Entropy-Based Methods for Motor Fault Detection: A Review

In the signal analysis context, the entropy concept can characterize signal properties for detecting anomalies or non-representative behaviors in fiscal systems. In motor fault detection theory, entropy can measure disorder or uncertainty, aiding in detecting and classifying faults or abnormal operation conditions. This is especially relevant in industrial processes, where early motor fault detection can prevent progressive damage, operational interruptions, or potentially dangerous situations. The study of motor fault detection based on entropy theory holds significant academic relevance too, effectively bridging theoretical frameworks with industrial exigencies. As industrial sectors progress, applying entropy-based methodologies becomes indispensable for ensuring machinery integrity based on control and monitoring systems. This academic endeavor enhances the understanding of signal processing methodologies and accelerates progress in artificial intelligence and other modern knowledge areas. A wide variety of entropy-based methods have been employed for motor fault detection. This process involves assessing the complexity of measured signals from electrical motors, such as vibrations or stator currents, to form feature vectors. These vectors are then fed into artificial-intelligence-based classifiers to distinguish between healthy and faulty motor signals. This paper discusses some recent references to entropy methods and a summary of the most relevant results reported for fault detection over the last 10 years.


Introduction
Recently, the pursuit of more reliable and accurate techniques for motor fault detection has increased, driving the critical role that electric machines play in various modern industrial applications.Entropy-based methods have gained significant attention among the many emerging methodologies due to their unique ability to capture complex system behaviors and anomalies based on mathematical algorithms.
Entropy, which is a foundational concept that was introduced by Rudolf Julious Emanuel Clausius, has been used as a fundamental tool in signal analysis by assessing the variability and sparsity of signals in different knowledge areas.This pioneering work laid the groundwork for new studies about entropy forms, like information entropy [1], fuzzy entropy [2], and sample entropy [3], which have become useful tools in fault diagnosis methodologies.In recent studies, there has been a growing emphasis on the application of entropy-based methodologies for motor fault detection; for instance, in [4], a feature extraction approach based on entropy was undertaken, where this paper introduced the "weighted multi-scale fluctuation-based dispersion entropy (wtMFDE)" method.Designed for condition monitoring in planetary gearboxes (PGB), wtMFDE harnesses the intricacies of entropy to discern fault signatures from mixed noisy signals.This entropy-based technique seamlessly integrates with adaptive and non-adaptive signal processing methodologies, positioning it ahead of the previously established multi-scale fluctuation-based dispersion entropy (MSFDE) method.When evaluated alongside advanced classifiers, such as multilayer perceptron (MLP), the wtMFDE approach capitalizes on entropy's power, achiev-ing an unparalleled 100% classification accuracy for specific fault types, as exemplified by sun chipping.
In [5], a fault diagnosis method for rolling bearings leveraging entropy-based techniques is presented.Ensemble empirical mode decomposition (EEMD) initially dissects training samples, with dispersion entropy (DE) quantifying their features.Principal component analysis (PCA) further refines these features, and the Gath-Geva (GG) clustering method categorizes them.When tested against various data sets, including the Case Western Reserve University (CWRU) data set, the method demonstrated its robustness, particularly with DE's superior stability over other entropy measures and GG's efficacy in clear sample categorization.
In [6], a method to detect sparking faults in DC motors using stray flux signals is proposed.It employs spectral entropy for signal analysis and introduces a severity indicator based on Mel frequency cepstral coefficients.Evaluations under various motor conditions highlight the method's consistent effectiveness, positioning it as a promising tool for integrating into DC motor diagnosis systems.
While entropy-based techniques have enriched our understanding of rotatory machine dynamics, there remains a challenge in effectively capturing temporal details.To overcome these temporal limitations, authors have developed multi-scale and multi-modal techniques in order to obtain reliable results [7][8][9].

Shannon Entropy
The first concept of entropy was introduced by Shannon in order to calculate the irregularity and self-similarity of signals.The Shannon entropy H(x) of a random signal x with n possible outcomes is defined by where p(x i ) is the probability density function of the signal x i [10,11].Shannon entropy can be used to measure a time series's complexity.By definition, Shannon entropy should be a monotonic increasing function and a continuous function.Lastly, if the probability can be divided into the sum of individual values, so should the Shannon entropy.
Thanks to its characteristics, Shannon entropy is a popular method, not only for fault detection but also for other applications, such as for the analysis of biological signals [12], computational applications [13], and environmental data [14].

