# A Method for Aileron Actuator Fault Diagnosis Based on PCA and PGC-SVM

Authors: Wei-Li Qin; Wen-Jin Zhang; Chen Lu
Journal: Shock and Vibration
 (2016)

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## Abstract

Aileron actuators are pivotal components for aircraft flight control system. Thus, the fault diagnosis of aileron actuators is vital in the enhancement of the reliability and fault tolerant capability. This paper presents an aileron actuator fault diagnosis approach combining principal component analysis (PCA), grid search (GS), 10-fold cross validation (CV), and one-versus-one support vector machine (SVM). This method is referred to as PGC-SVM and utilizes the direct drive valve input, force motor current, and displacement feedback signal to realize fault detection and location. First, several common faults of aileron actuators, which include force motor coil break, sensor coil break, cylinder leakage, and amplifier gain reduction, are extracted from the fault quadrantal diagram; the corresponding fault mechanisms are analyzed. Second, the data feature extraction is performed with dimension reduction using PCA. Finally, the GS and CV algorithms are employed to train a one-versus-one SVM for fault classification, thus obtaining the optimal model parameters and assuring the generalization of the trained SVM, respectively. To verify the effectiveness of the proposed approach, four types of faults are introduced into the simulation model established by AMESim and Simulink. The results demonstrate its desirable diagnostic performance which outperforms that of the traditional SVM by comparison.


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## Body

## 1. Introduction

The aileron actuator, which is used to control the aircraft’s rolling movement, is a pivotal component for the flight control system of aircraft [1]. The faults of aileron actuator, which include force motor coil break, sensor coil break, actuator cylinder leakage, and amplifier gain reduction, may cause a series of consequences from control system performance degradation to irretrievable economic loss and personal casualties. Therefore, it is utmost important to research on the fault detection of aileron actuators.Many fault diagnosis approaches have been used and proposed for classification of system health monitoring data, such as decision tree induction, Bayesian-based classification, neural networks, genetic algorithms, and fuzzy set classifiers [2]. Zhao and Su [3] proposed a novel fault diagnosis method for power transformer insulation based on a decision tree. Ozev et al. [4] presented a parametric fault diagnosis approach for analog/RF circuits based on a Bayesian framework. Zang and Imregun [5] performed structural damage detection via artificial neural networks. He et al. [6] used immune genetic algorithm to build a mathematical model for fault diagnosis of a modern power system. Altunok et al. [7] presented a damage pattern recognition approach based on fuzzy set theory. However, most of these methods are computationally expensive and their classification accuracy is highly depending on the sample size. Besides, with these methods, some faults such as hydraulic pump fault and external leakage fault can hardly be diagnosed.Model-based fault detection and diagnosis (FDD) scheme is another important way for FDD of aileron actuators. Henry et al. [8] built an aileron servo-loop model and presented an H-/H∞-based solution fitted with the structure of AIRBUS in-service monitoring systems. Vanek et al. [9] founded a reliable linear parameter-varying (LPV) model of the aircraft and performed two inherently different fault detection and isolation designs for aileron and elevator. Gheorghe et al. [10] presented a simple model-based approach for fault detection in both runaway case and jamming case and yielded a more than good performance under real flight test. Goupil and Marcos [11] built a generic aircraft model and representative fault scenarios and threw light upon both traditional and advanced model-based FDD approaches. Efimov et al. [12] presented a hybrid observer solution associated with the in-service A380 decision-making rules to solve oscillatory failure case in aircraft system.Support vector machines (SVMs) which were originally introduced by Vapnik have been successful for solving classification and function estimation problems. Characterized by convex optimization problems (typically quadratic programming), SVM models are capable of obtaining global minimum, avoiding the trap of local minimum brought by the greedy algorithm in other methods. And the ultimate decision function of SVM is determined by, instead of the whole sample, a few support vectors so that computational complexity is reduced and the curse of dimensionality is shunned. Besides, SVM has the advantage of dealing with nonlinear systems while aileron actuator happens to be a typical nonlinear system. SVM is a classical binary classifier and in order to solve multiclassification problem, which is common in fault diagnosis since there are generally more than two failure modes, many SVM algorithms are proposed to construct the multiclassification classifiers. The algorithms adopted in actual application can be divided into two types [13]: (1) the first type is one-time solution method; (2) combining many binary SVM subclassifiers to achieve multiclassification SVM, the second type includes one-versus-rest, one-versus-one, DDAGSVM, and binary-tree SVM. The research related shows that one-versus-one SVM can be more suitable to actual application because of its comparatively fast training speed and good classification accuracy [13–16].This paper adopts multiclassification algorithm of one-versus-one SVM to design the classifier for the aileron actuator fault diagnosis. In addition, the paper uses grid search to optimize two important parametersC and γ of one-versus-one SVM and Principle Component Analysis (PCA) to reduce dimension. In traditional SVM, the following procedures are usually used: (1) transform data to the format of SVM package; (2) randomly try a few kernels and parameters; (3) test the model. Due to the poor parameter selection and original data complexity, the classification accuracy is relatively unsatisfactory and the training speed is sometimes intolerable. However, the biggest problem is that there will be unclassifiable regions in traditional SVM. The one-versus-one SVM manages to avoid this problem and, with the help of PCA and grid search, the data complexity and parameter selection problems are solved. Hence, compared to the traditional SVM, the method proposed, through case study, yields a higher classification accuracy and a faster training speed while external leakage fault can be effectively diagnosed by the method. Generally, the most important thing to do in aeronautical engineering is to perform an early fault detection to switch as soon as possible on a redundant actuator. Once properly trained on ground using historical data, the proposed algorithm can achieve fault classification as fast as 0.1~1 s each time. With the development Flight Control Computer (FCC), its constraints such as low computational load and restricted symbol library will not be a problem for relatively complex algorithms in the future. On that basis, high fault classification accuracy will be a bonus since corresponding maintenance preparation can be done before the landing of the aircraft and thus efficiency is improved.The remainder of the paper is organized as follows. In Section2, a joint simulation model of aileron actuator based on AMESim and Simulink is set up. In Section 3, the faults of aileron actuator are analyzed and injected into the model. In Section 4, a detailed description of the proposed method is presented. In Section 5, the effectiveness of the proposed approach is demonstrated and the results of experimental are presented and discussed. Finally, the conclusion of the research will be given.


