Sensor fault detection with generalized likelihood ratio and correlation coefficient for bridge SHM

Journal of Sound and Vibration 442 (2019) 445e458

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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Sensor fault detection with generalized likelihood ratio and correlation coefficient for bridge SHM Lili Li a, b, Gang Liu a, b, *, Liangliang Zhang a, b, Qing Li c a

School of Civil Engineering, Chongqing University, Chongqing, 400045, China Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Chongqing, 400045, China c College of Computer Science, Chongqing University, Chongqing, 400044, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 February 2018 Received in revised form 23 October 2018 Accepted 30 October 2018 Available online 2 November 2018 Handling Editor: I. Trendafilova

Data is obtained from sensors in a structural health monitoring system for integrity assessment of the structure, and false alarm will be frequently triggered if a faulty sensor is present. A method based on the generalized likelihood ratio and correlation coefficient is presented to identify senor fault in this study. The acceleration response of a bridge is assumed Gaussian distributed when under operational condition, and evaluation of each sensor in the sensor network is accomplished via the minimum mean-squares-error algorithm. Multiple hypothesis test with the generalized likelihood ratio is then applied to the measured data with estimation to detect the sensor fault. Five common sensor fault types are studied with two correlation coefficients calculated between the estimation and measured data as the classification features. Unbalanced binary tree method is implemented to categorize the type of sensor fault. Numerical and experimental studies indicate that the proposed method is robust in the detection and classification of the sensor fault. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Structural health monitoring Sensor fault Likelihood ratio Correlation Classification MMSE Hypothesis test

1. Introduction Study with structural health monitoring (SHM) has gained much attention in recent years in the civil engineering industry with implementation of SHM system in many bridge structures. Performance of these systems depends on the quality of sensors and reliability of measurements. However, sensors may inevitably incur various types of faults during their service life. A faulty sensor may provide incorrect information for managerial decision on the bridge. Therefore, it is necessary to detect and identify sensor faults in the early stage of their development. This paper intends to provide an automated method for detecting and identifying the faulty sensor in the sensor network of a SHM system. Detection and identification of fault in instruments are widely studied in the areas of automatic control and mechanical engineering [1e3]. These approaches can be divided into three categories: physical model-based, knowledge-based and databased methods [4,5]. The data-based methods, or the data-driven methods, have become more popular in recent years because they do not require a physical model of the system which is usually not available. Zhang et al. [6] obtained the probability density function (PDF) of the sensor signal from kernel density estimation, and they successfully identified the mechanical fault from the Kullback-Leibler Divergence between signals from two sensor. Huang et al. [7] applied the generalized likelihood ratio to detect fault in power station sensors automatically. Dunia and Qin [8] studied the conditions

* Corresponding author. School of Civil Engineering, Chongqing University, Chongqing, 400045, China. E-mail address: [email protected] (G. Liu). https://doi.org/10.1016/j.jsv.2018.10.062 0022-460X/© 2018 Elsevier Ltd. All rights reserved.

