Chinese Journal of Aeronautics, (2016), 29(2): 462–469

Chinese Society of Aeronautics and Astronautics& Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cnwww.sciencedirect.com

A novel two-stage extended Kalman filter algorithm

for reaction flywheels fault estimation

* Corresponding author. Tel.: +86 451 86402357.

E-mail address: cxqhit@163.com (X. Chen).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.01.0081000-9361 � 2016 Chinese Society of Aeronautics and Astronautics. Published by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Chen Xueqin a,*, Sun Rui a, Jiang Wancheng a, Jia Qingxian b, Zhang Jinxiu a

aResearch Center of Satellite Technology, Harbin Institute of Technology, Harbin 150080, ChinabSchool of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China

Received 8 July 2015; revised 27 November 2015; accepted 15 December 2015Available online 23 February 2016

KEYWORDS

Fault estimation;

Reaction flywheels;

Satellite attitude control sys-

tems;

Separate-bias principle;

Two-stage extended Kalman

filter

Abstract This paper investigates the problem of two-stage extended Kalman filter (TSEKF)-based

fault estimation for reaction flywheels in satellite attitude control systems (ACSs). Firstly, based on

the separate-bias principle, a satellite ACSs with actuator fault is transformed into an augmented

nonlinear discrete stochastic model; then, a novel TSEKF is suggested such that it can simultane-

ously estimate satellite attitude information and actuator faults no matter they are additive or mul-

tiplicative; finally, the proposed approach is respectively applied to estimating bias faults and loss of

effectiveness for reaction flywheels in satellite ACSs, and simulation results demonstrate the effec-

tiveness of the proposed fault estimation approach.� 2016 Chinese Society of Aeronautics and Astronautics. Published by Elsevier Ltd. This is an open access

article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The separate-bias estimation algorithm is used to estimate the

state and constant bias of linear systems. The basic principle ofthis algorithm, which is also called two-stage Kalman filter(TSKF), is to estimate system states and constant biases sepa-

rately, then obtain the optimal estimate using the couplingrelationship between them. In 1969, Friedland firstly proposedthe separate-bias estimation algorithm1 and made a furtherinvestigation.2,3 During the past four decades, many research

achievements have been reported on this algorithm. At

present, the main research on this topic is concerned withestimation of constant/time-varying bias and all kinds ofengineering applications.

Recently, many scholars have paid considerable attentionto separate-bias estimation algorithm for linear systems.4–6

Keller and Darouach4 suggested an optimal solution of the

TSKF, which can be used to estimate optimal state and opti-mal random bias, and further a two-stage optimal strategywas developed for discrete-time stochastic linear systems sub-

ject to intermittent unknown inputs.5 Khabbazi and Esfanjani6

proposed a constrained TSKF for tracking control problem inan uncertain linear system and its main advantage is the

improvement of estimation accuracy.For fault/bias estimation of nonlinear systems, much pro-

gress has been made on EKF-based approaches. EKF-basedsub-optimal algorithm was proposed in Ref.7. Zhou et al.

investigated a pseudo separate-bias estimation algorithm fornonlinear time-varying stochastic systems.8,9 In Ref.10, fuzzy

A novel two-stage extended Kalman filter algorithm for reaction flywheels fault estimation 463

Kalman filter-based approach was proposed for nonlinearstochastic discrete time systems.

Based on the general TSKF, Hsieh11 proposed a general

two-stage extended Kalman filter (GTSEKF)-based constantparameter estimation approach. In Ref.12, an adaptive TSEKFlike Ref.11 was proposed to estimate the closed-loop position

and speed of sensorless control for permanent magnet syn-chronous motor. Kim13 proposed an adaptive TSEKF forINS-GPS loosely coupled systems and its main advantage

was it cost less computation time due to the introduction ofthe forgetting factor. In Ref.14, an adaptive TSEKF algorithmbased on Ref.13 was introduced to the application of geomag-netic aided inertial navigation filtering. By introducing strong

tracking multiple fading factors and embedding EKF into anoptimal TSKF, a novel robust filter-based bearings-onlymaneuvering target tracking problem was investigated, which

can provide an optimal estimation of the target state and theunknown statistical parameters of virtual noises.15

The faults of actuators and sensors in control systems can

be represented as biases via separate-bias estimation algo-rithm. Fault estimation for dynamic systems has attracted con-siderable attention during the past two decades. When

estimating the additive actuator/sensor faults, the algorithmcan be implemented easily since the biases representing faultsin these models have specific physical meanings and the princi-ples are clear.16 When estimating the multiplicative faults in

actuators, it is necessary to use other parameters to representthe fault models, such as control effectiveness factors, whichcan be used to indicate the fault degree of control systems.

