Rotating machinery Diagnostic Using Hidden Markov Models (HMMs)

Rotating machinery Diagnostic Using Hidden Markov Models (HMMs)

Miloud Sedira 1, Ahmed Felkaoui 2 1 LMPA labratory, Ferhat Abbas university, 19000 Se f, Algeria

[email protected] 2 LMPA labratory, Ferhat Abbas university, 19000 Se f, Algeria

[email protected]

Abstract In this article, we have implemented a system recognizing faults of rotating machines based on Hidden Mar‐kov Models (HMM). The HMMs are a modeling tool that has proven itself particularly in the field of speech processing, image processing and analysis of biological sequences. From three‐time indicators extracted from vibration signals, we constructed matrices that characterize each class (health system). Each health state of the machine is represented by a matrix or a set of matrices, whose values belong to a defined interval (supervised classification). Each health state is denoted by an attribute (observation). This also constitutes the observable state of the HMM considered. The hidden state of HMM, is determined by a probabilistic approach also called "Monrovian approach." Thus, each hidden state is cha‐racterized by an HMM, which in turn is defined by a maximum likelihood (max log‐likelihood), a transition probability matrix and a vector of probability distribution initial state or departure. These parameters were obtained after training conducted for each HMM, according to the Baum‐Welch algorithm (procedure) which is based on the principle of maximum likelihood EM (expectation maximization). The system obtained of HMMs and its computer processing, designed to end in the form of a toolbox labeled constituting the system targeted by this approach. The results testify to the reliability and efficiency of this system. Keywords – diagnostic, classification, hidden Markov models, the Baum‐Welch training...

1. Introduction

The industry is constantly in motion. It is perfected, relocates, develops and invents in order to keep or gain market share against fierce competition. This prompted maintenance to become a priority within the com‐pany as well, and in order to minimize the time for intervention or rehabilitation. Operational safety, maintenance cost effectiveness and asset availability has a direct impact on the competi‐tiveness of organizations and nations. Today’s complex and advanced machines demand highly sophisticated and costly maintenance strategies. An even more alarming fact is that one‐third to one‐half of this expendi‐ture is wasted through ineffective maintenance. Therefore, there is a pressing need to continuously develop and improve current maintenance programs. Current maintenance strategies have progressed from break‐down maintenance, to preventive maintenance, then to condition‐based maintenance (CBM) managed by experts, and lately towards a futuristic view of intelligent predictive maintenance systems [1]. The term con‐dition‐based maintenance (CBM) is used to signify the monitoring of an asset or equipment’s health for the purpose an early diagnostics, then, it allows to fix faults before catastrophic failure occurs. The diagnosis of equipment can be divided into two main parts: fault detection and fault identification. An important and indispensable, is still raised in diagnostics of machines, this is the classification. Fault detection classifies equipment as either normal or defective, whereas fault identification classifies the equipment under one of several possible defects that it might suffer from. Different types of classification models have been used for automa c diagnosis of faults in CBM [2]. Researchers in the area of maintenance continue to develop many techniques and models in order to recognize and classify faults, and this has brought to use all the tech‐niques, including artificial intelligence (AI). In this way, several studies have been performed, such as Support

Vector Machines (SVM), which have been used extensively for different diagnostic applications in Christian et al. [3], Xian, G.‐m. et al [4], and Widodo et al. [5]. Like SVM, wavelet decomposi on have been also large‐ly used in mechanical system, then it have applied to separate fault features in Hoonbin Hong et al [6]. In parallel, another interest model have been used to diagnose and predict fault in rotating machinery and mechanical system, it’s Bayesian network, which used in K.Medjaher [7] to localize fault in synchronous elec‐trical engine. The most is artificial neural networks (ANN), which have been used extensively in CBM, it has been applied with by R.ZOUARI et al in [8] to classify some unlearned levels of gradual faults such as partial flow or cavitations for Centrifugal Pumps. Many papers discuss combinations of different classifica‐tion tools in building a fault diagnosis decision model. For example, Xian, G.‐m., et al. [9] have inves gated the feasibility of detecting the time of damage to a 4‐storey shear structure building to harmonic excitation using both EMD and wavelet transform combined method.

