# Fault Diagnosis Method for Wind Power Equipment Based on Hidden Markov Model

Authors: Qun Zhao
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

The aging of mechanical equipment brings a lot of inconvenience to its application. Therefore, research on fault diagnosis technology of mechanical equipment is of great significance for maintaining equipment safety, improving production efficiency and reliability. Bearings and gears are one of the key components of mechanical equipment, and their working conditions seriously affect the changes in equipment performance. Therefore, the fault diagnosis and performance degradation assessment of bearings and gears have always been the research focus of equipment fault diagnosis. This paper proposes a multichannel information fusion method based on the coupled hidden Markov model, discusses the application of the coupled hidden Markov model in bearing fault diagnosis and performance degradation assessment, and studies the multichannel information fusion method based on the coupled hidden Markov model. Channel monitoring data performance degradation evaluation modeling and performance index calculation method are used, and the calculation method of adaptive alarm limit is given by using the performance index. Finally, the natural failure test data and accelerated fatigue test data of gears and rolling bearings are used to analyze and verify the effectiveness of the coupled hidden Markov model for evaluating the performance degradation of complete and incomplete data. The results prove that the selected performance indicators can be quantified, reflecting the degree of bearing performance degradation.


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## Body

## 1. Introduction

With the progress of science and technology and the modernization of industrial production, mechanical equipment, on the one hand, continues to have complex, high-speed, efficient, light, miniature, or large direction of development, but on the other hand, they face more demanding work and operation environment, so the dynamic problem is increasingly prominent. A variety of large rotating machinery equipment is the key equipment in many industrial productions, once the failure will inevitably lead to serious industrial accidents and even casualties, resulting in bad social impact [1]. With the continuous development and progress of mechanical equipment, it will challenge the operation in a more demanding working environment, and the problems in its dynamic process are attracting people’s attention. Many key parts in large rotating equipment (such as gear wheel and bearing) are vulnerable to damage and failure due to long-term continuous operation under high load and high speed. Sudden failure of mechanical equipment leads to increased maintenance burden, which will seriously affect the production cost and production efficiency of enterprises. The result will not only cause economic losses of enterprises but also may be accompanied by casualties, resulting in adverse social impacts [2].Trend prediction analysis is to estimate the development trend of equipment failure and predict the advantages and disadvantages of equipment based on vibration signal detection, which is an important link of equipment monitoring and fault diagnosis [1, 2]. The use of fault prediction analysis technology, more targeted and rational, greatly improves the unit operation cycle and efficiency; at present, there are many machine learning methods about fan equipment monitoring and diagnosis, but there are some limitations in use, for example, (1) neural network diagnoses the result of matching between information and template library at a certain moment, ignoring the context [3]; (2) the principle of empirical risk minimization is adopted, which requires a large number of fault training samples, and training samples are difficult to obtain [4].In order to avoid production accidents and ensure the safe and effective operation of rotating machinery, fault diagnosis technology arises at the historic moment. The research on gearbox fault diagnosis technology can be divided into the following three aspects: (1) gearbox fault diagnosis model research: the study of vibration diagnosis model can reveal the rule and mechanism of gearbox faults and find out the relationship between signal characteristics and faults. (2) Monitoring of operating state: the purpose of monitoring the gear box is to understand and master the running state of the gear box. On the premise of mastering the relevant parameters and operating conditions of the gearbox, a large amount of historical data of gearbox work under the same conditions are collected. When the gear box is abnormal, alarm information is sent to the operator so that the fault can be timely responded, and the status data of the gear box is analyzed by basic data analysis means, so as to prepare for the next fault diagnosis. (3) Fault diagnosis: as the core part of the gearbox fault diagnosis system, the purpose is to diagnose the faults that have occurred in the gearbox, find out the location, degree, and type of the faults, and timely replace the faulty parts or adjust the running state.The development of fault diagnosis technology for mechanical equipment greatly improves the reliability of the system and helps to make and perfect the maintenance plan. In view of the important role of fault diagnosis technology in production practice, more and more scientific research and technical personnel at home and abroad began to invest in this aspect of research; advanced technology and methods can be applied to the fault diagnosis of rotating machinery, promoting the continuous development of this new technical field. In gearbox fault diagnosis technology research object structure complex, the basic part of the study involves automation, machinery, mathematics, computer and other disciplines, the gearbox into fault diagnosis comprehensive use of the knowledge of these disciplines, engineering application background is very strong, with significant characteristics of the