AN APPLICATION OF THE PYTHAGOREAN FUZZY SETS IN THE FAULT DIAGNOSIS

: In this paper, a comprehensive review and critical analyses of methods based on the ordinary fuzzy set, Atanassov’s intuitionistic fuzzy set, and its extensions have been conducted to show their limitations and defects. Then, a novel similarity measure based on the generalized score function has been introduced that incorporates the significance (importance) of information, making it more intuitive to compare them. The proposed method is employed for the fault diagnosis of steam turbine generator unit under Pythagorean fuzzy environment. Ten fault types of rotating machines are established as failure patterns in nine different vibration frequency ranges, expressed in terms of Pythagorean fuzzy numbers. The superiority of the proposed method in dealing with uncertain and vague information is shown by comparing it with some existing measures in numerical examples.


Introduction
The fault diagnosis of a technical system is critical for its status monitoring in operation and maintenance.Signal-based fault diagnosis is the most popular approach in machine health monitoring.The typical signals used for diagnosing can be vibration signal [4], acoustic emission signal [13] or current signal [3].Among them, vibration signal analysis is the most widely used method to identify some particular faults because it is easy to measure and can provide highly accurate information about the machine's health condition [15].Nowadays, most of the fault diagnosis methods are usually based on the vibration signals or the signals transformed from the vibration signals based upon several popular signal processing techniques, including fast Fourier transform (FFT), wavelet transform (WT), and short-time Fourier transform (STFT).The fault diagnosis performance of the signal-based approach highly depends on the procedure of feature extraction in which characteristic features are extracted from vibration signals.After extracting the fault features from the fault signals, an intelligent decision-maker based on machine learning algorithms is used to determine the type of fault occurring.Traditionally, feature extraction exploits signal processing techniques to extract information from the fault signal in the time domain, frequency domain, and time-frequency domain [11].However, the diagnosing accuracy of the traditional approaches depends on the signal processing technique and requires expert knowledge [6].Due to the expert knowledge requirement, it is difficult to suggest a generalized framework for feature extraction.Although classical machine learning models, such as support vector machine (SVM) and k-nearest neighbour, have achieved remarkable progress over the past years, some drawbacks still exist when facing higher industrial requirements [5].
The first concept of intuitionistic fuzzy set (IFS) was introduced by Atanassov as a generalization of fuzzy set (FS) theory.The IFS can be viewed as an alternative approach to an ordinary fuzzy set to deal with imperfect information.The uncertainty of information is expressed by a membership function μ(x) in FSs.But in IFSs, it is represented by membership μ(x), and non-membership μ(x) functions, as shown in Fig. 1.As an extension of the IFS, the Pythagorean fuzzy set (PFS) originally proposed by Yager [14] was regarded to be more flexible in expressing vague and uncertain information due to its bigger domain, where the sum of the squared membership and non-membership degrees must be less than or equal to 1 (see Fig. 2).

Fig. 2. The domain comparison of the IFS and PFS methods
Since then, more and more methods for PFSs have been developed and widely used in many areas such as image clustering and partitioning, pattern recognition, medical diagnosis and multi-criteria decision-making (MCDM).For instance, Yager [14] developed a useful decision method based on Pythagorean fuzzy aggregation operators to handle Pythagorean fuzzy MCDM problems.Zhang and Xu [18] provided the detailed mathematical expression for PFS and introduced the concept of Pythagorean fuzzy number (PFN).Meanwhile, they also developed a Pythagorean fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) for handling the MCDM problem within PFNs.Peng and Yang [10] proposed the division and subtraction operations for PFNs.Also, they developed a Pythagorean fuzzy superiority and inferiority ranking method to solve multicriteria group decision-making problem with PFNs.
Several definitions of similarity measures between IFSs have been proposed recently.However, they have been proved to be unreasonable in some cases.For example, Li et al. [7] examined a comparative analysis of several existing similarity measures and pointed out their unreasonable cases in pattern recognition.Also, Papakostas et al. [9] studied the main properties of the existing distance and similarity measures for IFSs.Based on the transformation into the symmetric triangular fuzzy numbers, Zhang and Yu [16] presented a new similarity measure between IFSs.Similarly, Chen and Chang [2] presented a transformation-based similarity measure and adapted it to the pattern recognition problems.However, it has been shown to be also counterintuitive [8].
Inspired by these limitations, in this paper, we present a similarity measure for PFSs.The proposed measure is defined based on the amount and significance of information depicted in the PFSs.The rest of the paper is organized as follows.In Section II, basic concepts of PFSs are reviewed and some critical analyses of the existing methods for PFSs are conducted.In Section III, generalized knowledge measure, knowledge-based score function and similarity measure for PFSs with their axiomatic properties are introduced.
Section IV illustrates the advantages of the proposed generalized score function in differentiating and ranking the PFSs by some comparative numerical examples.Section V shows the effectiveness of the proposed similarity measure in applications for pattern recognition and medical diagnosis problems.Finally, Section VI provides concluding remarks.