This paper presents an application for detecting and classifying gear defects based on Hidden Mar‐

kov Models (HMMs). These are another type of stochastic decision model. This model has seen a largest

success in other domains, so it has attracted the attention of various research communities, including the ones in statistics, engineering, and mathematics. Firstly, HMMS have been used in speech processing [10] and in Mohamad Adnan Al‐Alaoui et al [12], then, in image processing in Wojciech Pieczynski [12] wich presents some aspects of Markov model based statistical image processing. Classical Markov models (fields, chains, and trees) used in image processing are developed. In bioinformatics, Stephen McCauley et al in [13] have applied HMMs to the single sequence annotation procedure (SSA) and applied it to incorporate evolu onary informa on 14 different strains of the HIV2 virus.

The application in mechanical domain is presented in following section, that we give summary re‐view for the most works in this context which consist of the principal motivation for us to initiate this work.

2. Related works

This section provides a brief overview on some work that is considered interesting and focuses on the appli‐cation of HMMs in the field of diagnosis, detection and classification of defects in rotating machines. thus,, Qiang Miao and Viliam Makis in [ 14 ] have proposed propose a modeling framework for the classification of machine (gearbox) conditions based on wavelet modulus maxima distribution and HMMs. A feature extrac‐tion approach based on wavelet modulus maxima, and an HMM‐based two‐stage machine condition classification system, are proposed. They also proposed an adaptive algorithm and validated it by three sets of real gearbox vibration data to classify two conditions: normal and failure. In addition, in condition classifica on (stage 2), three HMM models are set up to classify three different machine conditions, namely, adjacent tooth failure, distributed tooth failure and normal condition. In another context including the ab‐sence of machine tool wear and cu ng tool, Tony Boutros and Ming Liang in [15] applied hidden Markov model (HMM) to detect and diagnose mechanical faults. They have tested and validated them technique using two scenarios: tool wear/fracture and bearing faults. In the first case the model correctly detected the state of the tool (i.e., sharp, worn, or broken) whereas in the second application, the model classified the severity of the fault seeded in two different engine bearings. The rate obtained in them tests for fault severi‐ty classifica on was above 95%. In addi on to the fault severity, a loca on index was developed to detemine the fault location. This index has been applied to determine the location (inner race, ball, or outer race) of a bearing fault with an average rate of 96%. The training me required to develop the HMMs was less than 5 s in both the monitoring cases. Focus more on the implementation on the implementation method and the es ma on of model parameters with respect to previous research, Petar M. Djuric et al in [16] analyze non

stationary HMMs whose state transition probabilities are functions of time that indirectly model state dura‐tions by a given probability mass function and whose observation spaces are discrete. The objective of our work is to estimate all the unknowns of a non stationary HMM, which include its parameters and the state sequence. To that end, we construct a Markov chain Monte Carlo (MCMC) sampling scheme, where sam‐pling from all the posterior probability distributions is very easy. The proposed MCMC sampling scheme has been tested in extensive computer simulations on finite discrete‐valued observed data, and some of the simulation results are presented in the paper. Whereas, Xiaodong Zhang et al in [17] have integrated three fault diagnostic and prognostic algorithms for bearing health monitoring. The proposed unified framework which is capable of performing anomaly detection, fault detection and isolation, health/degradation estima‐tion, and remaining useful life prediction. Simulation results using some real bearing vibration data have been used. But more data intensive validation and performance evaluation are reserved for a future works. V. Purushothama et al in [18] have presented a method for detecting localized bearing defects based on wavelet transform. Bearing race faults have been detected by using discrete wavelet transform (DWT). Vi‐bration signals from ball bearings having single and multiple point defects on inner race, outer race, ball fault and combination of these faults have been considered for analysis. Wavelet transform provides a variable resolution time–frequency distribution from which periodic structural ringing due to repetitive force im‐pulses, generated upon the passing of each rolling element over the defect, are detected. It is found that the impulses appear periodically with a time period corresponding to characteristic defect frequencies. In the study, the diagnoses of ball bearing race faults have been investigated using wavelet transform. They com‐pared results with feature extraction data and results from spectrum analysis. They concluded that DWT can be used as an effective tool for detecting single and multiple faults in ball bearings. They just used hidden Markov Models (HMMs) for pattern recognition for bearing fault monitoring. Finally, Jihong Ya,et al in [19] have proposed a systemic prognostics scheme based on neural networks combined dynamic multi‐scale Markov model. A performance degradation indicator is designed by multi‐feature fusion technique based on neural network models. Based on this indicator, remaining life prediction is implemented by a dynamic mul‐ti‐scale Markov model. The specificity of them work consist of using an FCM algorithm to deal with state division and combining dynamic prediction method and multi‐scale theory with Markov model.

