This article was downloaded by: [York University Libraries]On: 11 August 2014, At: 09:56Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Click for updates

International Journal of Systems SciencePublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tsys20

Velocity-free fault-tolerant control allocation forflexible spacecraft with redundant thrustersQinglei Hu a b , Bo Li a , Danwei Wang b & Eng Kee Poh ca Department of Control Science and Engineering , Harbin Institute of Technology , Harbin ,150001 , Chinab School of Electrical and Electronic Engineering , Nanyang Technological University ,Singaporec DSO National Laboratories , SingaporePublished online: 06 Jun 2013.

To cite this article: Qinglei Hu , Bo Li , Danwei Wang & Eng Kee Poh (2013): Velocity-free fault-tolerant control allocation forflexible spacecraft with redundant thrusters, International Journal of Systems Science, DOI: 10.1080/00207721.2013.803634

To link to this article: http://dx.doi.org/10.1080/00207721.2013.803634

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

International Journal of Systems Science, 2013http://dx.doi.org/10.1080/00207721.2013.803634

Velocity-free fault-tolerant control allocation for flexible spacecraft with redundant thrusters

Qinglei Hua,b,∗, Bo Lia, Danwei Wangb and Eng Kee Pohc

aDepartment of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China; bSchool of Electricaland Electronic Engineering, Nanyang Technological University, Singapore; cDSO National Laboratories, Singapore

(Received 4 September 2012; final version received 21 April 2013)

This paper proposes a novel velocity-free nonlinear proportional-integral (PI) control allocation scheme for fault-tolerantattitude control of flexible spacecraft under thruster redundancy. More specifically, the nonlinear PI controller for attitudestabilisation without using body angular velocity measurements is first designed as a virtual control of the control allocatorto produce the three-axis moments, and can ultimately guarantee uniform boundedness of the closed-loop system in thepresence of external disturbances and possible faults. The associated stability proof is constructive and accomplished bythe development of passivity filter formulations together with the choice of a Lyapunov function containing mixed termsinvolving the various states. Then, a robust least-squares-based control allocation is employed to deal with the problemof distributing the three-axis moments over the available thrusters under redundancy, in which the focus of this controlallocation is to find the optimal control vector of the actuator by minimising the worst-case residual, under the condition ofthruster faults and control constraints like saturation. Simulation results using the orbiting flexible spacecraft model showgood performance under external disturbances and even in different thruster fault scenarios, which validates the effectivenessand feasibility of the proposed scheme.

Keywords: velocity-free fault-tolerant control; control allocation; flexible spacecraft; thruster

1. Introduction

Attitude control of a spacecraft in space is in general aprocess of reorienting it to a desired attitude or orientation,and plays an important role in achieving spacecraft oper-ational services, such as remote sensing, communicationand a variety of space-related research. This control prob-lem has been studied extensively in the existing literature.In Tsiotras (1996) and Tsiotras (1998), the authors showthat there exists a simple linear asymptotically stabilisingproportional-derivative (PD) control law for the attitudemotion of a rigid body. However, the effects of externaldisturbances on the attitude regulation are not investigated.To eliminate the offsets arising out of disturbances, theextended form of the PD control law is presented in Sub-barao and Akella (2004) by including a nonlinear termsuch that the global asymptotic stability is possibly guar-anteed. In the sense of optimal control, an optimal attitudecontrol law for the attitude control problem is presentedin Sharma and Tewari (2004) and Xin and Pan (2010).Lyapunov analysis based adaptive attitude tracking controlschemes are also presented to compensate for the unknownrigid body inertia matrix (Junkins, Akella, and Robinett1997; Schaub, Akella, and Junkins 2001; Xiao, Hu, andMa 2011). To cope with both model uncertainties and ex-ternal disturbances, variable structure control (VSC) wasemployed with different attitude representations to solve

∗Corresponding author. Email: huqinglei@hit.edu.cn

the robust attitude control problem (Hu and Ma 2005; Liu,Xia, Zhu, and Basin 2012; Zhao and Zou 2012). How-ever, actually, one problem is that, technically speaking,the results considered in these literatures solve the atti-tude stabilisation by full-state feedback that utilises bothattitude and angular velocity measurements, and this as-sumption of availability of the angular velocity measure-ment for use within the feedback signal is not always sat-isfied, because of either cost limitations or implementationconsiderations. With this in mind, several important re-cent results have utilised passivity-based attitude-controlstrategies to guarantee global asymptotic stability when thefeedback control signal is desired to be angular velocity-free (Akella 2001; Subbarao 2004; Akella, Valdivia, andKotamraju 2005; Zou and Kumar 2010). In Subbarao (2004)and Akella et al. (2005), a passivity approach was used todevelop an asymptotically stabilising set point controllerthat eliminated velocity measurements via the filtering ofthe unit quaternion. In Akella (2001), a modified controllaw was developed to the tracking problem using the mod-ified Rodrigues parameters, in which exact knowledge ofthe spacecraft inertia was required. Recently, this constraintwas overcome in Zou and Kumar (2010) via an adaptiveneural control law using only attitude measurements, whichcompensated for inertia-related uncertainty and externaldisturbances.

