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A fault tolerant model for multi-sensor measurement
Li Liang, Shi Wei
PII: S1000-9361(15)00084-9
DOI: http://dx.doi.org/10.1016/j.cja.2015.04.021
Reference: CJA 471
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Received Date: 25 April 2014
Revised Date: 13 April 2015
Accepted Date: 13 April 2015
Please cite this article as: L. Liang, S. Wei, A fault tolerant model for multi-sensor measurement, (2015), doi:
http://dx.doi.org/10.1016/j.cja.2015.04.021
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Contents lists available at ScienceDirect
C
hinese Journal of Aeronautics 2
8
(201
5
) xx
-
xx
A fault tolerant model for multi-sensor measurement
Li Lianga,*, Shi Weib
a Information and Ecommerce Institute, University of Electronic Science & Technology of China, Chengdu 610054, China
b China Gas Turbine Establishment, Mianyang 621703, China
Received 28 April 2014; revised 8 January 2015; accepted 13 March 2015
Abstract
Multi-sensor systems are very powerful in the complex environments. The cointegration theory and the vector er-
ror correction model, the statistic methods which widely applied in economic analysis, are utilized to create a fitting
model for homogeneous sensors measurements. An algorithm is applied to implement the model for error correction,
in which the signal of any sensor can be estimated from those of others. The model divides a signal series into two
parts, the training part and the estimated part. By comparing the estimated part with the actual one, the proposed
method can identify a sensor with possible faults and repair its signal. With a small amount of training data, the right
parameters for the model in real time could be found by the algorithm. When applied in data analysis for aero engine
testing, the model works well. Therefore, it is not only an effective method to detect any sensor failure or abnormali-
ty, but also a useful approach to correct possible errors.
Keywords: Multi-sensor; Fault tolerant; Cointegration; Turbine engine; Measurement
*Corresponding author. Tel.: +86 028 83201114
E-mail address: liliang@uestc.edu.cn
1. Introduction
Multi-sensor systems are widely applied in complex
environments to measure signals in many places. Al-
titude Test Facility of China is a huge laboratory for
aircraft engines, where hundreds of sensors for many
purposes are put inside an engine during testing. In
each cross-section of the engine, there are several
uniform linear sensor groups, e.g., seven sensors for
temperature, five sensors for pressure and two sensors
for vibration. When any unusual event happens, at
least one sensor’s signal inside the engine would be
abnormal. Therefore, by comparing the outputs from
the sensors, engineers can find out the possible fault.
Usually, there are two reasons for an abnormal signal.
One happens when a sensor itself breaks down, and
the other happens when something around a sensor
goes wrong. The traditional signal processing models
to fit a test signal are auto-regressive and moving av-
erage (ARMA) model and Wavelet etc. However,
their performances are not satisfactory for
non-stationary signals. That is why the multi-sensor
approach is becoming popular in many scenarios es-
pecially in engine testing.1-4 The redundant data from
multiple sensors can improve both robustness and
accuracy.5,6
The aim of this study is to establish a fault tolerant
mechanism by investigating the relationship among
the output signals of these engine testing sensors.
When one signal deviates too far from its normal po-
sition, we can identify it and give an alarm. Mean-
·2 · Chinese Journal of Aeronautics
while, we can restore a deviated signal to the right
position in which it “ought to be”. A fault tolerant
multi-sensor system is indispensable when one sensor
undergoes impairment or when an unusual event such
as oil leak happens.7-9 The output of one sensor could
be substituted by a combination of the outputs of oth-
ers in the same sensor group, in case that there is
something wrong around this sensor. In this paper, a
novel approach is presented that can repair the signal
from an abnormal sensor.
There are many papers discussing the data fusion
and relationship among sensors in a multi-sensor ar-
chitecture. In engine testing, each sensor in a sensor
group works individually and independently, and it is
impossible to put different sensors in exactly the same
place. Therefore, there must be some difference be-
tween the outputs of any two sensors in the same sen-
sor group no matter how close they are. Some papers
use an absolute value of the difference as the distance
between the outputs of two sensors, while more others
use a relative value as the distance by considering
their covariance.10-13 One of them uses an “arcot”
function to limit the distance within 0 to 1, and one
uses the Minkowski distance.14 All the papers above
have attempted to find a reasonable relationship for
two signals and use their distance for data fusion.
