vibration based condition monitoring of … · · 2016-07-031 vibration-based condition...

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VIBRATION-BASED CONDITION MONITORING OF COMPRESSORS BY PRINCIPAL COMPONENT ANALYSIS AND DISCRIMINANT ANALYSIS

Primož Potočnik

University of Ljubljana, Faculty of Mechanical Engineering, 1000 Ljubljana, Slovenia

email: [email protected]

Edvard Govekar

University of Ljubljana, Faculty of Mechanical Engineering, 1000 Ljubljana, Slovenia

Vibration-based condition monitoring and fault detection approach for compressors built in re-

frigeration appliances is proposed. The method combines feature extraction and principal com-

ponent analysis (PCA), and compares unsupervised k-means clustering and discriminant analy-

sis (DA). The method is demonstrated on a case study based on a large dataset of 10.000 indus-

trially acquired vibration measurements of compressors during the production of refrigeration

appliances. The initial step of the proposed method is feature extraction, based on statistics, sta-

tistical moments, and spectral analysis. A selected single feature was applied for statistical de-

tection of initial compressor faults, based on which the initial compressor classes were defined

as ‘normal’, ‘noisy’, and ‘inactive’. In the next step, extracted features were transformed by

PCA and only the first two principal components, contributing over 90% of variability, were re-

tained for subsequent analysis. Three initial classes were applied to initialize DA. The results of

linear DA revealed many additional ‘noisy’ and ‘inactive’ samples that were not evident from a

single extracted feature. Furthermore, an additional cluster defining new class ‘unstable’ was

detected, indicating a new type of defect characterized by high vibration transients. Results of

DA reveal decision boundaries between all classes, and confirm the efficiency of the proposed

method. Finally, the results are compared also with an unsupervised k-means clustering which

shows that unsupervised clustering doesn’t provide appropriate decision boundaries. The pro-

posed DA-based approach detects compressors with defects and has the potential to detect novel

classes of unusual or faulty operation. The method can be effectively applied for industrial con-

dition monitoring of compressors.

1. Introduction

Condition monitoring (CM) of machines and products is an established and important part of

successful modern industrial production. In order to manufacture fault-less products, various non-

destructive condition monitoring approaches can be applied during the production process. Vibra-

tion signal analysis [1, 2] continues to be one of the most useful and popular CM methods, beside

other acoustic and acoustic emission based approaches [3]. Methods for the analysis of vibration

signals include statistical methods, wavelets [4], psychoacoustic approaches [5], neural and fuzzy

logic based methods [6], and many modern machine learning methods, such as support vector ma-

chines [7, 8].

In this paper, vibration-based condition monitoring and fault detection approach for compressors

built in refrigeration appliances is discussed. Whereas particular CM solutions for different produc-

tion stages of reciprocating compressors have already been proposed [9-14], in this paper solutions

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International Congress on Sound and Vibration

2 ICSV23, Athens (Greece), 10-14 July 2016

for final vibration-based inspection of compressors [15-17] are discussed. The existing approaches

provided in the literature are based on supervised learning, where appropriate sets of samples repre-

senting normal and faulty operation are available. In this paper, semi-supervised approach is con-

sidered, which is based only on a few samples of faulty compressors. The method is therefore high-

ly relevant for industrial application where expert-based sets of labelled sets are difficult and expen-

sive to obtain.

The proposed method is based on feature extraction, principal component analysis (PCA), and

discriminant analysis (DA). The method is demonstrated on a case study comprising 10,000 vibra-

tion measurements of compressors acquired during the industrial production of refrigeration appli-

ances. The method is initiated with feature extraction and then statistical detection of initial com-

pressor faults based on selected single feature. Initial results are translated into PCA space which

reveals clusters of compressors with different operating conditions. Finally, discriminant analysis is

applied to determine decision boundaries between various categories of compressors. Results are

compared also with an unsupervised k-means clustering which confirms superior performance of

the proposed DA-based approach.

2. Measurements

An industrial vibration-based condition monitoring system (Figure 1) was developed for compa-

ny Gorenje in order to provide measurements of compressors built into refrigeration appliances.

