active fault-tolerant attitude control for ﬂexible spacecraft with …michael.friswell.com ›...

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2013; 27:925–943Published online 5 November 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.2363

Active fault-tolerant attitude control for flexible spacecraft withloss of actuator effectiveness

Bing Xiao 1, Qinglei Hu 1 and Michael I. Friswell 2,*,†

1Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China2College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK

SUMMARY

A theoretical framework for active fault-tolerant attitude stabilization control is developed and applied toflexible spacecraft. The proposed scheme solves a difficult problem of fault-tolerant controller design in thepresence of severe partial loss of actuator effectiveness faults and external disturbances. This is accomplishedby developing an observer-based fault detection and diagnosis mechanism to reconstruct the actuator faults.Accordingly, a backstepping-based fault-tolerant control law is reconfigured using the reconstructed faultinformation. It is shown that the proposed design approach guarantees that all of the signals of the closed-loop system are uniformly ultimately bounded. The closed-loop performance of the proposed control strategyis evaluated extensively through numerical simulations. Copyright © 2012 John Wiley & Sons, Ltd.

Received 19 May 2011; Revised 14 August 2012; Accepted 14 October 2012

KEY WORDS: fault-tolerant control; FDD; attitude stabilization; partial loss of actuator effectiveness;flexible spacecraft

1. INTRODUCTION

As attitude stabilization is one of the fundamental maneuvers that any spacecraft needs to performduring its operation, the past three decades have witnessed several important developments in thedesign of feedback control laws for spacecraft maneuver stabilization objectives. However, therestill remain certain open problems in this field that are of great theoretical and practical interest.One key difficulty is that the governing differential equations for the kinematics and dynamics ofspacecraft attitude motion are nonlinear. The attitude stabilization problem is further complicated bythe vibration of flexible appendages and the uncertainty of the spacecraft inertia properties becauseof onboard payload motion, rotation of solar arrays, and fuel consumption. Although a number ofcontroller design approaches, inspired from modern control theory, have been proposed by manyresearchers (see, for instance, [1–4]), most of the research deals only with uncertainties and exter-nal disturbances, assuming that there exists no actuator fault or failure during the entire attitudemaneuvers. This assumption is rarely satisfied in practice because some catastrophic faults mayoccur because of the malfunction of spacecraft components. As a result, if the attitude controlleris designed without any fault tolerance capability, an abrupt occurrence of an actuator fault couldultimately fail the mission. For instance, NASA’s Earth Orbiting Lewis Spacecraft suffered a totalloss of control on 28 September 1997, and the investigation board report stated that the spacecraftfailed mainly because of a technical flaw in the attitude control system design inducing excessivethruster firings that resulted in the shutdown of all thrusters [1]. Therefore, fault tolerance capabilityis one of the main issues that needs to be addressed in attitude control design.

*Correspondence to: Michael I. Friswell, College of Engineering, Swansea University, Singleton Park, SwanseaSA2 8PP, UK.

†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

926 B. XIAO, Q. HU AND M. I. FRISWELL

To achieve fault tolerance capability with satisfactory performance, a number of new fault-tolerantcontrol (FTC) approaches have been proposed in the literature; good overviews and extensivebibliographic references have been provided in [2–5]. As spacecraft attitude dynamics is modeledwith inherent nonlinearity, external disturbances, and system uncertainties, such intrinsic complexitymakes the straightforward application of many cited FTC schemes in [2–5] and the referencestherein infeasible for spacecraft attitude systems. Therefore, FTC design for spacecraft systemapplications is still an open problem for further research. At present, there are two approachesto synthesize controllers that are tolerant to system faults, known as passive FTC (PFTC), andactive FTC (AFTC). PFTC is designed and implemented using a fixed controller without any faultdetection and diagnosis (FDD) mechanism, as suggested in [6–11].

Passive FTC has the drawback that it is only reliable for the class of faults expected in the designprocess, and achieving robustness to certain faults is only possible at the expense of decreasednominal performance. In contrast to the PFTC, the AFTC can react to fault events and relieson the availability of an FDD mechanism that gives, in real-time, information about the natureand the intensity of the fault. This information is then used by a control configuration module toadjust the control effort online in such a way to maintain stability and to optimize the performanceof the faulty system. Hence, the work presented in this study falls into the AFTC category. Recently,FDD has become an ever increasingly important area of research activity, and many investigationson FDD have been conducted for linear and nonlinear systems [12–14]. Wang and Lum [15]developed an unknown input observer to detect and isolate actuator faults in aircraft by using anadaptive technique. Curry and Collins [16] proposed an alternative solution to the FDD problemwith the application of robust l1 estimation.

Unfortunately, most of the preceding FDD research is only feasible and applicable forlinear systems. The intrinsic complexities of spacecraft attitude dynamics mean these resultscannot be directly applied to spacecraft attitude systems. Although some investigations on FDD,together with methods for reconfiguring control systems, have been carried out for spacecraft,to the authors’ knowledge, there currently exists no unified framework to design simple controlstructures. A two-stage Kalman filtering algorithm was presented in [17] to estimate reac-tion flywheel faults, and two reconfiguration fault-tolerant controllers were then designed tocompensate for the faults on the basis of online statistical hypothesis tests. Two model-based diagnosis schemes were developed in [18] by using H1=H2 filters to address the faultdiagnosis problem of microscope thrusters in the presence of measurement noise and sensormisalignment. Williamson et al. [19] designed a set of fault detection filters for deep spacesatellites to detect and identify faults in each of the sensors or actuators on the basis of theShiryayev sequential probability theory. An iterative learning observer-based FDD mechanismwas reported in [20] to estimate time-varying thruster faults. However, only the disturbance-free case and bounded but additive unknown faults were considered. Patton et al. [21] explicitlyaddressed the problem of robust fault detection and isolation for faults affecting the thrustersof the satellite system Mars Express and considered attitude measurement errors, uncertainty,and disturbance.

In this paper, we use the modified Rodrigues parameters (MRPs) to represent the spacecraft atti-tude motion, and we provide an AFTC solution to the attitude stabilization problem for flexiblespacecraft under partial loss of actuator effectiveness. Also, the potential effects of external distur-bances and elastic vibrations on system performance are explicitly addressed. An observer-basedFDD mechanism is used to reconstruct the actuator faults, and the control law is reconfigured usingthe estimated fault information. The main contributions of this study, relative to other works, areas follows:

� The proposed observer-based FDI mechanism can achieve precise fault reconstruction within ashort period in the presence of external disturbances, in the sense that the reconstructed signalcan approximate the fault to any required accuracy.� The reconfigured control law can guarantee that all the signals of the closed-loop attitude

system are uniformly ultimately bounded, and can accomplish the attitude stabilizationmaneuver with high pointing accuracy and attitude stability. Compared with the passive attitude

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2013; 27:925–943DOI: 10.1002/acs

ACTIVE FAULT-TOLERANT ATTITUDE CONTROL 927

FTC approaches for spacecraft, the developed control scheme can compensate for actuatorfaults online in real time because of the effect of the FDD.

