0ex}}0.1555\phantom{\rule{1.em}{0ex}}0.1562\phantom{\rule{1.em}{0ex}}0.1540\phantom{\rule{1.em}{0ex}}0.1564\phantom{\rule{1.em}{0ex}}0.1557\phantom{\rule{1.em}{0ex}}0.1542\phantom{\rule{1.em}{0ex}}0.1546\phantom{\rule{1.em}{0ex}}0.1571\phantom{\rule{1.em}{0ex}}0.1537\phantom{\rule{1.em}{0ex}}0.1536].\hfill \end{array}$$ - Step 2
- Generate the SNS for the unknown fault sample based on the multi-stage data from each fault type under each feature. Each stage’s data collected is used to establish the normal distribution model. The generated normal distributions of fault types and the unknown fault sample are listed in Table 2. For instance, the normal distribution of ${{S}_{1}}_{{C}_{1}}$ data and ${{F}_{1}}_{{C}_{1}}$ with five stages of data is shown in Figure 4. As the figure shows, each stage’s data collected drift to a certain extent in a certain range. In particular, there were distinct differences between the fault types collected in each stage and the data of unknown fault samples. Therefore, it is significant to collect data in multiple stages and to use its integration with the neutrosophic set to deal with the uncertainty of fault information.Then, $\mu ,\pi ,\nu $ are calculated with Equations (6) and (7). For instance, the distribution of ${{S}_{1}}_{{C}_{1}}\phantom{\rule{4pt}{0ex}}Data$ was $N(0.1427,\phantom{\rule{4pt}{0ex}}0.0006)$, the normal distribution of ${{F}_{1}}_{{C}_{1}}$’s first stage of data was $N(0.1619,\phantom{\rule{4pt}{0ex}}0.0200)$, and the membership degree of SNN generated from the two distributions is shown in Figure 5. As the figure shows, the intersection points of distribution between the unknown fault sample (${S}_{1}$) and ${{F}_{1}}_{{C}_{1}}$’s first stage data are marked with X, and the peak point of ${S}_{1}$’s distribution is marked with X in the same way. Then, from the Equations (6) and (7), the SNN was generated and denoted as $(0.0197,\phantom{\rule{4pt}{0ex}}0.0969,\phantom{\rule{4pt}{0ex}}0.9803)$. The generated SNSs are listed in Table 3.
- Step 3
- Aggregate the generated SNSs based on each fault type under each feature. Fuse the five stages of SNNs for each fault type under each feature with the SNWA operator, Equation (5). It is assumed that the weights (w) of the five SNNs are $[0.20,\phantom{\rule{4pt}{0ex}}0.20,\phantom{\rule{4pt}{0ex}}0.20,\phantom{\rule{4pt}{0ex}}0.20,\phantom{\rule{4pt}{0ex}}0.20]$. For example, the SNNs based on the fault type ${F}_{1}$ under feature ${C}_{1}$ could be fused as follows:$$\begin{array}{c}{\alpha}_{11}=SNWA({\alpha}_{11}^{1},{\alpha}_{11}^{2},{\alpha}_{11}^{3},{\alpha}_{11}^{4},{\alpha}_{11}^{5})\hfill \\ =SNWA\left(\right(0.0197,\phantom{\rule{4pt}{0ex}}0.0969,\phantom{\rule{4pt}{0ex}}0.9803),(0.0092,\phantom{\rule{4pt}{0ex}}0.0521,\phantom{\rule{4pt}{0ex}}0.9908),\hfill \\ \phantom{\rule{1.em}{0ex}}(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}1),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}1),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}1))\hfill \\ =(0.0058,\phantom{\rule{4pt}{0ex}}0.0000,\phantom{\rule{4pt}{0ex}}0.9942).\hfill \end{array}$$The others are shown in Table 4.
- Step 4
- Aggregate the fused SNSs based on all features of each fault type. Fusing the SNNs is based on the four features of each fault type by the SNWA operator, Equation (5). In addition, it is supposed the weights (w) of the four SNNs are $[0.25,\phantom{\rule{4pt}{0ex}}0.25,\phantom{\rule{4pt}{0ex}}0.25,\phantom{\rule{4pt}{0ex}}0.25]$. For example, the SNNs based on fault type ${F}_{1}$ could be fused as follows:$$\begin{array}{c}{\alpha}_{1}=SNWA({\alpha}_{11},{\alpha}_{12},{\alpha}_{13},{\alpha}_{14})\hfill \\ =SNWA\left(\right(0.0058,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}0.9942),(0.72,\phantom{\rule{4pt}{0ex}}0.5868,\phantom{\rule{4pt}{0ex}}0.28),(0.0203,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}0.9797),\hfill \\ \phantom{\rule{1.em}{0ex}}(0.0252,\phantom{\rule{4pt}{0ex}}0.0008,\phantom{\rule{4pt}{0ex}}0.9748))\hfill \\ =(0.2633,\phantom{\rule{4pt}{0ex}}0.0000,\phantom{\rule{4pt}{0ex}}0.7367).\hfill \end{array}$$The others are shown in Table 5.
- Step 5
- Determine the fault type of the unknown fault sample. Finally, Table 5 can be regarded as an SNN fault diagnosis matrix which can be used to rank the three fault types via the defuzzification method (Equation (10)). The descendant ranks of the crisp numbers of the three fault types are shown in Table 6.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Fault Type | Stage | Feature | |||
---|---|---|---|---|---|