C© 2013 Taylor & Francis

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

2 Q. Hu et al.

Besides the obvious interest in ensuring closed-loopstability using only attitude information, there is a prac-tical motivation in introducing control saturations nestedinto the loop. It is well known the actuators are not ableto provide any requested joint torque due to the availableactuator control power limitation. Recognising these dif-ficulties, several solutions that take into account actuatorconstraints have been extensively studied (Boskovic, Li,and Mehra 2004; Wallsgrove and Akella 2005; Hu 2009;Ali, Radice, and Kim 2010; De 2010; Zhu, Xia, and Fu2011). Specifically, in Boskovic et al. (2004) and Zhu, Xia,and Fu (2011), using the VSC technique, the authors formu-lated robust sliding-mode controllers for global asymptoticstabilisation of spacecraft in the presence of control inputsaturation and disturbances. The smooth attitude stabilis-ing control containing hyperbolic tangent functions wasalso discussed in Hu (2009). Ali et al. (2010) presenteda method to design a bounded control for spacecraft atti-tude manoeuvring with backstepping control. Wallsgrove(Wallsgrove et al. 2005) investigated an adaptive attitudetracking control for a rigid spacecraft with linearly param-eterised disturbances.

To ensure fault tolerance in control performance andto increase onboard autonomy in fault management, thedesign of fault tolerant control (FTC) without ground inter-vention has attracted increasing attention in the aerospaceindustry and academia. Jin, Yang, and Peng (2012) pro-posed a robust adaptive tracking control of distributed de-lay systems with actuator and communication failures. Theproblem of automated attitude recovery for rigid and flex-ible spacecraft application was considered in Tafazoli andKhorasani (2006). Feedback linearisation control is used togenerate a control error signal based on quaternion addi-tion. Another fault-tolerant attitude control design was de-veloped for compensation of loss-of-effectiveness failuresof momentum wheels (Jin, Ko, and Ryoo 2008). Althoughthis approach is applicable to perform attitude tracking, thetechnique was developed in the absence of external distur-bances and uncertain inertia parameters. However, Li andYang (2012) presented a direct adaptive control schemeto accommodate actuator failures for multi-input, single-output linear systems with parameter uncertainties, and arobust adaptive fault-tolerant compensation control prob-lem for linear systems with parameter uncertainty, distur-bances and actuator faults was investigated in Li and Yang(2012). During the recent years, the concept of sliding modecontrol (SMC) has gained much attention in the field ofFTC due to its inherent robustness against some matcheduncertainties/disturbances; several researchers have appliedSMC to design FTC for spacecraft with component failures.In Liang, Xu, and Tsai (2007), passive and active SMClaws were developed for attitude stabilisation of space-craft with the actuator outage fault accommodated. In thework of Cai, Liao, and Song (2008), a control augmen-tation approach is proposed for a rigid spacecraft with

thruster failures. Another FTC strategy was developed tofollow the desired attitude of a flexible spacecraft (Xiao,Hu, and Zhang 2011). This scheme accommodates par-tial loss of actuator effectiveness without angular veloc-ity measurements. In a more recent work of Godard andKumar (2011), the authors focused on rapid reorientationof a rigid spacecraft under external disturbances and modeluncertainties, and uniformly boundedness was achievedin absence of control along either the roll or the yawaxis.

Control allocation (CA) is able to deal with distributingthe total control demand among the individual actuatorswhile accounting for their constraints under redundant ac-tuators (Harkegard 2003). Loosely speaking, it consists ofusing possibly desired control laws that specify only thetotal control effort that has to be produced to compensatethe system and, separately, the one of suitably distributingthe desired total control command over the available actu-ators, in which the actuator constraints like saturation andfault/failure can be taken into account explicitly. That is tosay, if one actuator saturates and fails to produce its nom-inal control effect, then another actuator may be used tomake up the difference. This way, the control capabilitiesof the actuator suite are fully exploited before the closed-loop performance is degraded. The general approaches ofCA have been deeply investigated in the last decade, andinclude: daisy chaining (Buffington and Enns 1996), lin-ear or nonlinear programming based on optimisation algo-rithms (Durham 1993), direct allocation (Tang, Zhang, andZhang 2011), dynamic CA (Harkegard 2004; Shi, Li, andHu 2012), adaptive CA (Alwi and Edwards 2008; Casavolaand Garone 2010; Fu, Cheng, Jiang, and Yang 2011), etc.Most or part of previous works studied linear CA by pro-gramming algorithms, which can be iteratively conductedto minimise the error between the commands produced bythe virtual control law and the moments produced by practi-cal actuator combinations. Recently, a robust least-squares-based CA (RLSCA) scheme applied for the flight controlsystem with an uncertain control effectiveness matrix isinvestigated in Cui and Yang (2011).