All the papers above did provide some useful tech-
niques, however, they have two drawbacks. Firstly,
their methods attempt to reconcile the difference by
using a non-existent “center”. It is thought that noises
and drifts make signals deviate from a right pathway
and this pathway is the “center” for all the signals.
Meanwhile, the goal of data fusion is thought to find
this center (right pathway) that all the signals ought to
have taken, so most of the sensor fusion papers have
used different methods to obtain the weights by
Eq. (1)with i
v the value and i
w the weight of
signal i.
E =
=
n
i
ii vw
1
(1)
Usually, E is the so-called “center value”, and a
signal that has a shorter distance from others is as-
signed a heavier weight in Eq. (1). Theoretically, this
might be true, but it is not true in reality. Due to limi-
tations in size, each sensor has to be put in different
locations. Each sensor measures its source in its own
location individually, and consequently, each output
signal goes its own way. Therefore, we believe there
is no center or “right pathway” for all the different
sensors. One cannot tell where the right pathway
suitable for all the sensors is. We believe that the rela-
tionship among sensors can be exploited for signal
correction or amendment, but there is no center. Sec-
ondly, many researchers do not treat signals from
sensor groups as time series. In their researches, mon-
itoring of gas turbine engines uses either snapshot
data at a time instant from various sensors or a win-
dow of time series data from selected sensor observa-
tions.15 Thus, in data fusion, engine studies only
compare the signals from different sensors at time t,
and ignore the signals at time t1, t2, t3 and so
on. Therefore, in multi-sensor engine testing, time
series analysis is usually not applied. Actually, the
signals at time t usually are more relevant to the sig-
nals at previous times. With previous time information,
people can find and handle fault propagation in an
aircraft engine test.16 In this paper, we try to build the
following model for an n-sensor group as Eq. (2).
1 2
others
( , , ,...)
( 1, 2, , ; 1, 2, )
t t t t
i k k
S f S S S
i n k
− −
=
= =
(2)
where f () is a transform function; t
i
S is the output
of sensor i at time t;
others
t
S represents the outputs of
other n1 sensors excluding sensor i at time t;
t k
k
S
−
represents the outputs of all sensors at time tk.
Fig. 1 shows the sensor signals from a sev-
en-sensor group, in which seven sensors (named sen-
sor 1, sensor 2,
, sensor 7, respectively) are put
on a cross-section of T23 in a jet engine. It is obvious
that the signal from sensor 3, goes very high and is
apparently abnormal. Is there any way to find the path
that the signal of sensor 3 ought to have taken? Or
whether we can restore the “original signal” of sensor
3? The studies which do not use time series analysis,
Chinese Journal of Aeronautics · 3 ·
provide no answers. So we will try a brand new way
in this study. To repair the signal from sensor 3, we
have to find a model that one signal can be substituted
by others. In the following sections, a statistical
method will be used to discover the possible relation
between sensor signals which will be used to build a
mathematical model.
Fig. 1 Signals from seven-sensor group with sensor 3
signal abnormal.
When dealing with non-stationary signals such as
those in Fig. 1, people tends to use difference calcula-
tion to make them “stationary”, such as ARIMA.
However, this way conceals the trend of the signal
data. In many cases, the data after difference calcula-
tion has nothing to do with the original ones. There-
fore, it is not easy to explain the phenomena as we
lose the important information hidden in the original
data.
Cointegration is an important statistical method to
describe the relation among multiple time-series data.
17 In the cointegration model, one signal could be re-
placed by others. Cointegration was first used in ex-
plaining economical phenomena, and now its applica-
tion has extended into many other fields. Though
cointegration is not used so often in engineering,
some papers can still be found. Kaufmann et al.18 used
cointegration for two sensors to analyze the relation
between the solar zenith angle and advanced very
high resolution radiometer data. Pan and Chen19 used
cointegration for four sensors in car engine testing. Lu
and Chen20 used cointegration for four sensors in a
hydraulic flap servo system. In those studies,
cointegration means a linear combination of variables
and eliminates the stochastic trend in data, so for mul-
tivariable, especially non-stationary signals,
cointegration is a powerful tool.
