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Research Article
International Journal of Distributed
Sensor Networks
2018, Vol. 14(10)
ÓThe Author(s) 2018
DOI: 10.1177/1550147718805938
journals.sagepub.com/home/dsn
Fault diagnosis method for closed-loop
satellite attitude control systems based
on a fuzzy parity equation
Hua Song
1,2
, Pengqian Han
1,2
, Junxiang Zhang
1
and Chunhua Zhang
1
Abstract
This article proposes a fault diagnosis method for closed-loop satellite attitude control systems based on a fuzzy model
and parity equation. The fault in a closed-loop system is propagated with the feedback loop, increasing the difficulty of
fault diagnosis and isolation. The study uses a Takagi-Sugeno (T-S) fuzzy model and parity equation to diagnose and iso-
late a fault in a closed-loop satellite attitude control system. A fully decoupled parity equation is designed for the closed-
loop satellite attitude control system to generate a residual that is sensitive only to a specific actuator and sensor. A T-S
fuzzy model is used to describe the nonlinear closed-loop satellite attitude control system. With the combination of the
T-S fuzzy model and fully decoupled parity equation, the fuzzy parity equation (FPE) of the nonlinear system can be
obtained. Then this article uses a parameter estimator based on a Kalman filter to identify deviations and scale factor
changes from information contained in the residuals generated by the FPE. The actuator and sensor fault detection and
isolation simulation of the three-axis stable satellite attitude control system is provided for illustration.
Keywords
Satellite attitude control system, fault diagnosis, closed-loop system, parity equation
Date received: 31 May 2017; accepted: 4 September 2018
Handling Editor: Amiya Nayak
Introduction
The closed-loop satellite attitude control system
(CLSACS) is the most critical subsystem guaranteeing
the normal operation of a satellite. However, it has a
complex structure, a poor working environment,
unknown disturbances and uncertain factors.
Furthermore, it is one of the most fault-prone subsys-
tems. For satellites in orbit, an attitude control system
fault often leads to disaster and is likely to result in sat-
ellite roll and attitude loss within a short period, ulti-
mately causing satellite mission failure.
1–7
Because the
faulty satellite component is irreparable, fault-tolerant
control methods for the CLSACS must be investigated,
and the operational status of the CLSACS must be
effectively monitored. Timely detection of a fault in the
CLSACS and the implementation of effective fault-
tolerant controls can improve the reliability and
security of satellites and also reduce risk. Incorporation
of reconfigurable or safety controls can protect the
CLSACS from catastrophic accidents as well as
enhance the resilience of the system.
8,9
Traditional fault diagnosis methods primarily con-
sider open-loop systems, and there are currently no in-
1
School of Automation Science and Electrical Engineering, Beihang
University, Beijing, P.R. China
2
Collaborative Innovation Center for Advanced Aero-Engine, Beihang
University, Beijing, P.R. China
3
School of Energy and Power Engineering, Beihang University, Beijing, P.R.
China
Corresponding author:
Pengqian Han, School of Energy and Power Engineering, Beihang
University, No. 37, Xueyuan Road, Haidian District, Beijing 100083, P.R.
China.
Email: hanpengqian@126.com
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
depth studies of fault diagnosis for closed-loop systems
in the literature. Although Adnane et al.
10
demonstrate
a new design version of the extended Kalman filter
which is effective and can be utilized as an alternative
LEO satellite attitude estimation system, the study did
not consider the sensor fault propagation problem in
closed-loop systems. Zhou et al.
11
proposed two main
reasons for a decline in fault diagnosis performance.
First, as the introduction of a feedback system generally
makes a system more robust to external disturbances,
the impact of the control variable may be oversha-
dowed when a fault is in its early stages or of a smaller
amplitude. The residual signal may still be within a
range of small fluctuations when a fault occurs, making
fault detection difficult and increasing the loss detection
rate. Second, feedback control may propagate faults
within a system, resulting in abnormal signals. For
example, in an open-loop system, a sensor fault occur-
ring in the system will not affect other sensors because
the measurement signal is still within the normal range,
while in a closed-loop system, with a feedback signal
introduced, the use of an abnormal measured value by
the feedback controller may cause the control signal to
deviate from the normal trend and subsequently cause
the entire system to depart from the normal operating
range. In this case, the signals measured by other sen-
sors are also abnormal. Overall, fault propagation
increases the difficulty of fault isolation. Research and
analysis related to closed-loop diagnosis are important
aspects of fault-tolerant control for a closed-loop sys-
tem, which improves the robustness of the system.
Yang and Zhang
12
achieved fault detection and isola-
tion for a closed-loop system by constructing a double
observer. However, the observer gain matrix design was
complex, the computational burden was large, and the
fault diagnosis performance was poor. Ruschmann
et al.
13
used fuzzy parity equations (FPE) to diagnose the
faults of actuators and sensors in flight control systems
and estimated the relevant fault parameters, though faults
in closed-loop systems were neither estimated nor ana-
lysed. Cheng et al.
14
proposed a combined method for
detecting and isolating minor actuator faults in closed-
loop control systems, yet they did not analyse the propa-
gation of faults in a closed-loop system. Wu and Liu
15
diagnosed the fault in the pitch actuator of wind turbines
with the method that combined the interval prediction
algorithm with the recursive subspace identification based
on the variable forgetting factor algorithm. Because the
fault in the pitch actuator will propagate in the closed-
loop system, the method cannot diagnose the fault in the
closed-system.