Reported Works That Used Shannon Entropy
The reported works that used Shannon entropy for fault detection are mostly devoted to analyzing vibration signals.Some of the most relevant works are listed in Table 1, where the methods, type of signals, type of faults, and accuracy of the classification are detailed.Notice that half of these works are proposed to detect bearing faults: inner race (IR), outer race (OR), and ball.

Approximate Entropy
Approximate entropy measures the probability of occurrence of a new pattern based on the observation of the embedding dimension m and the similarity coefficient r.ApEn is a scale-invariant indicator, given that it relies on the similarity coefficient, which is an equivalent of a standard deviation of a time series.ApEn can be defined as follows [10,18]: where ϕ m (r) is the mean value of the logarithm pattern mean count and r is the similarity coefficient; on the other hand, ϕ m (r) and ϕ (m+1) (r) can be calculated with the following expression: where c m i (r) can be defined as follows: Previous studies demonstrated the advantages of the ApEn, such as its insensitivity to inference and noise, its suitability for random and certain signals, and its stable estimation without requiring large amounts of data.

Reported Works That Used ApEn
ApEn emerged as an improvement of Shannon entropy, and its use in the fault detection area has been devoted mainly to analyzing vibration signals.In Table 2, some of the relevant works that used ApEn are listed.ApEn-improved methods, like refined composite multi-scale approximate entropy (RCMSAE), are commonly employed together with methods like empirical mode decomposition (EMD) and probabilistic neural network (PNN).

Permutation Entropy
Permutation entropy (PE) considers a signal's non-linear behavior and describes the time series's complexity by making a phase space reconstruction.PE only requires the order of the amplitude of the signal.In this regard, this type of entropy has a faster calculation time than others.PE can be expressed in terms of the relative frequency p(π) for each permutation π as follows [10,29]: PE is an adequate indicator of the complexity of signals from nonlinear processes; furthermore, PE's advantages have been highlighted in other works, such as its high calculation efficiency, its robust ability against noise, and its good complexity estimation [10].
Reported Works That Used PE Permutation-entropy-based methods for fault detection are listed in Table 3.Notice that all of these works are devoted to the detection of bearing faults by using vibration signals, as is common in most of the works that use PE [35][36][37][38][39][40].

Sample Entropy
Sample entropy (SE) measures the irregularity of a signal independent of the similarity coefficient r and the embedding dimension m.Consider a signal S of data length N expressed by S = {x 1 , x 2 , ..., x N }.A pattern is formed by m sequential points of the signal S; for example, X i = [x i , x i+1 , ..., x i+m+1 ] would represent the ith pattern.Hence, the pattern space X is defined as follows: SE can be calculated as follows: where B m (r) represents the mean value of the pattern mean count; B m (r) and B m+ (r) are calculated according to the following expression: where is the Heaviside function.In the context of SE, the suggestion of use for r is to select a value of 0.2 times the standard deviation of the data set [54].

Reported Works That Used SE
In the following Table 4, a summary of some of the most relevant works that utilized SE and improved methods, such as generalized refined composite multi-scale sample entropy (GRCMSSE) for motor fault detection, is presented.Most of them aim to detect bearing faults, but two of the cited works propose the detection of gear and impeller faults.

Fuzzy Entropy
Fuzzy entropy (FE) emerged as an improvement of the sample entropy because FE uses a Gaussian function for measuring the similarity between two time series instead of the Heaviside function that SE uses.Given a signal u(i), i = 1, 2, ..., N of N samples, a vector set {X m i , i = 1, 2, ..., N − m + 1} is formed.Each vector has m sequential elements from the signal u(i) in the form of where u o (i) represents the average of the vector X m i .
Then, the similarity FE for a time series is defined as follows: where d m ij is the distance between X m i and X m j , r represents the similarity tolerance, and µ(d m ij , n, r) is a fuzzy function.
On the other hand, the function φ m (n, r) is expressed as Finally, FE can be defined as follows [10]: FE considers the ambiguous uncertainties from the highly irregular time series, making it insensitive to background noise.