In this paper, cointegration is applied on sensor
signals in engine testing. As it is very difficult to
identify the pattern of non-stationary signals, the
proposed method does not examine the signals di-
rectly. Instead, we use the combination of the signals
to find out the relation among them, and then use the
relation to build a fault-tolerant model. Unlike in
other papers, cointegration is only used as a statisti-
cal tool for discovering the relation among the entire
range of data. In this study, it is also used for fore-
casting. We forecast each small fragment of a sensor
signal, and compare with the actual one in real time.
The results will be used to diagnose whether there is
any fault and, if so, to repair the signal. The way us-
ing diagnosis and prognosis together for mul-
ti-sensor equipment can also be found in some pa-
pers.15,21
2. Relation of homogeneous sensors
Take the assumption that the output of a sensor is a
linear function.22
ckxy
+
=
(3)
where x is the source intensity of the signal; y is the
observed output of the sensor; k is the sensitivity of
the sensor; c is the constant offset, which can be used
to explain the distance between the locations of the
sensor and the source. If the original source x is
non-stationary, its measurable output y is also
non-stationary. To simply Eq. (3), we do not include
the noise, which is usually added to the end of the
right side of Eq. (3). Suppose that each sensor’s noise
is white noise or can be counteracted with one another.
Since only y could be measured directly, theoretically
·4 · Chinese Journal of Aeronautics
the accurate value of x at time t could never be found.
Let’s take the assumption that we have two homoge-
nous sensors (sensor i and sensor j), which are put
close to each other, to measure a signal source. Then
we know that,
( , 1, 2, , ; )
i i i
j j j
y k x c
i j n i j
y k x c
= +
= ≠
= +
where the subscripts i and j represent the correspond-
ing physical meaning of sensor i and sensor j. The
relation of the two outputs can be written as
βα
+=+−= jijj
j
i
iyccy
k
k
y)( (4)
where and
i j i i j j
k k c k c k
α β
= = − are the re-
gression parameters.
In Eq. (4), one signal can be substituted by another
one. Furthermore, if there are more than two sensors
in the above scenario, Eq. (5) could be deduced.
≠
+=
ij
jji yy
αβ
(5)
where
j
α
is the regression parameter of sensor j. So
we can replace one signal with a linear combination of
others. Then rewrite Eq. (5) as
≠
+−=
ij
jjiii yy
αβε
(6)
where i
ε
is the residual error of the regression for
sensor i which can be regarded as the noise to the
sensor inside the engine;
i
β
is the regression param-
eter of sensor i. If Eq. (5) is right, then i
ε
should be
the white noise. Therefore, by examining i
ε
, we can
verify the assumption.
Now take a look at time series t
i
y, the measured
value of a non-stationary signal i from the ith sensor
at time t. It should be noted that in this paper, the su-
perscript is for time and the subscript is for the num-
ber of a sensor.
According to the cointegration theory, if several se-
ries are individually integrated but their linear com-
bination has a lower order of integration, then the
series are said to be cointegrated. A sensor group in
jet engine testing has several homogeneous sensors
that are put together, so signals from them must have
some long-term relations. These relations could be
expressed as a linear combination. For time series,
rewrite Eq. (5) for signals from n sensors as
1 1
m n
t t t p
i i j j pj j
j i p j
y y y
β α γ
−
≠ = =
= + +
(7)
where m is the autoregressive order;
pj
γ
is the re-
gression parameter. For the real-time performance, m
cannot be set to a large number. Suppose the vector of
the n-sensor signals at time t is
T
1 2
, , ...,
t t t t
n
y y y
=
Y
For the vector form, we can rewrite Eq. (7) as
0
1
m
t t t p
p
p
−
=
= + +
YYY
Π
ΠΠ
Π
(8)
where
T
1 2
[ , ,..., ]
n
β β β
= ,
0
and
p
are
transition matrices, with
p
a matrix for previous
time and
0
for current time. All the diagonal ele-
ments in
0
are zero.