The remainder of this paper is organized as follows: In Section 2, the flexible spacecraft attitudemathematical model and the control problem are summarized. An AFTC solution to accomplishthe attitude stabilization maneuver is presented in Section 3. Section 4 demonstrates the appli-cation of the derived control algorithm to a simulated flexible spacecraft. Conclusions are givenin Section 5.

NotationFor a square matrix A, �min.A/ denotes the minimum eigenvalue of A; jj � jj denotes the

Euclidean norm or its induced norm. The symbol In denotes the n � n unit matrix. For allx D .x1, x2, : : : , xn/T 2 <n, we define kxk1 D maxfiD1,2,:::,ng jxi j and the vector sgn.x/ D

Œsgn.x1/ sgn.x2/ : : : sgn.xn/�T 2 <n, where sgn.�/ denotes the sign function. Sat.x,�0/ 2 <n

represents a vector with its i th element defined by sat.xi /D sgn.xi /minfjxi j,�0g,i D 1, 2, : : : ,n.

2. FLEXIBLE ATTITUDE MODEL AND CONTROL PROBLEM FORMULATION

2.1. Mathematical model of flexible spacecraft

In this section, we briefly recall the mathematical model for the attitude of a flexible spacecraft. Thenonlinear equations of the attitude motion, in terms of components along the body-fixed controlaxes, are given by the attitude kinematics and the spacecraft dynamics.

2.1.1. Kinematics equation. Given the Euler rotation angle �.t/ 2 < about the Euler principle axisn 2 <3, the attitude orientation of the spacecraft in the body-fixed frame, B, with respect to aninertial frame, I, can be represented in terms of the MRPs � D

��1 �2 �3

�T2 <3, which are

given by

� D n tan

��.t/

4

�, �.t/ 2 Œ0, 360/ deg (1)

Now the differential equation for the kinematics described using MRPs can be summarized as [22]

P� D1

4

h�1� �T�

�I3C 2�

�C 2��Ti!D F.� /! (2)

where ! D�!1 !2 !3

�T2 <3 is the angular velocity of the spacecraft with respect to the

inertial frame, I, and expressed in the body-fixed frame, B. �� 2 <3�3 is a cross product matrixdefined by

�� D

24 0 ��3 �2

�3 0 ��1��2 �1 0

35 (3)

Property 1The matrix F .� / satisfies the following relationship [22]

F �1.� /D16

.1C �T� /2F T, kF .� /k6 1

2(4)

This definition highlights that the inverse F �1.� / always exists.



2.1.2. Spacecraft dynamics. When all of the actuators function normally, under the assumptionof small elastic displacements, the mathematical model of a flexible spacecraft with flexibleappendages is [23]

J P!C ıT R�D�!��J!C ıT P�

�C u.t/C d.t/ (5)

R�CC P�CK�C ı P!D 0 (6)

where J 2 <3�3 is the total inertia matrix of the spacecraft, u.t/D�u1 u2 u3

�T2 <3 denotes

the control torque that can actually be applied by the actuators, and d.t/D�d1 d2 d3

�T2 <3

denotes the external disturbance torque. � 2 <N is the modal coordinate vector relative to themain body, ı 2 <N�3 denotes the coupling matrix between the flexible and rigid dynamics,C D diagf2�iƒi , i D 1, 2, : : : ,N g 2 <N�N and K D diag

˚ƒ2i , i D 1, 2, : : : ,N

�2 <N�N

are the damping and stiffness matrices, N is the number of elastic modes considered, ƒi are thenatural frequencies, and �i are the associated damping ratios.

In this study, the considered spacecraft is controlled by three actuators without actuation redun-dancy, and they are mounted orthogonally, aligned with the spacecraft body axes X ,Y , and Z.Therefore, the output torque of each actuator is parallel to one axis of the spacecraft body frame,and thus the control inputs generated by the actuators are equal to the control torques applied to theattitude system. The flexible spacecraft attitude dynamics, given by (5) and (6), describe the actuatorfault-free condition. Consider the situation where the actuators experience a partial loss of actuatoreffectiveness. Then the attitude dynamics in (5) is modified to

J P!C ıT R�D�!��J!C ıT P�

�C �.t/u.t/C d.t/ (7)

where �.t/ D diagf�1, �2, �3g 2 <3�3 denotes the effectiveness factor matrix for the spacecraftactuators with 0 < �i .t/ 6 1 .i D 1, 2, 3/. The case when �i .t/ D 1 means that the i th actuatorworks normally, and 0 < �i .t/ < 1 corresponds to the case in which the i th actuator partially losesits effectiveness. Let terms d.t/, ıT R� and !�ıT P� be considered as lumped disturbances. Then (7)may be rewritten as

J P!D�!�J!C �.t/u.t/C T d .t/ (8)

with T d .t/, �ıT R��!�ıT P�C d.t/.

Remark 1In this paper, small elastic displacements for the flexible structures are considered, and themagnitudes of elastic vibration k�k, and its derivatives k P�k and k R�k, are assumed to be boundedduring the whole attitude maneuver. This is feasible in practice because there always exists dampingin flexible structures, even if this damping is small. The external disturbances, d.t/, in (8) includegravitational perturbations and atmospheric drag, and are also bounded [24]. Thus it is reasonableto assume that there always exists a constant NTd such that

kT d .t/k6 NTd (9)

Remark 2Because of physical limitations on the actuators, the control action generated is limited by thesaturation value, and for simplicity, we assume that all three control input torques have the sameconstraint value umax, that is, jui j6 umax for i D 1, 2, 3.

2.2. Control problem statement

The control objective can be stated as follows: Consider the flexible spacecraft attitude system givenby (2), (5), and (6). Determine an FDD mechanism and an FTC law, u.t/, such that the followinggoals are achieved in the presence of external disturbances, possible partial loss of effectivenessfault, �.t/, and actuator input saturation:



(a) All the signals of the resulting closed-loop attitude system are bounded and continuous.(b) The designed FDD can achieve precise fault reconstruction in the sense that the reconstruction

signal can approximate the fault to any required accuracy.(c) The attitude stabilization maneuver is accomplished with the attitude � and angular velocity!

converging to an arbitrary small set containing the origin in finite time T �, that is, k� .t/k6 "�1and k!.t/k6 "�2 for t > T �.

(d) The performance index

Ip D limt!1

1

t

Z t

0

hkek2Ck Qpk2

idt

is bounded with e D O! � ! and Qp D Op � p. Note that O! and Op are the estimates of ! and

p.t/D��1.t/ �2.t/ �3.t/

�T, respectively.