During the operation of an industrial production line, 10,000 vibration measurements of compres-

sors were acquired. Special custom-made flexible sensor head was developed for automated ad-

justment of accelerometer to various compressor types. Inside the sensor head, the accelerometer

PCB 352B was installed for high sensitivity vibration measurements. Each measurement was ac-

quired in the duration of 2 seconds with a sampling frequency 25.6 kHz. Figure 2 shows custom-

made flexible sensor head and its position on the compressor during vibration measurement. All

measurements were acquired without any labelled information about the quality of the measured

compressor. Therefore, this information has to be estimated based on the analysis of the measure-

ments.

Figure 1: Industrial data acquisition and condition monitoring system.

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ICSV23, Athens (Greece), 10-14 July 2016 3

Figure 2: Custom-made flexible sensor head and its position on the compressor during measurement.

3. Methods

The following methods were applied in this study: principal component analysis, discriminant

analysis, and k-means clustering.

3.1 Principal component analysis

The central idea of principal component analysis is to reduce the dimensionality of a data set

consisting of a large number of interrelated variables, while retaining as much as possible of the

variation present in the data set [18]. This is achieved by transforming initial variables to a new set

of variables, the principal components (PCs), which are uncorrelated, and which are ordered so that

the first few retain most of the variation in all of the original variables. In our case study, the set of

extracted features were first rescaled to zero mean and variance one, and then PCA analysis was

applied to extract principal components.

3.2 Discriminant analysis

The discriminant analysis (DA) is a classification method used to find discrimination boundaries

which separate (discriminate) two or more classes of objects [19, 20]. DA assumes that different

classes generate data based on different Gaussian distributions. In our research, linear discriminant

analysis (LDA) and quadratic discriminant analysis (QDA) [21] were applied. Both LDA and QDA

have been shown to rank among the top classifiers. The reason for such a good track record is most

probably the bias−variance trade-off where the data can only support simple decision boundaries

such as linear or quadratic, and the estimates provided via the Gaussian models are stable [22].

3.3 K-means clustering

The k-means algorithm is one of the most popular and commonly used clustering algorithms

employing a squared error criterion [23, 24]. The algorithm starts with a random initial partition and

keeps reassigning the patterns to k clusters based on the similarity between the pattern and the clus-

ter centres until a convergence criterion is met. The k-means algorithm is popular because it is easy

to implement. A major problem with this algorithm is that it is sensitive to the selection of the initial

partition and may converge to a local minimum of the criterion function value if the initial partition

is not properly chosen. The problem can be practically solved by several restarts of the algorithm

each time with new random seed and then taking the best solution as a final clustering result.

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4. Solution approach

The proposed solution approach is designed for industrial vibration-based condition monitoring

systems where no prior information regarding the quality of inspected products is available. There-

fore a supervised approach based on samples of normal and faulty products can not be applied, and

instead unsupervised or semi-supervised methods should be considered. Such situation is typical for

industrial condition monitoring where expert based acquisition of samples with normal and faulty

operation is difficult, expensive and time consuming. Consequently, we propose a semi-supervised

condition monitoring approach that consists of initial statistical isolation of representative faults,

followed by application of classification methods for more detailed recognition of faulty products.

The proposed solution approach, applied in our case study to the industrial condition monitoring of

compressors, consists of the following steps:

1. Feature extraction from vibration signals based on statistics, statistical moments, and spectral

analysis.

2. Application of a selected single feature z1 for statistical detection of initial compressor faults.

Based on statistical analysis and inspection of outliers, the initial compressor classes are defined

as ‘normal’, ‘noisy’, and ‘inactive’.

3. Transformation of extracted features by principal components analysis in order to reduce di-

mensionality. PCA results show that the first two principal components contribute over 90% of

variability and therefore only these two components are retained for subsequent analysis. The

visualization of measurements in 2-dimensional PCA space reveals clusters of compressors with

similar properties.

4. Application of discriminant analysis: initial statistically determined compressor faults are used

to initialize DA, and resulting decision boundaries determine regions of normal or faulty opera-

tion.