times. In order to further improve the accuracy and reliability of fault diagnosis, the current main problem is still as follows: in the complex operating conditions, how to accurately obtain the fault information of the gearbox, further reduce the failure rate, for enterprises and the public better service.The fault diagnosis method based on HMM is to observe, model, and evaluate the status of mechanical equipment in a dynamic environment and find the early signs of mechanical equipment failure through the online monitoring system, so as to take measures to maintain or replace as soon as possible and avoid greater losses. For example, in the diagnosis of the degradation process of rotating machinery equipment, the information of each stage of the whole process is first extracted, classified, and modeled. The observation information of a certain state can be analyzed and compared with the whole process, so as to finally determine the state of the equipment.Fault diagnosis of mechanical equipment is to use signal processing and analysis technology to find out the characteristic parameters related to the fault and use these characteristic parameters to distinguish the real-time technical status of the equipment. Here, two aspects of the problem are involved: one is the use of signal processing technology for feature extraction; the second is to use pattern recognition technology for fault diagnosis.As a time-frequency analysis method, wavelet analysis has been widely used in the field of fault diagnosis. Its application can be divided into two aspects: first, wavelet transform is used for filtering and envelope detection. The disadvantage of this method is that the characteristic frequency of the fault must be understood in advance, that is, the corresponding details of the equipment must be mastered in advance. At the same time, this method is also constrained by the signal-to-noise ratio, which undoubtedly limits its practical application. Another method is to determine the singularities of fault signals by using wavelet localization. In this method, it is assumed that the fault signal will produce certain mutation information in advance, and the fault detection is carried out by judging whether the signal has singularity. The disadvantage of this method is that it is impossible to judge whether the singularities belong to the signal itself or are caused by faults, so it is easy to lead to misjudgment. In order to solve the above problems, Zhengyou et al. proposed to use wavelet multiscale energy spectrum as a feature for fault diagnosis, which has been successfully applied in power transient signal classification and recognition. In order to further reduce the number of features, Zhang et al. proposed a feature method based on wavelet energy spectrum entropy and applied it to power system fault diagnosis. However, these two methods are based on the assumption that the coefficients of wavelet decomposition conform to Gaussian distribution and are independent of each other. The theoretical analysis and experiment in this paper show that wavelet decomposition coefficients are non-Gaussian and correlated, so it is unscientific to classify wavelet energy spectrum and energy spectrum entropy as features, and the conclusions obtained lack of credibility.As a statistical signal processing model, hidden Markov model in wavelet domain (WHMM) can effectively describe the non-Gaussian and nonstationary statistical characteristics of the wavelet coefficients of real signals. It has been preliminarily applied in image perception and recognition to compensate for the wavelet energy spectrum and energy spectrum. In this paper, a fault diagnosis method based on hidden Markov model parameters of wavelet decomposition coefficients of fault signals is proposed. The proposed method is compared with the method based on wavelet energy spectrum and energy spectrum entropy. The results show that the proposed method has higher accurate diagnosis rate than the method based on wavelet energy spectrum and energy spectrum entropy. In addition, the effects of wavelet basis, sliding window, and classifier on the diagnostic performance of the method are analyzed based on the experiments.Nuclear facilities have complex structure, strong continuity of production, and high safety requirements. Under the harsh environment of high temperature and high irradiation, any nuclear accident caused by failure will cause serious economic losses and significant social consequences. Therefore, the fan equipment is complex, with accumulated data and few fault samples and other special points. While the hidden Markov model is beneficial for processing continuous dynamic signals, the supported vector machine can separate different pattern samples with the maximum distance through the limited sample information and better express the difference between pattern categories.Based on the HMM for wind turbine gearbox fault diagnosis, it is based on the gear box under the same parameters and running conditions, in various stages of the gear box of whole life cycle to extract the state data, through collecting a large number of historical data, training out of the different stages of running state model, and then identifying the observation sequence in the model, so as to determine its status. In the process of gearbox system degradation, the matching degree of the extracted test data to the normal model decreases continuously, and the matching degree of the test data at each stage is the highest only with the model trained by the data at the corresponding stage. Therefore, the remaining service life of the gear box can be predicted according to a certain linear relationship.The method used in this paper verifies the effectiveness of gearbox fault diagnosis using HMM model and verifies the accuracy of fault diagnosis based on HMM by using multiple groups of test data. The results show that the powerful timing processing ability of HMM has important theoretical significance and practical value for fault diagnosis and prediction of wind turbine gearbox system.