In this paper, an attempt is made to provide a simpleand robust attitude control strategy for a flexible spacecraftwithout angular velocity measurements in the presence ofexternal disturbances, thruster faults and even control sat-uration constraints. This proposed control law is a novelcombination of the nonlinear proportional integral (PI) withRLSCA. For the former, it can achieve the desired rotationmanoeuvring using only attitude information and be ro-bust to the external disturbances. For the latter, it can dealwith the problem of distributing the total former commandinto the individual thrusters properly under fault/failure, inwhich the focus of this CA is to find the optimal controlvector of the thruster by minimising the worst-case resid-ual, under control constraints like saturation and possibleactuator fault/failure. A key feature of the designed

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

International Journal of Systems Science 3

controller is that it has a simple structure without directlyinvolving the inertia matrix or its estimation, thus simpli-fying the design process and online computation signifi-cantly such that it becomes user/designer-friendly, whichis of great interest for the aerospace industry for real-timeimplementation, especially when onboard space and com-puting power are limited, for instance. The benefits of theproposed control method will be analytically authenticatedand also validated via a simulation study.

This paper is organised as follows. The next sectionstates spacecraft modelling and control problems. Attitudecontrol laws are derived in Section 3. Next, the results ofnumerical simulations demonstrate various features of theproposed control law. Finally, the paper is completed withsome concluding comments.

2. Spacecraft modelling and problem formulation

2.1 Attitude dynamics

Attitude kinematics represented by four unitary quaternionsare given as follows (Sidi 1997):

[q0

q

]= 1/2QT (q0, q)ω, (1)

where q0 = cos(�/2) and q = [q1 q2 q3]T = l sin(�/2)

are subject to the constraint q20 + qT q = 1. Here� denotes

the rotation angle about the Euler axis, which is determinedby the unitary vector l, Q(q0, q) = [−q, q0I − S(q)] withS(q) denoting a skew-symmetric matrix which is given by

S(q) =⎡⎣ 0 −q3 q2

q3 0 −q1

−q2 q1 0

⎤⎦ (2)

and ω is the angular velocity of the unformed spacecraftin the body fixed frame. This description of the orienta-tion avoids the geometric singularities inherent with threeparameter descriptions (Euler angles, etc.).

2.2 Flexible spacecraft dynamics

Under the assumption of small elastic displacements, thedynamic equations of a spacecraft with flexible appendage,

Table 1. Actuator fault model.

Fault model ui ei

Normal 0 0Outage 0 1Loss of effectiveness 0 0 < ei < 1Stuck ui 1

actuated by gas jets (thrusters), can be found in Sidi (1997),and references therein, and are given by

J ω + δT η = −S(ω)(Jω + δT η) +Du(t) + d(t), (3a)

η + Cη +Kη + δω = 0, (3b)

where J ∈ R3×3 is the symmetric inertia matrix of thespacecraft, δ ∈ R3×N is the coupling matrix between theelastic structures and rigid body, η ∈ RN is the modal coor-dinate vector, u ∈ Rm denotes the force vector produced bythe m thrusters in which the constraint m ≥ 3 is required,D ∈ R3×l is the thruster distribution matrix and d(t) is theexternal disturbance (environmental disturbances, solar ra-diation, etc.);C ∈ RN×N andK ∈ RN×N denote the damp-ing and stiffness matrices, respectively, and are defined as

C = diag{

2ξi�12i

}and K = diag{�i}, i = 1, 2, . . . , N,

(4)

where N is the number of elastic modes considered, �12i is

the natural frequency and ξi is the corresponding dampingratio.

Remark 1: The above dynamics of the spacecraft are ob-tained by computing the kinetic and potential energies andthen applying the Lagrange equations with the assumptionof small elastic displacement approximation. This simpli-fied equation is easy to manipulate and more suitable forcontrol law design. Of course, the exact model, which istime-varying and more difficult to handle, can be used in-stead for verifying the effectiveness of the control law de-rived on the basis of the simplified model, to accomplish therational manoeuvre and vibration reduction for the closed-loop simulation later.

Let us now consider that the thruster faults occur, es-pecially the situation in which the thrusters lose the to-tal control power (or outage), the partial control effective-ness and, possibly, there exists a stuck failure. Generallyspeaking, these faults can occur due to various reasons, andhere, without going into the details of the possible natureof thruster faults, the following multimodel fault model isadopted for this study:

ui = uci + ei(ui − uci), (5)

Figure 1. Block diagram for spacecraft attitude control with CA.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

4 Q. Hu et al.

Table 2. Main parameters of one flexible spacecraft.