3. ACTIVE FAULT-TOLERANT ATTITUDE CONTROLLER DESIGN

In this section, an active fault-tolerant attitude controller design for flexible spacecraft is presented,including the design of the observer-based FDD mechanism and the design of the fault-tolerantattitude controller. The FDD design is first presented to achieve precise fault reconstruction. A back-stepping based FTC law is then reconfigured using the reconstructed faults from the FDD observerto perform the attitude stabilization maneuver.

3.1. Observer-based FDD mechanism design

As �.t/ is a diagonal matrix, the term �.t/u.t/ in (8) can be rearranged as

�.t/uD Up.t/ (10)

where U D diag.u1,u2,u3/. Using (10), the attitude dynamics (8), with faulty actuators, can bewritten as

J P!D�!�J!CUp.t/C T d .t/ (11)

The following state observer-based FDD mechanism is proposed to detect and reconstruct theactuator fault vector p.t/,

J PO!D� O!�J O!CU Op.t/��. O!�!/� l1sgn. O!�!/ (12)

Op.t/D l2 Op.t � T /C l3. O!�!/ (13)

where T is the updating interval, which is taken as the sampling time interval in this paper, � 2 <3�3

is a positive definite matrix, and li 2 < .i D 1, 2, 3/ are positive observer gains.In order to examine the performance of the proposed FDD scheme, one should develop an expres-

sion for the observer error dynamics. By using the definition e D O!�! and Qp D Op�p, the dynamicsof the error, e, between the actual state,!, and its estimate, O!, can be obtained from (11) and (12) as

J Pe D!�J!� O!�J O!CU Qp.t/� �e � l1sgn.e/� T d (14)

Remark 3As 0 < �i .t/6 1, kl2p.t � T /�p.t/k1 6 .1C l2/ always holds.

Remark 4For a sufficiently small sampling time, T , the assumption

!�J!� O!�J O! 6 kek holds forsome known constant > 0 [20].



Remark 5The term � . O! � !/ in (12) is introduced as a feedback term to guarantee the convergence of theobserver errors e.t/ and Qp.t/, as shown in (14) and used in the proof of Theorem 1. Furthermore,usually, the actuators are operating normally initially, that is, p.0/ D

�1 1 1

�T; in order to

achieve a fast reconstruction of the actuator faults, the initial value of Op.t/ in (13) is set to p.0/, thatis, Op.0/D

�1 1 1

�T.

Remark 6Note that (12) is a continuous differential equation, whereas (13) is a recursive equation. In practice,the observer will work in discrete time, and hence (12) would be transformed to its discrete timeequivalent. Alternatively, for small T , (13) may be considered as discrete approximation to afirst-order differential equation in Op.t/.

The preceding analysis leads to the following theorem, which is very useful in the FTC design.

Theorem 1Consider the estimation error (14), and suppose that the observer gains are chosen such that

1 D 1� �1

�u2max

2"3C 1C �

�> 0 (15)

2 D �min.�/� �"3

2� �2

�u2max

2"3C 1C �

�> 0 (16)

3 D l1 � NTd � 2l3.1C l2/

�u2max

2"3C 1C �

�> 0 (17)

where �1 D l22 .1C "1C "2/ , �2 D l23

1C 1

"1

�, for positive constants "i .i D 1, 2, 3/ and � . Then

the estimation errors e and Qp.t/ will converge to a small set containing the origin, and the perfor-mance index Ip will also be bounded by a small positive constant. Thus the control objectives (b)and (d) as stated in Section 2.2 are met with the designed FDD mechanism in (12) and (13).

ProofFrom (13), one has

Qp.t/D l2 Qp.t � T /C l3e C l2p.t � T /�p.t/ (18)

Define a new variable�D l2p.t � T /�p.t/. Then it follows that

QpT.t/ Qp.t/D l22 QpT.t � T / Qp.t � T /C l23e

Te C�T�C 2l2l3 QpT.t � T /e

C 2l2 QpT.t � T /�C 2l3e

T� (19)

Employing the well-known Young’s inequality, 2xTy 6 "ixTx C 1"iyTy for all x,y 2 <3, we

have, for arbitrary positive constants "i .i D 1, 2/,

2l2l3 QpT.t � T /e 6 "1l22 QpT.t � T / Qp.t � T /C

l23"1eTe (20)

2l2 QpT.t � T /�6 "2l22 QpT.t � T / Qp.t � T /C

1

"2�T� (21)



Substituting inequalities (20) and (21) into (19) gives

QpT.t/ Qpy.t/6 l22 .1C "1C "2/ QpT.t � T / Qp.t � T /C l23

�1C

1

"1

�eTe

C

�1C

1

"2

�„ ƒ‚ …

�3

�T�C 2l3kek k�k1

D �1 QpT.t � T / Qp.t � T /C �2e

Te C �3�T�C 2l3.1C l2/kek

(22)

Consider a candidate Lyapunov function of the form

V.t/D1

2eTJe C

Z t

t�T

QpT.s/ Qp.s/ds (23)

The inequality eTU Qp.t/ 6 "32eTe C

u2max2"3Qp.t/T Qp.t/ always holds for all "3 > 0, because

jui j 6 umax. After laborious yet relatively straightforward algebra, followed by the application of(15) to (17) and (22), the time derivative of V is given as

PV .t/D eT�!�J!� O!�J O!CU Qp.t/� � e � l1sgn.e/� T d

�C QpT.t/ Qp.t/� QpT.t � T / Qp.t � T /

6 kek2 � �min.�/ kek2 � .l1 � NTd / kekC e

TU Qp.t/C QpT.t/ Qp.t/� QpT.t � T / Qp.t � T /

6h � �min. /C

"3

2

ikek2 � .l1 � NTd / kek

C

�u2max

2"3C 1

�QpT.t/ Qp.t/� QpyT.t � T / Qp.t � T /

D

�u2max

2"3C 1C �

�h�1 Qp

T.t � T / Qp.t � T /C �2eTeC �3�

T�C 2l3.1C l2/keki

� QpT.t � T / Qp.t � T /Ch � �min. /C

"3

2

ikek2 � .l1 � NTd /kek � � Qp

T.t/ Qp.t/

6 ��1� �1

�u2max

2"3C 1C �

� QpT.t � T / Qp.t � T /

�

��min.�/� �

"3

2� �2

�u2max

2"3C � C 1

� kek2C �3

�u2max

2"3C 1C �

�.1C l2/

2

�

�l1 � NTd � 2l3 .1C l2/

�u2max

2"3C � C 1

� kek � � k Qp.t/k2

6 �2kek2 � � k Qp.t/k2C 4(24)

where � is a positive constant, 4 D �3

u2max2"3C 1C �

�.1 C l2/

2, and is given in Remark 4.