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required for fault detectability, reconstruction capability, and identifiability with analytical formulations. Both sensor and process faults could be identified, but the fault type has to be known. This information is, however, available only for a sensor fault but not for the process fault. Experimental verifications showed that the above method can detect the sensor fault satisfactorily. Other multivariate statistical analysis-based fault detection methods include the principal component analysis (PCA) [9e11], the parity equation-based methods [12], independent component analysis (ICA) [13] and etc. Comparisons [14] showed that the PCA method has excellent performance in modelling multivariate Gaussian distributed datasets. However, traditional PCA is inappropriate for modelling such data because it cannot capture the autocorrelations. The parity equationbased methods are not very sensitive to small sensor faults. The ICA method for detecting faults was originally proposed to solve the problem of linear instantaneous mixing blind source separation with the drawback that there is no standard criterion for sorting the independent components. There are different types of sensors in the SHM system but the nature of fault is the same with instruments for control and the mechanical system [15]. The data-based sensor fault detection methods have been implemented in infrastructural SHM system [16]. Kerschen et al. [17] applied the PCA to model the monitored data using the angle between the principal subspaces as feature for sensor-fault detection. Isolation of the sensor fault was realized by removing one sensor at a time, and the faulty sensor was the one with the minimum angle of the principal subspaces after its removal from the group. Kullaa proposed [18] the minimum mean-squares error (MMSE) method for sensor identification, isolation and correction. The spatial and temporal correlations of the sensor output data were combined to improve the identification capability. They also studied the quantification of sensor faults based on the MMSE method where seven different sensor fault types were investigated and modeled. Multiple hypothesis test approach using the generalized likelihood ratio was employed to identify and quantify the sensor faults [19]. MMSE method has also been applied to damage detection and localization in structures [20] and in distinguishing different sources of error that may lead to a change in the vibration data, i.e. environmental influences, sensor faults, and structural damage [21]. The above works indicated that the MMSE method could improve the fault detection capability compared to other methods in the field of civil engineering. Although the data-based methods have a great potential for sensor fault detection, there is still a need of data-based methodologies for detecting and identification of sensor fault in SHM system of civil structures. A practical sensor fault identification methodology is presented in this study with hypothesis test on the generalized likelihood ratio (GLR). The algorithm for combined detection and identification of sensor fault is introduced in the SHM system. An index on the classification feature is developed in the identification process. The method is based on an assumption that the structure is healthy, and only one sensor is faulty over a period of time. The rest of the paper is organized as following. The sensor network and models on different sensor faults are presented in Section 2. The detection of sensor fault with a hypothesis test using the GLR is conducted in Section 3. An index of classification is proposed in Section 4 to categorize the sensor fault. Numerical and experimental results are presented in Section 5 to validate the proposed method. Finally, concluding remarks are given in Section 6. 2. Sensor fault models and sensor network 2.1. Sensor fault models A sensor usually comprises of several components, i.e. the sensing device, transducer, signal processor and communication circuit, despite the different sensing principles. The sensor may have fault in any of these parts, and it can be faulty when it displays an unusual deviation from its characterizing performances [22]. There are different types of faults with different sensors. Mathematically, the variation of signal in faulty sensors may be described in five forms as follows: complete failure, gain, bias, linear drift and background noise. 2.1.1. Complete failure Complete failure refers to a case where the sensor gives a constant value with time along with noise regardless of the actual signal changes. The mathematics model can be given as

xo ðtÞ ¼ c þ dðtÞ

(1)

where x0(t) describes the sensor reading over time t, c is a constant andd(t) represents the white noise. Most cases of complete failure occur when the sensitive core insulation resistance of sensor drops or damaged. The sensor is therefore insensitive to the measurement, and the output remain constant with time. 2.1.2. Gain A sensor is known to have gain fault if the actual value x(t) of the sensor is associated with an excessive-variance. The model can be described as:

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xo ðtÞ ¼ bxðtÞ þ dðtÞ

447

(2)

where b is a coefficient indicating the gradient of the gain fault. The larger the gain coefficient, the greater the degradation of precision of the sensor. The sensor gain fault is often caused by unstable voltage supply or non-linearity of the sensor. Therefore, coefficient b may change many times compared with the actual values even in the same measurement. 2.1.3. Bias If the measured reading shifts by a constant value from the true value, it is defined as bias:

xo ðtÞ ¼ xðtÞ þ d þ dðtÞ

(3)

where d is a constant. Sensor bias fault occurs when the sensor is in creep or the sensor base is loose. 2.1.4. Linear drift A sensor is known as in linear drift if the difference between the sensor reading x0(t) and the actual value x(t) changes linearly with time shown as

xo ðtÞ ¼ xðtÞ þ a þ b,t þ dðtÞ

(4)

where a and b are constants. The main reason for this observation is the deformation of the transmission cable and friction. 2.1.5. Background noise When the sensor shows electrical noise interference, the sensor reading consists of the background noise only as:

xo ðtÞ ¼ dðtÞ

(5)

where d(t) may be a random signal with an unknown mean and covariance, and it is usually white noise in most cases. The gain coefficient in the gain fault model is assumed to be a random number between 1.3 and 2.0, with b ¼ 1.3 þ 0.7 h, where h is a random number between null and unity. Parameter d of the bias fault model and parameter c of the complete failure model are assumed to have the same value in this study. They are the mean of the signal between the maximum and minimum values but not inclusive. In order to have a continuous signal for this study, the linear drift fault coefficient a is null and coefficient b is twice the maximum value of the signal divided by its data length. The noise in signal is assumed white with zero mean and unit variance. The signals from sensors with the above faults are shown in Fig. 1. 2.2. Sensor network There are different types of sensors, such as accelerometer, displacement and strain transducers. There may be a few sensors of the same type mounted on the structure. In order to simplify the sensor fault detection, the same type of sensors with identical precision and sampling frequency are viewed as a sensor network. Only an accelerometer sensor network is discussed in this study. A structure is under ambient excitations in operation which can be modeled as Gaussian in most cases. The dynamic response for a linear stable system will also be a Gaussian process [23] described as


