Sensor Fault Detection Based on Particle Filter and ...

Jordan Journal of Civil Engineering, Volume 13, No. 4, 2019

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Sensor Fault Detection Based on Particle Filter and Mahalanobis Distance

Tianzhi Li 1), Gang Liu2) and Liangliang Zhang 3)

1) Master Student, School of Civil Engineering, Chongqing University, No.174 Shazhengjie, Shapingba, Chongqing, 400044, China. E-Mail: [email protected]

2) Professor, School of Civil Engineering, Chongqing University, No.174 Shazhengjie, Shapingba, Chongqing, 400044, China. E-Mail: [email protected]

3) Professor, School of Civil Engineering, Chongqing University, No.174 Shazhengjie, Shapingba, Chongqing, 400044, China. E-Mail: [email protected]

ABSTRACT

It is essential to evaluate the health condition of structure and sensors in structural health monitoring (SHM)

systems. Compared with the identification of structural damage, only a few papers investigated sensor fault

detection in civil engineering. This paper presents a model-based sensor fault detection approach utilizing

particle filter (PF) and Mahalanobis distance (MD). The discrete state-space model is constructed by a system

identification algorithm named N4SID instead of physical principles. A sensor fault is determined if

Mahalanobis distance between test state and training state is above the threshold. The experimental study

demonstrates the accuracy and efficiency of the proposed method.

KEYWORDS: Structural health monitoring, Sensor fault detection, Particle filter, Mahalanobis distance, System identification.

INTRODUCTION

During recent years, large-scale civil structures

equipped with multiple sensors are becoming popular as

a result of developing SHM systems. Since the structure

conditions are evaluated by the data measured by

sensors over the service period, it is essential to detect

the abnormal condition in sensors, which is called sensor

fault detection (FD). Sensor fault detection decides

whether there is an abnormal condition in the monitored

system (Gao, Cecati and Ding, 2015). In civil

engineering, a large list of relevant works are related to

the identification of structural damage. Only a few

papers focused on the detection of sensor faults (Kullaa,

2011; Li et al., 2018), which is solved in this study.

There are two major approaches for sensor fault

detection problems (Thomas, 2002): model-free and

model-based. In model-free methods, several physical

sensors at the same location measure the same quantity

to generate physical redundancy. The applications of

these methods are inevitably limited by many factors,

such as size, budget, weight,… etc. As to model-based

methods, a mathematical model of the monitoring

system is always employed to acquire redundancy.

Model-based methods with accurate models are able to

offer better performance compared with model-free

methods (Chen and Patton, 1999).

Generally, the residual between the measured output

from sensor and estimated output from the model is used

as detection feature for most model-based FD problems.

The residual generation can be broadly classified as

(Poon et al., 2016): observer -based methods, parameter

estimation methods or statistical approaches. Among Received on 3/12/2018. Accepted for Publication on 5/10/2019.

Sensor Fault Detection… Tianzhi Li, Gang Liu and Liangliang Zhang

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these approaches, the observer-based methods,

including Kalman filter, particle filter, multiple model

filter,… etc, are the most investigated fields in sensor

FD problems (Khan et al., 2017). These approaches are

also most promising due to their flexibility in choosing

proper observers to match performance requirements

(Cai and Wu, 2011).

In this paper, therefore, the observer-based approach

is selected and we apply the powerful algorithm in a

nonlinear system called particle filter. The residuals of

estimated output and measured output based on particle

filter (PF) are extensively employed in numerous studies

(V. Kadirkamanathan et al., 2002; Duan et al., 2008;

Tadić and Ðurović, 2014). Meanwhile, based on the fact

that the sum of unnormalized particle weights declines

if a sensor fault occurs, the decision rule can also be

developed by the weight of each particle (Li and

Kadirkamanathan, 2001; Yin and Zhu, 2015). However,

in a highly nonlinear system, the one-dimensional

residual or weight is usually unable to offer

comprehensive evaluation for system health condition.

For more accurate sensor FD results, PF is then

exploited to generate the state vector to construct the

relatively precise detection function (Wei, Huang and

Chen, 2009). Inspired from above, state vector is

selected as sensor FD index and then Mahalanobis

distance, powerful in measuring the correlation of data,

is applied to develop sensor FD rule.