## 2. Setup of Aileron Actuator

The aileron actuator consists of a hydraulic pump, an electrohydraulic servo valve, a cylinder, a PID controller, two electronic amplifiers, and two displacement sensors. The control loop includes two position feedbacks—direct drive valve displacement and actuator cylinder displacement, as is shown in Figure1. In this figure, the signals used for fault detection are marked with red ball.Figure 1
Closed-loop control system of aileron actuator.The simulation model of the aileron actuator is established with MATLAB Simulink and AMESim [17]. Simulink, developed by MathWorks, is the visual simulation environment in MATLAB. Thanks to its convenient graphic model modules such as linear/nonlinear modules, continuous/discrete modules, and advanced control toolboxes, it is quite fit for control loop modeling. However, it could not handle hydraulic modeling lacking corresponding modules. AMESim, developed by Imagine, is a hydraulic/mechanical system modeling, simulation, and analysis software. With abundant parameterized hydraulic modules, the hydraulic part of the aileron actuator can be easily founded. However, there are relatively few control modules in AMESim. With the combination of Simulink and AMESim, the advantages of these two can be fully utilized and thereby a relatively good model of aileron actuator is promising. The control part of the aileron actuator established in Simulink environment is shown in Figure 2; the mechanical part of the aileron actuator is shown in Figure 3. The mechanical part of the aileron actuator established in AMESim is converted to a Simulink S-Function, and the S-Function can be imported to Simulink. The physical parameters of the key components are described in Tables 1–9.Table 1
Elementary hydraulic properties.
FP04-1: all real parameters
Unit
Value
Density
kg/m3
850
Bulk modulus
Bar
17000
Slope of bulk modulus [bar] in function of pressure [bar] (in percentage)
Null
0
Absolute viscosity
cP
51
Absolute viscosity of air/gas
cP
0.02
Saturation pressure (for dissolved air/gas)
Bar
0
Air/gas content
%
0.1
Temperature
degC
40
TemperatureT1
degC
40
Kinematic viscosity at (Patm, T1)
cSt
56.47
TemperatureT2
degC
100
Kinematic viscosity at (Patm, T2)
cSt
9.71
Coefficient for temperature viscosity characteristic
Null
200
Polytropic index for air/gas/vapor content
Null
1.4
(Advanced user) High saturated vapor pressure
Bar
−0.5
(Advanced user) Low saturated vapor pressure
Bar
−0.6
(Advanced user) Absolute viscosity of vapor
cP
0.02
(Advanced user) Effective molecular mass of vapor
Null
200
(Advanced user) Air/gas density at atmospheric pressure 0 degC
kg/m3
1.2Table 2
Parameters of pressure source.
PS00-1: all real parameters
Unit
Value
Time at which duty cycle starts
s
0
Pressure at start of stage 1
Bar
210
Pressure at end of stage 1
Bar
210
Duration of stage 1
s
1e + 06Table 3
Parameters of servo valve.
SV00-1: all real parameters
Unit
Value
Ports P to A flow rate at maximum valve opening
L/min
150
Ports P to A corresponding pressure drop
Bar
20
Ports P to A critical flow number (laminar→ turbulent)
Null
1000
Ports B to T flow rate at maximum valve opening
L/min
150
Ports B to T corresponding pressure drop
Bar
20
Ports B to T critical flow number (laminar→ turbulent)
Null
1000