By introducing forgetting factors into the optimal TSKF inRef.4, an adaptive TSKF was exploited to estimate the abruptreduction of control effectiveness in dynamic systems by Wu

et al.17 This algorithm is applied for the identification ofimpairment in its control surfaces in an aircraft model. InRefs.18–22, a further investigation on this algorithm was made

and widely applied it on fault diagnosis and fault-tolerant con-trol. In Ref.23, control effectiveness factor estimation wasextended to the estimation of control distribution matrix ele-ments, and the TSKF was applied to actuator/surface fault

diagnosis and fault-tolerant control of F-16. In recent years,this method has been applied to the fault diagnosis andfault-tolerant control of sensors and actuators in satellite atti-

tude control system.16,24–26 The value of the effectiveness fac-tors can be derived via this method and system fault degreecan be analyzed to obtain biases for the fault-tolerant control

purpose. In addition, to the best of our knowledge, separate-bias principle never considers additive faults and multiplicativefaults simultaneously.

In view of this, this paper investigates TSEKF-based fault

estimation for reaction flywheels in satellites. A novel TSEKFis suggested such that it can simultaneously estimate satelliteattitude information and reaction flywheel faults no matter

they are additive or multiplicative. It is respectively appliedto estimating bias faults and loss of effectiveness for reactionflywheels in a satellite ACSs, and the simulation results demon-

strate the effectiveness of the proposed fault estimationapproach.

2. System fault model

Consider the following nonlinear discrete-time stochastic sys-tem with bias vector of unknown magnitude bk e R

p:

xkþ1 ¼ fkðxk; bkÞ þ wxk; bkþ1 ¼ bk þ wb

k

yk ¼ hkðxk; bkÞ þ vk

�ð1Þ

where xk e Rn is the system state; yk e R

m is the measurement

vector; the noise sequence wbk, w

xk and vk are zero-mean uncor-

related Gaussian random sequences with

E

wbk

wxk

vk

264

375 wbT

j wxTj vTj

� �0B@

1CA ¼

Wb

Wx

V

264

375dk;j ð2Þ

where Wb > 0, Wx > 0, V > 0 and dk,j is the Kroneckerdelta. The initial states x0 and b0 are Gaussian random vari-

ables with �x0 ¼ Eðx0Þ, b0 ¼ Eðb0Þ, �Px0 ¼ Eðx0 � �x0Þðx0 � �x0ÞT,

�Pb0 ¼ Eðb0 � bÞðb0 � b0ÞT and �Pxb

0 ¼ Eðx0 � �x0Þðb0 � b0ÞT. Thefunction fk(xk, bk) and hk(xk, bk) are assumed to be continuousand can be expanded by Taylor series expansion. If the higher

order terms can be neglected, the linear discrete-time stochasticsystem Eq. (1) appears in the following form:

xkþ1 ¼ Akxk þ Fakbk þ wx

k þMk; bkþ1 ¼ bk þ wbk

yk ¼ Ckxk þ Fskbk þ vk þNk

(ð3Þ

where Ak e Rn�n and Ck e R

m�n are state transition matrix andobservation matrix, respectively, and we have

Ak ¼ @f

@xT

����x ¼ xkjkb ¼ bkjk

;Fak ¼

@f

@bT

����x ¼ xkjkb ¼ bkjk

Ck ¼ @h

@xT

����x ¼ xkjk�1

b ¼ bkjk�1

;Fsk ¼

@h

@bT

����x ¼ xkjk�1

b ¼ bkjk�1

8>>>>>><>>>>>>:

ð4Þ

Mk ¼ fkðxkjk; bkjkÞ � Akðxkjk; bkjkÞxkjk � Fakðxkjk; bkjkÞbkjk

Nk ¼ hkðxkjk�1; bkjk�1Þ � Ckðxkjk�1; bkjk�1Þxkjk�1

�Fskðxkjk�1; bkjk�1Þbkjk�1

8><>:

ð5Þwhere xkjk and bkjk denote the optimal results of xk and bkrespectively.