The other parts of the paper are organized as follows:

firstly, the sensor FD problem is formulated; then, the

background of particle filter and Mahalanobis distance

is introduced; following that, the detection and

identification approach is described in detail; finally, the

experimental example illustrates the effectiveness of

this proposed method in sensor FD problems.

PROBLEM FORMULATION

In model-based methods, the model is obtained by

using either physical principles or system identification

techniques. It is difficult to construct a state-space model

of a large-scale civil structure by physical principles and

measure precise excitation, so the output-only space-

state model is established using system identification in

this study.

𝒙 𝑨𝒙 𝒘 (1)

𝑦 𝑯𝒙 (2)

where x and 𝑦 represent the state vector and true

output vector; A is the system state transmission with

nn dimension; H is the output influence matrix with

1n dimension; n is the dimension of the model; w is

the system noise vector. The subscripts k and k+1

describe the step time.

In the absence of sensor faults, the measured output

vector yk is described as 𝑦 𝑦 𝑣 , where v is the

measurement noise vector. Otherwise, there are

residuals between yk and 𝑦 . Three common types of

sensor faults are considered here, such that gain 𝑦𝑎 𝑦 𝑣 , bias 𝑦 𝑦 𝑏 𝑣 and constant 𝑦𝑐 𝑣 , where a, b, c are invariant constants, the

superscript i denotes the fault sensor channel. More

details about those three types of sensor faults can be

found in the reference (Li et al., 2018).

FAULT DETECTION AND ISOLATION

Background on Particle Filter

As a powerful method for the construction of

approximate probability density functions

(PDF) 𝑝 𝒙 |𝒚 : in linear and nonlinear systems, the

procedure of particle filter algorithm is expressed as

follows (Gordon, Salmond and Smith, 2002):

Predict: there are a set of particles 𝒙 : 𝑖 1, … , 𝑁

from known PDF 𝑝 𝒙 |𝒚 : and then a set of

samples 𝒙 ∗: 𝑖 1, … , 𝑁 is obtained by Equation

(1), where N is the amount of particles.

Update: considering measurements yk, a weight 𝑞 is

assigned to each 𝒙 ∗:

𝑞|𝒙 ∗

∑ |𝒙 ∗ (3)


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A discrete distribution over 𝒙 ∗: 𝑖 1, … , 𝑁 is

defined and the weight 𝑞 is related to state vector 𝒙 ∗.

Resample: a set of values 𝒙 : 𝑖 1, … , 𝑁

obtained by the above distribution 𝒙 ∗: 𝑖 1, … , 𝑁

after resampling finally construct approximate PDF

𝑝 𝒙 |𝒚 : .

𝑝 𝒙 |𝒚 : ∑ 𝑞 𝛿 𝒙 𝒙 (4)

Once p(xk|y1:k) is determined, we can develop the

minimum mean squared error (MMSE) estimation of the

state xk.

𝒙 𝒙 𝑝 𝒙 |𝒚 : 𝑑𝒙 (5)

If a fault arises, the change of measured output leads

to significant discrepancy between the state vector 𝒙

and 𝒙 .

Sensor Fault Detection Approach

There are four types of data utilized in this study;

namely, N4SID data y(n), training data y(r), threshold data

y(h) and test data y(t). The first three data types are

obtained in baseline condition, when the sensor is

healthy. We apply N4SID data to construct the state-

space model. It is worth noting that there will be an

individual output-only state-space model for each

sensor. Then, training data is applied to generate the

state vector 𝒙 by Equation (5). A fix-width (m=300)

sliding window of state ∑ 𝒙 serves as fault index:

∑ 𝒙 𝒙 𝒙 ⋯ 𝒙 (6)

In order to construct the threshold, the state matrix X

of training data formulated in Equation (7) is applied to

generate its mean vector μ and matrix covariance S.

𝑿 ∑ 𝒙 ∑ 𝒙 … ∑ 𝒙 (7)

The sum of state vector ∑ 𝒙 or ∑ 𝒙

formed by the threshold data or test data, μ and S,

constructs Mahalanobis distance.

𝑀𝐷 ∑ 𝒙 𝝁 𝑺 ∑ 𝒙 𝝁 (8)

𝑀𝐷 ∑ 𝒙 𝝁 𝑺 ∑ 𝒙 𝝁 (9)

Then, 𝑀𝐷 : 𝑗 1,2 … 𝑁 are sorted in a

descending order and the control limit h chosen is

determined.