  X1
1 Pðx0 Þ ¼

2p sj1=2 exp  ðx0  mÞT ðx0  mÞ 2

(6)

where m and s are the mean vector and covariance matrix respectively, and x0 is the vector of sensor reading or measured variable, which is typically a sample of data from several sensors in the sensor network. The sensor network is divided into two groups. One group includes only the missing sensors (with fault) u and another group includes all other observed sensors v. The latter serves as the observed set with

x0 ¼ ½ u

v T

(7)

The estimate of the partitioned covariance matrix S is obtained from the measured data when all sensors in the network are healthy as

 S¼

Suu Svu

Suv Svv



 ¼

Guu Guv Gvu Gvv

1

¼ G1

(8)

where the precision matrix G is defined as the inverse of the covariance matrix S which can also be written in the partitioned form. A linear MMSE estimate [24] for ujv(u given v) is given as

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(a) 40

10

20

Acceleration(m/s2)

Acceleration(m/s2)

30

Healthy Bias

15

Linear drift Complete failure Healthy

10

0

-10

5

0

-5

-10

(a) Complete failure and Linear drift

(c) Bias

-20 0

20

40

60

80

100

-15 0

Time/s

20

40

60

80

100

Time/s 15

Healthy Gain

20

Healthy Background noise

10

Acceleration(m/s2)

Acceleration(m/s2)

10

0

-10

5

0

-5

-10 -20

(b) Gain 0

20

40

60

80

(d) Background noise 100

-15 0

20

40

60

80

100

Time/s

Time/s Fig. 1. Healthy signal and signals with sensor fault.

b ¼ EðujvÞ ¼ mu  G1 u uu Guv ðv  mv Þ

(9)

where mu and mv are the mean of u and v, respectively, and E() is the expectation value. The error covariance matrix is

F ¼ covðujvÞ ¼ G1 uu

(10)

Assuming the measurement data takes up a Gaussian distribution, the conditional PDF of ujv can be obtained from Eqs. (9) and (10) as

  1 b ÞT F1 ðu  u bÞ pðujvÞ ¼ j2pFj1=2  exp  ðu  u 2

(11)

3. GLR test in the sensor fault detection When all sensors are in their healthy condition, the mean m0 and variance F0 of measured data from each sensor can be computed using Eq. (11). Similarly, the mean m and variance F of each sensor with unknown sensor condition can also be obtained from Eq. (11). In the case when the mean m or variance F of the latter set of sensors deviates from the mean m0 or variance F0 of the first set, the following hypothesis test can be used for the fault detection as



H0 : m ¼ m0 F ¼ F0 H1 : msm0 FsF0

(12)

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The above composite hypothesis test does not assume knowledge of the sensor condition in hypothesis H1. The generalized likelihood ratio (GLR) test [25] is applied to detect the faulty sensor, and the test statistic is the GLR for each sample as:

s ¼ ln

pðujv; H1 Þ pðujv; H0 Þ

(13)