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ⋯ | ${\mathit{C}}_{\mathit{n}}$ | ||

${F}_{1}$ | 1 | $({\mu}_{11}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{11}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{11}^{1})$ | $({\mu}_{12}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{12}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{12}^{1})$ | ⋯ | $({\mu}_{1n}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{1n}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{1n}^{1})$ |

2 | $({\mu}_{11}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{11}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{11}^{2})$ | $({\mu}_{12}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{12}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{12}^{2})$ | ⋯ | $({\mu}_{1n}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{1n}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{1n}^{2})$ | |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ | |

k | $({\mu}_{11}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{11}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{11}^{k})$ | $({\mu}_{12}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{12}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{12}^{k})$ | ⋯ | $({\mu}_{1n}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{1n}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{1n}^{k})$ | |

${F}_{2}$ | 1 | $({\mu}_{21}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{21}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{21}^{1})$ | $({\mu}_{22}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{22}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{22}^{1})$ | ⋯ | $({\mu}_{2n}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{2n}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{2n}^{1})$ |

2 | $({\mu}_{21}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{21}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{21}^{2})$ | $({\mu}_{22}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{22}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{22}^{2})$ | ⋯ | $({\mu}_{2n}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{2n}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{2n}^{2})$ | |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ | |

k | $({\mu}_{21}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{21}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{21}^{k})$ | $({\mu}_{22}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{22}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{22}^{k})$ | ⋯ | $({\mu}_{2n}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{2n}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{2n}^{k})$ | |

⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

${F}_{m}$ | 1 | $({\mu}_{m1}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{m1}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{m1}^{1})$ | $({\mu}_{m2}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{m2}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{m2}^{1})$ | ⋯ | $({\mu}_{mn}^{1},\phantom{\rule{4pt}{0ex}}{\pi}_{mn}^{1},\phantom{\rule{4pt}{0ex}}{\nu}_{mn}^{1})$ |

2 | $({\mu}_{m1}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{m1}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{m1}^{2})$ | $({\mu}_{m2}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{m2}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{m2}^{2})$ | ⋯ | $({\mu}_{mn}^{2},\phantom{\rule{4pt}{0ex}}{\pi}_{mn}^{2},\phantom{\rule{4pt}{0ex}}{\nu}_{mn}^{2})$ | |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ | |

k | $({\mu}_{m1}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{m1}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{m1}^{k})$ | $({\mu}_{m2}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{m2}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{m2}^{k})$ | ⋯ | $({\mu}_{mn}^{k},\phantom{\rule{4pt}{0ex}}{\pi}_{mn}^{k},\phantom{\rule{4pt}{0ex}}{\nu}_{mn}^{k})$ |

Fault Type | Stage | Feature | |||
---|---|---|---|---|---|

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ||

${F}_{1}$ | 1 | $N(0.1619,\phantom{\rule{4pt}{0ex}}0.0200)$ | $N(0.1538,\phantom{\rule{4pt}{0ex}}0.0112)$ | $N(0.1163,\phantom{\rule{4pt}{0ex}}0.0098)$ | $N(4.3057,\phantom{\rule{4pt}{0ex}}0.1124)$ |

2 | $N(0.1596,\phantom{\rule{4pt}{0ex}}0.0073)$ | $N(0.1509,\phantom{\rule{4pt}{0ex}}0.0052)$ | $N(0.1095,\phantom{\rule{4pt}{0ex}}0.0021)$ | $N(4.4143,\phantom{\rule{4pt}{0ex}}0.0226)$ | |

3 | $N(0.1644,\phantom{\rule{4pt}{0ex}}0.0009)$ | $N(0.1468,\phantom{\rule{4pt}{0ex}}0.0024)$ | $N(0.1063,\phantom{\rule{4pt}{0ex}}0.0037)$ | $N(4.2626,\phantom{\rule{4pt}{0ex}}0.6336)$ | |

4 | $N(0.1617,\phantom{\rule{4pt}{0ex}}0.0006)$ | $N(0.1519,\phantom{\rule{4pt}{0ex}}0.0316)$ | $N(0.1117,\phantom{\rule{4pt}{0ex}}0.0022)$ | $N(4.3138,\phantom{\rule{4pt}{0ex}}0.0249)$ | |