This article examines the system time redundancy
and principles of fault propagation in closed-loop sys-
tems and then establishes a set of fully decoupled parity
equations (FDPEs) for fault diagnosis of actuators and
sensors.
This study presents a fault diagnosis that incorpo-
rates both FDPEs and the Takagi-Sugeno (T-S) model,
a method in which the residual is sensitive to a specific
fault and is decoupled from system states, disturbance
inputs and other faults. The design methods of the
FDPEs for the fault control system (actuator faults,
sensor faults and multiple faults) is discussed; the fault
model is estimated through parameter estimation using
residual information; and the simulations of detection,
isolation and identification of actuator, sensor and
multiple faults are accomplished.
CLSACS dynamics model
The CLSACS is a typical closed-loop control system. A
fault in a member of a closed-loop system spreads with
the closed-loop feedback, and detection and isolation
are difficult to achieve.
16,17
In this study, a rigid three-
axis stabilized satellite is considered.
The satellite actuator consists of three orthogonal
flywheels, and the measuring device is a combination
of star sensors and gyros. The three-axis stabilized sat-
ellite attitude dynamics equations are as follows
18
Ix_vx+JzJy
vyvz=Tcx +HyvzHzvy+Tdx
Iy_vy+JxJz
ðÞvzvx=Tcy +HzvxHxvz+Tdy
Iz_vz+JyJx
vxvy=Tcz +HxvyHyvx+Tdz
8
<
:ð1Þ
where vx,vy,vzdenote the three angular velocities of
the satellite; Hx,Hy,Hz, the moment angular momen-
tum; Tdx,Tdy,Tdz, the space environmental disturbance
torques; Tcx,Tcy,Tcz, the control torques; and Ix,Iy,Iz,
the satellite inertia moments.
Assume that the rotational speed of the satellite orbit
system with respect to the inertial system is (0,v0,0).
The three attitude angles of the satellite with respect to
the tracking system are the roll angle u, pitch angle u
and yaw angle c. The rotation angular velocity of the
satellite relative to the orbital coordinate system is
½_
u_
u_
c.
When the three attitude angles are small, the conver-
sion matrix for converting from the satellite orbit coor-
dinate system to the body axes coordinate system is
CBO =
1cu
c1u
uu1
2
43
5ð2Þ
Then, vx,vy,vzcan be expressed as
vx
vy
vz
2
43
5=
_
f
_
u
_
c
2
43
5+
1cu
c1f
uf1
2
43
5
0
v0
0
2
43
5=
_
fv0c
_
uv0
_
c+v0f
2
43
5
ð3Þ
2International Journal of Distributed Sensor Networks
After applying differential calculus to equation (3),
we obtain
_vx
_vy
_vz
2
43
5=
€
fv0_
c
€
u
€
c+v0_
f
2
43
5ð4Þ
Without considering other disturbance torques, the
space environmental disturbance torques can be
expressed as follows when the three attitude angles are
small
Tdx ’3v2
0IzIy
f
Tdy’3v2
0IzIx
ðÞu
Tdz ’0
8
<
:
ð5Þ
Substituting equations (4) and (5) into equation (1),
we obtain
Ix€
f+v2
04Iy4Iz
v0Hy
f+v0Ix+v0Iyv0IzHy
_
c+Hz_
u+IzIy
_
u_
c+v0IzIy
_
uf =Tcx +v0Hz
Iy€
u+3v2
0IxIz
ðÞuHz_
f+v0Hzc+Hx_
c+v0Hxf+IxIz
ðÞ
_
c_
f+v0f_
fv2
0cf v0c_
c
=Tcy
Iz€
c+v2
0(Ix+Iy)v0Hy
c+v0Ixv0Iy+v0Iz+Hy
_
fHx_
u+IyIx
_
f_
uv0c_
u
=Tcz v0Hx
8
>
<
>
:
ð6Þ
The controller chosen for the CLSACS is the pro-
portional derivative control. The control torques are
Tcx =Kxtx_
u+uðÞ
Tcy =Kyty_
u+u
Tcz =Kztz_
c+c
8
<
:
ð7Þ
where Kx,Ky,Kzdenote the proportionality coefficients
of the controller and tx,ty,tz, the differential coeffi-
cients of the controller.
The state variable x=½u_
uu_
uc _
cT, and
the measurement equation is
y=Cx +vð8Þ
where C=
1 000 0 0
0 100v00
0 010 0 0
0 001 0 0
0 000 1 0
v0000 0 1
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
and vis white
noise.
If the state variable of the CLSACS is the same as in
equation (8), the CLSACS can be expressed as follows
according to equation (6)
_
x=fxðÞ+Bu +wð9Þ
where x2Rn31denotes the system state variable,
u2Rp31, the system input variable; f(x) is a smooth
nonlinear function and w2Rndenotes the system
white noise.
Suppose that the working point of the CLSACS is
o(o=1,2,...,P), where Pdenotes the number of
working points. Then, the linearized model of the sys-
tem equation at ocan be approximately expressed as
xok+1ðÞ=AoxokðÞ+BoukðÞ+wkðÞ
ykðÞ=CoxokðÞ+vkðÞ
ð10Þ
where Ao=(I+YMDt+Y2
MDt2)=2,Iis a unit matrix
636,Dtdenotes the sampling period,
YM=(∂f(x))=(∂xo(k)) is a Jacobian matrix and Co=C.
The satellite attitude control system (equation (6)) is
a nonlinear system, whereas the parity equation is suit-
able only for linear systems. El-Amrani et al.
19
intro-
duced the T-S model to describe a nonlinear system to
improve diagnostic accuracy.