Eq. (8) is very similar to the standard vector auto
regression model or the vector error correction model,
except that we have 0
t
Y
on the right side of Eq.
(8). 0
t
Y
is a characteristic of the model in this
paper, so it includes newer information than those in
other papers. The first-order difference is
1
t t t
−
∆ = −
Y Y Y
In Eq. (8), for example, when m = 1, we can get
1 1
0 0 1
( ) ( )
t t t t− −
∆ = − + + − +
YY Y I Y
Furthermore,
1
0 0 1
( )
t t t −
∆ = ∆ + + − +
YY I Y
(9)
where
1
0 1
( )
t
−
+ −
I Y
means that
t
∆
Y
depends
upon the signals at previous time. Without 0
t
∆
Y
,
Eq. (9) becomes the standard vector error correction
model. 0
t
∆
Y
is the non-lagged variables which is
a distinguishing feature which makes the study in this
paper different. When m > 1, one can also deduce
similar equation of Eq. (9), in which 0
t
∆
Y
is for
short-term variation and
1
0 1
( )
t
−
+ −
I Y
is for
long-term regression. As model proposed in this paper
considers both long and short terms, its adaptability is
relatively strong.
Chinese Journal of Aeronautics · 5 ·
3. Augmented Dickey-Fuller test for the model
The augmented Dickey-Fuller (ADF) test is a pop-
ular method to find a unit root for a time-series sam-
ple. Through the test, we can see whether the model
fits the data collected in the jet engine test. At the very
beginning, we pick up some normal data to see
whether the model works well.
To collect qualified data, the first step is to find an
appropriate group of sensors inside a engine. These
sensors, scattered in different locations, should have
much relevance. The noise from each location should
be low enough that the relations among the sensor
signals are not overwhelmed by the noise. From front
to back in a engine, there are air intake, air compres-
sor, combustor chamber, turbine and exhaust. In the
front end, the scenario is quite simple, and all signals
from a sensor group in a cross-section look the same,
so their relations are clear and easy to discover.
However, after the air compressor, the distribution of
the sensors’ locations is very complex, so their signals
are quite different. After balancing among the differ-
ent places, we choose the cross-section T23 just in
front of the inlet of the air compressor. Fig. 2 shows
the normal signals from a seven-sensor group put on a
cross-section of T23.
Fig. 2 Normal signals from cross-section T23.
In Fig. 2, all signals go through three segments. At
the first segment (initialization), all of them are
straight lines and close to each other. At the second
segment, the distances between the sensor signals
increase and we can see clearly the shapes and differ-
ences of the signals. At the last segment, each signal
goes its own way distinctly. It is clear that some have
big variances while some have small variances. The
variance of sensor 1 signal is the biggest and the var-
iance of sensor 7 signal is the smallest. The signals at
the first segment of the course are stationary while at
the last segment they are non-stationary.
Use the ARIMA function in SAS software to test
the signals in the third segment and see their features.
Use the SAS procedure as
proc arima data = myt23;
where (idt > 1000);
identify var = x07_02 stationarity = (adf) ;
run;
Table 1 (from SAS) is the ADF test results of
sensor 7 signal. In Tau statistic, its p-value is obvi-
ously bigger than 0.1, so the signal is non-stationary
and has at least one unit root. Thus, for the
non-stationary series, traditional models do not have
much competence.
Table 1 ADF test results of sensor 7 signal.
Type Lag
Statistic results
Rho Pr < Rho Tau Pr < Tau F Pr > F
Zero mean
0 0.0020 0.6825 0.06 0.6631
1 0.0028 0.6823 0.07 0.6608
2 0.0030 0.6823 0.07 0.6609
·6 · Chinese Journal of Aeronautics
Single mean
0 2.7270 0.6902 1.18 0.6858 0.70 0.8932
1 4.2326 0.5136 1.47 0.5512 1.08 0.7957
2 4.9997 0.4351 1.59 0.4883 1.26 0.7475
Trend
0 2.3590 0.9603 0.91 0.9538 0.74 0.9900
1 4.2098 0.8738 1.29 0.8885 1.07 0.9575
2 5.1751 0.8077 1.46 0.8432 1.27 0.9238
From Fig. 2, it seems that the last segment is more
valuable for regression than the others and we use this
segment to fit Eq. (7) to see how the model works.