Mission Imaging the earth

Mass (kg) 874.56Inertia moments (kg m2):

Principal moments of inertia J1 = 1543.9, J2 = 471.6, J3 = 1713.5Products of inertia J12 = −2.3, J23 = −35, J13 = −2.8

Orbit:Type CircularAttitude (km) 500The inclination (deg) 97.4The right ascension of the ascending node 10.30 am

AttitudeAttitude control type Three-axis control by six thrustersInitial attitude quaternion Q0 = [q0, qT]T = [0.1736, −0.5264, −0.2632, 0.7896]Initial angular velocity (rad/s) w0 = [0, 0, 0]

Thruster:The maximum force (N) 10The assembling location (m) l1 = 0.8, l2 = 0.8, l3 = 0.7; l4 = 0.7, l5 = 0.7, l6 = 0.7

where ei is the failure indicator for the ith thruster, ui repre-sents uncertain stuck failures for the ith thruster, uci is theith thruster’s desired control commanded by the controller,and ui is the control that can actually be applied by the iththruster for i = 1, 2, . . . , m.

In view of the actuator fault model in Equation (4), itcan be noted that the presented fault model can representoutage, loss of effectiveness and stuck faults, summarisedin Table 1.

Based on the above analysis, for the sake of convenienceof description, the following uniform actuator fault modelis established for all possible faulty modes:

u = uc + E(u− uc), (6)

Figure 2. Distribution schematics of the six thrusters.

where uc = [uc1, uc2, . . . , ucm]T ∈ Rm, u = [u1, u2, . . . ,

um]T ∈ Rm, u = [u1, u2, . . . , um]T ∈ Rm and E = diag

(e1, e2, . . . , em) ∈ Rm×m. Hence, the spacecraft dynamicequation in Equation (3a), incorporating possible thrusterfaults in Equation (6), can be rewritten as

J ω + δT η = −S(ω)(Jω + δT η) +Duc(t)

+DE (u− uc(t)) + d(t). (7)

Remark 2: In the high-performance spacecraft dynamicsystems, the total number of thrusters used usually is greaterthan the number of states to be controlled. That is to say, thiscontrol redundancy generally exists to achieve optimalitywith respect to control effort. In this case of redundantactuation, it is still possible to track closely the desiredstates, even if some of the actuators fail, as long as thenumber of active actuators is greater than or equal to thenumber of states to be tracked. In this work, the actuatorswith redundancy (m > 3) are considered for the attitudecontrol system design. While for a given orbiting spacecraft,the distribution matrix D is available and can be made offull-row rank by properly placing the thrusters at certainaxes and directions on the spacecraft.

For the synthesis of control system design, the followingreasonable assumptions are made:

Table 3. Controller parameters used for numerical analysis.

Control schemes Control gains

Proposed fault kx = 39.8, ki = 1.0, kp = 260tolerant controllerRLSCA ζ = 0.2, um = 50PD controller KP = 300, KD = 600

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

International Journal of Systems Science 5

Figure 3. Time responses of the quaternions under securitymode.

Assumption 1: The external disturbances, d(t), and theuncertain stuck failure u in Equation (6) are assumed tobe unknown but bounded. Thus, it is reasonable to assumethat there always exist constants ddist and ustk, respectively,such that

‖d(t)‖ ≤ ddist, (8a)

‖u‖ ≤ ustk. (8b)

Figure 4. Time responses of angular velocity under securitymode.

Assumption 2: Due to physical limitations on thethruster considered, the control action generated islimited by the saturation value, i.e. uc(t) ∈ :={uc ∈ Rm |ucmin ≤ uci ≤ ucmax, i = 1, 2, . . . , m }. Forsimplicity, we assume that actuator output torques havethe same constraint value um, i.e.

uc(t) ∈ := {uc ∈ Rm ||uci | ≤ um, i = 1, 2, . . . , m }.(9)

2.3 Control objective

Consider the spacecraft attitude system given by Equations(1) and (2) under the constraints from Equations (4) and(5). Design a control input law such that, for all physi-cally realisable initial conditions, the states of the closed-loop system can be stabilised, which can be expressed aslimt→∞ q = limt→∞ ω = 0.

3. Control system design

CA is useful for control of overactuated spacecraft systems,and deals with suitably distributing the desired total controlcommand among the individual actuators. Using CA, theactuator selection task is separated from the regulation taskin the control design. In this sense, it is possible to splitthe control design into the following two steps: (1) design acontrol law specifying which total control effort to be pro-duced (total combined thruster forces, etc.); and (2) designa control allocator that maps the total control demand onto individual actuator settings (commanded thrust forces,etc.). Figure 1 illustrates the configuration of the proposedoverall spacecraft control system.

For the synthesis of control system design, an equivalentrepresentation of Equation (7) can be written as

J ω + δT η =−S(ω)(Jω + δT η) + Buu(t) +DEu+ d(t) ,(10a)

Buu(t) = Bcuc(t), (10b)

whereu(t) is the virtual control input or called the combinedcontrol torque produced by the actuators, Bu is the virtualinput matrix andBc is used to describe the distribution of thephysical actuators with Bc = D (I − E) and representingthe influence of each actuator on the angular accelerationof the spacecraft considered. For the considered spacecraft,the virtual input matrix Bu as the identity Bu = I3×3 willbe defined, for convenience. With such a choice, the virtualinput u(t) represents exactly the total torques produced bythe actuators, and the following virtual equivalent plant canbe given:

J ω + δT η = −S(ω)(Jω + δT η) + u(t) + d(t), (11a)

u(t) = Duc(t) +Duc(t), (11b)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

6 Q. Hu et al.

Figure 5. Time responses of thruster forces under security mode.

with

d(t) = DEu+ d(t) andD = −DE.