A CLSACS can be expressed as a T-S fuzzy control
system constructed such that the three attitude angles
F=(uuc) function as the premise variable. Each
working point corresponds to a fuzzy rule.
20
Rule o(o=1,2,...,P): If Fis
Mo(Mo=(uouoco)), then the locally linear model
of a CLSACS can be expressed as equation (10) at the
working point o.
20
The global model of the CLSACS state equation can
be expressed as
xk+1ðÞ=P
P
o=1
joFðÞAoxokðÞ+BouokðÞ+wokðÞ½
P
P
o=1
joFðÞ
ð11Þ
where jo(F) is the membership degree, which is deter-
mined by the premise variable according to the mem-
bership function,jo(F)= Q
P
j=1
moj(F), Fis function of
time variable k.
Fault propagation analysis of a closed-loop
system
To effectively diagnose the faults of a control system,
it is necessary to understand the propagation of faults in
a closed-loop system and analyse the impact of faults in
the system. The approach taken in this study is to first
define the discrete equations of the system and then add
feedback; faults are then added to the actuators and sen-
sors. Finally, the propagation of the actuator and sensor
faults in the closed-loop system is diagnosed.
The discrete equations of the control system are
Song et al. 3
xk+1ðÞ=Ax kðÞ+Bu kðÞ+Ew kðÞ
ykðÞ=Cx kðÞ+Fw kðÞ
ð12Þ
where x(k)2Rn31denotes the system state variable;
y(k)2Rq31, the system output variable; u(k)2Rp31,
the actuator input; w(k)2Rr31, the system distur-
bance; and A,B,C,E,Fare matrices with the appropri-
ate dimensions.
We add a feedback loop to equation (12). Output
feedback is a common type of feedback control
method.
21
Thus, we consider output feedback as the
feedback loop
ucN kðÞ=u0kðÞ+Ky kðÞ=u0kðÞ+KCx kðÞ+KFw kðÞ
ð13Þ
where u0(k) is the system input.
After substituting (13) into (12), we obtain
xk+1ðÞ=Ax kðÞ+Bu kðÞ+Ew kðÞ
=A+BKCðÞxkðÞ+Bu0kðÞ+BKF +EðÞwkðÞ
ð14Þ
which is the control system state equation with
feedback.
According to Wang and Xiong,
22
the fault models of
actuators and sensors are considered as
uckðÞ=ucN kðÞ+bc
yskðÞ=ysN kðÞ+bs
ð15Þ
where ucis the actual actuator input, ucN is the ideal
actuator input, ysis the actual sensor input, ycN is the
ideal sensor input, bcis the actuator deviation fault, bs
is the sensor deviation fault, and bcand bscan be
abrupt or time-varying faults.
Next, the propagation of actuator and sensor faults
in a closed-loop system is analysed based on the closed-
loop system model and fault model.
1. Actuator fault propagation
When an actuator is faulty and bs=0, based on equa-
tion (13), the actuator can be expressed as
uckðÞ=ucN kðÞ+bc=u0kðÞ+KCx kðÞ+KFw kðÞ+bc
ð16Þ
By substituting equation (16) into equation (12), the
system state equation can be rewritten as
xk+1ðÞ=Ax kðÞ+BuckðÞ+Ew kðÞ
=A+BKCðÞxkðÞ
+Bu
0kðÞ+bc
+BKF +EðÞwkðÞ
ð17Þ
When an actuator is faulty, the form of the fault can
be expressed as in equation (17), and the coefficient
matrix of the fault is the same as the control matrix of
the given instruction.
According to equations (12) and (17), the effect of an
actuator fault on the sensor output is expressed as
yk+1ðÞ=Cx k +1ðÞ+Fw k +1ðÞ
=CA+BKCðÞxkðÞ+CB u0kðÞ+bc
+CE+BKFðÞwk+1ðÞ+Fw k +1ðÞ
ð18Þ
As noted above, an actuator fault affects both the
system state and system measurement output.
2. Sensor fault propagation
When a sensor is faulty and bc=0, by combining
equations (12) and (15), we can obtain the measured
output of the system as
yskðÞ=ysN kðÞ+bs=Cx kðÞ+Fw kðÞ+bsð19Þ
Due to the existence of the output feedback loop, the
fault of a sensor will be propagated with the feedback
loop and then affect the instruction input of the control
system actuator. The instruction input of the control
system can be expressed as
ukðÞ=u0kðÞ+Ky kðÞ=u0kðÞ+KCx kðÞ+KFw kðÞ+Kbs
ð20Þ
When the control instruction acts on the system state
equation and the sensor is faulty, the system state equa-
tion is
xk+1ðÞ=Ax kðÞ+BuskðÞ+Ew kðÞ
=A+BKCðÞxkðÞ+Bu
0kðÞ+Kbs
+E+BKFðÞwkðÞ
ð21Þ
The above analysis illustrates that a sensor fault can
be propagated with the feedback loop, affecting the
system input and the system state equation. A sensor
fault also affects the output of measurement and dis-
turbs the actuator input, and thus fault isolation is
affected. Regarding the actuator input and system
state, the coefficient matrix of the fault parameter bs,
including the feedback matrix K, indicates that the
effect of a sensor fault on a system is related to the
design of the feedback matrix K.
Fault diagnosis of an actuator in a
closed-loop system
FPE for the fault diagnosis of an actuator
After we establish the FDPE and obtain the residual
error generated by the FDPE, the residual state equa-
tion can be obtained by using the T-S model, which
4International Journal of Distributed Sensor Networks
means the method of FPE. The residual error is esti-
mated by using a Kalman filter based on the equation
of the residual. The estimated residual error can be
compared with the set threshold to determine whether
an actuator is faulty.