5 | $N(0.1598,\phantom{\rule{4pt}{0ex}}0.0010)$ | $N(0.1428,\phantom{\rule{4pt}{0ex}}0.0025)$ | $N(0.1182,\phantom{\rule{4pt}{0ex}}0.0017)$ | $N(4.3319,\phantom{\rule{4pt}{0ex}}0.0347)$ | |

${F}_{2}$ | 1 | $N(0.1696,\phantom{\rule{4pt}{0ex}}0.0096)$ | $N(0.3266,\phantom{\rule{4pt}{0ex}}0.0108)$ | $N(0.2772,\phantom{\rule{4pt}{0ex}}0.0250)$ | $N(4.9825,\phantom{\rule{4pt}{0ex}}0.1882)$ |

2 | $N(0.1742,\phantom{\rule{4pt}{0ex}}0.0045)$ | $N(0.3278,\phantom{\rule{4pt}{0ex}}0.0083)$ | $N(0.2726,\phantom{\rule{4pt}{0ex}}0.0095)$ | $N(4.5844,\phantom{\rule{4pt}{0ex}}0.1226)$ | |

3 | $N(0.1932,\phantom{\rule{4pt}{0ex}}0.0138)$ | $N(0.3384,\phantom{\rule{4pt}{0ex}}0.0115)$ | $N(0.2217,\phantom{\rule{4pt}{0ex}}0.0339)$ | $N(4.4358,\phantom{\rule{4pt}{0ex}}0.4015)$ | |

4 | $N(0.1916,\phantom{\rule{4pt}{0ex}}0.0037)$ | $N(0.3350,\phantom{\rule{4pt}{0ex}}0.0063)$ | $N(0.2131,\phantom{\rule{4pt}{0ex}}0.0053)$ | $N(5.0105,\phantom{\rule{4pt}{0ex}}0.6455)$ | |

5 | $N(0.1804,\phantom{\rule{4pt}{0ex}}0.0031)$ | $N(0.3187,\phantom{\rule{4pt}{0ex}}0.0041)$ | $N(0.2255,\phantom{\rule{4pt}{0ex}}0.0135)$ | $N(4.5631,\phantom{\rule{4pt}{0ex}}0.0678)$ | |

${F}_{3}$ | 1 | $N(0.3387,\phantom{\rule{4pt}{0ex}}0.0071)$ | $N(0.3413,\phantom{\rule{4pt}{0ex}}0.0207)$ | $N(0.1501,\phantom{\rule{4pt}{0ex}}0.0120)$ | $N(9.8483,\phantom{\rule{4pt}{0ex}}0.0709)$ |

2 | $N(0.3296,\phantom{\rule{4pt}{0ex}}0.0026)$ | $N(0.3511,\phantom{\rule{4pt}{0ex}}0.0090)$ | $N(0.1341,\phantom{\rule{4pt}{0ex}}0.0080)$ | $N(9.7652,\phantom{\rule{4pt}{0ex}}0.0953)$ | |

3 | $N(0.3247,\phantom{\rule{4pt}{0ex}}0.0074)$ | $N(0.3409,\phantom{\rule{4pt}{0ex}}0.0135)$ | $N(0.1341,\phantom{\rule{4pt}{0ex}}0.0113)$ | $N(9.7802,\phantom{\rule{4pt}{0ex}}0.0608)$ | |

4 | $N(0.3265,\phantom{\rule{4pt}{0ex}}0.0049)$ | $N(0.3357,\phantom{\rule{4pt}{0ex}}0.0098)$ | $N(0.1330,\phantom{\rule{4pt}{0ex}}0.0052)$ | $N(9.8739,\phantom{\rule{4pt}{0ex}}0.1267)$ | |

5 | $N(0.3275,\phantom{\rule{4pt}{0ex}}0.0023)$ | $N(0.3503,\phantom{\rule{4pt}{0ex}}0.0060)$ | $N(0.1295,\phantom{\rule{4pt}{0ex}}0.0048)$ | $N(9.7856,\phantom{\rule{4pt}{0ex}}0.1010)$ | |

${S}_{1}$ | 1 | $N(0.1427,\phantom{\rule{4pt}{0ex}}0.0006)$ | $N(0.1109,\phantom{\rule{4pt}{0ex}}0.0316)$ | $N(0.1337,\phantom{\rule{4pt}{0ex}}0.0022)$ | $N(4.0938,\phantom{\rule{4pt}{0ex}}0.0249)$ |

**Table 3.**The generated SNS for ${S}_{1}$ based on the multi-stage data from every fault type under every feature.

Fault Type | Stage | Feature | |||
---|---|---|---|---|---|

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ||

${F}_{1}$ | 1 | (0.0197, 0.0969, 0.9803) | (0.7400, 0.5731, 0.2600) | (0.0973, 0.3191, 0.9027) | (0.0841, 0.2888, 0.9159) |