The vector function x ¼ ½xx; xy; xz �T represents the

inertially referenced satellite angular rate vector of the satellite

body relative to the inertial coordinate system and the corre-sponding Euler angles are u, h and w. Define

x ¼ ½xx; xy; xz; u; h; w �T, the state equation of satel-

lite attitude control system based on the satellite attitudedynamics equation can be given as

_x ¼ gðxÞ þ BuðxÞ ð6Þwhere

gðxÞ ¼ I�1s ð�x� IsxÞUðxÞ

" #

UðxÞ ¼ 1

cosu

cosu cos h 0 cosu sin h

sinu sin h cosu � sinu cos h

� sin h 0 cos h

264

375x

B ¼ I�1s

0

" #

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð7Þ

uðxÞ 2 Rl is the known control input vector; B 2 Rn�l is thecontrol input matrix; Is is the moment of inertial matrix ofthe satellite.

464 X. Chen et al.

Observation equation is

y ¼ Cx ð8Þwhere C= I6 is the output matrix, and In is an n � n identitymatrix.

Discretize the combination of Eqs. (6) and (8), then the sys-tem model without fault is established like Eq. (1).

(1) Additive fault model

The bias bk represents additive faults of actuators, and the

system model with additive faults of actuators can be describedas

xkþ1 ¼ gk þ Bkðuk þ bkÞ þ wxk; bkþ1 ¼ bk þ wb

k

yk ¼ Ckxk þ vk

�ð9Þ

where gk, Bk, uk and Ck are the corresponding discrete systemmatrices of Eqs. (6) and (8), the corresponding matrix of

Eq. (1) is

Ak ¼ @gk@xT

����x¼xkjk

þ @Bkuk@xT

����x¼xkjk

; Fak ¼ Bk;F

sk ¼ 0 ð10Þ

(2) Multiplicative fault model

Here, bk represents the degree of the multiplicative faults ofactuators, and the bias Kkuk represents the multiplicative faults

of actuators, in which diagonal matrix Kk ¼ diagðbkÞ. The sys-tem model with multiplicative faults of actuators can bedescribed as

xkþ1 ¼ gk þ BkðI3 � KkÞuk þ wxk; bkþ1 ¼ bk þ wb

k

yk ¼ Ckxk þ vk

�ð11Þ

the corresponding matrix of Eq. (1) is

Ak ¼ @gk@xT

����x¼xkjk

þ @BkðI3 � KkÞuk@xT

����x¼xkjk

Fak ¼ �Bk diagðukÞ; Fs

k ¼ 0

8><>: ð12Þ

In Eq. (11), the vector function bk is also called the controleffectiveness factors, representing the possible loss of controleffectiveness in the model,21–23 whose value is 0 in the absence

of multiplicative faults.The objective of this paper is to design a TSEKF such that

it can estimate actuator faults no matter they are additive ormultiplicative. That is, it can give a solution for system

Eq. (1) with the unknown bias and system states, a solutionfor additive fault system Eq. (9) with additive faults and systemstates, and also a solution for multiplicative fault system

Eq. (11) with multiplicative faults and system states.

3. TSEKF-based actuator fault estimation approach

In this section, a novel TSEKF is designed to simultaneouslyestimate satellite attitude information and actuator faults.

Theorem 1. A discrete-time TSEKF is given by the followingcoupled difference equations when the nonlinear discrete-time

stochastic system with bias vector of unknown magnitude is givenby Eq. (1).

xkþ1jkþ1 ¼ ~xkþ1jkþ1 þ bkþ1jkþ1bkþ1jkþ1 ð13Þ

Pxkþ1jkþ1 ¼ ~Px

kþ1jkþ1 þ bkþ1jkþ1Pbkþ1jkþ1b

Tkþ1jkþ1 ð14Þ

where xkþ1jkþ1, ~xkþ1jkþ1 and bkþ1jkþ1 are the state vectors of the

TSEKF, the unknown bias free filter and the unknown bias filter,respectively; bkþ1jkþ1 is the result of coupling Eq. (27); Px

kþ1jkþ1

and Pbkþ1jkþ1 are the estimation error covariance matrices of

xkþ1jkþ1 and bkþ1jkþ1 respectively. ~Pxkþ1jkþ1 is the estimation error

covariance matrix of ~xkþ1jkþ1.