The fault model representing the case in which the
ith actuator is faulty is shown in equation (15). Assume
that the fault of the actuator does not change within a
data window, so
bckðÞ=0 bic kð Þ 0½
Ti=1,2,...,pðÞ
where pis the dimension for a given input u0(k) and
bic(k) is the fault parameter of the ith actuator. Thus,
the system equation at time ksis as follows
yksðÞ=Cx k sðÞ+Fw k sðÞ
uksðÞ=u0ksðÞ+Ky k sðÞ+bckðÞ=u0ksðÞ+KCx k sðÞ+KFw k sðÞ+bckðÞ
xks+1ðÞ=Ax k sðÞ+Bu k sðÞ+Ew k sðÞ=A+BKCðÞxksðÞ+Bu
0ksðÞ+bckðÞðÞ+BKF +EðÞwksðÞ
8
<
:
ð22Þ
At time ks+1, the system measurement equation
is
yks+1ðÞ=Cx k s+1ðÞ+Fw k s+1ðÞ
=CA+BKCðÞxksðÞ+CB u0ksðÞ+bckðÞðÞ
+CBKF+EðÞwksðÞ+Fw k s+1ðÞ
ð23Þ
Then, the measurement equation at time kis
ykðÞ=Cx kðÞ+Fw kðÞ
=CA+BKCðÞ
sxksðÞ
+CA+BKCðÞ
s1Bu
0ksðÞ+bckðÞðÞ
+CA+BKCðÞ
s2Bu
0ks+1ðÞ+bckðÞðÞ
+ +CB u0k1ðÞ+bckðÞðÞ
+CA+BKCðÞ
s1BKF +EðÞwksðÞ
+CA+BKCðÞ
s2BKF +EðÞwks+1ðÞ
+ +C BKF +EðÞwk1ðÞ+Fw kðÞ
ð24Þ
Let ^
A=A+BKC,^
B=B,^
E=BKF +E,^
C=C,
^
F=Fand ^
u(k)=u0(k), and assume that the data win-
dow length is s+1. Then, from time ksto time k,
the observation data y(ks), ...,y(k) satisfy
yksðÞ=^
Cx k sðÞ+^
Fw k sðÞ
yks+1ðÞ=^
C^
Ax k sðÞ+^
C^
B^
uksðÞ+bckðÞðÞ+^
C^
Ew k sðÞ+^
Fw k s+1ðÞ
.
.
.
ykðÞ=^
C^
AsxksðÞ+^
C^
As1^
B^
uksðÞ+bckðÞðÞ+^
C^
As2^
B^
uks+1ðÞ+bckðÞðÞ+
+^
C^
B^
uk1ðÞ+bckðÞðÞ+^
C^
As1^
Ew k sðÞ+^
C^
As2^
Ew k s+1ðÞ+ +^
C^
Ew k 1ðÞ+^
Fw kðÞ
ð25Þ
Equation (25) can be expressed as
YkðÞ=H0xksðÞ+HcUNkðÞ+b
ckðÞ
+HwWkðÞ
ð26Þ
where Y(k)=½yT(ks)yT(k)Tdenotes the mea-
sured quantity; UN(k)=½^
uT(ks)^
uT(k)T, the
actuator input; W(k)=½wT(ks)wT(k)T, the dis-
turbance input; b
c(k)=½bcT(k) bcT(k)T, the
actuator fault parameter vector; and H0,Hcand Hw,
the redundant measurement matrix, where
H0=
^
C
^
C^
A
.
.
.
^
C^
As
2
6
6
6
4
3
7
7
7
5
ð27Þ
Hc=
00 0
^
C^
B00 0
^
C^
A^
B^
C^
B00 0
.
.
.^
C^
B000
^
C^
As2^
B ^
C^
B00
^
C^
As1^
B^
C^
As2^
B ^
C^
A^
B^
C^
B0
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
,
Hw=
^
F0 0
^
C^
E^
F0 0
^
C^
A^
E^
C^
E^
F0 0
.
.
.^
C^
E^
F00
^
C^
As2^
E ^
C^
E^
F0
^
C^
As1^
E^
C^
As2^
E ^
C^
A^
E^
C^
E^
F
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
ð28Þ
Based on Song et al.,
23
we construct a fully
decoupled parity vector (FDPV), denoted as vT
i, that is
sensitive only to the ith actuator and satisfies
Song et al. 5
vT
iH0HwHci
½=0ð29Þ
where
Hci =
00 0
^
C^
B00 0
^
C^
A^
B^
C^
B00 0
.
.
.^
C^
B000
^
C^
As2^
B ^
C^
B00
^
C^
As1^
B^
C^
As2^
B ^
C^
A^
B^
C^
B0
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
ð30Þ
^
Bis ^
Bwithout the ith column which corresponds to
actuator i, that is, the actuator to which the parity
equations are applied. For example, for the first actua-
tor, ^
B=½b2b3 bp, where bl(l=1,2,...p)is
the column vector of ^
B.
Consider each row of vias vi,s, then the following
notation for vican be obtained
vi=
vi,1
vi,2
.
.
.
vi,s
2
6
6
6
4
3
7
7
7
5
Then, we can get vT
i=½vi,1vi,2 vi,s,vi,s2R.