2 | (0.0092, 0.0521, 0.9908) | (0.6576, 0.6426, 0.3424) | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | |

3 | (0.0000, 0.0000, 1.0000) | (0.6382, 0.6545, 0.3618) | (0.0000, 0.0000, 1.0000) | (0.0388, 0.1641, 0.9612) | |

4 | (0.0000, 0.0000, 1.0000) | (0.8108, 0.4851, 0.1892) | (0.0000, 0.0000, 1.0000) | (0.0001, 0.0006, 0.9999) | |

5 | (0.0000, 0.0000, 1.0000) | (0.7177, 0.5951, 0.2823) | (0.0004, 0.0032, 0.9996) | (0.0003, 0.0026, 0.9997) | |

${F}_{2}$ | 1 | (0.0021, 0.0152, 0.9979) | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) |

2 | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.0010, 0.0082, 0.9990) | |

3 | (0.0001, 0.0010, 0.9999) | (0.0000, 0.0000, 1.0000) | (0.0038, 0.0249, 0.9962) | (0.0486, 0.1944, 0.9514) | |

4 | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.0164, 0.0836, 0.9836) | |

5 | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | |

${F}_{3}$ | 1 | (0.0000, 0.0000, 1.0000) | (0.0001, 0.0008, 0.9999) | (0.1118, 0.3502, 0.8882) | (0.0000, 0.0000, 1.0000) |

2 | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.2525, 0.5650, 0.7475) | (0.0000, 0.0000, 1.0000) | |

3 | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.1847, 0.4785, 0.8153) | (0.0000, 0.0000, 1.0000) | |

4 | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.3815, 0.6648, 0.6185) | (0.0000, 0.0000, 1.0000) | |

5 | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.4364, 0.6850, 0.5636) | (0.0000, 0.0000, 1.0000) |

Fault Type | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|---|---|---|---|

${F}_{1}$ | (0.0058, 0.0000, 0.9942) | (0.7200, 0.5868, 0.2800) | (0.0203, 0.0000, 0.9797) | (0.0252, 0.0008, 0.9748) |

${F}_{2}$ | (0.0004, 0.0000, 0.9996) | (0.0000, 0.0000, 1.0000) | (0.0008, 0.0000, 0.9992) | (0.0134, 0.0038, 0.9866) |

${F}_{3}$ | (0.0000, 0.0000, 1.0000) | (0.0000, 0.0000, 1.0000) | (0.2836, 0.5332, 0.7164) | (0.0000, 0.0000, 1.0000) |

Fault Type | SNS |
---|---|

${F}_{1}$ | (0.2633, 0.0000, 0.7367) |

${F}_{2}$ | (0.0030, 0.0000, 0.9970) |

${F}_{3}$ | (0.0952, 0.0000, 0.9048) |

Fault Type | Crisp Number | Rank |
---|---|---|

${F}_{1}$ | 0.263335 | 1 |

${F}_{2}$ | 0.003040 | 3 |

${F}_{3}$ | 0.095221 | 2 |

Unknown Fault | Mehod | Rank of Fault Types | Diagnosis Result | Validity | ||
---|---|---|---|---|---|---|

F1 | F2 | F3 | ||||

${S}_{1}$ | The proposed method | 1 | 3 | 2 | ${F}_{1}$ | $Correct$ |

Xu’ method [54] | 1 | 3 | 2 | ${F}_{1}$ | $Correct$ | |

${S}_{2}$ | The proposed method | 2 | 1 | 3 | ${F}_{2}$ | $Correct$ |

Xu’ method [54] | 2 | 1 | 3 | ${F}_{2}$ | $Correct$ | |

${S}_{3}$ | The proposed method | 3 | 2 | 1 | ${F}_{3}$ | $Correct$ |

Xu’ method [54] | 3 | 2 | 1 | ${F}_{3}$ | $Correct$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, W.; Zhong, Y.; Deng, X.
A Neutrosophic Set Based Fault Diagnosis Method Based on Multi-Stage Fault Template Data. *Symmetry* **2018**, *10*, 346.
https://doi.org/10.3390/sym10080346

**AMA Style**

Jiang W, Zhong Y, Deng X.
A Neutrosophic Set Based Fault Diagnosis Method Based on Multi-Stage Fault Template Data. *Symmetry*. 2018; 10(8):346.
https://doi.org/10.3390/sym10080346

**Chicago/Turabian Style**

Jiang, Wen, Yu Zhong, and Xinyang Deng.
2018. "A Neutrosophic Set Based Fault Diagnosis Method Based on Multi-Stage Fault Template Data" *Symmetry* 10, no. 8: 346.
https://doi.org/10.3390/sym10080346