Clearly, if ke.t/k>p4=2 , N"a and k Qp.t/k>

p4=� , N"b , then PV < 0, which implies that V.t/

decreases monotonically. Using Theorem 4.18 (page 172) in [25], the estimated errors are ultimatelyuniformly bounded by

limt!1

�eT.t/ QpT.t/

�T2D1 ,

n�eT.t/ QpT.t/

�T ˇkek6 N"a, k Qpk6 N"b

o(25)

which defines a small set containing the origin�eT.t/ QpT.t/

�TD�0 0

�T. Moreover, the

larger the value of 2 and/or the smaller the values of 4 and � that are selected, the smaller errorset will be. Thus from (25), the precise faults reconstruction can be achieved with the designedobserver-based FDD given by (12) and (13). In addition, (24) shows that the following inequality isachieved:

limt!1

1

t

Z t

0

hkek2Ck Qpk2

ids 6 lim

t!1

�V.0/

tC

4

�min

�D

4

�min<1 (26)



where �min D minf2, �g. From (26), it can be seen that the larger �min and the smaller 4 are, thebetter the resulting performance. Thus the proof is completed. �

Remark 7It is worth mentioning that the proposed observer-based FDD given by (12) and (13) is synthesizedby using the previous information and the state estimation error information, as we can see in (13).From the proof of Theorem 1, the actuator fault reconstruction error and the angular velocity esti-mation error may be driven to zero as closely as required by tuning the observer gains. Thus theproposed FDD can achieve precise fault reconstruction, and it can also detect and reconstruct thepartial loss of actuator effectiveness in the more general cases of constant, periodic time-varying oreven random signals.

Remark 8In order to guarantee that the reconstructed faults, Op.t/, are bounded in the internal .0, 1�, the faultreconstruction law in (13) can be modified to

Op.t/D ProjŒpmin,1� fl2 Op.t � T /C l3. O!�!/g (27)

where pmin is a small positive constant (such as 0.0001), and Projf�g denotes the projectionoperator [26]. Note that the role of the projection operator is to project the reconstructed faultsonto the interval .0, 1�.

Remark 9Although the control input u.t/ is required to be bounded in the proof of Theorem 1, only theapplied control action generated by the actuators is involved, and the output torque of each actuatoris bounded by the physical limitations of the actuators. Therefore, precise fault reconstruction canalways be met with the proposed FDD scheme

Remark 10If large values of li .i D 1, 2, 3/ are chosen, the effect of measurement noise would be amplified in(12) and (13), and the fault estimates would be affected. Hence the choice of li in (15)–(17) shouldbe made carefully and their magnitudes limited to avoid amplifying the effect of the measurementnoise.

3.2. Fault-tolerant attitude stabilization controller design

The designed FDD mechanism, (12) and (13), requires the control signal u.t/ to be within thesaturation limit; otherwise, precise reconstruction of the actuator faults would not be guaranteed.In this section, on the basis of the reconstructed faults Op.t/ from the observer-based FDD given by(12) and (13), a novel fault-tolerant attitude stabilization control law will be reconfigured for thespacecraft, where its magnitude is bounded by the maximum torque generated by actuator. Beforederiving the specific control design, some new variables are introduced: x1 D

R�dt , x2 D � and

x3 D!. Accordingly, we may rewrite

Px1 D x2 (28)

Px2 D F .x2/x3 (29)

J Px3 D�x�3Jx3C O�.t/u� Q�.t/uC T d (30)

where O�.t/ D diag. Op1, Op2, Op3/ and Q�.t/ D diag. Qp1, Qp2, Qp3/. From Theorem 1, Qpi .i D 1, 2, 3/converges to an arbitrarily small value, and hence the following inequality can be achieved:

k�Q�.t/uC T dk6 umaxC NTd (31)



From (28) to (30), the spacecraft attitude system given by (2) and (8) has been written in triangularnonlinear form. Hence, the standard backstepping controller design can be employed, and thefollowing state transformation is performed in order to design the attitude controller to tolerateactuator faults:

´1 D x1, ´2 D x2 � ˛1, ´3 D x3 � ˛2 (32)

where ˛1 2 <3 and ˛2 2 <3 are virtual control inputs, which will be discussed later.On the basis of the preceding analysis, the design procedure of the active fault-tolerant controller

can be elaborated as the follows.

Step 1We start with (28) by considering x2 as the control variable. The time derivative of ´1 is thengiven as

P1 D Px1 D x2 D ´2C ˛1 (33)

The task in this step is to design a virtual control law ˛1 to make ´1 ! 0. Choose a candidateLyapunov function as V1 D 1

2´T1 ´1, and select an appropriate virtual control ˛1 as

˛1 D�c1x1 (34)

where c1 is a positive constant. Then it follows that

PV1 D ´T1 P1 D ´

T1 .´2 � c1´1/D�c1 k´1k

2C ´T1 ´2 (35)

Clearly if ´2 D 0, then PV1 D�c1 k´1k2 and ´1 is guaranteed to converge to zero asymptotically.

Step 2We now differentiate the second error, ´2, using (29) to give

P2 D Px2 � P 1 D F .x2/x3C c1x2 (36)

Choose a new Lyapunov function V2 D V1 C12´T2 ´2. Let us select the appropriate virtual control

law ˛2 as

˛2 D F�1.x2/.�´1 � c1x2 � c2´2/ (37)

where c2 is a positive constant, and F �1 is given explicitly in (4). Differentiating both sides of V2and inserting (37) yields

PV2 D�c1 k´1k2C ´T1 ´2C ´

T2 .F .x2/x3C c1x2/

D�c1 k´1k2C ´T1 ´2C ´

T2

˚F .x2/

�´3CF

�1.x2/.�´1 � c1x2 � c2´2/�C c1x2

�D�c1 k´1k

2 � c2 k´2k2C ´T2F .x2/´3

(38)

Again, if ´3 D 0, one has PV2 D �c1 k´1k2 � c2 k´2k

2, and thus both ´1 and ´2 will converge tozero asymptotically.

On the basis of the analysis in Steps 1 and 2, if ´3 can be driven to zero by designing a suitablecontrol law, then ´i .i D 1, 2/ will also be driven to zero, and thus the attitude orientation isstabilized according to (32). In the following theorem, we summarize our control solution to drive´3 to zero by incorporating backstepping based control action and the observer-based FDD.