For the signals of seven sensors, we use one signal as
the dependent variable (on the left side of Eq. (7)),
and use the other six signals as the independent varia-
bles (on the right side of Eq. (7)). Then we check the
residual error ii yy ˆ
− of the regression to see the
F-statistic, where i
y
ˆ is the estimated value of signal
i. Use the signals of seven sonsors in turn as the de-
pendent variable, and test the model seven times.
When we set m = 0, we could find that the value of
F-statistic is big enough, so generally, the regression
result is good. However, the Durbin-Watson (DW)
test statistic is lower than 0.5, so there must be auto-
correlation. Later, we set m = 2 and find that it im-
proves a lot. We choose two of them for illustration.
Fig. 3 shows the residuals from sensor 1 signal. Tables
2 and 3 (from SAS) show the results from sensor 4
signal with the DW test statistic being 2.04, with Ta-
ble 2 the autocorrelation check results for white noise
and Table 3 the ADF test results of sensor 4 signal.
Fig. 3 Regression residuals of sensor 1 signal.
From Fig. 3, it can be seen that the residual errors
are near zero with half negative and half positive
numbers. The dots in Fig. 3 have all the characteris-
tics of white noise except that the variances in the
front seem smaller than those in the rear.
Table 2 Autocorrelation check results for white noise
of sensor 4 signal.
Lag
2
χ
Degree of freedom Pr >
2
χ
6 10.60 6 0.1016
12 21.11 12 0.0488
18 23.17 18 0.1842
24 26.61 24 0.3227
Table 3 ADF test results of sensor 4 signal.
Type Lag
Statistic results
Rho Pr < Rho Tau Pr < Tau F Pr > F
Zero mean
0 1325.37 0.0001 36.77 < 0.0001
1 1163.78 0.0001 24.09 < 0.0001
2 980.297 0.0001 18.82 < 0.0001
Single mean
0 1325.37 0.0001 36.75 < 0.0001 675.34
0.0010
1 1163.78 0.0001 24.08 < 0.0001 289.98
0.0010
Chinese Journal of Aeronautics · 7 ·
2 980.293 0.0001 18.82 < 0.0001 177.03
0.0010
Trend
0 1326.02 0.0001 36.76 < 0.0001 675.50
0.0010
1 1165.38 0.0001 36.77 < 0.0001 290.18
0.0010
2 982.611 0.0001 24.09 < 0.0001 177.22
0.0010
From Tables 2 and 3, we know that the p-values of the most LP statistics in the last column are no less than 0.05,
so the residual error sequence may be regarded to be white noise. Besides, as all p-values in Table 3 are smaller than
0.05, we can say that there are cointegration relations among the signals of seven sensors. Therefore, we successfully
build a model for the signals of seven sensors from Eq. (7), in which one signal can be substituted by the combination
of others. The results are very helpful for sensor fault tolerant.
4. Real time forecasting algorithm
From above discussion, we see that for the entire segment, the signals of seven sensors are linearly correlated; how-
ever, how about a small piece of data? The real time analysis cannot allow too long data. If we use any fragment of
the data rather than the entire, can the signals of seven sensors still be replaced with each other? It is very important
because, in reality, people care more about online fault warrants than offline statistical results. For example, if we use
previous 2 s to forecast the next 1 s, the data length for regression is 100 when the sampling rate is 50 per second. On
one hand, each piece of the data is not allowed to be too long for the real time purpose. On the other hand, if the data
length is too short, there is no good fitting effect.
