1
15
th
International Congress on Sound and Vibration
610 July 2008, Daejeon, Korea
LEAST SQUARES SUPPORT VECTOR MACHINE BASED
CONDITION PREDICTION FOR BEARING HEALTH
Fagang Zhao
1
, Jin Chen
1
and Lei Guo
1
1
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University
Shanghai 200240, China
fagang@sjtu.edu.cn
Abstract
Due to the importance of condition maintenance, it is urgent to predict future condition in order to avoid
unexpected failure. So this paper presents a new scheme for the condition prediction of ball bearings
health based on least squares support vector machine (LSSVM). Simulation and the practical
application have been carried out to validate the method. In the practical application, vibration data
which was collected from equipment is used to predict the future condition
1. INTRODUCTION
The manufacturing and industrial sectors are increasingly required to produce more and higher
quality products but avoid accidents as less as possible. As manufacturing equipments become
more complex and sophisticated, machine breakdowns are common. However, failure
conditions are difficult to identify and localize in a timely manner, scheduled maintenance
practices tend to reduce machine lifetime and increase downtime, resulting in loss of
productivity. So in order to prevent unexpected failures from shutdown, and reduce the
economic loss, the abnormal condition should be found as early as possible. Therefore,
condition monitoring and trend prediction is important for condition maintenance [13]. It uses
the features extracted form the raw data to make sure of the machine condition, and to predict
the trend. Trend prediction and residual life prediction is meaningful to maintenance decision.
To fulfil prognostics, there are three steps. At first, the defect or abnormality should be able
to be detected at its early stage and it is better to know which part causes the fault. Secondly, the
part or machine should be monitored continuously, so that we can get the trend data to predict
the machine state in future. At last, a prediction needs to be generated estimating the trend or the
residual useful life (RUL). Above the three steps the third step is the most difficult.
There are many indicators to detect the fault of equipments, and these indicators can also be
used to track the trend and predict the future condition. But how to select useful indicators as
the prediction parameters is difficult for researchers. There are many indicators, such as
timedomain statistical indicators: PeakPeak (PP), Root Mean Squares (RMS), Crest Factor,
Skew and Kurtosis; wavelet index, energy factor etc, there are so many indicators that we can
not use any indicator to predict the condition or residual life. According to [4] [5], we choose
RMS as the indicator in this research. Furthermore, to select the proper model is difficult in the
prediction of residual life, some researchers constructed the prediction models based on crack
propagation models: namely Paris Law [6], [7]. Ref. [8] uses neural network to predict bearing
life, and compares with the real life. Wang et al. [9] compared the results of applying recurrent
neural networks and neural–fuzzy inference systems to predict the fault damage propagation
trend. Yan et al. [10] employed a logistic regression model to calculate the probability of failure
ICSV15 • 610 July 2008 • Daejeon • Korea
2
for given condition variables and an ARMA time series model to trend the condition variables
for failure prediction. Wang and Vachtsevanos [11] applied dynamic wavelet neural networks
to predict the fault propagation process and estimate the RUL as the time left before the fault
reaches a given value. Yam et al. [12] applied a recurrent neural network for predicting the
machine condition trend. Wang and Lee proposed a waveletneural network prediction
algorithm for performance evaluation, and evaluated and predicted the wearing condition of
machine spindle and cutting tools [13], and so on. In recent years, because the industry is in
urgent need of condition prediction and residual life prediction, the research in the field of fault
diagnosis is transferred to condition monitoring and prediction. So now researchers focus on
how to predict the future condition of machines intelligently and accurately, and reduce the