3.1 Nonlinear proportional-integral-based virtualfeedback control law design

From a practical point of view, the design of efficient andlow-cost attitude controllers is an important issue which isof great interest for the aerospace industry, for instance. Inthis section, on the one hand, due to either cost limitations(implementation constraints) or velocity sensor failures, anonlinear control law without velocity measurement is con-sidered for the attitude stabilisation problem of the flexiblespacecraft described by Equations (1) and (11a), while, onthe other hand, this nonlinear control law is simple anduser-friendly in that it does not involve a time-consumingdesign procedure and demands little redesigning or repro-gramming during orbiting operation. With these in mind,a PI-like controller for attitude stabilisation without usingbody angular velocity is discussed in the following.

Figure 6. Time response of the first three elastic modes undersecurity mode.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

International Journal of Systems Science 7

Figure 7. Time responses of attitude angles under security mode.

For simplicity of the development, let us first introducethe following variable:

ϕ = η + δω, (12)

which represents the total angular velocity expressed inmodal variables. This will simplify the following controllaw development. In view of Equation (3b), one obtains

ϕ = [−K −C ] [ ηϕ

]+ Cδω. (13)

Accordingly, the following equation can be obtained:

ψ =[

0 1−K −C

]ψ +

[−δCδ

]ω, (14)

where ψ is defined as ψ := [ηT ϕT ]T .Using this expression for the elastic modes,

Equation (11a) can be rewritten as

J0ω = −S(ω)(J0ω + δT ϕ

)+ u(t) + d(t)

+ δT [K C] [ ηϕ

]− δT Cδω, (15)

where J0 = J − δT δ. Note that here we assume that J0

remains positive definite even if there exist uncertaintiesor changes due to onboard payload motion for inertiamatrix J .

Following Subbarao (2004), a filter state χ (t) is definedas

χ(t) = −χ (t) + 2kxq(t) + ki

∫ t

0q(τ )dτ , (16)

where kx > 0 and ki > 0 are the design parameters chosenby the designer.

Accordingly, the nonlinear PI-like feedback control lawis computed by

u(t) = − ki(kx − ki/

2)(ST (q) − q0I

)χ︸ ︷︷ ︸

Nonlinear term

− (kp + 2kx(kx − ki)q0)q︸ ︷︷ ︸

Proprtional term

− ki(kx − ki/

2) (q0I + S(q))∫ t

0q(τ )dτ︸ ︷︷ ︸

Integral term

(17)

where kp > 0 with kp − 12k

2i > 0.

In the following theorem, the control solution to theunderlying flexible spacecraft attitude stabilisation problemis summarised.

Theorem 1: Consider the system defined in Equations (1)and (11)a under the auxiliary filter given in Equation (16).If the designed control law in Equation (17) is applied witha proper selection of the parameters kp, kx and ki , then

(1) all the trajectories of the resulting closed-loop sys-tem are guaranteed to be asymptotically stable un-der the assumption d(t) = 0;

(2) all the trajectories of the resulting closed-loopsystem are guaranteed to be globally uniformlybounded under d(t) = 0.

Proof: For case (1), consider the Lyapunov function can-didate

V = 1

2ωT J0ω + kp

[(q0 − 1)2 + qT q

]+ 1

2χT χ − kiq

T χ + 1

2ψT Pψ , (18)

Table 4.1. Comparison of control performance among three cases under security mode.

Control performance Settling time of q(s) Steady error of q Settling time of ω(s) Steady error of ω

FTC + RLSCA 50 3 × 10−4 50 3 × 10−4

FTC + PI 70 4 × 10−4 70 7 × 10−4

PD + PI 100 4 × 10−3 100 5 × 10−3

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

8 Q. Hu et al.

Figure 8. Time responses of the quaternions under outage failuremode.

where the matrix PT = P > 0 is the solution of

P

[0 1

−K −C]

+[

0 −K1 −C

]PT = −2Q (19)

for any QT = Q > 0.Let an auxiliary variable and matrix be defined as fol-

lows:

x = [ (1 − q0) qT ωT (χ − kiq)T ψT]T, (20a)

R = diag

[kp,

(kp − 1

2k2i

),

1

2J0,

1

2,

1

2P

]. (20b)

By selecting the proper parameters kp, ki and kp = 12ki ,

one has

c1 ‖x‖2 ≤ V = xT Rx ≤ c2(R) ‖x‖2 , (21)

where c1 = λmin(R) and c2 = λmax(R) are the minimumand maximum eigenvalues of matrix R, respectively, andthis shows that V is positive definite.