We adopt the second segment and the third segment in Fig. 2 to fit the model for real-time forecasting in this
section. As the first segment has only straight lines, it is impossible to make a full rank matrix, so it cannot be used for
Eq. (7) regression. The shape of the second segment is much better than that of the first segment. However, the last
segment can explain the model more clearly. We build a algorithm for real-time prediction. Firstly, define variable
STEPMATCH as the data length for regression, STEPFORCAST as the data length for forecasting and BEGIN as the
starting point somewhere in data for regression. When the sampling rate is 50 per second, for example, if
STEPMATCH = 50, it means each time the algorithm deals with data of 1 s. STEPFORCAST can be set to 1. How-
ever, to speed up calculation, it can be larger than 1. The algorithm has five steps as follows:
Step 1. Base upon Eq. (7), start the regression from BEGIN, with STEPMATCH being its regression length. After
regression, we get parameters
i
α
,
j
β
and
pj
γ
( , 1, 2, , ; 1, 2, , )
i j n i j p m
= ≠ =
；.
Step 2. Use
i
α
,
j
β
and
pj
γ
to forecast with STEPFORCAST times. The forecasting ranges from BEGIN +
STEPMATCH to BEGIN + STEPMATCH + STEPFORCAST, with the length of STEPFORCAST.
Step 3. BEGIN = BEGIN + STEPFORCAST.
Step 4. Compare the estimated values with the actual collected data in next round.
Step 5. If the test is not finished, go to Step 1.
The algorithm divides the signal series into two parts, the training part and the estimated part. The accuracy of the
predicted signal in the estimated part relies on the rightness of the training segment. In other words, we deduce the
value of one signal on the assumption that the value of the training part is correct. If everything is fine, the algorithm
moves both its training window and estimated window to the next. The algorithm forecasts the data with length
STEPFORCAST by using the previous data with length STEPMATCH. For q variables regression, it needs at least
q+1 equations. We have m×n + n 1 variables in Eq. (7). Therefore, STEPMATCH should be no smaller than n×
(m +1).
·8 · Chinese Journal of Aeronautics
Fig. 4 shows the measured signals of seven sensors and their estimated series (in the second segment and the
third segment) with STEPMATCH = 50 and STEPFORCAST = 25. The length of each signal is 1313. It is clear that
in Fig. 4, all the estimated series are similar to their original ones, so generally, the prediction is correct. Define the
estimated error as
ˆ
| |
PE
j j
j
y y
y
−
=
In Fig. 4, it seems that the estimated error increases with the variances of the original signals. The variance of
sensor 1 signal is much bigger than that of sensor 7 signal. Sensor 1 signal has the largest average predicted error
(4.8%), and its maximum prediction error is 12.7%. Sensor 7 signal has the smallest average predicted error (0.767%),
and its maximum prediction error is 3.04%. Fig. 5 is a clearer picture of sensor 1 signal and its estimated signal. In
the second segment, the variance is smaller and the estimated error is smaller, while in the third segment, the variance
is bigger and the estimated error is also bigger.
Fig. 4 Measured and estimated signals of seven sensors.
Fig. 5 Measured and estimated signals of sensor 1 with Eq. (7).
Now let’s see how the model fixes the error in Fig. 1. Use the first 50 correct data from the second segment in
sensor 3 signal along with the data from the other six signals to train the model and then forecast the data of sensor 3
signal from position 651 to position 1900, as shown in Fig. 6. The upper curve is the original signal in Fig. 1, which
is abnormal. The lower curve is its estimated one that is based on the model. The shape of the estimated one is similar
to those of the other six signals in Fig. 1. Therefore, the model can restore an abnormal signal and thus construct a
fault-tolerant mechanism for a sensor group.
Chinese Journal of Aeronautics · 9 ·
Fig. 6 Sensor 3 signal in Fig. 1 and its repaired signal.
