Faults Diagnosis for Vibration Signal Based on HMM...situation. In this situation, the Faults...

Sensors & Transducers, Vol. 165 , Issue 2, February 2014, pp. 8-15

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© 2014 by IFSA Publishing, S. L. http://www.sensorsportal.com

Faults Diagnosis for Vibration Signal Based on HMM

Shao Qiang, Song Peng, Feng Changjian Department of Automotive Engineering, Dalian Nationalities University,

Dalian 116600, China Tel.: 13555988100, fax: 86+411 87630449

E-mail: [email protected]

Received: 29 October 2013 /Accepted: 28 January 2014 /Published: 28 February 2014 Abstract: Faults behaviors of automotive engine in running-up stage are shown a multidimensional pattern that evolves as a function of time (called dynamic patterns). It is necessary to identify the type of fault during early running stages of automotive engine for the selection of appropriate operator actions to prevent a more severe situation. In this situation, the Faults diagnosis method based on continuous HMM is proposed. Feature vectors of main FFT spectrum component are extracted from vibration signals and looked up as observation vectors of HMM. Several HMMs which substitute the types of faults in automotive engine vibration system are modeled. Decision-making for faults classification is performed. The results of experiment are shown the proposed method is executable and effective. Copyright © 2014 IFSA Publishing, S. L. Keywords: Pattern recognition, Faults diagnosis, HMM. 1. Introduction

Many techniques in pattern recognition deal with static environments: the class distributions are considered relatively constant as a function of the time in which feature vectors are acquired. However, in some systems, time often plays a secondary role: it should be incorporated in the feature extraction procedure. For practical recognition tasks, the assumption of stationarity of the class distributions may not be hold. Alternatively, information in sequences of feature vectors may be used for recognition. We will call both groups of problems dynamic pattern recognition problems. A dynamic pattern is a multidimensional pattern that evolves as a function of time [1].

A set of feature vectors can be looked upon as the result of independent draws from a multi-dimensional distribution. All temporal information should now be present in each feature vector. Identification problem may then be based on the dissimilarity of a set of

newly measured feature vectors with respect to a set of known templates.

HMMs have been proved to be one of the most widely used tools for learning probabilistic models of dynamical time series. HMM can model dynamical behaviors variation existing in the system through a latent variable (hidden states). HMM method that is used for faults diagnosis is often seen in reference literature [1, 2, 5, 6].

For automotive engine, it is necessary to identify the type of faults during it’s early stage for the selection of appropriate operation actions to prevent a more severe situation, or to mitigate the consequences of the fault. It is not easy for an operator to identify the type of faults accurately, using the information given by instruments and alarms, with a limited time interval. Therefore, the use of a computer-based Fault diagnosis is recommended. This method is intended to support an operator’s decision-making, or to provide input signals from a computerized faults monitoring system

Article number P_1868

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and a computerized operating – procedure management system.

This paper is only focused on the faults diagnosis of automotive engine in the running-up stage based on continuous HMM. 2. Probability Theory of Faults Diagnosis

Faults diagnosis problem of automotive engine is

defined the classification of type of faults, j , given

sequential input pattern tX at time t. Input pattern

tX is mathematically defined as an object described

by a sequence of features at time t [2].

1 2( , , , )t dx x x X (1)

The space of input pattern tX consists of the set

of all possible patterns: dt RX , dR is a

d-dimensional real vector space. The k observed data up to time t is defined as,

1 1, , ,t k t k t t X X X (2)

The set of possible fault types j forms the

space of classes :

1 2( ) , , , ct , (3)

where c is the number of classes.

The faults diagnosis task can be considered to be the finding of function f, which maps the space of

input patterns t k to the space of classes .

The vibration signals of automotive engine in running-up stage often exhibit sequentially changing behaviors. If one short-time period is defined to a frame, the probability of a particular frame transition is different for each type of faults. Therefore, the probability of frame’s existence, and of a particular transition between frames, can be statistically modeled. The probability of specific signal is already known, and is called prior probability. When identifying a specific fault, a decision can be made only by selecting the type of signal with the

highest a prior probability P . The decision is

probably unreasonable. It is more reasonable to determine the type of time series after observing the trend of vibration signals of major variables, namely,

to get the conditional probability ( | )t kP . This

conditional probability is called a posterior probability. Decision-making based on the posterior probability is more reliable, because it employs a prior knowledge together with the observed fault

features. Classification of an unknown pattern Xt

corresponds to finding the optimal model ̂ that maximizes the conditional probability

( | )t kP over the whole time series of the type

. One can apply Bayes rule to calculate the a posterior probability,

( | ) ( )ˆ( | ) max

( )t k

t kt k

P PP

P

(1)

The conditional probability ( | )t kP comes

from comparing the shapes of fault models with the input observations, while the prior probability

( )P comes from the fault probability. Since

( )t kP is independent of ̂ ,

ˆ( / ) max{ ( | ) ( )}t k t kP P P

(5)

In fact it is difficult to calculate an a

priori probability )(P , which satisfy the following

equation.