## 2. Related Work

With the progress of computer technology, signal processing algorithm, and artificial intelligence, the fault diagnosis of equipment is becoming more automatic and intelligent. A large number of researchers have carried out active exploration in signal acquisition and storage, fault symptom and fault mechanism correlation, feature extraction and selection, fault identification methods, equipment performance evaluation and prediction, and other aspects [5]. As early as the 1960s, the American National Aeronautics and Space Administration (NASA) had set up a mechanical failure prevention group [6, 7]. In the UK, the Mechanical Health Monitoring Center (MHMC) was also established in the 1970s under the leadership of R.A. Collacott [8]. Japan’s University of Tokyo, Tokyo Institute of Technology, Nippon Steel, Mitsubishi Heavy Industries and other long-term fault monitoring and diagnosis of steel, chemical, vehicle, and ship application research. The domestic research and application of fault diagnosis technology that started from the 1980s have carried out the research on equipment fault diagnosis technology. In addition, abundant achievements have been made in the fields of wavelet analysis theory, research on rotating equipment failure mechanism and symptoms, and development and application of equipment online monitoring and diagnosis system [9, 10].Because the data directly collected contains environmental noise and information irrelevant to the analysis target, in order to reduce the error and improve the diagnostic accuracy, it is necessary to preprocess the monitoring data and extract features. Signal pretreatment mainly includes normalized processing of data and removal of “wild points” in data [11]. Feature extraction is to use a variety of linear or nonlinear transformation to map the original data into different representations, so as to remove the interference of noise and explain the change of equipment state from different angles. The diversity of mapping methods determines that features are usually multidimensional. In order to give consideration to computing efficiency and full description of equipment status, features can be reduced before actual modeling and analysis [12].A variety of observation signals, including vibration and sound signals, ultrasonic signals, current signals, and optical signals, can be obtained through the state monitoring of the equipment, which is used for fault diagnosis [13, 14]. Among them, vibration and sound signals are widely used in the field of equipment fault diagnosis due to the convenience of acquisition and the completeness of theory. This paper also takes such signals as the analysis object [15].The main purpose of mechanical equipment fault diagnosis is to use the data obtained from state monitoring and extracted features through modular recognition (machine learning) method to identify and classify the state of equipment, diagnosis conclusions; the whole process is completed by machine or computer, without expert analysis and interpretation. How to establish the model and identify the working state of the equipment through the model is always the focus of the research in the field of fault diagnosis.ANN is a kind of information processing paradigm, whose idea and theoretical basis are derived from the study of the structure of neural network and information processing. By simulating and abstracting biological neurons, Rosenblatt and Widrow give ANN model structure and computational reasoning method, so as to realize the training, learning, and recognition of samples. ANN obtains a complex layer structure by connecting simple processing units (neurons), thus establishing a nonlinear mapping between input and output space. Because of its nonlinear, adaptive, self-learning, fault tolerance, evidence response ability, and parallel processing ability, ANN has been a research hotspot and has been widely used. In the field of device state monitoring and fault diagnosis, there are many types of ANN for signal separation, feature extraction, fault classification, and fault detection. Feedforward neural network (FFNN) has been widely used in mechanical fault diagnosis. Based on FFNN, the back propagation neural network (BPNN) with feedforward training algorithm is the most commonly used, but BPNN is difficult to determine the network structure and the number of nodes at each layer during modeling. And the disadvantage of slow convergence speed in model training limits its application.Ge et al. proposed an AR-HMM diagnosis method combining autoregressive (AR) models with HMM. In this method, AR models were used to model monitoring signals at different stages, and residual errors of AR models were used as features of input HMM. Thus, the fault diagnosis in the stamping process is realized, and the accuracy rate of this method reaches about 90% in the actual test [16]. Ocak [17] and Purushotham [18], respectively, used HMM to diagnose bearing faults and proved the effectiveness of the method through bearing tests. Li et al. combined independent component analysis (ICA) and factorial HMM (FHMM) to diagnose possible faults in the process of speed up and down of rotating machinery. Multichannel monitoring signals are fused and redundant removed by ICA, and the fused features are used as the input of FHMM. The results show that ICA-FHMM has higher diagnostic accuracy than ICA-HMM [19]. Dong [20] used hidden semi-Markov model (HSMM) with time component to diagnose the state of the hydraulic pump and estimate the remaining life of the pump. The results show that the diagnostic accuracy of HSMM is 29.3% higher than that of traditional HMM, and the existence of time structure also enables HSMM to be used to estimate pump life. Xiao et al. discussed the application of coupled hidden Markov model (coupled HMM) in multichannel information fusion and fault diagnosis, and the results proved that coupled HMM has better accuracy than single-channel data. In addition, there are other machine learning methods such as fuzzy logic, decision tree, expert system, and Gaussian mixture model.HMM is also widely used to evaluate device performance due to its rigorous mathematical reasoning structure. Geramifard introduced a segmented HMM with continuous output to predict the degree of tool damage, and the comparison with the multilayer sensory neural network and Elman network shows that the HMM has higher prediction precision [21].At present in our country, there are also a lot of scientific research workers in the study of gearbox fault diagnosis technology; in the field of industry in our country, gearbox fault diagnosis system and some automation-based equipment online and offline detection system have been put into application. At present, many researchers have been engaged in the fault diagnosis research of wind turbine gearbox in China; among which the most popular and mature fault diagnosis method is vibration measurement method. For example, Tang and others made a preliminary diagnosis of the gear operating state by analyzing the time domain statistical characteristics of the gearbox vibration signal and further confirmed the diagnosis results by using fast Fourier transform (FFT) and power spectrum analysis [22]. For example, Feng discussed the application of HMM in the fault diagnosis of rotating machinery, Li discussed spectrum analysis in the application of gearbox fault diagnosis, and Ocak and Purushotham applied HMM to gearbox fault diagnosis. Teng, Xiao, and Zhu applied the continuous hidden Markov model (CHMM) to state recognition and performance degradation evaluation of different objects, respectively. In addition to time domain characteristics, frequency domain characteristics are also commonly used to judge the running state of gears. Analyzing the frequency of fault signals can accurately find the fault location, thus improving the accuracy of fault diagnosis [23]. The time-frequency analysis method developed later is a fault diagnosis method combining time domain and frequency domain characteristics, which is suitable for processing nonstationary vibration signals [24]. The commonly used time-frequency analysis methods include wavelet analysis, short-time Fourier transform, and empirical mode decomposition. Later, some researchers put forward the temperature measurement method, which is based on the temperature change of parts to identify whether the operation state is abnormal. Its advantage is that the measurement is convenient and the operation is simple. In the control system of the current wind turbine, the temperature has become an important standard to detect the health status of the gearbox. However, temperature measurement method, single measurement factors, applicable structure also has certain limitations. At present, vibration measurement method is still the main method used in domestic equipment fault detection, which has been widely used in the fault diagnosis of wind turbine gearbox [25].