where pðujv; Hi Þ is the probability according to hypothesis Hi, (i ¼ 0,1). The hypothesis H0 corresponds to the situation when the new set of parameters is equal to that of the training data (healthy state). The hypothesis H1 denotes that the parameters are different to those of the training data (anomaly). The distribution pðujv; H1 Þ is obtained by estimating the parameters from the current measurement, while those of pðujv; H0 Þ are estimated from the training data. A threshold of the likelihood ratio gs can be selected for determining the hypothesis H1 or H0. For example, when jsj > gs , pðujv; H1 Þ > pðujv; H0 Þ and hypothesis H1 will be accepted and vice versa. The following description is for any sensor selected in the sensor network. The recorded time history of any healthy sensor in the network is subdivided into m smaller equal samples and they are grouped as SetR. One data sample is randomly selected. The generalized likelihood ratio, s, can be obtained from all m samples in SetR with Eq. (13). The above calculation is repeated for the next selected sample from SetR, such that m  (m-1) GLR values are obtained. Then the simple 3s criterion can be applied to all the GLR values to compute the threshold value gs for the selected sensor. Similar computation is applied to the response data record from other sensors in the network. Similar procedures may also be applied to the recorded data sample grouped as SetF from any potential sensor with fault to calculate the modulus of the GLR values for comparing with the threshold value in the hypothesis tests. 4. Sensor fault classification 4.1. Index of classification features Classification Feature (CF) extraction is an important step in sensor data fusion for sensor fault detection. This involves the determination of a feature vector from a pattern with minimal loss of important information. The measured sensor reading and the estimated value from MMSE method carry much information of the system. Therefore, the correlation coefficient r which measures the degree of similarity between different signals is adopted as the CF as n P

ðxi  xÞðyi  yÞ r ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n P P ðxi  xÞ2 ðyi  yÞ2 i¼1

i¼1

(14)

i¼1

where x and y are the measured sensor reading and estimated time series using MMSE algorithm respectively. n corresponds to the number of sample data. x and y are the mean of x and y respectively. When the time series x and y are perfectly correlated, the absolute value of coefficient r is unity. If there is no relevancy between x and y, the coefficient r is equal to null. The absolute value of r is larger when the degree of correlation between two time series is higher. When measured data of the sensor network with healthy and unknown sensor conditions are available, the output with the healthy sensor is described as yh and that with the faulty sensor is yg . The estimated output from the MMSE algorithm for the case with faulty sensor is denoted as yM . The difference between the output of the system with healthy sensor and the faulty sensor is yhg . The difference between the estimated output and the output with faulty sensor is yMg . The correlation coefficient between yh and yg is denoted as rtrain , and it is calculate from Eq. (14). The correlation coef1 . The correlation coefficient between yM and yg is denoted as rtest ficient between yh and yhg is denoted as rtrain 2 1 . The cor-

relation coefficient between yM and yMg is denoted as rtest 2 . The classification model based on the MMSE estimation is

and rtrain for distinguishing the type of sensor fault in the test data. constructed from the classification index rtrain 1 2 The index of classification is significantly different for different types of fault. Since all signals contain white noise, r1 and r2 are close to null in the cases of complete failure and bias faults respectively. Theoretical values of the classification feature are shown in Table 1 as

Table 1 Theoretical value of classification feature. Fault type

r1

r2

Complete failure Gain Bias Linear drift Background noise

z0 1 1 0.5e1.0 1~þ1

1 1 z0 0 1~þ1

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4.2. The classifier The classification model described above is adopted in dealing with the present Two-Category classification problem. The solution of the multi-classification problem commonly employs the binary tree method in Relevance Vector Machine where the Gauss kernel function is used. In this paper, sensor fault type classifier is developed by unbalanced binary tree method. According to the basic principle of the unbalanced binary tree method, four classifiers are constructed to identify five kinds of sensor faults listed as complete failure, gain, bias, linear drift and background noise which associated with the classifiers MS1, MS2, MS3 and MS4 as shown in Table 2. The MS1 classifier associates with the complete failure. When the sensor exhibits complete failure, the output of MS1 is unity, and null if otherwise. The latter case demonstrates that the sensor generates one of the four types of fault: gain, bias, linear drift and background noise. Similarly, the MS2, MS3 and MS4 classifier are associated with the gain fault, the background noise fault and the linear drift fault respectively. The selection and identification of sensor fault algorithm is summarized in the following steps (Fig. 2): (1) (2) (3) (4) (5) (6) (7) (8) (9)

Acquire the acceleration data when the sensor is healthy. Compute pðujv; H1 Þ under H1 condition and pðujv; H0 Þ under H0 condition for all sensors Compute the fault threshold (gs ) of each sensor. Compute the likelihood ratios of all sensors by Eq. (13). The likelihood ratios are plotted in curves to compare with the fault threshold. If an acceptable number of exceedance of the threshold occurs, it is an indication of a sensor fault as shown in Fig. 2. Extract the index of classification features from the faulty sensor data. Construct the Classifier for sensor fault classification The classifier and index of classification features are used to classify the faults and to identify the fault type.