Ports P to B flow rate at maximum valve opening
L/min
150
Ports P to B corresponding pressure drop
Bar
20
Ports P to B critical flow number (laminar→ turbulent)
Null
1000
Ports A to T flow rate at maximum valve opening
L/min
150
Ports A to T corresponding pressure drop
Bar
20
Ports A to T critical flow number (laminar→ turbulent)
Null
1000
Working density for pressure drop measurement
kg/m3
850
Working kinematic viscosity for pressure drop measurement
cSt
60
Valve rated current
mA
1
Valve natural frequency
Hz
500
Valve damping ratio
Null
0.8
Deadband as fraction of spool travel
Null
0Table 4
Parameters of leakage and viscous friction module.
BAF11-1: all real parameters
Unit
Value
External piston diameter
mm
90
Clearance on diameter
mm
1e − 05
Length of contact
mm
30Table 5
Parameters of piston modules.
BAP11-1: all real parameters
Unit
Value
Piston diameter
mm
90
Rod diameter
mm
30
Chamber length at zero displacement
mm
150Table 6
Parameters of displacement sensor.
DT000-1: all real parameters
Unit
Value
Offset to be subtracted from displacement
M
0
Gain for signal output
1/m
1Table 7
Parameters of mass and displacement module.
MAS005-1: all real parameters
Unit
Value
Mass
kg
10
Coefficient of viscous friction
N/(m/s)
5000
Coefficient of windage
N/(m/s)2
0
Coulomb friction force
N
1000
Stiction force
N
1000
Lower displacement limit
m
−0.15
Higher displacement limit
m
0.15
Inclination (+90 port 1 lowest, −90 port 1 highest)
Degree
0Table 8
Parameters of spring damper.
SD000-1: all real parameters
Unit
Value
Spring rate
N/m
1e + 06
Displacement giving zero spring force
m
0
Damper rating
N/(m/s)
10000Table 9
Parameters of flow control valve.
OR0000: all real parameters
Unit
Value
Characteristic flow rate
L/min
1
Corresponding pressure drop
bar
1
Equivalent orifice diameter
mm
1e − 05
Maximum flow coefficient
null
0.7
Critical flow number (laminar→ turbulent)
Null
1000Figure 2
Control part of aileron actuator in Simulink.Figure 3
Mechanical part of aileron actuator in AMESim.One leakage/viscous friction module, which is utilized for cylinder internal leakage fault injection [18], and two piston modules constitute the simulation model of the hydraulic cylinder of the aileron actuator.In this paper, the aileron actuator works at normal temperature, 40°C, which is shown, as a part of elementary hydraulic properties, in Table1. And the kinematic viscosity will decrease when temperature increases.Under normal condition, the pressure of the pump as shown in Table2 is 210 bar.Under normal condition, the pressure drops and the flow rate at maximum valve opening as shown in Table3 is 20 bar and 150 L/min.Under normal condition, the clearance diameter as shown in Table4 is set to 1e-05 mm. And it will be increased in order to inject the internal leakage fault.The chamber length at zero displacement, the rod diameter, and the piston diameter as shown in Table5 are 150 mm, 30 mm, and 90 mm.The gain for signal output as shown in Table6 is set to be 1.The mass and displacement module, whose parameters are shown in Table7, is adopted to confine the hydraulic cylinder’s movement scope.The spring damper, whose parameters are shown in Table8, is adopted for simulation of the damp of aerodynamic loads.In Table9, list the parameters of flow control valve.