At the end of this section, we conclude that the majority of researches that makes use of hidden Markov models (HMMs) focus on: ‐ The extract manner of the vibration characteristics or indicators, where one often tends to make it more complex. ‐ The choice of indicators, which often use indicators relatively complex, knowing beforehand that there is no indicator (s) or single universal one hand, on the other hand it has used certain assumptions which in our view remains too virtual for the location of faults like the work of Tony Boutros and Ming Liang in [15] and Xiaodong Zhang et al in [17]. ‐ The component fault detection and diagnosis of bearings had a lot of interest from the researchers in rela‐tion to gears and other mechanical components, as in the case of Tony Boutros and Ming Liang in [15], Xiao‐dong Zhang et al in [17] and V. Purushotham et al in [18]. Our motivation for making this work stems from three observations above. Indeed, we had the objective of bridging the remarks by this contribution. Thus, we opted for an application of HMMs in the classification of gear faults, and the choice of simple indicators vibration in a combination of three temporal indicators which are: RMS, Kurtosis, and the factor peak. This, in part based on practical considerations including the defini‐tion of classes or health status of the machines. In the next section, we present the experimental equipment and the running of the event.

3. Experimentation 3.1 System studied

The system studied is a reduc on gear, consis ng of two gears with 20 and 21 teeth spur gears, preceded by a gear loop comprising two gears with 40 and 41 teeth spur gears as shown in figure 1. The system belongs to the CETIM (Engineering Industries Technology Center, France); data from this system are:

The speed of rotation of the gear shaft is Vrot = 1000 rpm or a frequency of fr=16,67Hz

Meshing frequency is fe = 330Hz

Figure 1. Experimental equipment

3.2 Expertise of test

For a period of 13 days, there was a daily vibra on signal on the test bench, comprising 60160 samples with a sampling frequency of 20 KHz. The data are samples of a signal emi ed by the vibra on reducing system every day. Temporal representa on of the vibra on signal (Figure 2).The vibra on signal remains unchanged un l the 12th day the error occurred. A shock occurs at a period corresponding to the period of rotation of the gear system and having very high amplitude compared to that of the signal collected during the other days (Figure 2). The various observa ons made during this experiment were summarized in Table 1

Table 1. Exper se report Day Observation

2 1st of acquisition, no anomaly

3,4,5,4,6 No anomaly, normally state

7 Sapling of tooth surface of the tooth 1/2

8 No evolu on of the tooth 1/2

9 No evolution of the tooth ½ state, beginning of sapling on the surface of the tooth 15/16

10,11 No evolution

12,13 sapling over the en re surface of the tooth 15/16

Figure 2. Vibration signal for gear

4. Selecting Indicators

Various indicators are used in vibration monitoring of gears include for this purpose, the effective value RMS, Kurtosis, Crest Factor ... The presence of an abnormality can be detected, if an indicator exceeds a pre‐determined threshold (or range) [20]. The thresholds are defined either by a standard vibration severity (eg ISO 10816, ISO 2372), or by experience. In general, there is hardly a universal indicator can detect all types of faults in rotating machines, however, for a specific type of defects, one can choose or adapt one or more indicators that are appropriate present. For our application, we opted for the combination of a triplet of temporal indicators in a single matrix which has been called "indicator matrix», denoted by IM, it’s the RMS, the crest factor and the kurtosis such de‐scribed below. IMi is a 13X3, with: 3 columns represent three indicators: RMS, crest factor and Kurtosis 13 lines represent signal group of a sample which characterizes a given state health i.