5. An additional cluster, defining a new class ‘unstable’ is visible in the projection in the PCA

space, indicating a new type of defect characterized by high vibration transients. DA is applied

with this new class included, and the final result of DA shows decision boundaries between all

four classes, namely ‘normal’, ‘noisy’, ‘inactive’, and ‘unstable’.

6. Finally, the results of discriminant analysis are compared also with an unsupervised k-means

clustering which shows that unsupervised clustering doesn’t provide appropriate decision

boundaries.

5. Results

The results of the proposed solution approach to the industrial vibration-based condition moni-

toring of compressors are presented in this section.

5.1 Feature extraction

Feature extraction step provides initial data compression from time domain vibration signals into

the set of features characterizing each measurement. In our study the following features were ex-

tracted from vibration signals:

a) 10 statistical features:

log M1 (logarithmic mean absolute value of the vibration signal)

log M2, log M4, log M6 (logarithmic moments),

kurtosis,

run var M2 (running variance of M2, calculated across 0.1 second segments),

perc 75, perc 90, perc 95, perc 98 (percentiles of absolute value of the signal).

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b) 17 features based on spectral analysis (components of power spectrum):

P1 (0 Hz), P2 (800 Hz), P3 (1600 Hz), …, P17 (12800 Hz).

The set of 27 extracted features (z1… z27) for each compressor provided initial information for

subsequent analysis.

5.2 Statistical detection of initial faults

Figure 3 shows selected single feature z1 (log M1) for n = 10,000 compressors with marked ini-

tial outliers representing ‘noisy’ and ‘inactive’ faults, and a set of median samples representing

‘normal’ compressors. These samples represent initial classes of compressors of different qualities.

Figure 3: Selected single feature z1 with marked initial compressor classes.

5.3 PCA transform

Extracted features were compressed into the principal components, and the first two components

contribute over 90% of variability, therefore only these two components were used for subsequent

analysis. Figure 4 shows measurements in resulting 2-dimensional PCA space which reveals clus-

ters of compressors with similar properties.

5.4 Application of discriminant analysis

Based on initially defined compressor classes, LDA is applied to determine decision boundaries

between ‘normal’ class and each faulty class. Figure 5 presents initial samples, defining classes

‘normal’, ‘noisy’, and ‘inactive’, and resulting LDA boundaries that reveal many additional ‘noisy’

and ‘inactive’ samples that are not evident from a single extracted feature.

Figure 4: Samples in the first two PCA axes with

marked initial compressor classes.

Figure 5: Linear discriminant analysis with class

borders for three classes of compressors.

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5.5 Detecting additional ‘unstable’ class

Figure 6 presents initial sample of a new class ‘unstable’, and the resulting decision boundaries

defined by LDA. This final result represents decision strategy capable of classifying the quality of a

compressor according to one of the four classes: ‘normal’, ‘noisy’, ‘inactive’, and ‘unstable’. It is

also possible to apply QDA to isolate just two classes of compressors, namely ‘normal’ and ‘faulty’

that includes all irregular compressor operations, as presented in Figure 7.

Figure 6: Linear discriminant analysis with class

borders for four classes of compressors (inclu-

ding the additional ‘unstable’ class).

Figure 7: Quadratic discriminant analysis with class

border between ‘normal’ and ‘faulty’ compressors.

5.6 Comparison of results with k-means clustering

For comparison with DA, k-means clustering results are shown in Figure 8. Results represent un-

supervised approach which does not provide appropriate decision boundaries, as demonstrated by

semi-supervised approach based on DA described above.

Figure 8: Classification results obtained by k-means clustering.

6. Conclusions

The paper presents a semi-supervised approach to industrial condition monitoring based on anal-

ysis of vibration signals. The proposed method is appropriate for industrial situations where very

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limited or no prior information about product classes is available, and therefore statistical methods

have to be applied first in order to acquire initial faulty conditions. Based on initial defined classes

of normal and abnormal operation, the methods proposed in this paper can be applied to determine

condition of products by defining decision boundaries using principal component analysis and dis-

criminant analysis. The method has been succesfully applied to an industrial case study, namely the

condition monitoring of compressors built in refrigeration appliances. The method detects compres-

sors with defects and has the potential to detect novel classes of unusual or faulty operation.

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