## 3. Fault Analysis and Injection

According to statistical maintenance data, main faults of an aileron actuator include amplifier fault, sensor fault, leakage fault, external leakage fault, pump fault, and valve fault, which are listed in Table10.Table 10
Fault analysis of aileron actuator.
Object
Fault mode
Fault cause
Fault phenomenon
Direct driving valve
Force motor coil shutoff
Force motor coil break
Force motor has no power output; the position of valve element is constantly in the neutral position.
Blockage
Oil pollution
Null bias increases; frequency response decreases; system is unstable.
Valve element stuck
Oil pollutionSlide valve distortion
The direct driving valve output constant flow; the system pressure decreases.
Valve leakage
Edge abrasionRadial valve element abrasion
Null bias increases; gain of system decreases; system pressure decreases; flow noise increases.
Null bias out-of-tolerance
Inaccurate zero setting
Bias exists when DDV is at null position.
Displacement sensor
Coil break
Cable break
Sensor has no output.
Constant bias
Iron core of sensor loose
Displacement bias.
Actuator cylinder
External leakage
Sealing washers failHydraulic leakage
The speed of moving parts decreases; system pressure decreases.
Internal leakage out-of- tolerance
Cylinder abrasionOil in high pressure area flow into low pressure area
The speed of moving parts decreases; system pressure decreases; vibration and noise appear.
Electronic amplifier
Gain reduction
Internal faults of electronic amplifier
Magnification notably decreases.Failure mode, effects, and criticality analysis (FMECA) is a bottom-up, inductive analytical method for fault analysis. According to the FMECAs of hydraulic system made by Li et al. [19] and Balaban et al. [20], these faults can be roughly divided into four quadrants depending on their criticality and frequency. As shown in Figure 4, the first-quadrant faults are high-frequency and high-criticality so that normally they have to be considered in design phase. The second-quadrant faults, currently dealing with visual inspection, are high-frequency but low-criticality. The third-quadrant faults are low-frequency and low-criticality and thus are dismissed taking cost into consideration. The fourth-quadrant faults, which need constant monitoring, are low-frequency but high-criticality.Figure 4
Aileron actuator fault quadrantal diagram.To demonstrate the approach presented in this paper, four faults including electronic faults and mechanical faults were introduced into the simulation model and listed in Table11.Table 11
Fault mode and details.
Number
Fault mode
Fault type
Details
Quadrant
1
Force motor fault
Mechanical fault
Force motor coil break
I
2
Sensor fault
Electronic fault
Sensor coil break
I
3
Leakage fault
Mechanical fault
Actuator cylinder leakage run out of tolerance
II
4
Amplifier fault
Electronic fault
Amplifier gain reduction
IVThe faults listed in Tables10 and 11 were introduced into the simulation model by changing several specific parameters of the fault component, and these components were marked with red box in Figures 2 and 3. The details of fault injection were listed in Table 12. The parameter of force motor fault is set to 0, indicating force motor coil break; the parameter of sensor fault is set to 0, indicating sensor coil break; the parameter of leakage fault is set to 5 instead of the default normal value 1e-5, indicating 5.56% leakage since the diameter of the valve is 90 mm; the parameter of amplifier fault is set to 15 instead of the default normal value 50, indicating 70% signal transmitting loss. All the faults were introduced into the simulation model in advance of stimulation.Table 12
Fault injection details.
Test number
Fault mode
Fault component
Changed parameter for fault injection (unit)
Parameter(normal)
Parameter(fault)
1
Force motor fault
Force motor
FMC gain
1
0
2
Sensor fault
Displacement sensor
Signal output (1/m)
1
0
3
Leakage fault
Actuator cylinder
Flow control valve equivalent orifice diameter (mm)
1
e
-
5
5
4
Amplifier fault
Electronic amplifier
Gain
50
15