Now, the time derivative of V taken along trajectoriesof the closed-loop system can be evaluated through labou-rious yet relatively straightforward algebra followed by ap-plication of the control torque Equation (17), and can bestated as

V = ωT J0ω + kp(2(q0 − 1)q0 + 2qT q

)+ χT χ − ki qT χ

− kiqT χ + ψT P ψ = − (χ − kiq)T (χ − kiq)

−ωT δT Cδω − ψTQψ + ωT δT[K−P C+CP ]ψ.

(22)

In addition, if the selection of Q and P satisfies

� =⎡⎣ δT Cδ − δT

[K − P C + CP

]2

−[K − P C + CP

]δ

2 Q

⎤⎦

≥ 0, (23)

then one has

V = − (χ − kiq)T (χ − kiq) − [ωT ψT]�

[ω

ψ

]≤ − (χ − kiq)T (χ − kiq) ≤ 0 (24)

and thus V , ω, χ and ψ are all uniformly bounded. Also,due to V ≥ 0 and V ≤ 0, one has lim

t→∞V (t) = V (∞) for

some finite V (∞) ∈ (0,∞), and further we have

∫ ∞

0‖(χ − kiq)‖2 = V (0) − V (∞) (25)

which implies that (χ − kiq) ∈ L2; then, using Bar-balat’s lemma, we can conclude that (χ − kiq) → 0and also (χ − ki q) → 0 as t → ∞, and then we fur-ther have limt→∞ q = 0 and limt→∞ q0 = 0 whenever(χ − ki q) → 0.

In view of Equation (1), premultiplying both sidesof Equation (1) by the matrix QT (q0, q), it follows thatlimt→∞ ω = 0. Next, Equation (1) can be differentiatedwith respect to time to show the fact that ω ∈ L∞, whichimplies that ω → 0 as t → ∞. This result together withthe designed control law can be used in Equation (10) todemonstrate limt→∞ q = 0 and limt→∞ ψ = 0, leading to

Figure 9. Time responses of angular velocity under outage fail-ure mode.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

International Journal of Systems Science 9

the following results:

limt→∞ q = 0, lim

t→∞ q0 = ±1, limt→∞ω = 0 and

limt→∞ψ = 0, (26)

thereby completing the proof of achieving the stated attitudestabilisation objective.

Case (2) For case (2), i.e. d(t) = 0, considering the sameLyapunov function candidate as defined in Equation (18),it follows by application of the control torque Equation (9)that

V = ωT d(t) − [ωT (χ − kiq)T ψT]�

⎡⎣ ω

(χ − kiq)ψ

⎤⎦ ,(27)

where

� =

⎡⎢⎢⎣ δT Cδ 0 − δT

[K − P C + CP

]2

0 I 0

−[K − P C + CP

]δ

2 0 Q

⎤⎥⎥⎦.

∥∥∥∥1 − q0

q

∥∥∥∥2

≤∥∥∥∥1 − q0

q

∥∥∥∥ , (28a)

‖x‖2 =

∥∥∥∥∥∥∥∥∥∥

(1 − q0)q

ω

(χ − kiq)ψ

∥∥∥∥∥∥∥∥∥∥

2

≤∥∥∥∥1 − β0

β

∥∥∥∥+∥∥∥∥∥∥

ω

(χ − kiq)ψ

∥∥∥∥∥∥2

.

(28b)

In view of Assumption 1, one rewrites Equation (27) asfollows:

V ≤ −k ‖x‖2 + ‖ω‖ sup0≤τ≤t

∥∥d(τ )∥∥+ k

∥∥∥∥1 − β0

β

∥∥∥∥≤ −k ‖x‖2 + ‖x‖

(k + sup

0≤τ≤t

∥∥d(τ )∥∥) (29)

with

k = min{1, λmin(�)}.

Figure 10. Time responses of thruster forces under outage failure mode.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

10 Q. Hu et al.

Figure 11. Time response of the first three elastic modes underoutage failure mode.

Hence, taking θ ∈ (0, 1), one has

V2 ≤−θk ‖x‖2 + ‖x‖(k + sup

0≤τ≤t

∥∥d(τ )∥∥− (1 − θ ) k ‖x‖

).

(30)

Then, for ‖x‖ >

(k+ sup

0≤τ≤t‖d(τ )‖

)(1−θ )k , one obtains

V2 ≤ −θk ‖x‖2 ≤ −θkc2V (t). (31)

Finally, using similar arguments to that in Khalil (2002),one works out for the following bound of x:

‖x‖ ≤ max

{2

√c2

c1e− θκ

2c2t ‖x0‖ ,

√c2

c1

2

(1 − θ ) k

×(k + sup

0≤τ≤t

∥∥d(τ )∥∥)} (32)

and the results follow.

3.2 Control allocation

In view of Equations (1) and (15), it can be clearly seen thatthe developed control in Equation (17) can be utilised asthe virtual control to achieve ‘velocity-free’ attitude sta-bilisation even in the presence of external disturbancesand possible unknown stuck failures, like the analysis inSection 3.1. In what follows, we shall develop a controlallocator for the actuator to distribute the virtual control re-quirements to individual actuator settings in the best possi-

ble manner while accounting for their constraints defined inEquation (9) and possible fault scenarios given in Table 1.