Theorem 2Consider the closed-loop system consisting of the faulty attitude system given by (2) and (7) underthe designed FDD mechanism (12) and (13). The following FTC law is implemented

uD Sat.v,umax/ (39)



with v designed as

v.t/D O�.t/�1

(x�3Jx3 � J

"dF �1.x2/

dt.´1C c1x2C c2´2/CF

�1.x2/. P1C c1 Px2C c2 P2/

#

�c3´3 �K1xa �FT.x2/´2 �

´3�2Td

k´3k�Td C " exp.�ˇt/

)(40)

where �Td D umax C NTd , " is a sufficiently small positive scalar; K1,ˇ, and c3 are positive controlgains; and xa 2 <3 is the output of the following auxiliary system:

Pxa D�K2xa �kO�.t/k2 k�uk2 xa

kxak2

� O�.t/�u (41)

�uD u�v, andK2 is a positive constant. Suppose that the control parameters are chosen such that

c3 � 1 > 0, K2 �K212�1

2> 0 (42)

Then the closed-loop attitude system is stable in the sense that all of the trajectories of the resultingclosed-loop attitude system are uniformly ultimately bounded. Moreover, the attitude orientationand velocity converge to a small set containing the origin, that is, k�.t/k 6 "�1 and k!.t/k 6 "�2 for8t > T �. Thus the control objectives (a) and (c) as stated in Section 2.2 are achieved.

ProofIt is obvious from (30) and (37) that

P3 D Px3 � P 2

D J�1��x�3Jx3C �.t/uC T d

�C

dF �1.x2/

dt.´1C c1x2C c2´2/

CF �1.x2/. P1C c1 Px2C c2 P2/

(43)

where

dF �1.x2/

dtD16�1C xT

2 x2�2 h dF .x2/

dt

iT� 64

�1C xT

2 x2�xT2F .x2/x3F

T.x2/�1C xT

2 x2�4 (44)

dF .x2/

dtD1

2

˚�xT

2 ŒF .x2/x3� I3C .F .x2/x3/�CF .x2/x3x

T2 C x2x

T3F

T.x2/�

(45)

Consider another candidate Lyapunov function V3 as follows:

V3 D V2C1

2´T3 J´3C

1

2xTa xa (46)

From (41),

xTa Pxa D�K2 kxak

2 � kO�.t/k2 k�uk2 � xTa O�.t/�u (47)

Differentiating (46) and inserting (43) and (47) leads to

PV3 D PV2C ´T3 J P3C x

Ta Pxa

D�c1 k´1k2 � c2 k´2k

2C ´T2F .x2/´3 �K2 kxak2 � kO�.t/k2 k�uk2 � xT

a O�.t/�u

C ´T3

(�x�3 Jxy3C O�.t/u� Q�.t/uC T dCJ

hdF�1.x2/

dt .´1C c1x2C c2´2/CF�1.x2/. P1C c1 Px2C c2 P2/

i )

D�c1 k´1k2 � c2 k´2k

2C ´T2F .x2/´3 �K2 kxak2 � kO�.t/k2 k�uk2 � xT

a O�.t/�u

C ´T3

(�x�3Jx3C O�.t/vC O�.t/�u� Q�.t/uC T d

CJh

dF�1.x2/dt .´1C c1x2C c2´2/CF

�1.x2/. P1C c1 Px2C c2 P2/i )

(48)



Substituting the control law (39) into (48) results in

PV3 D�c1 k´1k2 � c2 k´2k

2 � c3 k´3k2 �K2 kxak

2 � kO�.t/k2 k�uk2 � xTa O�.t/�u

C ´T3

(O�.t/�u�K1xa � Q�.t/uC T d �

´3�2Td


)(49)

Using inequality (31), it follows that

´T3

"�Q�.t/uC T d �

´3�2Td


#6 k´3k�Td �

k´3k2 �2Td


6 " exp.�ˇt/

(50)

Further, using the specific case of Young’s inequality xTy 6 12xTxC 1

2yTy for all x,y 2 <3, the

following inequalities may be established:

´T3 O�.t/�u61

2kO�.t/k2 k�uk2C

1

2k´3k

2 (51)

�K1´T3 xa 6

1

2k´3k

2CK212kxak

2 (52)

� xTa O�.t/�u6

1

2kO�.t/k2 k�uk2C

1

2kxak

2 (53)

Therefore, inserting inequalities (50)–(53) into (49) yields

PV3 6 �c1 k´1k2 � c2 k´2k2 � .c3 � 1/ k´3k2 ��K2 �

K212�1

2

�kxak

2C " exp.�ˇt/ (54)

From (46), 2V3 6 k´1k2Ck´2k2CJmax k´3k2Ckxak

2, where Jmax is the largest eigenvalue of J .Hence,

PV3 6 �c1 k´1k2 � c2 k´2k2 � .c3 � 1/ k´3k2 ��K2 �

K212�1

2

�kxak

2C " exp.�ˇt/

6 � Nmk´1k

2Ck´2k2C Jmax k´3k

2Ckxak2�C "

6 �2 NmV3C "

(55)

where NmDmin

�c1, c2, c3�1

Jmax,K2 �

K21

2� 12

�, and all terms in the definition of Nm are positive from

the inequalities given by (42). Using Theorem 4.18 (page 172) in [25], inequality (55) means thatV3 is uniformly ultimately bounded together with the states ´i ,i D 1, 2, 3 and xa. More precisely,

there exists a finite time T � > 0 such that for all "� >q

"Nm

and for all t > T �, we have

k´ik< "�.i D 1, 2, 3/ and kxak< "

�

In other words, the closed-loop attitude system converges in finite time to the ball

D"� Dn�´T1 ´T2 ´T3

�T2 <9 j k´1k< "

�, k´2k< "�, k´3k< "

�o

and t > T �

Note that under the state transformation (32) and the definition of xi .i D 1, 2, 3/, it is clear that

kx1k D k´1k< "�, 8t > T � (56)



and thus one has

k�k D kx2k D k´2C ˛1k D k´1 � c1´1k< .1C c1/"� , "�1 , 8t > T � (57)

It is easily concluded from (19) that the spacecraft attitude is ultimately uniformed bounded.Moreover, using (4), one has

F �1.x2/6 16.1C�T�/2

kF k6 8. Therefore

k!k D kx3k D k´3C ˛2k

< "�CF �1.x2/ Œ1C c1.1C c1/C c2� "�

< Œ9C 8c1.1C c1/C 8c2� "� , "�2 , 8t > T �

(58)

which implies that the spacecraft angular velocity is ultimately uniformly bounded.Thus the result as stated in Theorem 2 is established, and hence the proof is completed. �

Remark 11The structure of the corresponding new control law (39) is independent of the order of the truncatedspacecraft model. This is important because theoretically flexible structures have an infinite numberof elastic modes. Moreover, the proposed fault-tolerant controller (39) is also robust with respect tounmodeled dynamics describing the flexibility because modal vibration information is not needed.Therefore, from the standpoint of disturbances and flexible vibration rejection, the reconfiguredcontrol law has great stability robustness.

Remark 12A graphical representation of the whole attitude control plant with the proposed AFTC scheme ispresented in Figure 1. Note that the FDD in (12) and (13) is developed for the open-loop attitudesystem and does not require the stability of the closed-loop attitude system. As shown in Figure 1,the FDD (12) and (13) and the fault-tolerant controller (39) are derived separately; the FDD isdeveloped to reconstruct the actuator faults, whereas the controller is reconfigured to compensatefor the effect of the actuator faults.