## 4. Methodology

The proposed method in the paper consists of two major parts, the model training using historical data and the real-time diagnosis using real-time data. In model training part, three steps are conducted. Firstly, historical data which include DDV input, force motor current, DDV displacement, and actuator cylinder displacement are corrupted by white noise with signal-noise ratio to be 20 dB in MATLAB. Then, the corrupted data are truncated into a number of data segments according to the data period. And the mean, root mean square (RMS), peak-to-peak value (ppV), and kurtosis of these data segments are calculated and normalized, respectively. Hence 16-dimensional primitive inputs, shown in Table13, are obtained. It is worth mentioning that the dimensions of the primitive inputs are not specially chosen since the key fault information will be automatically extracted by PCA. The flow chart of first step is shown in Figure 5. Secondly, PCA is utilized to conduct dimension reduction and noise reduction and thus the reduced inputs for the SVM model are got. Thirdly, the reduced inputs are input to the SVM model for parameter optimization using grid search. Once the optimal parameters for the SVM model are searched, the trained model is prepared for the real-time diagnosis. In Real-time diagnosis, real-time data are acquired by sensors deployed on the air craft. The same data preprocess as in model training part is conducted and the classification result is returned by the trained SVM model. Fault report would be generated for the pilot or ground control station if the classification results meet a certain fault criterion. The flow chart of the method is shown in Figure 6.Table 13
Primitive inputs for SVM model.
Model input
Features
Object
Input 1
Mean
Direct drive valve input
Input 2
Force motor current
Input 3
Direct drive valve displacement
Input 4
Actuator cylinder displacement
Input 5
RMS
Direct drive valve input
Input 6
Force motor current
Input 7
Direct drive valve displacement
Input 8
Actuator cylinder displacement
Input 9
ppV
Direct drive valve input
Input 10
Force motor current
Input 11
Direct drive valve displacement
Input 12
Actuator cylinder displacement
Input 13
Kurtosis
Direct drive valve input
Input 14
Force motor current
Input 15
Direct drive valve displacement
Input 16
Actuator cylinder displacementFigure 5
Flow chart of data preprocess.Figure 6
Flow chart of fault diagnosis.


### 4.1. PCA

Invented by Karl Pearson, PCA adopts orthogonal mapping to map a set of possibly correlated variables to principal components that are linearly uncorrelated. The greatest variance lies on the first principal component, the second greatest variance on the second principal component, and so on.Here is a data matrix,X, whose n rows are n different repetition of the experiment and whose p columns are p different parameters: X=(X(1),…,X(i),…,X(n))T, X(i)=(x1,…,xp)(i). The scores of new vector of principal components t(i)=(t1,…,tp)(i), i=1,2,…,n, where i is ordinal number of the row, are given through a mathematical transformation defined by p-dimensional vectors of weights w(k)=(w1,…,wp)(k)T,  k=1,2,…,p, where k is the ordinal number of the principal component. The equation is shown as below:(1)tki=Xi·wk,where tk(i) is the kth component of X(i), and thus the maximum possible variance from X is inherited by t with each vector of weight w constrained as a unit vector.The first component of a data vectorX(i) can then be given as a score t1(i)=X(i)·w(1) in the transformed coordinates, where w(1) has to satisfy(2)w1=argmaxw=1⁡∑it1i2=argmaxw=1⁡Xw2=argmaxw=1⁡wTXTXw,and the kth component can be found by subtracting the first k-1 principal components from X:(3)X^k=X-∑s=1k-1XwswsT,and then finding the weight vector which extracts the maximum variance from this new data matrix(4)wk=argmaxw=1⁡X^kw2=argmaxw=1⁡wTX^kTX^kTw,the kth component of a data vector X(i) can then be given as a score tk(i)=X(i)·w(k) in the transformed coordinates.The full principal component decomposition ofX can therefore be given as(5)T=XW,where W is a p-by-p matrix whose columns are the eigenvectors of XTX.