1

1c

jj

(6)

The HMM can successfully treat an identification

of faults of dynamic pattern under a probabilistic or statistically framework.

In this faults diagnosis problem, the HMM is used

to estimate the conditional probability ( | )t kP .

3. Design of Faults Diagnosis Based

on HMM 3.1. Vibration Feature Vector Extract

The vibration signals are sampled with various frequencies according to the velocity of automotive engine running in the method of complete alternation. Each running alternation we can gain 64 points sample of vibration signal. We repeat this process with 8 complete alternations. Thus 512 points sample is available and saved to disk. For the 512 points sample, we can get FFT spectrum with the frequency that is the multiple or half of running frequency of automotive engine. Fig. 1 shows the vibration wave and FFT spectrum of samples.

In the Fig. 1 the note “x” is the running frequency of the automotive engine. From the FFT spectrum we can get the Xtas following:

1

{ , , 2 ,3 , 4 ,5 }2t x x x x x xX


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Fig. 1. Vibration wave (a) and FFT spectrum (b)

of vibration. 3.2. The Component of HMM

By using the HMM, the pattern variability in the parameter space and time can be modeled effectively. HMM uses a Markov chain to model the changing statistical characteristics that exist in the actual observations of dynamic process signals. The Markov process is therefore a “double” stochastic procedure that enables the modeling of not only spatial phenomena, but also time-scale distances.

The Hidden Markov Model is a finite set of states, each of which is associated with a (generally multidimensional) probability distribution [4]. Transitions among the states are governed by a set of probabilities called transition probabilities. In a particular state an outcome or observation can be generated, according to the associated probability distribution. It is only the outcome, not the state visible to an external observer and therefore states are “hidden” to the outside, hence the name Hidden Markov Model.

The following parameters are needed to define a mixed-density continuous HMM [3, 4, 7]:

The number of states of the model, N. The number of observation symbols in the

alphabet, M. If the observations are continuous then M is infinite.

A set of state transition probabilities { }ijaA .

1( | ),1 ,ij t ta P q j q i i j N , (7)

where tq denotes the current state.

Transition probabilities should satisfy the normal stochastic constraints,

0,1 ,i ja i j N (8)

and,

1

1, 1N

ijj

a i N

(9)

A probability distribution in each of the states,

{ ( )}jb kB .

( ) { | },

1 ,1

j t k t jb k p v q

j N k M

o, (10)

where kv denotes the thk observation symbol in the

alphabet, and to the current parameter vector.

Following stochastic constraints must be satisfied,

( ) 0,1 ,1jb k j N k M (11)

If the observations are continuous then we will

have to use a continuous probability density function, instead of a set of discrete probabilities. In this case we specify the parameters of the probability density function. Usually the probability density is approximated by a weighted sum of M Gaussian distributions G,

1

( ) ( , , )M

j t jm jm jm tm

b c G

o μ o , (12)

where jmc is the weighting coefficients, jm is the

mean vectors, jm is the covariance matrix, the

initial state distribution, { }i .

Where,

1( ), 1i p q i i N (13)

Therefore we can use the compact notation,

( , , ) A B to denote an HMM with discrete

probability distributions, while,

( , , , , )jm jm jmc A μ π (14)

to denote one with continuous densities.

In this paper, we mainly discussed the continuous densities HMM as equation (14). 3.3. Training of HMM

According to the above definition of an HMM, there are three problems of interest [4].

(i) The Evaluation Problem. Given an HMM and a sequence of

observations, 1 2 3 1, , ,T T O o o o o o , what is the

probability that the observations are generated by the model, ( | )P O ?

Sample points

Am

plit

ude

Frequency


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(ii) The Decoding Problem. Given a model and a sequence of

observations, 1 2 3 1, , ,T T O o o o o o what is the

most likely state sequence in the model that produced the observations?