Reported Works That Used FE
In Table 5, a summary of some of the most relevant works that utilized FE for motor fault detection is presented.Most of them aim to detect bearing faults, but two of the cited works propose the detection of gear and impeller faults.Some of the improved methods based on FE are multi-scale fuzzy entropy (MSFE), refined composite multi-scale fuzzy entropy (RCMSFE), generalized composite multiscale fuzzy entropy (GCMSFE), multi-scale refined composite standard deviation fuzzy entropy (MSRCSDFE), and multivariable multi-scale fuzzy distribution entropy (MMSFDE).Although these methods extend the scope of FE by adding, for example, the multi-scale or the generalized analysis, all of them are still driven by FE principles [73][74][75][76][77][78][79][80][81][82][83].

Energy Entropy
Energy entropy (EE) estimates a signal's complexity based on its intrinsic mode functions (IMFs).Its calculation starts with the energy of the ith IMF as follows: where m is the length of the IMF.Then, the total energy of the n IMFs is given by Finally, the energy entropy H en of the signal is calculated based on the following expression: where p i = E i /E represents the percentage of the ith IMF relative to the total energy entropy [10].
The energy entropy provides very good results when analyzing non-stationary and nonlinear complex signals; for example, if a fault in the motor provokes a change in the signal's frequency, the energy distribution will change.Hence, energy entropy can be used to effectively portray the signal's characteristics [89].
Other fields besides fault detection where the EE has been applied are milling chatter detection [90], computational chemistry [91], and thermomechanics applications [92].
Reported Works That Used EE Some of the latest relevant works that used energy entropy for fault detection are listed in Table 6.Unlike the previously mentioned methods, by using EE, more types of faults have been detected, such as misalignment, imbalance, and bearing faults.It is also important to recall that one of these works relied on current signals for the analysis [93,94].Improved methods for EE are also proposed for fault detection, such as characteristic frequency band energy entropy (CFBEE) and improved energy entropy (IEE).

Dispersion Entropy
The dispersion entropy (DE) of a signal x of n samples can be calculated with the following steps [97,98]: First, the signal x is normalized between 0 and 1.To do so, a sigmoid function is usually employed for this mapping.Some works have reported using normal cumulative distribution functions (NCDF) for this step [97,98].Hence, the time series y is obtained from the NCDF of the signal x, which is defined as follows: where σ represents the standard deviation and µ is the mean of the signal x.
The second step consists of mapping the time series y to c classes by multiplying y i by c, then adding 0.5 and rounding to the nearest integer, as follows: where z c i represents the ith term of the classified time series z c .In the third step, the time series z m,c j is constructed based on the embedding dimension m and the time delay d: Then, z m,c j is mapped into a dispersion pattern π v0v1...v m−1 : Here, the number of feasible dispersion patterns is c m given that each z m,c j is conformed by m elements, which can be an integer from to c.
The fourth step corresponds to the calculation of the relative frequency of each dispersion pattern π v0v1...v m−1 , which is given by Finally, the DE is calculated as follows: where m represents the embedding dimension, c is the number of classes, and d is the time delay.Some works prefer to express the DE in its normalized form, which is given by The advantages of DE have been used for other applications, such as the analysis of biomedical signals [70] and image processing [99].

Reported Works That Used DE
In Table 7, relevant works that used DE for motor fault detection are listed.Notice that DE has become popular, especially during the last few years; authors rely on this method due to its high stability.

Multi-Scale Entropy
The multi-scale version of any type of entropy method consists of the calculation of the entropy at different scales.To this end, a coarse-grained data sequence y (s) j should be obtained by a coarse-grained process of the original signal x.Then, y (s) j can be expressed as follows [64]: where s represents a scale factor.Therefore, the signal x is transformed into a coarse grain sequence of length N/s.The multi-scale entropy (MSE) accuracy is constrained by the single-scale method; however, it is usually preferred over the one-scale analysis because it provides more information despite the increase in the calculation time.
The type of applications where the MSE can be used are as vast as the applications of each single-scale method, such as the analysis of time series [106,107]; biological signals, such as heartbeats and encephalographics [108][109][110]; image processing [111]; and hydrologic applications [112].There are some interesting works on improvements around the MSE, as presented in [113], where the authors successfully diagnosed gearbox and milling tool faults.The method utilizes a novel technique that combines MPE with contrastive learning (LE), yielding results that improve the accuracy of traditional entropy-based methods.
Finally, in Table 8, a summary of each method's advantages and disadvantages is presented to provide a wider panorama of its characteristics.