A. Pythagorean Fuzzy Sets
In 2013, R.R. Yager [14] provided two basic representations for PFS by polar coordinates and membership degrees, respectively.In the following, we recall the general definition of PFS, which is similar to the definition of IFS.Definition 1. [1] A PFS  in a finite universe of discourse  = {  ,   , … ,   } is an object with the following form: where   (  ),   (  ) ∈ [, ] denote the membership degree and the non-membership degree of an element   to PFS , respectively, such that   ∈ , The hesitancy degree of   to PFS  is given by: For convenience, we denote a single-valued PFS as Pythagorean fuzzy number (PFN) by  = 〈  ,   〉.
Afterwards, on the basis of relationship between IFS and PFS, Zhang and Xu [18] defined some operation for PFS as follows: Definition 2. Let   = 〈   ,    〉 and   = 〈   ,    〉 be two PFNs and  > , then the operations on PFNs are defined in [18] as: The operating results were proved to be PFNs as well.
They also utilized a relation between two PFNs defined in [14] as: In order to compare the magnitude of PFNs, Zhang and Xu [18] defined the score function for PFNs applied as a score-based ranking method as follows: Definition 3. [18] Let   = 〈   ,    〉 and   = 〈   ,    〉 be two PFNs, the score function of PFN is defined as follows: Then, a comparison law for PFNs is introduced as: If (  ) < (  ) then   ≺   , if (  ) = (  ) then   ∼   .

Application
In this section, the proposed similarity measure is employed for the fault diagnosis of steam turbine generator unit under Pythagorean fuzzy environment.The vibration of the steam turbine generator unit suffers the influence of varying factors, such as mechanical load, vacuum degree, fluctuation of network load, the temperature of lubricant oil and defects of mechanical structure.Interaction effects of these factors result in the vibration of the generator unit.Ten fault types in rotating machines are established as failure patterns, i.e.  1 -unbalance,  2 -pneumatic force couple,  3 -offset center,  4 -oil-membrane oscillation,  5 -radial impact friction of rotor,  6 -symbiosis looseness,  7 -damage of antithrust bearing,  8 -surge,  9 -looseness of bearing block and  10 -non-uniform bearing stiffness.The knowledge of fault types and symptoms in the vibration frequencies of the turbine is adopted from [12] by transforming the vague values into the PFVs.The vibration frequency of turbine is divided into 9 different frequency ranges, in which the failure patterns are represented by PFVs, as shown in Table 1.

Table 1
The knowledge of system fault pattern [12] Frequency range (f-operating frequency) Suppose that there are two fault-testing samples A and B expressed in PFVs as shown in Table 2.

Table 2
The characteristic of the fault-testing samples [12] Frequency range (f-operating frequency) Our goal in the fault diagnosis analysis is to classify the fault-testing samples into one of the known fault patterns   , ( = 1,2, . .,10), adopting the proposed similarity measure Sim N (13).The calculation results of similarity measures between the fault-testing samples and the known fault patterns are summarized in rows of Table 3.According to the principle of maximum similarity measure, we can decide that the fault-testing sample A is most similar to the known fault pattern  7 -damage of antithrust bearing, consistent with the results obtained in [12]12.In the same manner, we derive that the fault-testing sample B is most similar to the known fault pattern  5 -radial impact friction of rotor, which is also consistent with the results obtained in [12].

Conclusions
The paper discusses some limitations of the existing methods for PFSs.Also, a novel similarity measure based on knowledge measure for PFSs has been proposed, which incorporates the knowledge conveyed by PFSs and significance related to positive and negative information to distinguish their meanings.Thanks to that, it can overcome the drawbacks of the existing methods.The superiority of the proposed method in dealing with uncertain information has been shown in numerical examples.The effectiveness and applicability of the proposed method have been demonstrated in applications of fault diagnosis of the turbine generator.The results obtained by the proposed methods were consistent with that of the existing methods, which though have shown to be failing in some other cases.The limitation of the paper is the lack of experimental verification by own reallife applications or statistical test.Thus, future work will be focused on these issues.

Fig. 1 .
Fig. 1.Example triangular membership and non-membership functions of the IFS