For the synthesis of the control allocator, the followingreasonable assumption is made.

Assumption 3: The uncertainty D due to possible faultscenarios given in Equation (11b) is an unknown butbounded matrix satisfying

‖D‖∞ ≤ ς (33)

for some constant ς . Note that if there is no uncertaintyD due to actuator faults, this kind of CA, like the pseudo-inverse CA, can be easily realised by

uc = D†u(t), (34)

where

D† = DT (DDT )−1.

However, when considering this kind of uncertaintyD, the problem becomes more challenging. Fortunately,this kind of CA with an uncertain control distribution ma-trix is solved by robust least-squares schemes like Ghaouiand Lebret (1997), Ghaoui, Oustry, and Lebret (1998) andCui and Yang (2011,). Then, the considered CA problemcan be stated as:

Given a configuration matrix D with its uncertain termD, and virtual control u(t) designed by Section 3.1, theoptimal actuator control vector uc can be described by

uc = arg min|uci |≤um

max‖D‖≤ς

‖(D +D)uc − u‖ , (35)

subject to the following constraints:

(1) The uncertain term D is an unknown matrix sat-isfying ‖D‖∞ ≤ ς .

(2) The control vector uc is bounded by um.

Figure 12. Time responses of attitude angles under outage failuremode.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

International Journal of Systems Science 11

Table 4.2. Comparison of control performance between two cases under outage failure mode.

Control performance Settling time of q(s) Steady error of q Settling time of ω(s) Steady error of ω

FTC + RLSCA 50 3 × 10−4 50 3 × 10−4

PD + PI 140 5 × 10−3 140 2 × 10−3

To solve this problem, the following derives RLSCA toa second-order cone programming (SOCP) problem. For avariable uc under the boundedness of um, the worst-caseresidual is

γ (uc) = max‖D‖∞≤ς

‖(D +D) uc − u‖ . (36)

Using the triangle inequality

γ (uc) ≤ max‖D‖∞≤ς

(‖Duc − u‖ + ‖Duc‖)

= ‖Duc − u‖ + max‖D‖∞≤ς

‖Duc‖ . (37)

Assume that

D = ς

||uc||εuTc (38)

where

ε =⎧⎨⎩

Duc − u

‖Duc − u‖ , ifDuc = u

any unit norm vector, otherwise(39)

Figure 13. Time responses of the quaternions under stuck failuremode.

Then, in the direction of ε, the worst-case residual is

γ (uc) = ‖Duc − u‖ + ς ‖uc‖ . (40)

The worst-case residual in Equation (34) satisfies thefollowing:

‖Duc − u‖ + ς ‖uc‖ ≤ κ, (41)

where κ is the upper bound of the residual to be minimisedby finding the optimal τ in the interval of (τ , τ ).

Thus, the RLSCA problem can be written as an SOCPproblem:

minuc,μ,κ

κ (42)

subject to ‖Duc − u‖ ≤ κ − μς ‖uc‖ ≤ μ |uci | ≤ um(43)

with the variables τ , μ and κ .

Theorem 2: The optimal solution uc to the RLSCA prob-lem is given by

uc ={(ρI +DTD

)−1DT u, ifρ

= (κ−μ)μς2

μ2+2ς2u2m> 0

D†u else(44)

where ρ > 0, κ andμ are the optimal solutions to the aboveproblem.

Proof: The dual problem of the problem in Equation (43)is

minz1,z2,z3,z4

uT z1 + uTmz3 − uTmz4 (45)

subject to DT z1 + ςz2 = 0‖z1‖ ≤ 1‖z2‖ ≤ 1zT3 ≥ 0

−zT4 ≥ 0

(46)

Both the primal and dual problems are feasible; then,there exists an optimal point for each. According to optimi-sation, the optimal point of the primal problem is equal tothat of the dual problem.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

12 Q. Hu et al.

If κ = μ is at the optimum, then Dτ = u, and

κ = μ = ς ‖τ‖ . (47)

Under this condition, τ = D†u is the optimal solution.It is similar to the pseudo-inverse CA.

If κ > μ is feasible, the primal and the dual optimalobjectives are equal:

‖Dτ − u‖ + ς ‖τ‖ = κ = uT z1 + uTmz3 − uTmz4

= − (Dτ − u)T z1 − [ τT −uTm uTm]⎡⎣−DT z1

z3

z4

⎤⎦ .(48)

The following is derived:

z1 = − Dτ − v

‖Dτ − v‖ , (49a)⎡⎣−DT z1

z3

z4

⎤⎦ = − ς ‖τ‖

‖τ‖2 + 2u2m

⎡⎣ τ

−uTmuTm

⎤⎦ . (49b)

From Equation (49b), the following is obtained:

DT z1 = − ς ‖τ‖‖τ‖2 + 2u2

m

τ. (50)

Substituting Equation (49a) into Equation (50) yields

DT Dτ − u

‖Dτ − u‖ + ς ‖τ‖‖τ‖2 + 2u2

m

τ = 0. (51)

Figure 14. Time responses of angular velocity under stuck fail-ure mode.