Practical Example: Applied Entropy Methods for Broken Bar Detection
To provide an example of the use of different entropy methods and their effects on the classification accuracy, an implementation of three of the methods presented in this paper was conducted: Shannon entropy, approximate entropy, and energy entropy.These three methods were applied to the same signals from a motor with a healthy bar (HB) and with a broken bar (BB), without any preprocessing.A set of 50 current signals in the steady state were analyzed, as shown in Figure 1.Further explanation about the experimental setup to acquire these signals can be found in [114], where the authors performed an early broken bar detection.As can be observed from Figure 1, the signals were quite similar; therefore, an entropy method could be helpful to discern between the two conditions of the motor.Results are displayed in Figure 2 comparing the entropy for the two conditions of the motor.Notice that the use of entropy allowed for a separability of data in a similar way that other traditional methods could provide, such as motor current signature analysis.According to the nature of the signal and the aim of the analysis, a certain method of entropy could be more useful than others.For example, energy entropy could be more suitable for this application since it only depends on the intrinsic characteristics of the signal.It is worth noticing from Figure 2 that the separability of data using this type of entropy is better than using Shannon or approximate entropy.Actually, approximate entropy is commonly employed for vibration signals, which are usually more irregular than current signals.Also, according to the characteristics of the phenomenon, certain entropy methods could be discarded; for example, when analyzing a high-frequency phenomenon, dispersion entropy is not adequate.The selection of the type of entropy is also dependent on its application.A signal with higher separability, such as the comparison between a healthy motor and a motor with a medium level of damage, could be successfully classified with more straightforward methods, such as Shannon entropy, or a faster method, such as permutation entropy.But regarding a more complex analysis, a multi-scale analysis could be necessary.

The Role of Entropy in the Fault Diagnosis of Electromechanical Systems: Challenges and Advances
As a statistical measure, entropy is capable of quantifying the complexity of signals, which is closely related to the functional status of an electromechanical system.Conse-quently, entropy emerges as a promising non-parametric tool to extract characteristics from a system.Recently, several studies applied entropy indices for fault diagnosis, detection, and prediction in electric machines.Some of them employed more than one entropy index to obtain a multi-modal analysis.Despite the existence of several entropy-based algorithms for fault detection, most of them are based on Shannon entropy for random or deterministic behavior detection in signals from electric machines.The different forms of entropy employed for fault detection are usually based on the assessment of aleatory and complexity metrics of the signals, and any change in these indices could be related to important changes in the system behavior.
Depending on the nature of the signals, a specific index may be more useful than other; for this reason, it is necessary to apply different entropy metrics in combination with different classification methods, with the aim to cover all the possible faults.The entropy indices described in Sections 2.1-2.7 are commonly used for fault detection.Unfortunately, the classical models of these entropy-based indices are only useful for analyzing signals at one level (monoscale analysis), which does not provide the complete feature extraction of the signal.
To overcome the limitations of a monoscale analysis, multiscale-entropy-based methods were proposed, such as the method presented in Section 2.8.Despite their advantages, there exist some problems with this kind of method, like indeterminacy problems and instability for short signals, in addition to its low sensibility for high-frequency systems.Based on these, the main challenge and the current research status on entropy-based methods is a multiresolution analysis, which is needed to obtain indices that entirely describe the dynamics of the signal under study based on all of its oscillatory components [115][116][117][118].In general, new entropy-based methods aim to provide information about the signal's state at various levels of oscillation, and thus, better extract the characteristics of the signals under study in order to detect a fault.It is important to mention that the actual trend is the combination of entropy indices with artificial intelligent methods to improve the accuracy of the control systems and fault classification.Another important aspect about entropy based methods is the computational complexity, which allows for online hardware implementations.
However, the advantages of entropy-based methods are evident, in contrast with other methodologies, due the capability of the entropy indices to give information about the dynamics at different abstraction levels of the electromechanic systems.Some of the information aspects provided by entropy indices are systems complexity, stability and regularity, changes detection, resilience to disturbances, hidden patterns and structures, anomalies detection, future events prediction, and model validation, among others.In contrast to other methods, the calculation of entropy indices does not require a large amount of data, nor does it depend on the model and parameters of electric machines.

ShanEn
Allows for the assessment of the quantity of information in a signal.It is the basis of the following methods.
Its value only depends on the elements with probability ̸ = 0; therefore, some elements could be neglected.

ApEn
Uncertainty estimation regarding future observations based on past observations.Dependent on the selection of the hyperparameters.Dependent on the length of the signal.Self-similarity feature [87].

SE
Better performance and less sensitivity to data length compared with ApEn Dependent on selecting the hyperparameters.Similarity criteria dependent on the Heaviside function [50].