5. Numerical and experimental examples 5.1. Numerical verification The structure is a 20 m long simply supported beam with a 0.25 m  0.6 m uniform cross-section. The elastic modulus and density of material are respectively 3  1010 N/m2 and 2500 kg/m3, and the poisson ratio is 0.3. Structure analysis of the system adopts Beam188 Element in ANSYS. The beam is made up of two hundred elements, with each 0.2 m long. It was excited with white noise at the supports. The sampling rate is 100 Hz and the response time duration studied is 0.01s. Newmark-b method is used to calculate the acceleration response of the beam. Four sensors, A, B, C and D, were mounted on the beam at 2 m, 5 m, 10 m and 12 m from the left end as shown in Fig. 3. The five different types of sensor faults discussed above are studied. Sensor A is assumed to have only one type of fault at one time for a period of time, and the sensor faults are described in Table 3. The thresholds for each type of fault obtained from the procedure described in section 3 are shown in Table 4. A total of 6 scenarios were studied including the 5 scenarios with one faulty sensor and scenario 6 without fault. 90 sets of independent responses for scenario 6 are grouped as SetR and 1 set of responses is grouped as SetF for each of scenarios 1 to 5. Each response sample consists of 100 data. The threshold value and the GLRs are calculated and shown in Figs. 4 and 5. Since only sensor A has fault in scenarios 1 to 5, the GLR pattern for sensor A may varies but those of other sensors remain the same as for the healthy scenario as shown in Fig. 4(b). The signal from faulty sensor A is noted to have different mean and variance compared to those from the healthy sensor as shown in Figs. 4 and 5. All the GLR values are much larger than 1.11 for the case with a malfunction sensor. All the GLR values of sensors B, C and D are less than their threshold when there is fault in sensor A (Fig. 4(b)). This would indicate that sensor A is faulty and other sensors are healthy. This provides clear evidence that a comparison of the GLR values between scenarios with and without fault could distinguish the faulty sensor.

Table 2 MMSE classifier for sensor faultv. classifier

output

sensor fault type

MS1

1 0 1 0 1 0 1 0

Complete failure

MS2 MS3 MS4

gain

bias

linear drift

bias

linear drift

background noise

bias

linear drift

Gain Background noise Linear drift bias

background noise

L. Li et al. / Journal of Sound and Vibration 442 (2019) 445e458

Sensor Faul t Det ect i on St age

451

Sensor Faul t I dent i f i cat i on St age

Acquire the sensor data

Extracted index of classification features Compute the fault threshold

Calculate the healthy sample PDF

Calculate the test sample PDF

Constructed classifier

MS for sensor fault identification

Compute the log-likelihood ratios

MS1 Exceeded the threshold

No

Sensor health MS2 Complete failure

Yes

MS3 Exceeds the allowable number

No

Sensor heath

Gain MS4 Background noise

Yes

Linear drift

Sensor fault

Bias

Fig. 2. Flowchart for detection and identification of sensor faults.

20m 2m

3m

A

5m

B

2m

C

8m

D Accelerometers

Fig. 3. Simply supported beam model.

In order to verify the reliability of the proposed method under different levels of sensor fault, the detection of different amplitude of the gain fault is illustrated in Fig. 6. The excess threshold ratio is defined as the percentage of the GLR value above the threshold. The results clearly indicate that the method shows better performance with the increase of the fault amplitude, and the fault can be successfully detected by the proposed method with the fault amplitude greater than 1.6. The classification indices r1 and r2 are adopted in the four MS classifiers to determine the fault type in sensor A, and they are shown in Fig. 7. The green square dots in Fig. 7(a) denote the case of complete failure of sensor, and the other four fault types are marked by purple circular dots. It is noted that the MS1 classifier is effective to differentiate the complete failure fault. The gain fault of sensor is marked by the green square dots in Fig. 7(b) while the bias, linear drift and background noise are marked by purple circular dots. The MS2 classifier is noted to correctly distinguish the gain fault. Similarly, Fig. 7(c) and (d) show that the fault of background noise and linear drift can be accurately distinguished in MS3 and MS4. This comparison shows that the four classifiers based on MMSE method can differentiate the five types of sensor faults with the correlation coefficient r1 and r2 as classification indices.