## 3. Basic Theory and Algorithm of HMM

### 3.1. Working Principles of HMM

Hidden Markov model (HMM) describes the observed value sequence with statistical model and rigorous mathematical structure, which can express the behavior characteristics of the entire observed value sequence more completely.HMM’s dual random structure is very suitable for modeling and describing real systems. In the field of engineering signal processing and fault diagnosis, the state of the observed object cannot be obtained directly. For example, in the bearing fault diagnosis, because the bearing is running at high speed, its state is difficult to be directly observed, that is, the state of the bearing is “hidden.” However, through the state monitoring system, bearing vibration signals that change with time can be observed. This characteristic is very consistent with the structure of HMM, so HMM is very suitable for engineering to estimate the actual state of the system by using observed signals.Secondly, HMM has a rich mathematical theoretical basis, so it can describe the research object more clearly and conveniently. Traditional intelligent diagnosis methods such as expert system, neural network, and fuzzy diagnosis have their own advantages and disadvantages [175]. For example, in expert system, the complex inference algorithm of each system has to be completed by experienced engineers. In a neural network system, all possible fault modes of the system must be trained, and when new fault types are added, the system will have to be retrained. In addition, how to choose the appropriate membership function for the system is also a difficult problem to be solved in fuzzy diagnosis. Compared with these models, HMM does not need to determine complex inference algorithms in advance. If a new fault class needs to be added, it only needs to be trained and added to the fault library, rather than retraining the entire system. And HMM can also analyze and describe the variable-length system, with better robustness, so it is more suitable for modeling and analysis of the actual system.