### 4.2. Grid Search and CV

Grid search executes exhaustive searching through an artificially selected parameter set of certain learning algorithms. A typical soft-margin SVM classifier equipped with an RBF kernel has two parameters that need to be tuned: a regularization constantC and a kernel hyper parameter γ. The goal of grid search is to identify good pair (C,γ) so that the classifier can accurately predict unknown data. Exponentially growing sequences of C and γ (e.g., C=2-5,2-3,…,215, γ=2-15,2-13,…,23) are recommended by Hsu et al. [21].In contrast with other optimization algorithms such as genetic algorithm and particle swarm algorithm, the grid search is straightforward but seems naive. However, there are two motivations why I prefer the simple grid-search approach. One is that, psychologically, we may not feel safe to use methods which avoid doing an exhaustive parameter search by approximations or heuristics. The other reason is that the computational time required to find good parameters by grid search is not much more than that by advanced methods since there are only two parameters in this case. Furthermore, the grid search can be easily parallelized because each (C,γ) is independent. Many of advanced methods are iterative processes, for example, walking along a path, which can be hard to parallelize.And the performance of the pair is assessed by cross validation on the training set. The training set is divided intov equal-sized subsets in v-fold cross validation. In proper sequence, each subset is used for test while other v-1 subsets are used for classifier training. Hence, v prediction results are obtained and the percentage of data correctly classified is the final cross validation accuracy. Rodriguez et al. [22] conducted a sensitivity analysis for cross validation and found 10-fold cross validation is a practical method.


### 4.3. One-versus-One SVM

One-versus-one SVM is proposed by Knerr et al. [23] that transform the n-classification problem into n(n-1)/2 two-classification problem. One-versus-one SVM adopts the voting method to get, respectively, the number of votes that the sample x belongs to each classification. In the end, x belongs to the classification in which the number of votes is the largest. Hsu and Lin [24] compared one-versus-one SVM, one-versus-all SVM, and DAG-SVM and the results showed that one-versus-one SVM may be more suitable for practical use.In order to construct the subclassifier for classi and j, take the sample of class i and class j from the original sample as the training sample for two-classification problem; the optimal problem is shown as follows:(6)minωij,bij,ξij12ωijTωij+C∑j=1iξiijωijT.Corresponding decisive plane is(7)ωijTϕxt+bij≥1-ξiij,ifyt=i,ωijTϕxt+bij≤-1+ξiij,ifyt=j,ξiij≥0,where ωij is the coefficient of the hyperplane between classes i and j, bij is the intercept of the hyperplane between classes i and j, ϕ(xt) is the map of the sample xt in the high-dimension space, C is error penalty factor which reflects the valued degree of sample outliers and adjusts the proportion between the incredible range and empirical risk of SVM network model, and ξiij is the fitting error variable.


### 4.4. Alarm Criterion

With the continuous development of FCC, the limitation of FCC will be kept pushing. Once the computational limits were broken and complex algorithm could also achieve practically fast fault detection, then in order to reduce false alarm rate to the greatest extent, three-alarm criterion listed below can be attempted.Criterion 1 (two consecutive classification results concur).
If two consecutive classification results concur, then the classification results are validated and the corresponding fault can be reported.Criterion 2 (two out of three consecutive classification results concur).
If the first two consecutive classification results differ and yet two out of three consecutive classification results concur, then the classification results are validated and the corresponding fault denoted by the two same classification results can be reported.Criterion 3 (none of three consecutive classification results concur).
If three consecutive classification results all differ, then the diagnosis fails and the diagnosis-fail report will be submitted.


## 4.1. PCA

Invented by Karl Pearson, PCA adopts orthogonal mapping to map a set of possibly correlated variables to principal components that are linearly uncorrelated. The greatest variance lies on the first principal component, the second greatest variance on the second principal component, and so on.Here is a data matrix,X, whose n rows are n different repetition of the experiment and whose p columns are p different parameters: X=(X(1),…,X(i),…,X(n))T, X(i)=(x1,…,xp)(i). The scores of new vector of principal components t(i)=(t1,…,tp)(i), i=1,2,…,n, where i is ordinal number of the row, are given through a mathematical transformation defined by p-dimensional vectors of weights w(k)=(w1,…,wp)(k)T,  k=1,2,…,p, where k is the ordinal number of the principal component. The equation is shown as below:(1)tki=Xi·wk,where tk(i) is the kth component of X(i), and thus the maximum possible variance from X is inherited by t with each vector of weight w constrained as a unit vector.The first component of a data vectorX(i) can then be given as a score t1(i)=X(i)·w(1) in the transformed coordinates, where w(1) has to satisfy(2)w1=argmaxw=1⁡∑it1i2=argmaxw=1⁡Xw2=argmaxw=1⁡wTXTXw,and the kth component can be found by subtracting the first k-1 principal components from X:(3)X^k=X-∑s=1k-1XwswsT,and then finding the weight vector which extracts the maximum variance from this new data matrix(4)wk=argmaxw=1⁡X^kw2=argmaxw=1⁡wTX^kTX^kTw,the kth component of a data vector X(i) can then be given as a score tk(i)=X(i)·w(k) in the transformed coordinates.The full principal component decomposition ofX can therefore be given as(5)T=XW,where W is a p-by-p matrix whose columns are the eigenvectors of XTX.