(iii) The Learning Problem (Training Problem). Given a model and a sequence of observations,

1 2 3 1, , ,T T O o o o o o how should we adjust the

model parameters A , jmc , jmμ , jm andπ in order

to maximize ( | )P O ?

The Evaluation Problem can be solved by Forward Algorithm and Backward Algorithm. The problem also can be solved by Viterbi Algorithm. The Decoding Problem can be solved by Viterbi Algorithm. The Training Problem can be solved by Baum-Welch Algorithm [4, 7].

The Training Problem is discussed in this paper. The all steps for training of an HMM is as following:

Input the numbers of hidden states, initial probability distribution , initial state transition probabilities, the error e of iteration, the maximization number of iteration L and observations O.

Initial Gauss function parameters, jmc , jm and

jm are estimated by k-means algorithm. Then the

initial HMM 0 is gained.

According to the Baum-Welch Algorithm, the

each parameters A , jmc , jmμ and jm of HMM are

re-estimated. In this step, Multi-observation (from various load conditions) can be considered. Therefore

the HMM i re-estimated by the i-th iteration.

According to the Viterbi Algorithm, the output

probability ( | )iP O of re-estimated HMM i with

the observations O is calculated. Then increasing error of output probability between the twice iterations. If the error is satisfied the given error e in the first step, the re-estimated HMM is taken as the final HMM. But if the error is not satisfied given the condition error e, then calculation flows to the step (3).

If the number of iterations exceeds maximization number L, then the training exits.

The training flowchart is shown in Fig. 2. It is important for initial HMM. The description

on the initial HMM is detail in Reference [4] and [8]. 3.4. Faults detection by HMM

HMMs can be trained by the multiple training data, set detail description about this is in reference [4, 5, 7]. For instance, if we have several load conditions for the operation of automotive engine, we can train a single HMM using vibration data set for various load conditions. This work we mainly discuss

the detection of automotive engine in running-up process. Therefore, we can pick up the vibration by repeating the running-up process in order to obtain the multiple training data sets.

Fig. 2. Flowchart for training of HMM.

For the purpose of detecting the presence of a fault of faults, it is sufficient to train a single HMM for the normal running state of automotive engine. The HMM is trained with features extracted from vibration data of normal running. Given the unknown feature matrix, the probability of the HMM for the normal condition is calculated. If the probability is above a predetermined threshold, then there is no fault in the engine system. Otherwise, faults present in the automotive engine system. This is summarized in Fig. 3.

The threshold value can be chosen as follows [5],

min [ ( | )]normal

thr P O

O

(15)

3.5. Faults Diagnosis by HMM

After detecting the presence of a fault in automotive engine, the next goal is diagnosis of fault identification. Diagnosis of a fault, therefore, identifies the source of the fault. For such purpose, a HMM for the normal condition is necessary, but is not sufficient. We also need to train HMMs to represent the automotive engine faults that are of

Input N，M，O, e, L

K-meansalgorithm

jmc jmμjm

Start i=1

Baulm-Welch reestimate

Viterbi algorithm

( | )ip O

1( | ) ( | )i iP P e O O

1i i

?i L

0

i

Exit Final i

Y

N

Y

N


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interest. Once the models are trained, an automotive engine fault can be diagnosed by following the steps shown in Fig. 4.

Fig. 3. HMM based detection.

Fig. 4. Faults diagnosis by HMM.

The c types of faults are considered, the probabilities of the observation sequences given all the HMMs in the previously modeled by history database. The HMM which the probability is maximum, determines the types of fault. 4.Experiments and results 4.1. Experiments Setup

The engine vibration belongs to rotating machinery vibration. The implementation of rotating machinery vibration monitoring and fault diagnosis technology includes four steps: vibration signal acquisition, extraction of feature information, state

identification and diagnosis decision-making. Vibration signal acquisition is the premise of fault diagnosis, and the extraction of information feature is the key to fault diagnosis; in the process of vibration monitoring and fault diagnosis of civil aviation engine, the feature extraction of vibration fault has played an increasingly important role, so the effective and accurate feature extraction will directly determine the implementation of the latter two steps. To accurately and effectively utilize sensors to acquire vibration information of the rotating machinery is the basic requirement to the abnormal findings as well as quick and accurate fault diagnosis. At the same time, how to extract the feature information which can reflect the healthy state of the rotating machinery using the vibration signal measured by sensors is the key factor to ensure the accuracy of fault diagnosis.