The unknown bias-free filter is

~xkþ1jkþ1 ¼ ~xkþ1jk þ ~Kxkþ1ðykþ1 � Ckþ1~xkþ1=k �Nkþ1Þ ð15Þ

~Pxkþ1jkþ1 ¼ ðI� ~Kx

kþ1Ckþ1Þ~Pxkþ1jk ð16Þ

~Kxkþ1 ¼ ~Px

kþ1jkCTkþ1ðCkþ1

~Pxkþ1jkC

Tkþ1 þ VÞ�1 ð17Þ

~xkþ1jk ¼ fkðxkjk; bkjkÞ � bkþ1jkbkþ1jk ð18Þ

~Pxkþ1jk ¼ Ak

~PxkjkA

Tk þWx þ rkP

bkjkr

Tk � bkþ1jkP

bkþ1jkb

Tkþ1jk ð19Þ

The unknown bias filter is

bkþ1jk ¼ bkjk ð20Þ

Pbkþ1jk ¼ Pb

kjk þWb ð21Þ

bkþ1jkþ1 ¼ bkþ1jk þ Kbkþ1½ykþ1 � hkþ1ðxkþ1jk; bkþ1jkÞ� ð22Þ

xkþ1jk ¼ fkðxkjk; bkjkÞ ð23Þ

Kbkþ1 ¼Pb

kþ1jkHTkþ1jk Hkþ1jkP

bkþ1jkH

Tkþ1jkþCkþ1

~Pxkþ1jkC

Tkþ1þV

� ��1

ð24Þ

Pbkþ1jkþ1 ¼ ðI� Kb

kþ1Hkþ1jkÞPbkþ1jk ð25Þ

with the coupling equations

Hkþ1jk ¼ Fsk þ Ckbkþ1jk ð26Þ

bkþ1jkþ1 ¼ bkþ1jk � ~Kxkþ1Hkþ1jk ð27Þ

rk ¼ Akbkjk þ Fak ð28Þ

bkþ1jk ¼ rkPbkjkðPb

kþ1jkÞ�1 ð29Þ

To facilitate the derivations, the following notations areused:

Xk ¼xk

bk

� ; �Ak ¼

Ak Fak

0n�N IN

�

�Ck ¼ Ck Fsk½ �; C ¼ In

0N�n

�

�wk ¼wx

k

wbk

� ; �W ¼ Wx 0

0 Wb

�

8>>>>>>>><>>>>>>>>:

ð30Þ

then the model Eq. (1) could be written as

Xkþ1 ¼ �AkXk þ �wk þ CMk

yk ¼ �CkXk þ vk þNk

(ð31Þ

A novel two-stage extended Kalman filter algorithm for reaction flywheels fault estimation 465

A common approach to estimate the system state of systemEq. (31) is using the following well-known KF:

Xkþ1jkþ1 ¼ Xkþ1jk þ Kkþ1ðykþ1 � �Ckþ1Xkþ1jk �Nkþ1Þ ð32Þ

Xkþ1jk ¼ �AkXkjk þ CMk ð33Þ

Pkþ1jk ¼ �AkPkjk �ATk þ �W ð34Þ

Kkþ1 ¼ Pkþ1jk �CTkþ1ð�Ckþ1Pkþ1jk �C

Tkþ1 þ VÞ�1 ð35Þ

Pkþ1jkþ1 ¼ InþN � Kkþ1�Ckþ1

�Pkþ1jk ð36Þ

where

Pkjk ¼Px

kjk Pxbkjk

Pbxkjk Pb

kjk

" #ð37Þ

where Pxbkjk and Pbx

kjk are nondiagonal matrices.