Remark 13Summarizing the analysis in Theorems 1 and 2, the observer and the control gains of the presentedAFTC scheme could be chosen according to the following gain tuning procedure.

Step 1: Choose small � , l2, l3,"i .i D 1, 2/; and also select a large "3.Step 2: Choose � and l1 such that (15) to (17) are satisfied.Step 3: Choose positive c1 and c2, and then select c3, K1, and K2 such that (42) is guaranteed.

ReferenceAttitude

Trajectory

Flexible SpacecraftAttitude ModelEqs. (2), (5)-(6)

Feedback Loop

ExternalDisturbances

ActuatorFaults

The ProposedControl Law

Eq. (39)

d

,Eq. (40) v u

AuxiliarySystem Eq. (41)

u +

ax

+

Observer basedFDI Mechanism

Eq. (12)-(13)

p

Figure 1. Active fault-tolerant control for the flexible spacecraft attitude system.



4. SIMULATED EXAMPLE

To verify the effectiveness and performance of the proposed attitude stabilization control scheme,numerical simulations have been performed using the flexible spacecraft system (2), (5), and (6).The same physical parameters considered in [23] are used, which are

J D

24 350 3 4

3 270 10

4 10 190

35 kg m2, ı D

264

6.45637 1.27814 2.15629�1.25819 0.91756 �1.672641.11687 2.48901 �0.836741.23637 �2.6581 �1.12503

375 kg1=2 m/s2

and the first four elastic modes have been taken into account, that is,N D 4. The natural frequenciesare ƒ1 D 0.7681, ƒ2 D 1.1038, ƒ3 D 1.8733, and ƒ4 D 2.5496 rad/s; and damping ratiosare �1 D 0.0056, �2 D 0.0086, �3 D 0.013, and �4 D 0.025. For an on-orbit spacecraft, theexistence of external disturbances is inevitable. Therefore, the external disturbances acting on thespacecraft are considered for all of the numerical examples presented in this section and are givenby d.t/D .k!k2C 0.05/

�sin 0.8t cos 0.5t cos 0.3t

�TN m [11]. The control input saturation

amplitude is assumed to be 5 N m, that is, umax D 5 N m. To better represent the engineering appli-cation, measurement noise is added to the attitude and angular velocity sensors outputs, modeled asa zero-mean Gaussian random variable with standard deviation 0.0001. An extended Kalman filteris used in the attitude determination sub-system.

At present, there are few examples of the design of active fault-tolerant attitude controllers forspacecraft, and most of the existing literature on the design of FTCs for spacecraft attitude focuson PFTC control laws. Thus a comparison between the designed controller and existing AFTCs forspacecraft cannot be made. Hence, in this section, the proposed AFTC law (39) is compared withconventional control methods: a proportional integral derivative (PID) controller and the passivefault-tolerant controller proposed in [11]. Using these three controllers, we simulate two differentcases: (1) constant loss of actuator effectiveness and (2) time-varying loss of actuator effectiveness.To implement the developed control strategy, according to the procedures in Remark 12, the observergains of the presented FDD mechanism, (12) and (13), were selected by trial-and-error until a goodperformance was obtained. The observer gains were ultimately chosen as l1 D 50, l2 D 0.2, l3 D 2,� D diag.800, 850, 1000/, and � D 10; whereas the control gains of the control law given by (1)were chosen as c1 D c2 D 0.5, c3 D 1.25, K1 D 0.55, K2 D 1.5, " D 0.001, and ˇ D 0.25. Theseparameters remain unchanged for all of the simulation cases in order to permit a fair and meaningfulcomparison. Further, in the context of simulation, at time t D 0, the orientation of spacecraft is setto be � .0/D

��0.3 �0.4 0.2

�Twith a zero initial body angular velocity, and the initial modal

displacements and velocities are given by �i .0/D 0 and P�i .0/D 0 for i D 1, 2, 3, 4.

4.1. Response under constant loss of actuator effectiveness fault

In this case, a constant loss of actuator effectiveness fault is introduced to the flexible spacecraftattitude system, and the effectiveness factors are described by8<

:�1.t/D 0.4, t > 8 s�2.t/D 0.6, t > 5 s�3.t/D 0.7, t > 10 s

(59)

We first present the simulation results when applying AFTC. Figure 2 shows the time response ofthe reconstructed fault from the observer-based FDD mechanism (12) and (13) incorporated in theAFTC. It is shown from the response of the first 5 s in Figure 2(a) that the proposed FDD canreconstruct the effectiveness factor in 3 s when the actuator is fault free. After the occurrence ofthe actuator fault at 5 s, within a short period, the actuator fault is successfully reconstructed. Thisreconstruction happens with j�i � O�i j 6 1.2 � 10�3.i D 1, 2, 3/ even in the presence of externaldisturbances. Figure 2(b) shows that the observed velocity state O! converges to the actual angularvelocity ! in 15 s with high accuracy. Moreover, as shown in Figure 2(c), the performance indexIp.t/ is bounded for all time and ultimately converges to a small value. From the results obtained,



0 10 20 30 40 50

0.5

1

0 10 20 30 40 500.5

1

0 10 20 30 40 500.6

0.8

1

ρ 1ρ 2

ρ 3

Time(sec)

Actual faultEstimated fault



0 10 20 30 40 50-0.2

0

0.2

e 1(rad

/s)

0 10 20 30 40 50-0.2

0

0.2

0.4

e 2(rad

/s)

0 10 20 30 40 50-0.2

0

0.2

0.4

e 3(rad

/s)

Time(sec)

(a) The reconstructed faults (b) The angular velocity error

0 20 40 60 80 100-0.2

0

0.2

0.4

0.6

0.8

Time(sec)

The

per

form

ance

inde

x I p

(c) The performance index Ip (t)

Figure 2. Fault reconstruction with the proposed FDD with constant loss of actuator effectiveness faults.

it is clear that the designed FDD can accurately reconstruct the fault values even if no knowledgeof the faults is available in advance, despite the presence of disturbances. Hence, the conclusions ofTheorem 1 are verified.

Figure 3 (solid line) shows the attitude control performance with the application of the AFTC inthe given fault case (59). We see clearly that the proposed controller has accomplished the attitudestabilization maneuver. Figure 3(a and b, solid line) also shows that the desired system performanceis achieved, and the attitude orientation is governed to near zero within 15 s, even in the presenceof external disturbances. As the proposed FDD mechanism can reconstruct the actuator fault, (59),in a short period, the associated control law, (39), can tolerate the fault in 7 s when a constantloss of effectiveness fault occurs. Thus large and even maximum control torque is demanded inorder to compensate for the lost power, as we see from the corresponding control input shown inFigure 3(c, solid line). However, the control input is within its maximum allowable limit, and themodal displacements of flexible appendages are completely damped out within 70 s, as shown inFigure 3(d, solid line).