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Table 3 The fault in Sensor A for different scenarios. Scenarios

1

2

3

4

5

Fault type Parameters

Complete failure c; d

Gain b; d

Bias d; d

Linear drift a, b and d

Background noise noise

Table 4 Fault thresholds. Sensor

Fitted Mean ms

Fitted Variance ss

Threshold gs

A B C D

0 0 0 0

0.3686 0.3432 0.3548 0.3940

1.11 1.02 1.05 1.17

30

Threshold S

25

GLR(S)

20

15

10

5

0

0

100

200

300

400

Test point

(a) Sensor A

(b) Sensor B, C and D

Fig. 4. Pattern of generalized likelihood ratio (scenario 1).

5.2. Experimental study An experiment was conducted on the Dong-shui-men Yangtze River Bridge which is a cable-stayed bridge both for lightrailway and highway as shown in Fig. 8. This bridge has a double tower and single cable plane with 222.5 m þ 445 m þ 190.5 m spans and a total length of 858 m. Fourteen accelerometers were mounted on the main deck to measure the vertical responses. Detailed layout of the measured deck sections and the sensor locations are shown in Fig. 9. The monitoring system of the bridge collected data since October 2014. The acceleration response from accelerometer ZL51-S1 on the bridge deck is shown in Fig. 10. Acceleration data recorded on October 5, 2014 is studied with the assumption that the bridge structure and all the sensors are healthy. Ten minutes recorded data was divided into 300 samples of 2s each. The procedure described in Section 3 was adopted to calculate the threshold of the GLR. The GLR from each sensor are noted different. A total of 300  299 ¼ 89700 GLRs were calculated for each sensor. The threshold value is obtained with 99.74% probability with the 3s criterion with a normal distribution assumption. Next, 10 min test data collected on November.5.2014 and October 5, 2015 were used for the evaluation of the sensor conditions. Similar to the previous set of data collected on the healthy structure and sensors, 89700 GLR values were computed for each sensor. The probability distribution of these GLR values are plotted and curved fitted with a normal distribution, and the probability of exceedance of the threshold value is noted for each sensor. The results are plotted in Fig. 11. It is noted that no sensor has the ||s|| value exceeding the threshold by more than 0.26% probability. Therefore, it may be concluded that the accelerometers on the bridge deck are in healthy condition, which is consistent with results from manual inspection.

L. Li et al. / Journal of Sound and Vibration 442 (2019) 445e458 10

Threshold S

453

Threshold S

8

8 6

GLR(S)

GLR(S)

6

4

2

2

0

4

0

100

200

Test point

300

0

400

0

100

400

(b) scenario 3 12

Threshold S

S Threshold

9

GLR(S)

8

GLR(S)

300

Test point

(a) scenario 10

200

6

6

4 3

2

0

0

100

200

Test point

300

(c) scenario

400

0

0

100

200

300

400

Test point

(d) scenario 5

Fig. 5. Generalized likelihood ratio pattern of Sensor A under scenarios 2 to 5.

Since all the accelerometers are healthy, the performance of detection and classification of sensor fault type is further studied with man-made fault. Again, Sensor ZL51-S1 was simulated to have only one type of fault at one time, and the sensor faults are described as shown in Table 3. The faulty parameters are consistent with descriptions in section 2.1. The threshold of Sensor ZL51-S1 is 1.87 which is obtained from the procedure described in section 3, and the detection result is shown in Fig. 12. Obviously, when the aforementioned five kinds of faults occur in sensor ZL51-S1, the GLR values of the sensor are much larger than the threshold (1.87). It is therefore confirmed that the sensor condition can be correctly evaluated when one of the aforementioned faults occurs. The classification indices r1 and r2 are adopted in the four MS classifiers to determine the fault type of sensor ZL51-S1 similar to those adopted for the numerical study. Results from the four classifiers are shown in Fig. 13. It is noted that the MS1 can distinguish the complete failure fault, and the MS2 can distinguished the gain fault. The background noise fault and the linear drift fault can also be identified via the MS3 and MS4 classifiers. The successful discrimination rate is 100%. It may therefore be concluded that the correlation coefficient r1 and r2 serving as classification indicators can correctly be used to distinguish the five types of sensor faults studied.