### 3.2. Markov Chains

Markov process is a kind of random process without aftereffect proposed by A. A. Markov, in which the “future” state of processXt is only related to the current state, but has nothing to do with the “past” state under the condition that the “present” state of the random process is known. Its mathematical definition is as follows.We know the random processXt,t∈T, and for any n,n=1,2,⋯,n∈R, we have t1<t2<⋯<tn, the probability PXt1=S1,⋯,Xtn−1=Sn−1>0, which satisfies the condition.
(1)PXtn≤SnXtn−1=Sn−1,Xtn−2=Sn−2,⋯,Xt1=S1=PXtn≤SnXtn−1=Sn−1.Then,Xt is called a Markov process. The random sequence Sn−1,Sn−2,⋯,S1 is called Markov sequence, and the conditional distribution function Xt,t∈T is often called transition probability distribution.When the state space and time parameters of a Markov process are discrete, it can also be called a Markov chain. The mathematical definition of Markov chain is as follows: the possible state of a random processXt at time t must be one of N different states S1,S2,⋯,SN, and the state transition of a random process only occurs at the enumerable moments t1,t2,⋯,tn. If the probability that the random process Xt transforms into any state Sj at tm+k is only related to the state Si at tm moment and has nothing to do with the state before tm moment, then the probability that the corresponding state of the random process may occur between these two moments is
(2)PXtm+k≤SjXtm=Si,Xtm−1=Sp,⋯,Xt1=Sq=PXtm+k=SjXtm=Si.This probability is also known as the state transition probability, i.e.,(3)Pijtktm=Pijtm,tm+k=PXtm+k≤SjXtm=Si.Equation (3) illustrates the probability that the state of Markov chain appearing at tm moment is Si and that the state of Markov chain appearing at tm+k moment is Sj. In addition, when tk=1, the one-step transition probability aij can be obtained.
(4)aij=Pij1=Pijtm,tm+1=PijXtm+1=SjXtm=Si.For all possible transition probabilities betweenN states, it can be denoted as probability transition matrix A, so
(5)A=aij=a11a12⋯a1Na21a22⋯a2N⋯⋯⋯⋯aN1aN2⋯aNN,in whichaij satisfies the following attributes:
(6)aij≥0,∑j=1Naij=1.Equation (6) shows that aij is nonnegative and the sum of probabilities of all possible states at any time is 1.In whichtk=n, the transition probability of n steps is
(7)aijn=Pijn=Pijtm,tm+n=PijXtm+n=SjXtm=Si.The correspondingn step transition probability is
(8)An=aijn=a11na12n⋯a1Nna21na22n⋯a2Nn⋯⋯⋯⋯aN1naN2n⋯aNNn.Similarly,aijn has the following properties:
(9)aijn≥0,∑j=1Naijn=1.According to the Chapman-Kolmogorov equation,m+R step transfer probability can be expressed by m step and R step transfer probability, namely,
(10)Pijm+rn=∑kPikmnPkjrn+m,1≤n≤N.Markov chains also have some characteristics:(1)
Periodicity: a Markov chain is aperiodic if its state is triggered and immediately returns to its current state or if there are states in the chain that are connected to states with self-loops(2)
Interoperability: if the probability distribution of Markov chainXn satisfies Pijn≥0, it is said that the Markov chain can reach Sj from state Si. If Pjin≥0 also exists, it is said that the two states of the Markov chain are intercommunicating. The interoperability of Markov chains guarantees the reachability between different states(3)
Ergodicity: if a Markov chain has a limit for all statesSi and Sj independent of state Si


## 3.1. Working Principles of HMM

Hidden Markov model (HMM) describes the observed value sequence with statistical model and rigorous mathematical structure, which can express the behavior characteristics of the entire observed value sequence more completely.HMM’s dual random structure is very suitable for modeling and describing real systems. In the field of engineering signal processing and fault diagnosis, the state of the observed object cannot be obtained directly. For example, in the bearing fault diagnosis, because the bearing is running at high speed, its state is difficult to be directly observed, that is, the state of the bearing is “hidden.” However, through the state monitoring system, bearing vibration signals that change with time can be observed. This characteristic is very consistent with the structure of HMM, so HMM is very suitable for engineering to estimate the actual state of the system by using observed signals.Secondly, HMM has a rich mathematical theoretical basis, so it can describe the research object more clearly and conveniently. Traditional intelligent diagnosis methods such as expert system, neural network, and fuzzy diagnosis have their own advantages and disadvantages [175]. For example, in expert system, the complex inference algorithm of each system has to be completed by experienced engineers. In a neural network system, all possible fault modes of the system must be trained, and when new fault types are added, the system will have to be retrained. In addition, how to choose the appropriate membership function for the system is also a difficult problem to be solved in fuzzy diagnosis. Compared with these models, HMM does not need to determine complex inference algorithms in advance. If a new fault class needs to be added, it only needs to be trained and added to the fault library, rather than retraining the entire system. And HMM can also analyze and describe the variable-length system, with better robustness, so it is more suitable for modeling and analysis of the actual system.