The vT
icomputed according equations (27)–(30) is
now only sensitive to the ith actuator and not sensitive
to the other actuators and system disturbances. Thus,
the residual error is affected by actuator iand is
decoupled from the other actuators and disturbances.
When an actuator is faulty, only the corresponding resi-
dual error is nonzero, thus, fault isolation is achieved.
The necessary and sufficient condition for the exis-
tence of the nonzero vT
iof equation (29) is
s.Nc
q1ð31Þ
where Ncis the number of columns that are not related
to matrix H0HwHci
½and qdenotes the dimension
of the output variable.
The FDPE residual ri(k), which is sensitive only to
the ith actuator, is defined as
rikðÞ=vT
iYkðÞHcUNkðÞ½ð32Þ
where UN(k)=½u0T(ks)u0T(k)Tdenotes the actua-
tor input.
By substituting equation (26) into equation (32), resi-
dual ri(k) can be rewritten as
rikðÞ=vT
iH0xksðÞ+HcUNkðÞ+b
c
+HwWkðÞHcUNkðÞ
=vT
iH0xksðÞ+Hcb
c+HwWkðÞ
ð33Þ
Equation (26) indicates that vT
isatisfies vT
iH0=0
and vT
iHw=0, meaning that the FDPV is decoupled
from both the system state and disturbance. Therefore,
the residual ri(k) can be expressed as
rikðÞ=vT
iHcb
ckðÞ ð34Þ
Because the designed FDPV is sensitive to the ith
actuator but decoupled from the other actuators, only
the fault of the first actuator should be considered in
the fault diagnosis. We assume that
b
ckðÞ=bic kðÞEcð35Þ
where Ec=½1 1Tis a (s+1)pcolumn vector
and bic(k) denotes the fault parameter of the ith
actuator.
By substituting equation (35) into equation (34), the
parity equation residual ri(k) can be rewritten as
rikðÞ=vT
iHcEcbic kðÞ ð36Þ
The above discussion indicates that bic(k)=0and
ri(k)=0when the actuator is normal and that
bic(k)6¼ 0and ri(k)6¼ 0when the actuator is faulty; vT
i
should satisfy
vT
iHc6¼ 0ð37Þ
Otherwise, the FDPE residual will always be zero
and cannot reflect the fault characteristics. When vT
iis
selected, it should increase the value of vT
iHcEcbic(k),
ensuring that the projection of the fault feature in the
direction of vT
ican be fully reflected. Otherwise, the
fault characteristic information can be easily over-
whelmed by noise, which cannot be detected by the
FDPE.
By applying the T-S model presented in the section
‘CLSACS dynamics model’, we can express the output
of the T-S model as
rik
ðÞ
=P
P
o=1
joFðÞvT
oiHoc Eoc
P
P
o=1
joFðÞ
bic k
ðÞ ð38Þ
where jo(F)= Q
p
j=1
moj(F).
The correspondence between the fault parameters
and the residual error is used as the observation equa-
tion. Thus, we obtain
rikðÞ=fkðÞbic kðÞ+hkðÞ ð39Þ
where
6International Journal of Distributed Sensor Networks
fkðÞ=P
P
o=1
joFðÞvT
oiHoc Eoc
P
P
o=1
joFðÞ
and h(k) is the zero-mean white noise with a covariance
matrix of R(k).
Because the change trend of bic(k) is unknown, the
dynamic model is set up as a random walk process
bic k+1ðÞ=bic kðÞ+ekðÞ ð40Þ
where e(k) is the independent zero-mean white noise
with a covariance matrix of Q(k).
The actuator fault parameter bic(k) is estimated by
using a Kalman filter according to equations (39) and
(40).
The fault parameters are statistically determined,
and the fault threshold is TCD.If bic (k)jj.TCD , the
actuator is faulty; if bic(k)jj\TCD, the actuator operates
correctly.
The fault diagnosis of an actuator in a closed-loop
system is completed. When multiple actuators are
faulty, multiple actuator fault diagnoses and isolations
can be achieved. Then, the fault parameters can be esti-
mated after establishing the FDPE for each actuator.
Simulation results and analysis
The actuator fault of a three-axis stabilized CLSACS is
simulated. The output feedback loop and controller are
designed simultaneously. The controller is replaced with
the output feedback loop to ensure the stable operation
of the system. The parameters which are provided by
the team’s predecessors are shown in Table 1.
When the CLSACS is described, the triangle mem-
bership function is chosen for the premise variables.
The working point and membership function are
selected as shown in Figure 1.
An abrupt fault at the Y-axis actuator is expressed
as
bic =0t\600 s
23103tø600 s
ð41Þ
After establishing the fully decoupled fuzzy parity
equation (FDFPE) for the three-axis actuator of a con-
trol system, the fault parameter can be estimated as
follows.
Figures 2 and 3 illustrate that when the FDFPE is
established for the Z-axis actuator, the fault deviation
estimation result is a zero-mean sequence that is
affected only by noise. For the Y-axis, the FDFPE is
established to estimate the true fault error. Some errors
exist in the estimation results before or after the time at
which the fault is added; for the remainder of the time,
the fault parameter can be estimated accurately. Thus,
the FDFPE is effective in the detection and isolation of
actuator faults.
Table 1. Satellite simulation parameters.
Name Parameters
Controller parameters tx=160:019, ty=124:185, tz=197:183,
Kx=9:523, Ky=7:391, Kz=11:7348
Satellite inertia moment Ix=1849:3765 kg m2,Iy=1435:234kg m2,Iz=2278:8824kg m2
Measurement mechanism noise ss=600 star sensor noiseðÞ,sg=0:0058=h gyro noiseðÞ
Tracking parameters 08orbital inclinationðÞ,v0=0:0012 rad=sorbital angular velocityðÞ
Step 0.5 s
Figure 1. Schematic of the working point selection and
membership function of the antecedent variables.