The one-step prediction value Xk+1|k is then transformed to

promote the one-step prediction variance into a diagonalmatrix. The orthogonal transformation matrix is chosen as

T ¼ In �bkþ1jk0 IN

� ; bkþ1jk ¼ Pxb

kþ1jkðPbkþ1jkÞ

�1 ð38Þ

thus

TXkþ1jk ¼xkþ1jk � bkþ1jkbkjk

bkjk

" #

TPkþ1jkTT ¼

Pxkþ1jk � bkþ1jkP

bkþ1jkb

Tkþ1jk 0

0 Pbkþ1jk

" #8>>>>><>>>>>:

ð39Þ

Define new variables

~xkþ1jk ¼ xkþ1jk � bkþ1jkbkþ1jk ð40Þ

~Pxkþ1jk ¼ Px

kþ1jk � bkþ1jkPbkþ1jkb

Tkþ1jk ð41Þ

Then, from Eqs. (33) and (34), the corresponding covariance

matrix ~Pxkþ1jk of Eq. (19) can be obtained and Eq. (40) can

be changed to

~xkþ1jk ¼ Ak~xkjk þMk þ rkbkjk � bkþ1jkbkþ1jk ð42Þ

Then from Eqs. (38) and (41), we can get Eq. (29). Eq. (21)

can be derived from Eq. (34) directly. From Eqs. (34)–(36) andEq. (4), we have

Pkþ1jkþ1 ¼ InþN � Kkþ1�Ckþ1

�Pkþ1jk ¼

a1 a2

a3 a4

� �1

ð43Þ

in which

a1 ¼ ~Pxkþ1jk

� ��1

þ CTkþ1V

�1Ckþ1

a2 ¼ ða3ÞT ¼ � ~Pxkþ1jk

� ��1

bkþ1jk þ CTkþ1V

�1Fskþ1

a4 ¼ Pbkþ1jk

� ��1

þ bTkþ1jk ~Px

kþ1jk� ��1

bkþ1jk þ Fskþ1

�TV�1Fs

kþ1

8>>>>><>>>>>:

ð44Þ

According to the rule of finding the inverse of partitionedmatrix and Eq. (37), we can get

Pkþ1jkþ1 ¼~Pxkþ1jkþ1þbkþ1jkþ1P

bkþ1jkþ1b

Tkþ1jkþ1 bkþ1jkþ1P

bkþ1jkþ1

Pbkþ1jkþ1b

Tkþ1jkþ1 Pb

kþ1jkþ1

" #

ð45ÞThen it can be changed to Eq. (14), and in Eq. (45), we have

~Pxkþ1jkþ1 ¼ a�1

1 ð46Þ

bkþ1jkþ1 ¼ �a�11 a2 ð47Þ

Pbkþ1jkþ1 ¼ ða4 � a3a

�11 a2Þ�1 ð48Þ

From Eqs. (44) and (46), we can get Eq. (17). Eq. (27) canbe derived from Eq. (47) directly.

From Eqs. (43) and (45), Eq. (35) can be changed to

Kkþ1 ¼~Pxkþ1jkþ1 þ bkþ1jkþ1P

bkþ1jkþ1b

Tkþ1jkþ1 bkþ1jkþ1P

bkþ1jkþ1

Pbkþ1jkþ1b

Tkþ1jkþ1 Pb

kþ1jkþ1

" #

� CTkþ1

ðFskþ1ÞT

" #V�1 ð49Þ

then we can calculate Kxkþ1 and Kb

kþ1:

Kxkþ1 ¼ ~Px

kþ1jkþ1CTkþ1V

�1þbkþ1jkþ1 Pbkþ1jkþ1b

Tkþ1jkþ1C

Tkþ1V

�1h

þPbkþ1jkþ1ðFs

kþ1ÞTV�1i

Kbkþ1 ¼Pb

kþ1jkþ1HTkþ1jkþ1V

�1

8>>><>>>:

ð50Þ

Hkþ1jkþ1 ¼ Ckþ1bkþ1jkþ1 þ Fskþ1 ð51Þ

Then Eq. (26) can be obtained from Eq. (51).Defining

~Kxkþ1 ¼ ~Px

kþ1jkþ1CTkþ1V

�1 ð52ÞSubstituting Eq. (51) into Eq. (50), we have

Kxkþ1 ¼ ~Kx

kþ1 þ bkþ1jkþ1Kbkþ1 ð53Þ

Eq. (17) can be obtained from Eqs. (16) and (52).