The application of the conventional PID control leads to the attitude control performance shownin Figure 3 (dash-dot line). It is clear that the constant fault (59) is propagated to the attitude andthat the controller, shown in Figure 3(c, dash-dot line), is unable to compensate for this fault within



0 20 40 60 80 100-0.4

-0.2

0

0.2

σ 1

0 20 40 60 80 100

-0.4

-0.2

0

0.2

σ 2

0 20 40 60 80 100

-0.4

-0.2

0

0.2

σ 3

Time(sec)

AFTC PFTC PID

0 20 40 60 80 100

-4

-2

0

2

4

0 20 40 60 80 100-4-202468

0 20 40 60 80 100-4

-2

0

2

4

ω3(

rad/

s)ω

2(ra

d/s)

ω1(

rad/

s)

Time(sec)

AFTC PFTC PID

0 20 40 60 80 100-5

0

5

u 1(N

m)

u 2(N

m)

u 3(N

m)

0 20 40 60 80 100-5

0

5

0 20 40 60 80 100-5

0

5

Time(sec)

AFTC PFTC PID

0 20 40 60 80 100-0.04

-0.02

0

0.02

χ 1

0 20 40 60 80 100-5

0

5 x 10-3

χ 2

0 20 40 60 80 100-6-4-202

x 10-3

χ 3

0 20 40 60 80 100-5

0

5

10 x 10-4

χ 4

Time(sec)

PID PFTC AFTC

(a) The attitude orientation (b) The angular velocity

(c) The control input (d) The vibration displacement

Figure 3. Attitude control performance with constant loss of actuator effectiveness faults.

100 s. As a result, a time-critical aerospace mission would not be performed. This control schemenot only fails to perform the attitude maneuver but also excites large oscillations in the flexibleappendages, which may further deteriorate the system control performance. Although there existssome room for improvement with different control parameter sets, there is not much improvementin the attitude and velocity responses.

The attitude control objectives can be achieved with the PFTC proposed in [11] because of its faulttolerance capability to certain faults. However, the time required to perform the attitude stabilizationmaneuver is much longer than the proposed controller (39), and there also exists severe oscillationsin the flexible appendages as shown in Figure 3(d, dash line). Because of its great conservativeness,the PFTC cannot compensate for the fault immediately once the partial loss of actuator effectivenessfault occurs. In contrast, the AFTC developed in (39) can accommodate the fault as soon as possible



because of the effect of the incorporated FDD mechanism. Thus the PFTC in [11] requires moretime than AFTC to stabilize the attitude in the presence of actuator faults. Also, as the PFTC controlin [11] always acts on the spacecraft whether faults occur or not, it usually requires more controlpower, as shown in Figure 3(c, dash line), especially in the time interval from 20 to 50 s. The AFTCand PFTC controllers can protect the control torque from actuator saturation, as shown in Figure 3(c, solid and dashed line).

These results completely support the theoretical result that the desired performance of the systemcan be achieved with the proposed controller and FDD even if the faults are unknown.

4.2. Response under time-varying loss of actuator effectiveness fault

To further verify the effectiveness of the proposed FTC, a time-varying loss of actuator effectivenessfault is considered, given by8<

:�1.t/D 0.2C 0.1 sin.0.2�t/, t > 15 s�2.t/D 0.2C 0.1 cos.0.3�t/, t > 12 s�3.t/D 0.2C 0.1 sin.0.4�t/, t > 10 s

(60)

0 10 20 30 40 500

0.5

1

0 10 20 30 40 500

0.5

1

0 10 20 30 40 500

0.5

1

ρ 1ρ 2

ρ 3

Time(sec)




0 10 20 30 40 50-1

-0.5

0

0.5

e 1(ra

d/s)

0 10 20 30 40 50

-0.4

-0.2

0

0.2

e 2(ra

d/s)

0 10 20 30 40 50-1

-0.5

0

0.5

e 3(ra

d/s)

Time(sec)

(a) The reconstructed faults (b) The estimated angular velocity error

0 20 40 60 80 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time(sec)

The

per

form

ance

inde

x I p

(c) The performance index Ip (t)

Figure 4. Fault reconstruction by the proposed FDD with time-varying loss of actuator effectiveness faults.



In this case, the previous simulation is repeated for the fault (60). When the AFTC is implementedin the spacecraft attitude control system, under the effect of the incorporated FDD mechanism (12)and (13), the time response of the reconstructed fault, angular velocity estimation error, and theperformance index are as shown in Figure 4. After the occurrence of the time-varying fault (60), theproposed FDD can successfully estimate these faults in 5 s, as shown in Figure 4(a, dashed line).Figure 4(a–c) clearly shows that the proposed observer-based FDD successfully achieves precisefault reconstruction with fairly good performance even for time-varying faults.

Figure 5 shows the attitude control performance under the effect of PID, PFTC, and AFTC con-trol. It is clear that the PID controller does not have the capability to accommodate the actuator fault(60) with inferior attitude pointing accuracy and also a lower slew rate, as shown in Figure 5(a and b,

0 50 100 150 200-0.4

-0.2

0

0.2

0 50 100 150 200-0.5

0

0.5

0 50 100 150 200

-0.4

-0.2

0

0.2

Time(sec)

AFTC PFTC PID

0 50 100 150 200-5

0

5

ω1(

rad/

s)

0 50 100 150 200-4

0

4

ω2(

rad/

s)

0 50 100 150 200-4

0

4

ω3(r

ad/s

)

Time(sec)

AFTC PFTC PID

0 50 100 150 200-5

0

5

0 50 100 150 200-5

0

5

0 50 100 150 200-5

0

5

σ 1σ 2

σ 3u 1(

Nm

)u 2(

Nm

)u 3(

Nm

)

Time(sec)

AFTC PFTC PID

0 50 100 150 200-0.02

0

0.02

χ 1

0 50 100 150 200-4

-2

0

2 x 10-3

χ 2

0 50 100 150 200-3-2-101

x 10-3

χ 3

0 50 100 150 200-2

0

2

4 x 10-4

χ 4

Time(sec)

PID PFTC AFTC

(a) The attitude orientation (b) The angular velocity

(c) The control input (d) The vibration displacement

Figure 5. Attitude control performance with time-varying loss of actuator effectiveness faults.



dash-dot line). However, the application of AFTC and PFTC leads to better results (Figure 5,solid and dash line). Although both fault-tolerant controllers, AFTC and PFTC, managed to com-pensate for the severe time-varying fault in the presence of external disturbances and control inputsaturation, the PFTC control developed in [11] requires 150 s to stabilize the attitude, whereas theproposed AFTC needs just 20 s to perform the attitude maneuver (Figure 5(a), solid and dashedline). This is because the developed AFTC includes an FDD mechanism given by (12) and (13) toprecisely reconstruct the actual actuator faults online and in real time. Consequently, the controllaw (39) can react to the faults immediately when they occur, because the controller is synthesizedby using knowledge of the reconstructed faults. The control input for the AFTC and PFTC is shownin Figure 5(c, solid and dashed line), and we see clearly that both control schemes have the abilityto protect the control effort from actuator input saturation.