## 3.2. Markov Chains

Markov process is a kind of random process without aftereffect proposed by A. A. Markov, in which the “future” state of processXt is only related to the current state, but has nothing to do with the “past” state under the condition that the “present” state of the random process is known. Its mathematical definition is as follows.We know the random processXt,t∈T, and for any n,n=1,2,⋯,n∈R, we have t1<t2<⋯<tn, the probability PXt1=S1,⋯,Xtn−1=Sn−1>0, which satisfies the condition.
(1)PXtn≤SnXtn−1=Sn−1,Xtn−2=Sn−2,⋯,Xt1=S1=PXtn≤SnXtn−1=Sn−1.Then,Xt is called a Markov process. The random sequence Sn−1,Sn−2,⋯,S1 is called Markov sequence, and the conditional distribution function Xt,t∈T is often called transition probability distribution.When the state space and time parameters of a Markov process are discrete, it can also be called a Markov chain. The mathematical definition of Markov chain is as follows: the possible state of a random processXt at time t must be one of N different states S1,S2,⋯,SN, and the state transition of a random process only occurs at the enumerable moments t1,t2,⋯,tn. If the probability that the random process Xt transforms into any state Sj at tm+k is only related to the state Si at tm moment and has nothing to do with the state before tm moment, then the probability that the corresponding state of the random process may occur between these two moments is
(2)PXtm+k≤SjXtm=Si,Xtm−1=Sp,⋯,Xt1=Sq=PXtm+k=SjXtm=Si.This probability is also known as the state transition probability, i.e.,(3)Pijtktm=Pijtm,tm+k=PXtm+k≤SjXtm=Si.Equation (3) illustrates the probability that the state of Markov chain appearing at tm moment is Si and that the state of Markov chain appearing at tm+k moment is Sj. In addition, when tk=1, the one-step transition probability aij can be obtained.
(4)aij=Pij1=Pijtm,tm+1=PijXtm+1=SjXtm=Si.For all possible transition probabilities betweenN states, it can be denoted as probability transition matrix A, so
(5)A=aij=a11a12⋯a1Na21a22⋯a2N⋯⋯⋯⋯aN1aN2⋯aNN,in whichaij satisfies the following attributes:
(6)aij≥0,∑j=1Naij=1.Equation (6) shows that aij is nonnegative and the sum of probabilities of all possible states at any time is 1.In whichtk=n, the transition probability of n steps is
(7)aijn=Pijn=Pijtm,tm+n=PijXtm+n=SjXtm=Si.The correspondingn step transition probability is
(8)An=aijn=a11na12n⋯a1Nna21na22n⋯a2Nn⋯⋯⋯⋯aN1naN2n⋯aNNn.Similarly,aijn has the following properties:
(9)aijn≥0,∑j=1Naijn=1.According to the Chapman-Kolmogorov equation,m+R step transfer probability can be expressed by m step and R step transfer probability, namely,
(10)Pijm+rn=∑kPikmnPkjrn+m,1≤n≤N.Markov chains also have some characteristics:(1)
Periodicity: a Markov chain is aperiodic if its state is triggered and immediately returns to its current state or if there are states in the chain that are connected to states with self-loops(2)
Interoperability: if the probability distribution of Markov chainXn satisfies Pijn≥0, it is said that the Markov chain can reach Sj from state Si. If Pjin≥0 also exists, it is said that the two states of the Markov chain are intercommunicating. The interoperability of Markov chains guarantees the reachability between different states(3)
Ergodicity: if a Markov chain has a limit for all statesSi and Sj independent of state Si


## 4. Experimental Analysis

### 4.1. Test Scheme and Basic Idea

Bearing is a key part of rotating equipment, and its failure is one of the main causes of rotating machinery failure. Therefore, state monitoring and fault diagnosis of bearing vibration signals have been an important research field. This chapter verifies the feasibility and effectiveness of the method through bearing fault test. The test bearings were gb6205-2RS deep groove ball bearings, and the test data came from the literature. Four bearing states are simulated; among which three pitting faults were obtained by EDM, with fault size of 0.18 mm and load of 0-3HP (corresponding to speed of 1797 rpm, 1772 rpm, 1750 rpm, and 1730 rpm), and sampling frequency of 12 kHz.(1)
Data collection: vibration signals of bearings in different states were collected, respectively. A total of 160 groups of data were collected in the test, 40 groups of bearing data of each state, and the length of each group of data was 1 s(2)
Feature extraction: the selected time domain and frequency domain features include effective value, peak-to-peak value, zero peak value, skew index, kurtosis index, peak value index, margin index, waveform index, pulse index, spectrum total value, and amplitude spectrum entropy(3)
Feature reduction: the reduction algorithm of multidimensional features has been introduced in detail in the previous chapter, and the LPP algorithm is used here(4)
Model training: the samples in different states of the training data were trained by CHMM and saved in the model library(5)
Fault diagnosis: diagnostic results can be obtained by comparing the output probabilities of test samples calculated in different modelsBearing fault diagnosis flowchart based on feature reduction and CHMM is shown in Figure1.Figure 1
Bearing fault diagnosis flowchart based on feature reduction and CHMM.Figure1 shows the flowchart of bearing fault diagnosis based on feature reduction and CHMM. The flowchart introduces the whole process of bearing fault diagnosis, which has an important guiding role for the research of the paper.