ℎ 𝑀𝐷 ∗ (10)

The empirical coefficient α is selected as 0.01

according to the reference (Verdier and Ferreira, 2011).

A fault is detected while 𝑀𝐷 formed by Equation

(8) is above the threshold h. All the sensor fault

detection procedure is presented in Figure 1.

Figure (1): Sensor fault detection procedure

Training DataOutput-onlyState-space

Model

N4SID Data

N4SID

Threshold Data

μ S

Test Data

0ˆm h

k iix

0ˆm t

k iix

Threshold h

There is a fault

Yes

tkMD

?tkMD h


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EXPERIMENTAL STUDY

Experimental Details

In order to assess the performance of the proposed

approach in reality, a two-floor frame described in

Figure 2 was built in the laboratory. The acceleration of

each floor was measured at X direction with a sampling

frequency of 200Hz. There were four circle columns

with 7mm diameter on each floor. Data representing

sensor faults was artificially created by healthy data. We

assumed three typical sensor faults; namely, gain, bias

and constant, to have occurred in sensor 1 and sensor 2.

The acceleration data of the two sensors was

collected under environmental noise, which meant that

the structural dynamic behavior was mainly controlled

by stochastic excitation and the signals should remain

relatively steady all the time. The 90s data (18000 steps)

presented in Figure 3 was under baseline condition when

the two sensors were healthy. We defined the data from

time step 1 to 3000 (15s) as N4SID data, data from time

step 3001 to 6000 (15s) as training data, data from time

step 10001 to 13000 (15s) as threshold data and data

from time step 15001 to 18000 (15s) as test data. We

assumed that there was one faulty sensor at one time and

data representing sensor faults was artificially created by

the healthy test data (Case 1). The seven test cases listed

in Table 1 were included to verify the feasibility of this

FD rule.

Figure (2): Experimental equipment

Figure (3): Healthy data utilized in this study


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Table 1. Seven test cases

Test Case Faulty Sensor Faulty Type Parameter 1 All healthy2

1 Gain a=1.5

3 Bias b=0.0003 4 Constant c=0.0003 5

2 Gain a=1.5

6 Bias b=0.0003 7 Constant c=0.0003

The N4SID algorithm, widely used for system

identification and easily achieved by MATLAB system

identification box (Ljung, 1991), formed the optimal

parameters of A, H, w and v by the N4SID data. It is

worth noting that there is an individual state-space

model for each sensor. The training data from step 3001

to 6000 was employed to generate μ, S. Then, the

threshold data between time step 10001 and 13000

provided ∑ 𝒙𝒌𝒉 to construct the threshold by

Equation (8). Figure 4 presents 2700 samples

𝑀𝐷 : 𝑘 1,2, … ,2700 and the control limits

h1=8.18, h2=6.20 calculated by Equation (10). The

thresholds are applied to FD.

Figure (4): Mahalanobis distances and control limits calculated by threshold data

Sensor Fault Detection Results ∑ 𝒙 : 𝑘 15001,17002, … ,17700 from test

data and μ, S were applied to calculate Mahalanobis

distances 𝑀𝐷 : 𝑘 1,2, … ,2700 by Equation (9).

Figure 5 and Figure 6 represent the detection

performance for the three types of faults in sensor 1 and

sensor 2, respectively. In case 1, all Mahalanobis

distances are below the threshold, which means that

there is no sensor fault. As to the other test case, sensor

faults, including gain, bias and constant, are successfully

detected, since their MDs are above the threshold.

Moreover, this fault detection method shows its

sensitive detection ability to bias and constant, since the

MDs in cases 3, 4, 6 and 7 are largely above the

thresholds.


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Figure (5): Fault detection in sensor 1

Figure (6): Fault detection in sensor 2

CONCLUSION

A sensor fault detection approach for SHM systems

is presented in this paper. PF is employed to generate

system states of training data and test data. Then,

Mahalanobis distance is utilized to detect sensor faults.

Three typical sensor faults; namely, gain, bias and

constant, are used to verify the proposed method. The

results show that this method is very sensitive to these

faults, especially bias and constant. The discrimination

of structural damage and sensor faults is not investigated

in this paper. Our future work intends to solve this

problem.

ACKNOWLEDGEMENT

The authors would like to thank the support from the

Natural Sources Foundation of China (Grant No.

51578095).


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