## 4.2. Grid Search and CV

Grid search executes exhaustive searching through an artificially selected parameter set of certain learning algorithms. A typical soft-margin SVM classifier equipped with an RBF kernel has two parameters that need to be tuned: a regularization constantC and a kernel hyper parameter γ. The goal of grid search is to identify good pair (C,γ) so that the classifier can accurately predict unknown data. Exponentially growing sequences of C and γ (e.g., C=2-5,2-3,…,215, γ=2-15,2-13,…,23) are recommended by Hsu et al. [21].In contrast with other optimization algorithms such as genetic algorithm and particle swarm algorithm, the grid search is straightforward but seems naive. However, there are two motivations why I prefer the simple grid-search approach. One is that, psychologically, we may not feel safe to use methods which avoid doing an exhaustive parameter search by approximations or heuristics. The other reason is that the computational time required to find good parameters by grid search is not much more than that by advanced methods since there are only two parameters in this case. Furthermore, the grid search can be easily parallelized because each (C,γ) is independent. Many of advanced methods are iterative processes, for example, walking along a path, which can be hard to parallelize.And the performance of the pair is assessed by cross validation on the training set. The training set is divided intov equal-sized subsets in v-fold cross validation. In proper sequence, each subset is used for test while other v-1 subsets are used for classifier training. Hence, v prediction results are obtained and the percentage of data correctly classified is the final cross validation accuracy. Rodriguez et al. [22] conducted a sensitivity analysis for cross validation and found 10-fold cross validation is a practical method.


## 4.3. One-versus-One SVM

One-versus-one SVM is proposed by Knerr et al. [23] that transform the n-classification problem into n(n-1)/2 two-classification problem. One-versus-one SVM adopts the voting method to get, respectively, the number of votes that the sample x belongs to each classification. In the end, x belongs to the classification in which the number of votes is the largest. Hsu and Lin [24] compared one-versus-one SVM, one-versus-all SVM, and DAG-SVM and the results showed that one-versus-one SVM may be more suitable for practical use.In order to construct the subclassifier for classi and j, take the sample of class i and class j from the original sample as the training sample for two-classification problem; the optimal problem is shown as follows:(6)minωij,bij,ξij12ωijTωij+C∑j=1iξiijωijT.Corresponding decisive plane is(7)ωijTϕxt+bij≥1-ξiij,ifyt=i,ωijTϕxt+bij≤-1+ξiij,ifyt=j,ξiij≥0,where ωij is the coefficient of the hyperplane between classes i and j, bij is the intercept of the hyperplane between classes i and j, ϕ(xt) is the map of the sample xt in the high-dimension space, C is error penalty factor which reflects the valued degree of sample outliers and adjusts the proportion between the incredible range and empirical risk of SVM network model, and ξiij is the fitting error variable.


## 4.4. Alarm Criterion

With the continuous development of FCC, the limitation of FCC will be kept pushing. Once the computational limits were broken and complex algorithm could also achieve practically fast fault detection, then in order to reduce false alarm rate to the greatest extent, three-alarm criterion listed below can be attempted.Criterion 1 (two consecutive classification results concur).
If two consecutive classification results concur, then the classification results are validated and the corresponding fault can be reported.Criterion 2 (two out of three consecutive classification results concur).
If the first two consecutive classification results differ and yet two out of three consecutive classification results concur, then the classification results are validated and the corresponding fault denoted by the two same classification results can be reported.Criterion 3 (none of three consecutive classification results concur).
If three consecutive classification results all differ, then the diagnosis fails and the diagnosis-fail report will be submitted.