frequency of sudden accidents.
In this paper, we propose a new scheme for the condition prediction of a ball bearing health
based on least squares support vector machine (LSSVM). This scheme can effectively research
equipments’ whole life cycle from the first time it comes into use to the final failure. In order to
validate the model, we carry out an experiment to test the new method. Fig. 1 is the whole flow
chart of this research.
Figure
.1.
the overall flow diagram of this research
2. THEORETICAL BACKGROUND OF LSSVM [14]
Vapnik proposed support vector machines (SVM) method based on statistical learning [15].
Traditional support vector machine gets the solutions with optimal quadratic function. In the
process of optimal solution, the dimension of the matrix is related with the number of training
samples directly, and it is feasible to use inner product to solve the mediumscale optimal
solution. But to largescale, the matrix should be decomposed or trimmed to reduce the
complexity. Much research has been done in the largescale optimal solution. However, they
still use quadratic inequality constraints, which cost much time and can not process realtime
data. So it usually has to be used to process offline data, which constricts the application of
SVM. Suykens [14] introduced variance term in the optimal function of SVM, and changed
constraints from inequality to equality, and then proposed SVM based on the equality
constraints, which is called Least Squares Support Vector Machine (LSSVM). Since the
variance term was introduced into LSSVM, optimal function of traditional SVM changed into
equality constraints, which the solution has changed from optimal quadratic function to linear
function, simplified the complexity of solution.
ICSV15 • 610 July 2008 • Daejeon • Korea
3
The LSSVM algorithm is as follows. Suppose the training set:
( )
{
}
, 1,2,....,
k k
D x y k N= =
,
n
k
x
R∈
,
m
k
y R∈
Where
k
x
is the input data,
k
y
is the output data, in the primal space (
w
space), the
optimization problem can be describe as:
( )
2
,,1
1 1
,,
2 2
min
M
T
L
S i
w B i
L w B w w
ε
ε
γ ε
=
= +
∑
(1)
Subject to the equality constraints:
( )
,1,...,
T
i i i
y w x B i M
ϕ ε= + + =
(2)
Where the nonlinear mapping
ϕ
:
n m
→
maps the input data into a high dimensional feature
space, which can be infinite dimensional. In the high dimensional feature space, the super
classification face is defined by
n
w R∈
,
B
∈
.
w
is weight vector in the high dimensional
feature space,
B
is the bias term, and
i
ε
is the error variable,
γ
is the adjusting factor, and Eqn.
(1) is formula of the least squares support vector machine, which has been investigated by
Saunders et al [16] and Suykens & Vandewalle [17].
According to the optimal function Eqn. (1), we can define the Lagrange function:
( ) ( ) ( )
( )
1
,,;,,
M
T
LS LS i i i i
i
L w B J w B w x B yε α ε α ϕ ε
=
= − + + −
∑
(3)
Where
i
α
denotes Lagrange multiplier, and the KTT optimality function is
( )
( )
1
1
0
0 0
0,1,...,
0 0,1,...,
M
i i
i
M
i
i
i i
i
T
i i i
i
LS
w x
w
LS
B
LS
i M
LS
w x B y i M
αϕ
α
α γε
ε
ϕ ε
α
=
=
∂⎧
= → =
⎪
∂
⎪
∂
⎪
= → =
⎪
∂
⎪
⎨
∂
⎪ = → = =
∂
⎪
⎪
∂
⎪
= → + + − = =
∂
⎪
⎩
∑
∑
(4)
After eliminating of
w
and
i
ε
, we can get the following set of linear equations.
1
0
0 1
1
T
B
y
K I
α
γ
−
⎡ ⎤
⎡
⎤ ⎡ ⎤
=
⎢ ⎥
⎢
⎥ ⎢ ⎥
+
⎣
⎦ ⎣ ⎦⎢ ⎥
⎣ ⎦
(5)
Where
[ ]
1
...
M
x
x x=
,
[
]
1
;...;
M
y y y=
,
[
]
1 1;...;1
=
,
[
]
1
;...;
M
α
α α=
,1,...,
i j N
=
. As in the SVM
theory, according to Mercer’s condition, the matrix
K
can be written as
( )
( )
(
)
,
T
ij i j i j
K K x x x xϕ ϕ= =
(6)
Then the function estimation of LSSVM is
( )
1
N
i ij
i
y
x K b
α
=
=
+
∑
(7)
Where
i
α
and
b
can be computed by Eq. (5). RBF kernels one can take [15]
(
)
2
2
exp
ij i j
K x xη= − −
(8)
We can see from above that all the constraints have changed to be the equations, and we can
solve the linear equations to get the results. Obviously, linear equations can solved by least
ICSV15 • 610 July 2008 • Daejeon • Korea
4
square, which makes the computation easy and reduces computation time, So LSSVM has
strong adaptability.
Furthermore, we choose normalized rooted mean squares error (NRMSE) as the index to
decide whether the prediction result is good or not. The expression is:
( )
2
,,
1
1
1
N
i pre i obs
i
obs
O O
N
NRMSE
S
=
−
−
=
∑
(9)
Where
N
is the number of prediction data;
obs
S
is the standard deviation of samples;
,
i pre
O
is
the predicted value; and
,
i obs
O
is the true value at the time of
i
.
3. METHOD
In this research, we propose the method of LSSVM to predict machines condition. The flow
chart of the proposed method is given in Fig.2.
Figure
.2. Flowchart of predict method with LSSVM
There are many basis functions for LSSVM, such as radial basis function (RBF) kernel,
linear function, polynomial function, wavelet function. RBF based LSSVM has a good
adaptability to vibration signals, its robustness is better than the other basis functions based
LSSVM, and its prediction preciseness is better than traditional SVM and neural network,
computation time is very small, but high efficiency. So In this Paper, RBF kernel will be used,
which is defined as:
2
2
(,) exp
2
K
σ
⎛ ⎞
−
=
⎜ − ⎟
⎜ ⎟
⎝ ⎠
x
y
x y
(10)
In order to get the more precise result, we utilize the leaveoneout cross validation approach.
The kernel width and the regularization parameter must be decided when we use the RBF
kernel. In this paper, we adopt a method to determine these parameters based on the
crossvalidation idea. We define two data sets, namely the training set and the validation set
from the observed time series, respectively. The prediction error is estimated via cross
validation and when the model provides the lowest estimated error,
σ
is chosen. It can be
shown that for large data sets, cross validation is asymptotically equivalent to analytical model
selection. In this case, the computational cost of cross validation in terms of computational time
and training time is high.
ICSV15 • 610 July 2008 • Daejeon • Korea
5
4. SIMULATION
In this section we present the results of the simulations and compare with the traditional
LSSVM method. In the research of time series prediction, sunspot series and MackeyGlass
time series are often used to test the algorithm. Here we use sunspot series. The sunspot data
extracted from Matlab toolbox. It is a sample of size
280m
=
. The first 200 values of the
sunspot data is used to train the model and the remaining values are used to predict. The
NRMSE yielded by LSSVM is 1.758. For the sunspot dataset (normalized) we see that the
LSSVM model provides us a good result. This method is significant when compare with
traditional LSSVM. To illustrate the performance of the LSSVM, the predicted time series are
shown in Fig. 3.
5
10
15
20
25
30
35
40
−
0.4
−
0.2
0
0.2
0.4
0.6
0.8
1
Predicted Serial
Real Serial
Figure
. 3.
The predicted result of the sunspot
5. EXPERIMENT
An experiment of condition monitoring is set up for validate the model. Fig. 4 is the position
where the sensor is installed.
Figure
. 4. Photo of equipment installed sensors
Then through data acquisition, preprocessing, feature extraction and feature reduction,
training samples and test samples are obtained. After that, train LSSVM with the training
samples which are the time series. At last the model is employed to predict future condition and
compared with the result of traditional LSSVM. Fig. 5 is the sketch of data acquisition system,
which includes sensors and signal conditioner, antialiasing filter, data acquisition computer,
oscilloscope and
dynamic analyzer
. Signals are probed by sensors. Then after signal conditioned
ICSV15 • 610 July 2008 • Daejeon • Korea
6
and antialiasing filtered, the information is collected by computer. Oscilloscope and online
monitoring system are employed to analyze the validity of the signals. Fig. 6 is the result of the
predicted experiment series (normalized) used LSSVM.
0
5
10
15
20
25
30
35
40
4
5
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
RMS Value
Predicted Serial
Real Serial
Figure 5. Sketch of data acquisition system Figure
. 6.
The predicted result of the experiment
6. CONCLUSIONS AND FUTURE WORK
According to the nonlinearity of bearings vibration, LSSVM model is introduced into times
series prediction of vibration in this paper to predict bearing condition.
To provide a more reliable and realtime prognostic tool for the bearing condition, we
developed LSSVM prediction approach to predict the behaviour of dynamic systems in this