FuzzyEn
Better consistency and less dependent on the signal length compared with SE.Reflects the complexity and self-similarity features of a signal in a better way than SE and ApEn.
Dependent on the selection of parameters.
PerEn High computational speed.Suitable for stationary and non-stationary signals.
Low discrimination capacity given that it does not consider amplitude values.

DE
Faster calculation speed than PerEn.High stability.
Only analyzes the low-frequency part of the signal.

MSE Analyzes the signal in multiple scales
Efficiency dependent on the single-scale entropy method.Slower method given the entropy calculation within a range of scales.

Future Trends
Over the years, the use of entropy methods has evolved, with the aim to obtain more accurate and robust results.To this end, improved methods were proposed, such as generalized, multi-scale, composite, hierarchical, and multivariable entropy methods.Some works also proposed combined methods in order to overcome the drawbacks of using only one type of entropy.But most importantly, entropy methods are usually employed together with signal processing techniques, such as PCA, EMD, and EWT.Artificial-intelligence-based classifications are also commonly used with entropy methods to achieve good classification accuracies when more than one type of fault is being analyzed.
As a summary, some of the trends observed during the elaboration of this work are listed below:

•
Most of the entropy methods are applied to vibration signals.This can be attributed to the nature of the signal and the straightforward acquisition.The presence of a fault in a motor usually increases the complexity of the vibration signal, given that it would introduce abnormal components in the spectrum.In this regard, it is expected that vibration analysis remains the preferred type of signal for entropy-based fault detection techniques.It is important to mention that aspects such as algorithmic optimization and hardware implementation are fundamental areas of study.These areas aim to adapt fault detection technology to the emerging trends in electrical systems, particularly in line with the philosophy of smart systems that embrace trends like Industry 4.0 and the Internet of things.

Conclusions
Different entropy methods were proposed over the years, with some of them aiming to improve the performance of the older ones.In general, the entropy methods are used for extracting characteristics of the motor's signal to provide a classification that is commonly based on artificial intelligence.
Vibration analysis stands out as the preferred signal type among all the entropy methods reported in this work.In the future, it would be valuable for the state of the art to propose analysis based on other physical variables, such as the current or flux.
In the same regard, the analysis of a wider range of faults would be valuable given that over the years; the focus has been maintained on bearing fault detection.
Fuzzy entropy and dispersion entropy are some of the most reliable methods for entropy-based fault detection thanks to their high stability and reliability, and they are not dependent on the selection of parameters, like the sample entropy and approximate entropy.Permutation entropy is another popular method, and it has shown very good classification accuracies when applied with a classification method like SVM or ELM.
Multi-scale entropy has been preferred in recent years given that it provides more accurate results than a one-scale entropy analysis.Although selecting a multi-scale analysis could have the drawback of a slower calculation, usually, this is not relevant given that the progression of a fault, such as a bearing fault, is rather slow compared with the computation times of the method.
As machinery includes more sophisticated technologies and the demand for uninterrupted services by society increases, it is imperative to find new efficient and accurate mechanisms for fault detection and classification.Entropy-based methods are poised to play a pivotal role in the next generation of monitoring and control systems in conjunction with machine learning methods due to their capability to detect changes in dynamic systems over time.

Figure 1 .
Figure 1.Current signals from the two conditions of a motor: healthy and one broken bar.

Figure 2 .
Figure 2. Comparison of three different entropy methods for a practical case of broken bar fault detection.

Table 1 .
Motor fault detection using Shannon entropy.

Table 3 .
Motor fault detection using PE.

Table 4 .
Motor fault detection using SE.

Table 5 .
Motor fault detection using FE.

Table 6 .
Motor fault detection using EE.

Table 7 .
Motor fault detection using DE.

Table 8 .
Advantages and disadvantages of different entropy methods.
• Bearing fault detection is the type of fault that is mostly covered in entropy-based works.Other faults analyzed with entropy methods are gearbox faults, misalignment, and broken rotor bars, among other less common faults.However, these types of fault represent less than 10% of the work compared with those that analyze bearing faults.•PE and FE are the most popular methods for motor fault detection.During the last few years, DE has also gained attention.Therefore, it is expected that these would remain the preferred methods, along with their variations, such as composite, weighted, refined, generalized, and multi-variable approaches.
• Hardware implementation of entropy-based methodologies for online monitoring.