## 5. Fault Diagnosis and Result Analysis

In this simulation case, the amplitude of system input is 2 mm, the frequency of system input is 0.5 Hz, the sampling rate is 200 S/s, and the sampling time is 20 s. Accordingly, the features, which include mean, RMS, ppV, and kurtosis, are extracted every 200 points. The data obtained are considered as the historical data to train the model. Once trained, the model can be used to detect faults every 200 points (less than 1 s depending on the real-time sampling rate) in real-time diagnosis. The extracted features at different working conditions are listed in Figure7, from which it is clear to see that different working conditions result in different feature amplitude so that fault classification using these features is positively tenable.Figure 7
Features extracted.The results of PCA for the 16-dimensional primitive inputs are shown in Table14 and Figure 8, from which it is clear to see that the first three principal components occupy up to 100% cumulative contribution rate and that the inputs dimension may even be reduced to just 1-dimension depending on the ultimate test rate of the trained model.Table 14
Contribution rate of principal component.
Principal component
Eigenvalue
Contribution rate
Cumulative contribution rate
1
4.0114
83.44%
83.44%
2
0.1995
4.15%
87.59%
3
0.1339
2.79%
90.38%
4
0.1257
2.61%
92.99%
5
0.1048
2.18%
95.17%
6–16
<0.1
<2%
>95.17%Figure 8
Result of PCA.As shown in Table15, using just the first principal component to train the model, the trained model’s cross validation rate is 76.73% and test rate is 76.75%, which implies overreduction and information loss. Using first five principal components to train the model, the trained model’s cross validation rate is increased up to 98.88% and test rate to 99.01%, close to 1, which is more than good so the reduced inputs can be determined as three-dimensional.Table 15
The cross validation rate and test rate with regard to different dimension of inputs.
Dimension
CV rate
Test rate
1
76.73%
76.75%
2
82.08%
83.05%
3
89.37%
89.87%
4
96.63%
96.50%
5
98.88%
99.01%There are two parameters, penalty factorC and kernel parameter γ, to be optimized in one-versus-one SVM using RBF kernel which is shown below:(8)Kx,x′=exp⁡γx-x′2.The exponentially growing sequences of parameter pairs (C,γ) are adopted in grid search and the contour of parameter pairs is shown in Figure 9. The best pair (C=4096, γ=1), whose cross validation accuracy was up to 98.87%, was obtained.Figure 9
Grid search for model with PCA.The final results, shown in Table16, indicated that the PGC-SVM proposed in the paper outperformed traditional SVM in both time cost (Pentium(R) Dual-Core CPU T4500 @ 2.30 GHz) and classification accuracy.Table 16
Comparison between traditional SVM and PGC-SVM.
Method
Time cost
Classification accuracy
Parameters optimization
Training
General SVM
days even months
1.5884 s
81.87%
PGC-SVM
8 h 20 mins
0.8989 s
98.88%


## 6. Conclusions

This paper presents an aileron actuator fault diagnosis approach combining principal component analysis (PCA), grid search (GS), 10-fold cross validation (CV), and one-versus-one support vector machine (SVM). The classification accuracy is good enough for the diagnosis of the main faults of aileron actuators which include force motor coil break, sensor coil break, actuator cylinder leakage, and amplifier gain reduction. Compared to the traditional SVM, the PGC-SVM demands less time for both parameters optimization and model training. The performance of the proposed algorithm is fast enough for fault detection to switch as soon as possible on a redundant actuator. The high classification accuracy of actuator faults gives the algorithm a bonus that maintenance efficiency can be promoted since maintenance preparation can be done before the landing of aircraft. Hence, the algorithm presented in the paper shows a great potential once it passed the rigorous test of real FCC.Obviously, the future work lies in the field validation of the proposed algorithm. And in practice, the field data are often severely corrupted by various data noise that may influence the performance of the algorithm so that noise immunity of the proposed algorithm should also be considered. Besides, the method still has some room for improvement—the computational resource consumption can be compressed further and detection delay can be minimized.



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*Source: 4807250-2016-01-20.xml*