Three types Faults simulation experiments are implemented on automotive engine (Santana 2000). Vibration signals are extracted evenly from engine. Then feature vectors are extracted from these vibration signals by FFT spectrum method (described in section 3.1). Finally a sequence of feature FFT vectors is formed into the observation O. Where,

1 2 150{ , , , }O = X X X

During the experiment, vibration data was

collected using three 5 mm piezoelectric sensors. A high-speed computer-based analog-to-digital converter was used to convert and store the acquired vibration data including the optical triggering signal into the computer memory. The positions of sensor mounted are shown in Fig. 5. The vibration data of this paper is collected from the sensor. The accelerometer was mounted on the automotive. On the other words, the faults diagnosis is implemented based three sensors.

Data was collected for four different fault conditions: (i) normal (N), (ii) bolt loose fault (L), (iii) output axis whirling fault (W), and (iv) input air channel fault (I). Vibration signal changes because of above faults. The signal of faults was introduced into computer to analyze its character.

Fig. 5. Positions of measuring sensor.

Automotive Engine

Vibration Pick up

Feature Extraction

1( | )P O

1( | )P O 1( | )P O

{ ( | )} , (1, 2, , )ii m ax P i c O

Vibration pick-up

Vibration pick-up

Vibration pick-up

Vibration pick-up

O

Normal state

Run.1 Run.2

Feature Extraction

Run.3

HMM Training

Unknown state

Run.1

Feature Extraction

Feature Extraction

Feature Extraction

( | )P O

1O 2O 3O

( | ) ,

( | ) ,

P O thr normal

P O thr faulty


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4.2. Faults Diagnosis Result The HMMs which represent four conditions of

(i) normal (N), (ii) bolt loose fault (L), (iii) output axis whirling fault (W), and (iv) input air channel fault (I) are modeled from the initial Model in Fig. 6.

The number of the model density used M=5 mixtures.

As seen from the Fig. 7-13, the probabilities of a given data set is largest correspond to the given HMM, which represents the condition of the data.

Fig. 6. The time series and power spectrum of normal vibration stage.

Fig. 7. The time series and power spectrum of bolt loose fault.

Fig. 8. The time series and power spectrum of output axis whirling fault.

Fig. 9. The time series and power spectrum of input air channel fault.


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Fig. 10. Initialized HMM with 6 hidden states.

Fig. 11. HMM probabilities under output axis whirling.

The results verified that the proposed method in this paper is able to detect and identify the faults of automotive engine in running-up stage. (i) normal (N), (ii) bolt loose fault (L), (iii) output axis whirling fault (W), and (iv) input air channel fault (I). The recognition result is shown in Table 1.

Table 1. Result of Vibration recognition by CHMM.

Engine stage NormalBolt loose

Output axiswhirling

Input air channel

Normal 20 0 0 0 Bolt loose 1 18 1 0 Output axiswhirling

0 0 19 1

Input air channel

1 0 0 19

As seen from the above figures and the recognition results, the probabilities of a given data set is largest correspond to the given HMM, which represents the condition of the data. The results verified that the proposed method in this paper is able to detect and identify the faults of automotive engine in running-up stage. 5. Conclusions

We introduced the continuous density HMM into the detection and faults diagnosis of automotive engine in running-up stage based on vibration signal. In this method, HMM models were trained to represent various running conditions of engine. These models were test with experimental data collected from the automotive engine. It was shown that this method is executable and effective. Acknowledgements

This paper is supported by the "Fundamental Research Funds for the Central Universities", State ethnic Affairs Commission of China, and University of Dalian Nationalities talent import fund (20116202).

References [1]. Alexander Ypma, Learning methods for machine

vibration analysis and health monitoring, Ph.D. Thesis, Delft University of Technology, 2001.

[2]. Shao Qiang, Feng Changjian, Kang Jing. The study and application of dynamic recognition patterns based on HMM, International Journal of Advancements in Computing Technology (EI), Vol. 5, No. 7, April 15, 2013.

[3]. Jose Carmona, Jon L. Barker, Angel M. Gomez, Ma Ning. Speech spectral envelope enhancement by HMM – based analysis/resynthesis, IEEE Signal

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Faults Diagnosis for Vibration Signal Based on HMM...situation. In this situation, the Faults...

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