Based on Eq. (32), bkþ1jkþ1 and xkþ1jkþ1 can be calculated as

bkþ1jkþ1 ¼ bkþ1jk þKbkþ1 ykþ1 �Nkþ1 �Ckþ1~xkþ1jk �Hkþ1jkbkþ1jk

� �ð54Þ

xkþ1jkþ1 ¼ ~xkþ1jkþ1þbkþ1jkþ1bkþ1jkþ1 þ bkþ1jk� ~Kxkþ1Hkþ1jk�bkþ1jkþ1

� �bkþ1jk

ð55ÞThen Eq. (22) is obtained from Eqs. (54) and (23). Eq. (15)

can be derived from Eq. (32) directly. From Eqs. (15) and (47),Eq. (55) can be simplified into Eq. (13).

From Eq. (48), we have

ðPbkþ1jkþ1Þ

�1 ¼ ðPbkþ1jkÞ

�1 þHTkþ1jk ~G

�1kþ1Hkþ1jk

~Gkþ1 ¼ Vþ Ckþ1~Pxkþ1jkC

Tkþ1

8<: ð56Þ

Eq. (25) can be derived from Eq. (29) directly. Eq. (24) is

obtained by substituting Eq. (56) into Eq. (51). Eq. (42) isobtained by substituting Eq. (4) into Eq. (18).

466 X. Chen et al.

Remarks.

(1) The TSEKF, which is given by Eqs. (13)–(29), is equiv-alent to the TSKF when Mk = 0 and Nk = 0 in Eq. (5).

(2) System states and bias of Eq. (1) can be estimated basedon Theorem 1.

(3) System states and additive faults of Eq. (9) can be esti-mated based on Theorem 1.

(4) System states and multiplicative faults of Eq. (11) can beestimated based on Theorem 1.

(5) The basic idea of this theorem is TSKF, so the TSEKF

has the same advantages of low computational cost andhigh estimation precision as TSKF.

(6) The TSEKF processes the nonlinear terms in system

model Eq. (1) with the same way of EKF. This waycan motivate the generalization of the linear TSKF tononlinear systems.

4. Fault simulation on closed-loop satellite attitude control

systems

Actuators in closed-loop ACSs are three reaction flywheels.Sensors in closed-loop ACSs are three gyros and two star sen-sors. Without loss of generality, we assume that the fault hap-pens in the reaction flywheel along x axis.

Here, we consider two simulation backgrounds: attitudestabilization control and attitude tracking control. The formeris considered for additive faults estimation. The latter is mainly

considered for multiplicative fault estimation. To estimatemultiplicative fault for reaction flywheels, satellite ACSs mustsatisfy the persistent excitation condition. In other words, reac-

tion flywheels should be activated to ensure satellite attitudemaneuver or tracking.

4.1. Faults conditions and simulation parameters

The conditions of the additive faults and the multiplicativefaults are given as follows.

Condition 1. The first fault b1 ¼ ½ 0:005; 0; 0 �T N � m exists

during the time interval 200 s 6 t1 < 400 s, and the second

fault b2 ¼ ½ 0:010; 0; 0 �T N�m exists during the time interval400 s 6 t2 < 600 s.

Condition 2. The control effectiveness factors of the first fault

b1 ¼ ½ 0:3; 0; 0 �T exist during the time interval200 6 t1 < 400 s, and the control effectiveness factors of the

second fault b2 ¼ ½ 0:5; 0; 0 �T exist during the time interval

400 6 t2 < 600 s.

The conditions of the simulation and initial values are cho-

sen as follows.

(1) The maximum output torque of the flywheels is0.15 N � m and the maximum angular momentum is

15 N � m � s.(2) The moment of inertial of the satellite is

Is ¼ diagð200; 100; 300Þ kg _s m2.

(3) The PD controller is chosen with its gains:

u ¼ �½KP; KD �ðx� xdÞ, with KD ¼ ½ 36 18 54 �Tand KP ¼ ½ 2 1 3 �T.