### 4.2. Fault Diagnosis Example Based on CHMM

After collecting bearing monitoring data in different states, the bearing fault diagnosis is carried out according to the process shown in Figure2. Firstly, the multidimensional features of training data and test data were extracted, with a total of 160 groups of samples, and 40 groups of samples for each bearing state.Figure 2
Results of LPP feature reduction.A total of 15 features, including 8 time domain indicators, 2 frequency domain indicators, and the amplitude of 5 feature frequencies of rolling bearings, were selected as feature sets for fault diagnosis. These 15 feature indicators were extracted from each group of samples, and the total feature vector sampleR15×160 was obtained. Half of the samples were randomly selected as training data and the rest as test data to obtain training feature matrix XTrain15×80 and test feature matrix XTest15×80. The training samples included 4 states, 20 samples in each state, a total of 80 training samples. The test sample was 80. LPP method is used for feature reduction, k adjacency is used for adjacency graph construction, the optimized value is k=17, and the weight is simple weight. If two-point adjacency is used, Sij=1; otherwise, it is 0. In order to ensure that the contribution rate of principal components in each state reaches more than 90%, the first three principal components are selected as the features after dimensionality reduction. The reduced LPP training features and test features were denoted as XLPPTrain15×80 and XLPPTest15×80, respectively. The distribution of the first three principal elements in training and testing is shown in Figure 2, respectively. It can be seen that, whether training samples or test samples, the reduced features can effectively identify samples in bearing 4 under different states, which can be used for bearing fault diagnosis. For comparison, all features before feature reduction are used as input observation vectors to train the model and make diagnosis. Here, the parameter of CHMM is selected as the number of states N=2, the number of Gaussian elements in each state M=2, O=15 when all features are used as input for the observation vector, and O=3 when features after feature reduction are used for the observation vector.When all features are used as observation vectors, the classification results of bearing test samples in different states are shown in Figure2. It can be seen that logarithmic likelihood can intuitively express the actual state of samples. Each test sample was input into four pretrained models to calculate logarithmic likelihood. Taking the 1-12 normal samples in Figure 3(a) as an example, all the samples get the maximum probability after input NC, so it can be judged that these samples are in the normal state, which is consistent with the actual state of these samples. In the inner ring fault test samples, the 5 samples 36-40 could not be correctly identified, and the outer ring fault samples could also be all identified. However, the rolling body fault samples 67, 72, and 79 in (d) were also misclassified, and the recognition accuracy of all test samples was 92.5%. Bearing fault diagnosis results using all features and CHMM are shown in Figure 3. Bearing fault diagnosis results based on LPP and CHMM are shown in Figure 4.Figure 3
Bearing fault diagnosis results using all features and CHMM.Figure 4
Bearing fault diagnosis results based on LPP and CHMM.The diagnosis results using the first three features after LPP reduction as the observation vector are shown in Figure4. Except for the misclassification of sample 79 of the rolling body, all the other samples can be correctly classified, and the total recognition accuracy is 98.8%. By Figures 2 and 4 as you can see, both use all the features and characteristics after the subtract, which have very good identification of four kinds of condition of bearing fault, but after about reducing the characteristics of the classification accuracy is higher than the former, this may be due to LPP method features about reduction after eliminating the original feature set of information between conflict and redundancy. In addition, LPP feature classification is better than the former. Taking sample no. 30 as an example, when all features are used for diagnosis, the logarithmic likelihood distance between maximum likelihood rate and submaximum likelihood rate is 12.8, while when LPP feature is used, the distance is 29.2. In addition, when using LPP features for fault diagnosis, the training time of the model is 73.3% of the former, and the time required for fault diagnosis is only 80.0% of the former.


## 4.1. Test Scheme and Basic Idea

Bearing is a key part of rotating equipment, and its failure is one of the main causes of rotating machinery failure. Therefore, state monitoring and fault diagnosis of bearing vibration signals have been an important research field. This chapter verifies the feasibility and effectiveness of the method through bearing fault test. The test bearings were gb6205-2RS deep groove ball bearings, and the test data came from the literature. Four bearing states are simulated; among which three pitting faults were obtained by EDM, with fault size of 0.18 mm and load of 0-3HP (corresponding to speed of 1797 rpm, 1772 rpm, 1750 rpm, and 1730 rpm), and sampling frequency of 12 kHz.(1)
Data collection: vibration signals of bearings in different states were collected, respectively. A total of 160 groups of data were collected in the test, 40 groups of bearing data of each state, and the length of each group of data was 1 s(2)
Feature extraction: the selected time domain and frequency domain features include effective value, peak-to-peak value, zero peak value, skew index, kurtosis index, peak value index, margin index, waveform index, pulse index, spectrum total value, and amplitude spectrum entropy(3)
Feature reduction: the reduction algorithm of multidimensional features has been introduced in detail in the previous chapter, and the LPP algorithm is used here(4)
Model training: the samples in different states of the training data were trained by CHMM and saved in the model library(5)
Fault diagnosis: diagnostic results can be obtained by comparing the output probabilities of test samples calculated in different modelsBearing fault diagnosis flowchart based on feature reduction and CHMM is shown in Figure1.Figure 1
Bearing fault diagnosis flowchart based on feature reduction and CHMM.Figure1 shows the flowchart of bearing fault diagnosis based on feature reduction and CHMM. The flowchart introduces the whole process of bearing fault diagnosis, which has an important guiding role for the research of the paper.