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

KurtosisFactorCrestRMS

IM i

5. Design of the data set

This operation is established, to characterize each class by giving it a mathematical sense so that it is inte‐grated in the modeling of global classifica on system, C. Lacour in [21].Indeed, for each day of experimenta‐on, a collec on of indicators has been completed and put in a matrix 13x3 indicated by a capital If the index

i represen ng the day, and having 13 people in the lines of the day column in the order, the RMS value, crest factor and kurtosis. This brings us back to the form table 2. Table 2. Classes characterizing Designation of classes of fault observation Indicator matrix (IM)based learning 1st class New IM2 –IM6

2nd class Good IM7, IM8

3rd class Acceptable IM9‐IM11

4th class Alarm IM12, IM13

The above decisions have been taken from the array of expertise and vibration signals emitted by the stan‐dards and prac ces including the ISO 10816 standard.

6. Design of the data test set and Learning data set

Table 3. Data test set and learning data set Designation of classes of fault observation Test data Learning data 1st class New IM3,IM4,IM5 IM2 ,IM6

2nd class Good IM7 IM8

3rd class Acceptable IM10, IM11 IM9

4th class Alarm IM12 IM13

7. Hidden Markov Models implementation 7.1 Definition

The hidden Markov models (HMMs) are based on two stochastic processes dependent on one another. In‐deed, the state of the system is not directly observable and is hidden by a process of observation as illu‐strated by Figure 3. The word “hidden” means the HMM states are not directly observable. In other words, the HMM states can only be observed through a set of stochastic processes that produce the sequence of observations [10].

7.2 Element of HMM

An HMM is denoted by l and characterized by the following parameters:

‐ The number of states of the model N, in our case N=4 (figure 3) ‐The number of symbols of observations M, in our applica on M=4 observa on, which are; New, Good, Ac‐ceptable and Alarm. Every observation i is characterized by an indicator matrix IMi

‐ The distribution of emission probabilities of observations in each state j Bj, for our application;

, with 14

1=∑

=kjkb stochastic property ( 1)

)]([ ObB tj= (2)

)()( 1 iqOqPOb ttktj === + (3)

‐The probability distribution of state transitions ][aA ij= , in our case it is defined as:

⎥⎥⎥⎥

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⎤

⎢⎢⎢⎢

⎣

⎡

=

aaa

aaaa

Aij

44

3433

2322

1211

00000

0000

, with 14

1=∑

=jija stochastic property (4)

)( 1 iqjqPa kkij === + (5)

‐The probability distribution of initial state or departure p0, in our case

, with 14

1=∑

=jjπ stochastic property ( 6)

The parameters of the HMM, A, B and p remain subject to the conditions for stochastic processes, namely:

),,( πλ BA= (7)

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=

0001

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1

)(

Figure 3. Left‐Right HMM modeling for gear reducer fault

8. Choice of the architecture of HMM

The most appropriate architecture for this application is that of "left‐right", as our model takes into account the following conditions: ‐The wear of the material causing the degradation of components of the system (including gear) is a natural‐ly occurring irreversible (progress in one direction). ‐For each state of degradation, there is a characteristic observation).The design of hidden Markov model has led to the diagram shown in figure 3, with: S1, S2, S3 and S4, define the hidden states of the HMM. Y1, Y2, Y3 and Y4, refer to the observa ons of HMM. Aij, denote the transition probabilities between states of the model. Bk(j)Represent the emission probabilities of observation. In the left‐right model (Figure 3) there is a communica on from le to right in an evolving system irreversi‐ble. The first observation is produced while the Markov model is in an initial state S1, with [10]:

⎩⎨⎧

≠===

MkMk

kb j ,0,1

)( (8)

1=π i , 1=i (9)

0=aij , 1pj (10)

0=π i , Ni ≤≤2 (11)

8.1 Markov properties

A Markov chain is a dynamic Bayesian network the current state depends only on the condition that precedes it (Markov assumption)

)(),......,(121 SSSSS t tttt PP−−− = (12)

To know St (the system state at time t), it’s necessary to consider P (St) Every moment we have a set of observations about the current state of the system

)( SOP tt (13)

The probability of the observations Y is conditionally dependent on state S,

( ) ∏=

=K

k

kOPOP1

)( λλ (14)

8.2 Three basic problems of HMMs The treatment of HMMs leads often to deal with three problems, also called features of HMMs which are :

• Evaluation (or recognition) of the probability of observing a sequence O • Estimate (or decoding) from a sample set (observation sequence) sequence hidden • Learning (or training) of a sequence (learning the parameters of the MMC). The latter problem is the subject of our application.