(4) For attitude stabilization control, xd = 06�1.(5) For attitude tracking control, � �

xd ¼ ½01�3 rad=s;p3sin p

200t rad; p

18sin p

400t rad; 0 rad �T

(6) The initial state is x0 = 06�1.(7) The initial transfer matrix is P0 = I6.(8) The process noise covariance matrix is W ¼

diagðr2xI3; r

2UI3Þ, where rx ¼ 2� 10�4 rad=s, rU =

5 � 10�5 rad.(9) The measurement noise covariance matrix is

V = 0.0012I6.(10) The sampling period k is 0.01 s.

4.2. Simulation results

(1) Fault estimation under attitude stabilization control.

The TSEKF algorithm proposed in this paper is applied forfault estimation of attitude stabilization under Condition 1.

The simulation results are shown in Fig. 1.For attitude stabilization control, the fault estimation

results using TSEKF algorithm and TSKF algorithm are the

same: 0.005 N � m and 0.010 N � m.The estimation results of multiplicative faults cannot be

obtained under the same attitude stabilization control back-ground under Condition 2. That is, the precondition of esti-

mating the multiplicative faults of actuators is that thecontrol torque output of actuators must not be approximatelyequal to zeroes. So, we have to estimate the multiplicative

faults of actuators under attitude tracking control with a per-sistent excitation.

(2) Fault estimation under attitude tracking control.

The system model of attitude tracking control is nonlinear,which makes the normal TSKF useless. The TSEKF algorithm

proposed in this paper is applied to fault estimation of attitudemaneuver under Condition 1 and Condition 2. The simulationresults are shown in Figs. 2 and 3.

For attitude tracking control, the fault estimation resultsusing TSEKF algorithm are 0.005 N � mand 0.010 N � m.Whenthe fault occurs, the trough and peak of the attitude angle curve

are �58.942� and 59.372� respectively, while the trough andpeakof the attitude angle curve are�59.085� and59.085� respec-tively without faults. That is, after the first fault occurs, a bias of

�0.143� is added to the attitude angle, and after the second faultoccurs, a bias of �0.287� is added to the attitude angle.

For attitude tracking control, the estimation results of con-trol effectiveness factors using TSEKF are 0.3 and 0.5 respec-

tively, and the MSE is 7.8 � 10�4.According to the simulation results, there is no obvious dif-

ference between the TSEKF and the TSKF in fault estimation

of linear system (such as attitude stabilization control). Whilein fault estimation of nonlinear system (such as attitude track-ing control), in which the TSKF cannot be applied, we can get

good performances on the basis of TSEKF.

Fig. 2 Simulation results of attitude tracking control under Condition 1.

Fig. 1 Simulation results of attitude stabilization under Condition 1.

A novel two-stage extended Kalman filter algorithm for reaction flywheels fault estimation 467

Fig. 3 Simulation results of attitude tracking control under Condition 2.

468 X. Chen et al.

5. Conclusions

This paper has investigated the problem of two-stage extended

Kalman filter-based reaction flywheel fault estimation.

(1) Based on the separate-bias principle, motivated by the

optimal TSKF and EKF, employed EKF to processnonlinear terms in nonlinear system model, a novelTSEKF algorithm is designed such that it cannot onlyestimate satellite attitude angular rates and Euler angles,

but also estimate reaction flywheel faults no matter theyare bias ones or loss of effectiveness.

(2) It is respectively applied to estimating bias faults and

loss of effectiveness for reaction flywheels in a satelliteACSs. Simulation results validate the feasibility andeffectiveness in the cases of both attitude stabilization

and attitude tracking control.

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Chen Xueqin received her B.S. degree in Automation, M.S. degree in

Spacecraft Design and Ph.D degree in Control Science & Engineering

from Harbin Institute of Technology in 2003, 2005 and 2008, respec-

tively, and now she is an associate professor there. Her main research

interests are fault diagnosis and fault-tolerant control.

Zhang Jinxiu received his B. S. and M. S. degree in Spacecraft Design

and Ph. D degree in Aerospace Science and Technology from Harbin

Institute of Technology in 2001, 2003 and 2006, respectively. He

worked from 2008 to 2013 as an associate professor there. From 2014

to 2015, he visited in Ecole polytechnique federale de Lausanne

(EPFL) Swiss Space Center. Now he is a professor of Harbin Institute

of Technology from 2013. His main research interests are dynamics

and control of formation flying or satellite swarm.