Figure 2. Y-axis actuator deviation estimation result.
Song et al. 7
A time-varying fault at the Y-axis actuator is
expressed as
bic =0t\600 s
1:531063t600ðÞtø600 s
ð42Þ
In the same manner, after establishing the FDFPE
for the three-axis actuator of a control system, the fault
parameter can be estimated as follows.
Figures 4 and 5 show that when the actuator fault is
time varying, the fault parameter can be accurately esti-
mated after the FDFPE for the system is established.
Thus, fault detection and isolation can be completed,
and the goal of the actuator fault detection and isola-
tion can be achieved.
To verify the ability of the FDFPE to diagnose the
faults of multiple actuators, at 600 s, the abrupt fault
shown in equation (41) is added to the X-axis, and the
time-varying fault shown in equation (42) is added to
the Y-axis. After establishing the FDFPE for the three-
axis actuator of the control system, the fault parameters
can be estimated as follows.
Figures 6–8 show that when multiple actuators are
faulty, the FPEs established for different actuators
reflect only the fault characteristics of the correspond-
ing actuator. The fault parameters can be estimated
using the residual error information of the parity equa-
tion and the Kalman filter. Identification of the fault
parameters can result in the detection and isolation of
the system fault.
Fault diagnosis of a sensor in a closed-loop
system
FPE for the fault diagnosis of a sensor
When a sensor is faulty, the fault not only will directly
affect the measurement results of the sensor but can
Figure 3. Z-axis actuator deviation estimation result.
Figure 4. Residual of the Y-axis actuator.
Figure 5. Residual of the Z-axis actuator.
Figure 6. Residual of the X-axis actuator.
8International Journal of Distributed Sensor Networks
also propagate through the feedback loop in a closed-
loop system, thereby affecting the input of the actuator
and system state. The FPE for the fault diagnosis of a
sensor is established based on the same method pre-
sented in the ‘Fault diagnosis of an actuator in a closed-
loop system’ section. A double parity equation diagno-
sis method is designed based on the propagation char-
acteristics of a sensor fault in a closed-loop system. A
set of parity equations is used for fault detection and
isolation, and another set is used to estimate the fault
parameters.
The fault model representing the case in which the
ith sensor is faulty is shown in equation (15). Assume
that the fault deviation of the same sensor does not
change within a data window and the fault vectors
bs(k)=½0 bsi(k) 0Ti=1,2,...,qðÞ,
where qis the dimension of output y(k) and bsi(k) is the
fault parameter of the ith sensor. The system equation
at time ksis as follows
yksðÞ=Cx k sðÞ+Fw k sðÞ+bskðÞ
uksðÞ=u0ksðÞ+Ky k sðÞ=u0ksðÞ+KCx k sðÞ+KFw k sðÞ+KbskðÞ
xks+1ðÞ=Ax k sðÞ+Bu k sðÞ+Ew k sðÞ=A+BKCðÞxksðÞ+Bu
0ksðÞ+KbskðÞðÞ+BKF +EðÞwksðÞ
8
<
:
ð43Þ
At time ks+1, the system measurement equation
is
yks+1ðÞ=Cx k s+1ðÞ+Fw k s+1ðÞ+bskðÞ
=CA+BKCðÞxksðÞ+CB u0ksðÞ+KbskðÞðÞ
+C BKF +EðÞwksðÞ+Fw k s+1ðÞ+bskðÞ
ð44Þ
Hence, the measurement equation at time kis
ykðÞ=Cx kðÞ+Fw kðÞ+bskðÞ
=CA+BKCðÞ
sxksðÞ+CA+BKCðÞ
s1Bu
0ksðÞ+KbskðÞðÞ+CA+BKCðÞ
s2Bu
0ks+1ðÞ+KbskðÞðÞ
+ +CB u0k1ðÞ+KbskðÞðÞ+CA+BKCðÞ
s1BKF +EðÞwksðÞ
+CA+BKCðÞ
s2BKF +EðÞwks+1ðÞ+ +C BKF +EðÞwk1ðÞ+Fw kðÞ+bskðÞ
ð45Þ
With the same method in the ‘Fault diagnosis of an
actuator in a closed-loop system’ section, the equation
(46) is computed as follows
YkðÞ=H0xksðÞ+HcUNkðÞ+GðÞ+HwWkðÞ+b
skðÞ
ð46Þ
where Y(k)=½yT(ks)yT(k)T,UN(k)=½^
uT(ks)
^
uT(k)T,W(k)=½wT(ks)wT(k)T,b
s(k)=
½bsT(k) bsT(k)Tis the sensor fault parameter
vector, and G(k)=½Kbs(k)ðÞ
T (Kbs(k))TTis the
function vector of the sensor fault with the feedback
loop with respect to the system state. H0,Hcand Hw
are the same as in section ‘Fault diagnosis of an actua-
tor in a closed-loop system’.
Figure 7. Residual of the Y-axis actuator Figure 8. Residual of the Z-axis actuator.
Song et al. 9
Li et al.
21
introduced a method for constructing the
FDPEs of a sensor, which works well in open-loop sys-
tems. However, because of the existence of the feed-
back loop, a sensor fault in closed-loop systems can
degrade the actuator; as the time redundancy measure-
ment equation is related only to a specific sensor, the
sensor fault cannot be obtained.