5. Modification of the model
Though from the above results, the model seems like a good one, it still has some problems. After regression for i
y,
we check the coefficient pi
γ
(p = 1, 2, … , m). We could find out that it tends to be much larger than other parame-
ters. In common sense, it is not good when the autoregressive coefficient is too large. In the calculation of Eq. (7), as
the sampling speed is pretty high, there is much relevancy between the dependent variable on the left side of Eq. (7)
and its lagged variables on the right side of Eq. (7). This is the reason that induces the coefficient to be large. Because
of the autoregressive correlation, the model would become unstable. For example, when we change BEGIN in the
algorithm to 1000, or when we start from position 1000 rather than the previous beginning, the forecasting becomes
uncontrollable. Fig. 7 shows the estimating for sensor 7 signal when starting from position 1000. After trying many
times, we could find out that to have a stable model, the autoregressive correlation of the dependent variable has to be
eradicated.
Fig. 7 Estimation of sensor 7 signal with Eq. (7).
Improve the model by removing the autoregressive items from the right side of Eq. (7) and thus it becomes
= ≠
−
≠
++=
m
p ij
pt
jpj
ij
t
jji
t
iyyy
1
γαβ
(10)
In Eq. (10), for any dependent variable on the left side, there is no lagged variable on the right, so the left variable
only depends on the other signals but itself. That means one signal can be replaced totally by other signals. Theoreti-
cally, the improved way is more stable and much better than the old one, because it can get rid of any impact from
auto-regression. We use this model to do the same thing in Fig .7 and then get Fig. 8, which shows sensor 7 signal and
its estimated one with Eq. (10). After using the first 50 points of sensor 7 signal as well as all the data of the other
signals for Eq. (10), we have the estimated curve of the signal. In contrast to Fig. 6, it is improved a lot. Its average
·10 · Chinese Journal of Aeronautics
predicted error is 1.05% and the maximum is 3%. As noises are counteracted by sensors with each other, the estimat-
ed signal looks smoother than the original one.
Fig. 8 Measured and estimated signals of sensor 7 with Eq. (10).
When using Eq. (10) to calculate, we get the estiamed signals for all seven original ones in Fig. 9. The estimated
error of each signal looks smaller than that in Fig. 3. Though the DW test statistic from the new model is not as good
as that from the old model, on the whole, the results from the new model are better.
Fig. 9 Original and estimated signals of seven sensors with Eq.(10)
Fig. 10 illustrates a clear picture of sensor 1 signal and its estimated one. When comparing this figure with Fig.
5, we can see that the modified model works much better. For example, the average estimated error for sensor 1 sig-
nal is 1.26% and the maximum is 6.6%, so from the estimaed error perspective, the new model improves a lot.
Chinese Journal of Aeronautics · 11 ·
Fig. 10 Measured and estimated signals of sensor 1 with Eq. (10).
6. Evaluation
The experiment in this study uses the signals of the first second to forecast those in (up to) the next 25 s. To cope with
the 50 per second sampling rate, we use the LU decomposition fast algorithm to invert the matrix in C# program. It
runs good and stable. The program calculates the substitute for each signal in real time, so when one sensor in the
sensor group fails, it can be replaced by its substitute.
When evaluating the results of our model, there are two things to consider. The first one is the speed. To forecast
and give alarm in real time, we not only need fast calculation but also keep the length of the regression data as short
as possible. The second one is accuracy. We have defined and used the estimated error as the most important factor
for our model in the previous section. From the estimated error point of view, our model works well.
A factor often used to evaluate accuracy is to detect the presence of autocorrelation. Our autocorrelation detection
uses residual analysis which is a very common way for regression evaluation. The sequence of residual errors can
bring a lot of information that is worth thinking. Usually, people use the DW test or Eq. (10) to test residual errors.
When DW statistic equals 2, it indicates no autocorrelation, which is the best.
2
1
2
2
1
( )
DW
T
t t
t
T
t
t
ε ε
ε
−
=
=
−
=
(11)
where T is the number of observations; t
ε
is the residual associated with the observation at time t.
In Table 2, we can see that for the entire signals, the sequence of the residual errors from our model looks like the
white noise. When forecasting for small segments in real time, DW statistics vary in different positions. Though DW
statistics for small segments are not as good as those in Table 2, on the wholethey are close to 2.0 and acceptable.