9. HMM Learning Learning the HMM established, the procedure was performed by the Baum‐Welch, on the basis of the actual model classes taken and included in Table 3. The different phases of the learning opera on are summarized in the following. Each of the four classes mentioned above, was characterized by an HMM, a convergence curve of the log‐likelihood (Figure) to its maximum value.

Table 4 . Learning results

State Class HMM Log‐likelihood Transition matrix S1

New

l1

‐102.4511

0.8276 0.1724 0 0 0 0.2157 0.7843 0 0 0 0.9653 0.0347 0 0 0 1.0000

S2

Good

l2

‐103.3498

0.0139 0.9861 0 0 0 0.0642 0.9358 0 0 0 0.3856 0.6144 0 0 0 1.0000

S3

Acceptable

l3

‐104.1902

0.0751 0.9249 0 0 0 0.5958 0.4042 0 0 0 0.6872 0.3128 0 0 0 1.0000

S4

Alarm

l4

‐ 109.3608

0.0078 0.9922 0 0 0 0.0048 0.9952 0 0 0 0.2099 0.7901 0 0 0 1.0000

In interpreting the above results, we may state the following: For the choice of insertion of three indicators simultaneously for the characterization of a fault condition, the results were very significant because we found that the difference between two successive states has been detected or felt each time by at two indicators. The HMMs have reacted favorably to the changes which their effectiveness is affirmed. The training of HMMs represented by matrices of indicators and rec‐orded in hidden Markov models has been very consistent with the results of the expert report of the expe‐rience, knowing that it is a supervised classification. In the end we record these HMMs with their parameters as a basis for recognition. The result of the decision issued by the recognition is the "diagnosis" of the ma‐chine seen in an automatic way by using the HMMs. For an enhancement of the efficiency and reliability of the classification system, we proceed in the following section to test it to verify its ability to recognize poten‐tial suspects states.

Figure 4. HMMs curves characterizing classes fault of gear reducer

Below, we give the procedure that lead to HMMs learning

Figure 5. Procedure of modeling reducer gear fault diagnostic by HMMs

10. Evaluation tests modeling (recognition capabilities of the Model) In order to determine the effectiveness and reliability of the system for the classification of defects gear by the hidden Markov models, it is essential to test a meaningful way. For this, we conduct a series of tests, which involve the injection in the toolbox "HMM" matrices previously recorded in a database of recognition, then compare the decision of the model with the findings of Table of expertise (supervised classification). The capacity of the system to recognize the observation or the state of the system is deduced from the com‐parison of training results and testing results (recognition). We present the following tests made, their re‐sults and comments related there.

Table 5. Evaluation tests results

Transition matrix : 0.0347 0.9653 0 0 0 0.4170 0.5830 0 0 0 0.9833 0.0167 0 0 0 1.0000

Log‐likelihood: ‐ 102.4070 Recognotion ratio :0. 9979


Log‐likelihood: ‐103.2072 Recognotion ratio : 0.9923


Log‐likelihood: ‐104.1347 Recognotion ratio : 0.9834


Log‐likelihood:‐109.5645 Recognotion ratio : 0.9347

Figure 6. HMM’s curves of evaluation tests

11. Conclusion

At the end of this work, we confirm that the system is implemented based on hidden Markov models is as effective, as the other techniques mentioned in section I, this also says that the classification of defects ro‐tating machinery by the MMCs is also possible, based on a triplet of time indicators (RMS, crest factor and kurtosis), extracted from vibration signals and inserted into appropriate matrices, under pre‐established assumptions. The system designed (toolbox) can be adapted to any rotating machinery it; simply repeat the training with the vibration signals of the machine targeted at all.

Despite the positive results, our system remains in need to be tested with a maximum signal of the same system on the one hand, on the one hand, his bank of learning should also be enhanced by signal overload for example, able to simulate all the maximum operating hazard that may be encountered in practice. This will be shown in the futures works.

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Rotating machinery Diagnostic Using Hidden Markov Models (HMMs)

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