Some scholars have suggested enhancing the fault
information in the residual of the parity equation by
optimizing the design of the feedback controller of the
closed-loop system.
11
Following their work, the double
parity equation diagnosis method is designed based on
the propagation characteristics of a sensor fault in a
closed-loop system. A set of parity equations is used to
detect and isolate the fault, and another set is used to
estimate the fault parameters.
1. Parity equations for fault detection and isolation
The FDPV vT
di, which is used for sensor fault detection
and isolation, should satisfy
vT
di H0HwHc
½=0ð47Þ
Equation (47) can ensure that the parity equation is
decoupled from the system state, system disturbance
and feedback vector G(k).
After the FDPVs are constructed, the parity equa-
tion residual rdi(k) can be expressed as
rdi kðÞ=vT
di YkðÞHcUNkðÞ½ð48Þ
By substituting equation (46) into equation (48), we
obtain
rdi kðÞ=vT
di H0xksðÞ+HcUNkðÞ+GkðÞðÞ+HwWkðÞ+b
skðÞHcUNkðÞ
=vT
di H0xksðÞ+HcG+HwWkðÞ+b
skðÞ
ð49Þ
Based on equation (47), the parity equation residual
rdi(k) can be simplified to
rdi kðÞ=vT
dib
skðÞ ð50Þ
In this manner, the residual error rd(k) related only
to the parity equation is obtained. However, this error
can only be used to detect the fault; the location of the
fault is not determined, and fault isolation is not
completed.
When only the ith sensor is faulty, bs(k)=
½0 bsi(k) 0Ti=1,2,...,pðÞand b
s=
½bsT bsTT. Decoupling matrix Lis constructed
to satisfy
b
skðÞ=LbskðÞ ð51Þ
where Lis ((s+1)p)3((s+1)p). The diagonal ele-
ments of Lin the diagonal of
(j1)p+i(j=1,2,...,s+1,i=1,2,...,p)are1,
which corresponds to the fault parameters of the ith
sensor, whereas the remaining elements of Lare 0.
For example, let s=3and p=6, if the matrix Lis
corresponding to the first sensor fault. Then,
L=diag½10 010 0, with the 1st,
7th, 13th and 19th diagonal elements of Lbeing equal
to 1, which reflects the fault position of the first faulty
sensors in b
s.
Let vT
di satisfy
vT
diL=0ð52Þ
Then, the parity equation residual can be expressed
as
rdi kðÞ=vT
dib
skðÞ=vT
diLb
skðÞ=0ð53Þ
vT
di, which satisfies equation (52), is only decoupled
from the ith sensor fault and is sensitive to the remain-
ing sensor faults.
Considering the effect of noise, the fault parameters
are statistically determined, and the fault threshold is
TSD.If rdi(k)
jj
\TSD, the parity equation residual
includes fault information. If rdi(k)jj.TSD , the parity
equation residual does not include fault information.
When only one sensor is faulty, after designing mul-
tiple FDPVs that are decoupled from each sensor in the
aforementioned manner, we can determine that a sen-
sor parity equation residual equal to zero represents a
fault. If multiple sensors are faulty, the FDPV can be
constructed such that many diagonal elements of Lare
equal to 1. Then, the faults can be detected and isolated
by using the parity equation residual.
Because vT
di is designed to be decoupled from the
redundant measurement matrices H0,Hcand Hwand
the decoupling matrix L, the degrees of freedom of the
FDPV design are reduced, and the projection of the
fault in the direction of vT
di can be less noticeable. An
FDPV with a high sensitivity to faults should be
selected to improve the diagnostic reliability.
2. Parity equation for fault parameter estimation
If the observation equation for the Kalman filter is con-
structed based on equation (54), the fault parameter
will be an (s+1)p-dimensional state variable, and the
computational burden will be large.
10 International Journal of Distributed Sensor Networks
To reduce the computational burden of the filter esti-
mation, another FDPV, denoted as vT
ei, is designed to
estimate the fault parameter.
The vT
ei that satisfies equation (47) is constructed; the
corresponding parity equation residual can then be
expressed as
rei kðÞ=vT
eib
skðÞ ð54Þ
where rei(k) is a one-dimensional vector and b
s(k)is
multidimensional. The computational burden of the fil-
ter estimation is reduced by reducing the dimensions of
the fault parameters. The location of the fault sensor in
b
s(k) can be determined based on fault isolation. In the
case of a single fault mode, the fault parameters corre-
sponding to faulty sensor b
s(k) are nonzero, and the
remaining elements of b
s(k) are 0.
Assume that the sensor fault does not change within
a data window.
A new fault parameter vector is composed of the
fault parameters of the ith sensor in b
s(k)
b
es kðÞ=Ebsi kðÞ ð55Þ
where E=½1 1Tis an s+1-dimensional vec-
tor, all the elements of which are equal to 1.
Considering fault parameter vector b
s(k), only the
fault parameters corresponding to the faulty sensor are
nonzero. When the ith sensor is faulty, the value of the
new FDPV vT
ei corresponds to the position of the ith
sensor fault for vT
ei.
The other elements of vT
ei are multiplied by the zero
element of b
s(k) and thus do not affect the result of the
residual. Therefore, equation (50) can be expressed as
rei kðÞ=vT
ei b
es kðÞ=vT
ei Ebsi kðÞ ð56Þ
Equation (50) illustrates that the unknown fault
parameter changed is b
s(k), which is a one-dimensional
scalar. Thus, the dimension of the estimated state is
reduced.