Summarizing the two simulated cases, it is noted that the proposed AFTC scheme has bettercontrol performance and also has greater fault tolerance capability than the PID control and PFTC.In addition, extensive simulations were performed using different control parameters, externaldisturbances, and even random loss of actuator effectiveness faults. The results show that the pro-posed controller performed very well and accomplishes the attitude stabilization maneuver despitethese undesired effects in the closed-loop system. High attitude pointing accuracy and high attitudestability were achieved.

5. CONCLUSION

In this paper, an active fault-tolerant attitude stabilization control scheme has been developed forflexible spacecraft. The proposed control approach incorporates an observer-based FDD mechanismand a fault-tolerant controller. The FDD was derived using the previous state estimation information,and it was shown that the proposed FDD achieves precise reconstructions of actuator faults. Thefault-tolerant controller was reconfigured on the basis of the reconstructed fault. Through a rigorousLyapunov analysis, all of the signals of the resulting closed-loop attitude system are guaranteedto be uniformly ultimately bounded in the presence of external disturbances and actuator faults.Numerical implementation of the new controller was also presented to confirm the advantagesand improvements over existing controllers. The case of partial loss of actuator effectiveness hasbeen considered here, and other forms of actuator faults, such as the complete loss of controlpower or stick-type faults, were not considered. The latter case should be investigated by controlreconfiguration together with actuator hardware redundancy and management, and is the subject offuture research.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support from the Scientific Research Foundation of theNational Natural Science Foundation of China (61004072, 61273175), the National Key Basic ResearchProgram of China (2013CB035605), and the Program for New Century Excellent Talents in University(NCET-11-0801).

REFERENCES

1. NASA. Lewis spins out of control. System Failure Case Studies 2007; 1:1–8.2. Patton RJ. Fault-tolerant control systems: the 1997 situation. Proceedings of the IFAC Symposium—SAFEPROCESS,

Hull, UK, 1997; 1033–1054.3. Zhang YM, Jiang J. Bibliographical review on reconfigurable fault-tolerant control systems. Annual Reviews in

Control 2008; 32(2):229–252.4. Benosman M. A survey of some recent results on nonlinear fault tolerant control. Mathematical Problems in

Engineering 2009:1–25.5. Blanke M, Izadi-Zamanabadi R, Bogh SA, Lunau CP. Fault-tolerant control systems—a holistic view. Control

Engineering Practice 1996; 5(5):693–702.6. Tafazoli S, Khorasani K. Nonlinear control and stability analysis of spacecraft attitude recovery. IEEE Transactions

on Aerospace and Electronic Systems 2006; 42(3):825–845.



7. Liang YW, Xu SD, Tsai CL. Study of VSC reliable designs with application to spacecraft attitude stabilization. IEEETransactions on Control Systems Technology 2007; 15(2):332–338.

8. Jin J, Ko S, Ryoo CK. Fault tolerant control for satellites with four reaction wheels. Control Engineering Practice2008; 16(10):1250–1258.

9. Hu QL, Xiao 0B, Friswell MI. Robust fault tolerant control for spacecraft attitude stabilization subject to inputsaturation. IET Control Theory and Applications 2011; 5(2):271–282.

10. Varma S, Kumar KD. Fault tolerant satellite attitude control using solar radiation pressure based on nonlinear adaptivesliding mode. Acta Astronautica 2009; 66(3–4):486–500.

11. Cai WC, Liao XH, Song YD. Indirect robust adaptive fault-tolerant control for attitude tracking of spacecraft. Journalof Guidance Control and Dynamics 2008; 31(5):1456–1463.

12. McNeill SS, Zimmerman DC. Fault detection in open-loop controlled structures. Journal of Guidance Control andDynamics 2007; 30(5):1378–1385.

13. Chen RH, Speyer JL. Sensor and actuator fault reconstruction. Journal of Guidance Control and Dynamics 2004;27(2):186–196.

14. Floquet T, Edwards C, Spurgeon SK. On sliding mode observers for systems with unknown inputs. InternationalJournal of Adaptive Control and Signal Processing 2007; 21(8-9):638–656.

15. Wang D, Lum KY. Adaptive unknown input observer approach for aircraft actuator fault detection and isolation.International Journal of Adaptive Control and Signal Processing 2007; 21(1):31–48.

16. Curry TD, Collins EG. Robust fault detection and isolation using robust l1 estimation. Journal of Guidance Controland Dynamics 2005; 28(6):1131–1139.

17. Hou Q, Cheng YH, Lu NY, Jiang B. Study on FDD and FTC of satellite attitude control system based on theeffectiveness factor. 2nd International Symposium on Systems and Control in Aerospace and Astronautics, IEEE:Shenzhen, China, 2008; 1096–1101.

18. Henry D. Fault diagnosis of microscope satellite thrusters using H-infinity/H-filters. Journal of Guidance Controland Dynamics 2008; 31(3):699–711.

19. Williamson WR, Speyer JL, Dang VT, Sharp J. Fault detection and isolation for deep space satellites. Journal ofGuidance Control and Dynamics 2009; 32(5):1570–1584.

20. Chen W, Saif M. Observer-based fault diagnosis of satellite systems subject to time-varying thruster faults. Journalof Dynamic Systems Measurement and Control-Transactions of the ASME 2007; 129(3):352–356.

21. Patton RJ, Uppal FJ, Simani S, Polle B. Robust FDI applied to thruster faults of a satellite system. ControlEngineering Practice 2010; 18(9):1093–1109.

22. Tsiotras P. Stabilization and optimality results for the attitude control problem. Journal of Guidance Control andDynamics 1996; 19(4):772–779.

23. Di Gennaro S. Output stabilization of flexible spacecraft with active vibration suppression. IEEE Transactions onAerospace and Electronic Systems 2003; 39(3):747–759.

24. Yang CD, Sun YP. Mixed H2/H -infinity state-feedback design for microsatellite attitude control. ControlEngineering Practice 2002; 10(9):951–970.

25. Khalil HK. Nonlinear Systems. Prentice Hall: Upper Saddle River, 2002.26. Boskovic JD, Bergstrom SE, Mehra RK. Robust integrated flight control-design under failures, damage, and

state-dependent disturbances. Journal of Guidance Control and Dynamics 2005; 28(5):902–917.


active fault-tolerant attitude control for ﬂexible spacecraft with …michael.friswell.com ›...

Documents