paper. According to the example given above, we can see that it is useful to implement for both
bearing residual life prediction and condition prediction. Test results of this study showed that
the LSSVM model is a reliable forecasting tool. It can capture the system’s dynamic behaviour
quickly and track the system’s features accurately. It is also a robust forecasting tool in terms of
its capabilities to accommodate different system operation conditions and variations in
system’s dynamic characteristics.
There are two aspects for the further research: one is to implement the predictor in other
complex industrial facilities and to develop new strategies for multistep predictions, the other
one to find out if there is a better method to predict the time series or not.
7. ACKNOWLEDGEMENTS
The research was supported by the National Natural Science Foundation of China (Approved
Grant: 50675140) and the National High Technology Research and Development Program of
China (863 Program, NO. 2006AA04Z175).
REFERENCES
[1]
Vichare, N, and Pecht, M., "Prognostics and Health Management of Electronics", Trans. on
Components and Packaging Technologies, IEEE 29, 222229 (2006).
[2]
Wang, W., “A twostage prognosis model in condition based maintenance”. European Journal of
Operational Research 182(3), 11771187 (2007).
[3]
[W. Wang, “A model to predict the residual life of rolling element bearings given monitored
condition information to date”, IMA Journal of Management Mathematics 13, 316 (2002).
ICSV15 • 610 July 2008 • Daejeon • Korea
7
[4]
T. Williams, X. Ribadeneira, S. Billington, T. Kurfess, “Rolling element bearing diagnostics in
runtofailure lifetime testing”, Mechanical Systems and Signal Processing 15 979–993 (2001).
[5]
Runqing Huang, Lifeng Xi. “Residual life predictions for ball bearings based on selforganizing
map and back propagation neural network methods”, Mechanical Systems and Signal Processing
21, 193–207 (2007).
[6]
Yawei Li, Dynamic prognostics of rolling element bearing condition, Ph. D. dissert, Georgia
Institute of technology, 1999.
[7]
Tara Reeves Lindsay. Applying adaptive prognostics to rolling element bearings, Master Dissert,
Georgia Institute of technology, 2005.
[8]
N. Gebraeel, M. Lawley, R. Liu, V. Parmeshwaran, “Residual life predictions from
vibrationbased degradation signals: A neural network approach”, IEEE Transactions on
Industrial Electronics 51, 694–700 (2004).
[9]
W.Q. Wang, M.F. Golnaraghi, F. Ismail, “Prognosis of machine health condition using
neurofuzzy systems”, Mechanical Systems and Signal Processing 18, 813–831 (2004).
[10]
J. Yan, M. Koc, J. Lee, A prognostic algorithm for machine performance assessment and its
application, Production Planning and Control 15, 796–801 (2004).
[11]
P. Wang, G. Vachtsevanos, “Fault prognostics using dynamic wavelet neural networks”, AI
EDAMArtificial Intelligence for Engineering Design Analysis and Manufacturing 15, 349–365
(2001).
[12]
R.C.M. Yam, P.W. Tse, L. Li, P. Tu, “Intelligent predictive decision support system for
conditionbased maintenance”, International Journal of Advanced Manufacturing Technology 17,
383–391(2001).
[13]
Wang X, Yu G, Koc M, Lee J. “Wavelet neural network for machining performance assessment
and its implication to machinery prognostic”. Proceedings of MIM 2002: 5th International
Conference on Managing Innovations in Manufacturing (MIM), Milwaukee, Wisconsin, USA,
150156 (2002).
[14]
Suykens J.A .K, Vandewalle J, and De Moor B. “Optimal Control by Least Squares Support
Vector Machines”, Neural Networks 14, (2001) 2335.
[15]
V. N. Vapnik, “Statistical Learning Theory”. John Wiley and Sons Inc., New York, 1998.
[16]
Saunders C., Gammerman A., Vovk V., “Ridge Regression Learning Algorithm in Dual
Variables”, in Proceedings of the 15th International Conference on Machine
Learning,MadisonWisconsin, 515521 (1998)
[17]
Suykens,J. Vandewalle. “Least squares support vector machine classifiers”. Neural Process,
Letters 9, 293300 (1999).
Enter the password to open this PDF file:
File name:

File size:

Title:

Author:

Subject:

Keywords:

Creation Date:

Modification Date:

Creator:

PDF Producer:

PDF Version:

Page Count:

Preparing document for printing…
0%
Comments 0
Log in to post a comment