Based on the T-S model presented in the ‘CLSACS
dynamics model’ section, we obtain
rikðÞ=P
P
o=1
joFðÞvT
ei E
P
P
o=1
joFðÞ
bsi kðÞ ð57Þ
We can estimate the fault parameter via the optimal
estimation method. Assume that measurement noise
h(k) is a zero-mean white-noise covariance matrix, that
is, R(k). Then, we obtain
rei kðÞ=fei kðÞbsi kðÞ+hkðÞ ð58Þ
where
fei kðÞ=P
P
o=1
joFðÞvT
ei E
P
P
o=1
joFðÞ
The dynamic model is set up as a random walk pro-
cess because the change trend of bsi(k) is unknown
bsi k+1ðÞ=bsi kðÞ+ekðÞ ð59Þ
where e(k) is the independent zero-mean white noise
and Q(k) is the covariance matrix. The fault parameter
bsi(k) can be estimated using the Kalman filter accord-
ing to equations (58) and (59).
The method for reducing the dimensions of a sen-
sor’s fault parameters is suitable only when a single sen-
sor is at fault. Equation (56) cannot be obtained when
multiple sensors are at fault.
Simulation results and analysis
The first sensor fault in the CLSACS is simulated and
verified. The simulation parameters are shown in
Table 1.
An abrupt fault at the first sensor is expressed as
follows
bsi =0t\600 s
23103tø600 s
ð60Þ
Based on the time redundancy measurement infor-
mation, the FPE is established according to equations
(54) and (50) to diagnose the sensor fault.
Select simulation results are shown below (a 5-s
smoothing window is applied to the residual results).
Figures 9 and 10 illustrate that if the sensor that the
residual is decoupled from is the faulty sensor, the
Figure 9. Residual decoupled from the first sensor.
Song et al. 11
residual will be the sequence that is affected only by
noise. If the sensor that the residual is decoupled from
is not the faulty sensor, the fault will be reflected in the
residual.
The aforementioned method is used to detect and
isolate a sensor fault under the single fault condition.
Because the measurements of different sensors and the
projections of the measurement results in different
FDPV directions are different, the residual form of the
parity equation may be different.
After fault detection and isolation, the fault para-
meters are estimated using equations (54) and (58). The
simulation results are shown below.
Figure 11 shows that estimation of the fault para-
meter can be achieved by designing a reduced-
dimension FDPV. Some errors exist in the estimation
results before or after the time at which the fault is
added, and there is a jump in the residual of the parity
equation. The fault parameters can be estimated accu-
rately during the remaining period.
A time-varying fault at the first sensor is expressed
as equation (60). The simulation results are shown
below
bsi =0t\600 s
1:531063t600ðÞtø600 s
ð61Þ
Figures 12 and 13 show that this method can also be
used to detect and isolate time-varying faults. When the
first sensor is faulty, the residual decoupled from the
first sensor is a white-noise sequence. The parity resi-
dual that is decoupled from the second normal sensor
goes awry because it is affected by the fault.
The estimation results for a time-varying fault are
shown in Figure 14.
Figure 10. Residual decoupled from the second sensor. Figure 11. Estimation results for an abrupt fault at the first
sensor.
Figure 12. Residual decoupled from the first sensor.
Figure 13. Residual decoupled from the second sensor.
12 International Journal of Distributed Sensor Networks
Figure 14 shows that based on the fault isolation,
the reduced dimension of the FDPV can be used to esti-
mate the parameter of a time-varying fault.
To verify the fault diagnosis of the decoupled FDPV
for multiple sensors, the results for the case in which a
fault is added at the first sensor are shown in equation
(60); those for the case in which a fault is added at the
third sensor are shown in equation (61). The simulation
results are shown in Figures 15 and 16.
Figures 15 and 16 show that when multiple sensors
are faulty, the parity residual result of the zero-mean
white noise can be obtained only by designing the
FDPV for two faulty sensors; otherwise, the fault infor-
mation will be reflected in the residual. Thus, fault diag-
nosis and isolation of multiple sensors are achieved. In
addition, the projection in the direction of the FDPV is
different for different sensor faults. The residual shown
in Figure 16 indicates that the projection in the direc-
tion of the residual of the abrupt sensor fault is more
obvious.
Conclusion
This article introduces a fault diagnosis method for the
actuators and sensors in a closed-loop control system.
Closed-loop control system equations and fault modes
are presented. An actuator fault can affect both the sys-
tem measurement output and system state. The fault
will be propagated with the feedback loop, which will
affect the system input and the system state equation.
Sensor faults affect the system measurement output
while disturbing the actuator input, which impacts fault
isolation.
The residual generated by the FPE is sensitive only
to a specific actuator and is decoupled from the system
state and other actuators. The residual is obtained by
establishing the FPE for each actuator, and then the
fault model parameters can be estimated. In this man-
ner, the fault detection and isolation of actuators in a
closed-loop system are achieved. A double parity equa-
tion diagnosis method for sensor faults in a closed-loop
system is used to detect and isolate sensor faults and to
achieve fault parameter estimation that allows the sin-
gle and multiple fault diagnosis isolation of sensors in a
closed-loop nonlinear system.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Figure 14. Estimation results for a time-varying fault at the
first sensor.
Figure 15. Decoupling vectors for the first and third sensors.
Figure 16. Decoupling vectors for the second and fourth
sensors.
Song et al. 13
Funding
The author(s) disclosed receipt of the following financial sup-
port for the research, authorship, and/or publication of this
article: This work was supported by the National Natural
Science Foundation of China (No. 61573059).
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