## 4.2. Fault Diagnosis Example Based on CHMM

After collecting bearing monitoring data in different states, the bearing fault diagnosis is carried out according to the process shown in Figure2. Firstly, the multidimensional features of training data and test data were extracted, with a total of 160 groups of samples, and 40 groups of samples for each bearing state.Figure 2
Results of LPP feature reduction.A total of 15 features, including 8 time domain indicators, 2 frequency domain indicators, and the amplitude of 5 feature frequencies of rolling bearings, were selected as feature sets for fault diagnosis. These 15 feature indicators were extracted from each group of samples, and the total feature vector sampleR15×160 was obtained. Half of the samples were randomly selected as training data and the rest as test data to obtain training feature matrix XTrain15×80 and test feature matrix XTest15×80. The training samples included 4 states, 20 samples in each state, a total of 80 training samples. The test sample was 80. LPP method is used for feature reduction, k adjacency is used for adjacency graph construction, the optimized value is k=17, and the weight is simple weight. If two-point adjacency is used, Sij=1; otherwise, it is 0. In order to ensure that the contribution rate of principal components in each state reaches more than 90%, the first three principal components are selected as the features after dimensionality reduction. The reduced LPP training features and test features were denoted as XLPPTrain15×80 and XLPPTest15×80, respectively. The distribution of the first three principal elements in training and testing is shown in Figure 2, respectively. It can be seen that, whether training samples or test samples, the reduced features can effectively identify samples in bearing 4 under different states, which can be used for bearing fault diagnosis. For comparison, all features before feature reduction are used as input observation vectors to train the model and make diagnosis. Here, the parameter of CHMM is selected as the number of states N=2, the number of Gaussian elements in each state M=2, O=15 when all features are used as input for the observation vector, and O=3 when features after feature reduction are used for the observation vector.When all features are used as observation vectors, the classification results of bearing test samples in different states are shown in Figure2. It can be seen that logarithmic likelihood can intuitively express the actual state of samples. Each test sample was input into four pretrained models to calculate logarithmic likelihood. Taking the 1-12 normal samples in Figure 3(a) as an example, all the samples get the maximum probability after input NC, so it can be judged that these samples are in the normal state, which is consistent with the actual state of these samples. In the inner ring fault test samples, the 5 samples 36-40 could not be correctly identified, and the outer ring fault samples could also be all identified. However, the rolling body fault samples 67, 72, and 79 in (d) were also misclassified, and the recognition accuracy of all test samples was 92.5%. Bearing fault diagnosis results using all features and CHMM are shown in Figure 3. Bearing fault diagnosis results based on LPP and CHMM are shown in Figure 4.Figure 3
Bearing fault diagnosis results using all features and CHMM.Figure 4
Bearing fault diagnosis results based on LPP and CHMM.The diagnosis results using the first three features after LPP reduction as the observation vector are shown in Figure4. Except for the misclassification of sample 79 of the rolling body, all the other samples can be correctly classified, and the total recognition accuracy is 98.8%. By Figures 2 and 4 as you can see, both use all the features and characteristics after the subtract, which have very good identification of four kinds of condition of bearing fault, but after about reducing the characteristics of the classification accuracy is higher than the former, this may be due to LPP method features about reduction after eliminating the original feature set of information between conflict and redundancy. In addition, LPP feature classification is better than the former. Taking sample no. 30 as an example, when all features are used for diagnosis, the logarithmic likelihood distance between maximum likelihood rate and submaximum likelihood rate is 12.8, while when LPP feature is used, the distance is 29.2. In addition, when using LPP features for fault diagnosis, the training time of the model is 73.3% of the former, and the time required for fault diagnosis is only 80.0% of the former.


## 5. Conclusion

iHMM is a new dynamic pattern recognition method. Compared with the traditional hidden Markov model, it can effectively avoid the defects left by HMM in the initial stage of modeling and can determine the number of hidden states and mathematical structure of the model adaptively. Based on this, the hidden Markov model was introduced into the fault diagnosis of rotating machinery, and a new fault recognition method based on iHMM was proposed, which was compared with the traditional HMM recognition method. The experimental results show that both the traditional HMM recognition method and the proposed iHMM recognition method can effectively distinguish different fault types. However, in traditional HMM, the model structure must be determined in advance. Such artificially specified number of states often cannot best describe the state of the system, which will inevitably affect the accuracy of diagnosis. In addition, overlearning problem is inevitable in the process of maximum likelihood estimation. In the iHMM recognition method, the number of HMM states is extended to an infinite number, and the established model effectively avoids the inevitable disadvantages of HMM in the modeling process through data integration. Therefore, the proposed method has obvious advantages.



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