6. Conclusions Sensors in a SHM system collect information of the structure for different managerial decision on the operation and performances of the infrastructure. Incorrect data from a faulty sensor will most likely mislead further structural safety evaluation causing false alarms and misinterpretation of the signal which are not the purpose of the installed SHM system. The proposed strategy uses the linear minimum least-squares-error to predict the signal of the faulty sensor. A hypothesis test is conducted against a new threshold based on the generalized likelihood ratio. An index to classify different types of

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Excess threshold ratio(%)

100

80

60

40

20

0 1.2

1.4

1.6

1.8

2.0

2.2

Gain fault amplitude ( ) Fig. 6. Detection results of different gain fault amplitude.

1.0

1.0

Complete failure Gain,Bias,Linear drift,Background noise Decision boundary

0.5

Gain Bias,Linear drift,Background noise Decision boundary

2

2

0.5

0.0

0.0

-0.5

-0.5 -0.2

0.0

0.2

0.4

0.6

0.8

-0.2

1.0

0.0

0.2

0.4

Background noise Bias,Linear drift Decision boundary

0.1

0.8

1.0

(b)MS2

(a)MS1 0.2

0.6

1

1

0.2

0.1

0.0 -0.1

Linear drift Bias Decision boundary

2

2

0.0

-0.2

-0.1

-0.3 -0.4

-0.2

-0.5 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.75

0.80

0.85

0.90

1

1

(d)MS4

(c)MS3 Fig. 7. Classification diagram of MMSE.

0.95

1.00

1.05

1.2

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455

Fig. 8. Dong Shui Men Yangtze River Bridge.

L1

L2

L3

L4

L5

L6

L7

L8

L9

L10

L1

L2

L3

L4

L5

L6

L7

L8

L9

L10

(a) Monitored deck section

ZL11-S1

ZL21-S1

ZL31-S1

ZL51-S1 ZL71-S1 ZL61-S1 ZL81-S1 ZL41-S1

ZL42-D1 ZL52-D1 ZL62-D1

ZL91-S1

ZL101-D1

ZL102-D1

(b) Location of accelerometers on deck Fig. 9. Arrangement of accelerometers.

0.8

Acceleration(m/s2)

0.6 0.4 0.2 0.0

-0.2 -0.4 -0.6

0

5000

10000

15000

20000

25000

Data point Fig. 10. Acceleration signal of real bridge on October 5, 2014.

30000

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Data on Oct.5,2015 Data on Nov.5,2014 Threshold

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malfunction is proposed based on MMSE method. This strategy is experimentally verified with a real set of data from a cablestayed bridge. Numerical and experimental studies show that: 1. The proposed method to detect faulty sensors with hypothesis test based on generalized likelihood ratio can quickly reveal the type of sensor fault and position of the faulty sensor. 2. Classification method on the type of malfunction of sensor is proposed, and the classification indices proposed based on the relevance between the predicted and real signals from sensors. Numerical example with a simply supported beam and experimental data from a cable-stayed bridge show that the five common types of faults in acceleration sensors can be classified properly with four classifiers. 3. Experimental application of the proposed method to a real set of acceleration data from the structural health monitoring system of a cable-stayed bridge show that all the sensors are working properly, which agrees with results from manual inspection. The illustration of the GLR-based detection method is limited to only five kinds of sensor faults. Since sensor fault is not common in newly installed SHM systems, recorded data from such a sensor is not available for verification of the proposed

L. Li et al. / Journal of Sound and Vibration 442 (2019) 445e458

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approach. More evaluation of the method with in-situ faulty sensor is needed. Further research is also needed to check on its applicability to other fault patterns. Though the proposed classification indices are found applicable to the five fault types studied, yet a better and more robust classification index may be necessary to cover other types of fault patterns. Acknowledgments This research work was supported by the Fundamental Research Funds for the Central University (Grant Nos. 2018CDYJSY0055), Graduate Scientific Research and Innovation Foundation of Chongqing, China under the grant number CYB18058, the National Natural Science Foundation of China (Grant Nos. 51578095, 51778093) and the 111 project (Grant Nos. B18062). The comments and work by Prof. S.S. Law in improving the English standard of this paper is fully acknowledged. References [1] [2] [3] [4] [5]

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