Another topic that needs discussion is heteroscedasticity. Basically, the ordinary least square (OLS) regression re-
quires a strictly stationary process. From Figs. 2 and 3, the variances at different locations are different. In this case,
the OLS regression may not work well. In the example above, heteroscedasticity is not an issue. However, in many
other engine tests, we can find big variations in different positions from the time series data, so heteroscedasticity is
an important factor to consider in time series regression. Basically, homoscedasticity or the same of variance is a re-
quirement for regular regression. Many non-stationary sequences do not have such characteristics. One of the mostly
often adopted tests for heteroscedasticity is the autoregressive conditional heteroskedasticity (ARCH) test in Eq.
·12 · Chinese Journal of Aeronautics
(12), in which
(
)
0,1, ,
i
i p
∂ =
is the regression parameter.
=
−
∂+∂=
p
i
itit
1
2
0
2
εε
(12)
After performing the ARCH test regression, one can check the 2
)( Rpn − statistic to see whether there is
heteroscedasticity, with 2
R
the degree of similarity. From Figs. 2 and 3, we can see that there exist some different
variances in different locations of the data sequence, though the heteroscedasticity is small. Therefore, we can further
improve our model by eliminating the heteroscedasticity. There are various approaches to do so and we tend to
choose the flexible generalized least squares (FGLS) technique. In FGLS, the first step is to do a common OLS re-
gression and get the residual error i
e for each calculation. Next, use
2
ln ( )
i
e
as the left side of the equation to do
another regression and get the residual error i
g. Finally, we use i
g
e/1 as the weights for both sides of the equa-
tion and regress again.
Fig. 11 illustrates sensor 1 signal and its estimated signal by using the FGLS technique. We can see the im-
provement in contrast to Fig. 4. In Fig. 11, the estimated curve and the original one twist together and the estimated
error is much lesser than that of the previous model. By introducing FGLS, the overall precision increases about 19%.
Since the fitting of our old model in the second segment of the data is already good enough, in the last segment, the
precision rate increases more than 30%. Meanwhile, the DW test statistics in most cases are close to 2.0.
At the jet engine testing site, when one sensor fails or around it something unusual happens, there must be a big
difference (for example, 3threshold) between the estimated signal and the original one. At that time, our program
will give alarms.
Fig. 11 Improvement of Fig. 5 by FGLS.
7 Conclusions and discussion
This study adopts a statistical method to analyze signals from a sensor group in engine testing and find the
cointegration relation among the signals. The original signal and our estimated one should twist with each other, un-
less there is something wrong in the engine. Therefore, any sensor signal can be substituted by the combination of
others. This paper takes the cointegration concept which is widely used in economic analysis and implements it for
signal processing. In the engineering field, few studies have adopted this way so far. By comparing the estimated sig-
nal and the original one, people can find whether there is any exception and furthermore restore a signal from the
combination of other sensor signals in real time. Besides, in contrast with other papers that adopt cointegration, this
study uses a non-lagged item which brings more information to the model.
Our approach is a novel and effective method to restore a signal from the combination of other sensor signals in
real time. We are fully aware of the limitations of our method. Firstly, our approach works well when one sensor goes
Chinese Journal of Aeronautics · 13 ·
wrong. However, when more than one sensor fail, it would not handle, so there is much work to do and it might be
improved in our future research. Secondly, the signal in our model should be divided into two parts, the training part
and the predicted part, so the accuracy of the estimated signal relies on the rightness of the training part, and we de-
duce the value of one signal on the assumption that the value of the training segment is correct. In the future, we plan
to use numerical interpolation to generate the values of a whole line through the values of several sensor signals. In
addition, we might even deduce the values for all the locations in a cross-section, if possible.
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Acknowledgement
This study was supported by the Aeronautical Science Foundation of China (No. 20101024006).
Li Liang received the Ph.D. in Computer Engineering from University of Electronic Science & Technology, China in 1999, and
then became a teacher there. His main research interests are data mining and algorithm design.
Shi Wei received his M.S. degree in computer engineering from University of Electronic Science & Technology, China. He is a
senior engineer in China Gas Turbine Establishment. His main research interests are data mining and computer application for gas
turbine experiment.