Then, the optimal control vector τ is represented as

τ = (ρI +DTD)−1

DT u (52a)

with ρ= (κ−μ)μς2

μ2+2ς2u2m

. Thus, the conclusion is achieved.

4. Simulation and comparison results

To study and verify the effectiveness and performanceof our proposed control scheme, detailed responses arenumerically simulated using the flexible spacecraft atti-tude system Equations (1)–(3) in conjunction with thedeveloped velocity-free FTC law Equation (17) and theoptimal RLSCA solution Equation (44). The orbital pa-rameters of a flexible spacecraft used in the numericalsimulations are given in Table 2. The coupling matrixbetween flexible appendages and rigid dynamics is given

by δ =(−9.4733 −15.5877 0.0052

−0.5331 0.4855 18.01400.5519 4.5503 16.9974

−0.0289 0.0199 6.2378

)kg1/2 m/s2. Although the

derived control law is applicable to spacecraft modes ofany order, for simplicity, only the first four elastic modeshave been taken into account with natural frequencies�1 =0.7400,�2 = 0.7500,�3 = 0.7600 and�4 = 1.1600 rad/sand damping ξ1 = 0.004, ξ2 = 0.005, ξ3 = 0.0064 andξ4 = 0.0085, respectively. External disturbances from grav-itational perturbation, solar radiation pressure, electromag-netic force, atmospheric drag, etc., are also considered, andthey are assumed as

d(t) = 10−3 ×⎡⎣ 3 cos(10ω0t) + 4 sin(3ω0t) − 1.0

−1.5 sin(2ω0t) − 3 cos(5ω0t) + 1.52 sin(10ω0t) − 1.5 cos(4ω0t) + 1.0

⎤⎦

× N m, ω0 = 0.01 rad/s . (52b)

In order to accomplish attitude manoeuvres, sixthrusters are mounted. Figure 2 shows the schematic con-figuration of those six thrusters.

As a result, the distribution matrix is calculated as

D =⎡⎣ l1 −l2 0 0 0 0

0 0 l3 −l4 0 00 0 0 0 l5 −l6

⎤⎦ . (53)

Remark 3: Suppose that gas jets (thrusters) produce on–off control actions, while the control signals commandedby the designed controller in Equations (17) and (44) areof continuous type. Thus, the control signals need to beimplemented in conjunction with the on–off actuators. Forthe discrete-type actuators, continuous signals can be con-verted into equivalent discrete signals by pulse-width pulse-frequency (PWPF) modulation (Alwi and Edwards 2008).The key idea of a PWPF modulator is to produce a pulse

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

International Journal of Systems Science 13

Figure 15. Time responses of thruster forces under stuck failure mode.

command sequence to the thruster by adjusting the pulsewidth and pulse frequency. In its linear range, the averagetorque produced equals the demanded torque input. In thiswork, we do not go into the details of the characteristics andoperation principle of PWPF modulators [refer to Alwi andEdwards (2008) for more details]. Therefore, in this work,the PWPF modulation is implemented such that it can beapplicable in practice. Furthermore, simulations have beenrendered more realistic by considering a thruster limit, andit is assumed that the maximum value of the control forceof the thruster (gas jet) is 10 N.

The control parameters for the proposed velocity-freeFTC with the optimal RLSCA scheme, and the widely usedPD controller with the traditional pseudo-inverse CA aregiven in Table 3. For the purpose of comparison, the controlperformance is evaluated by using the following three cases:

Case(1): Security mode. In this case, the flexible space-craft attitude stabilisation is actuator fault free, i.e. all thethrusters T1–T6 are working well.

Case(2): Outage failure mode. Some thrusters undergopartially loss of effectiveness and even outage faults (i.e.the thruster totally fails). The detail fault scenario for each

thruster is assumed as: T1 and T4 are working well, butwhen the system starts to work, there occur faults for T3and T6 with the faults 0.15 sin(t) + 0.4 and 0.07 sin(0.2t) +0.4; in addition, after 10 s, T3 and T6 lose 50% and70% effectiveness, respectively; finally, T4 outages in20 s.

Case(3): Stuck failure mode. Some thrusters lose effec-tiveness partially and even stuck in some special value.Fault mode II is assumed as: T1 and T4 are working well;when the system starts to work, there occur the time-varyingfaults for T3 and T6 with the faults 0.15 sin(t) + 0.4 and0.07 sin(0.2t) + 0.4; then, after 10 s, T3 and T6 lose 50%and 30% effectiveness, respectively; then, T1 gets stuck in–5 N and T4 in 2 N in 20 s simultaneously.

4.1 Response of security mode

In this case, the proposed control scheme FTC + RLSCAis first applied to the flexible spacecraft attitude stabilisa-tion. Also, the FTC + PI and PD + PI schemes are studiedfor the purpose of comparison. Figures 3 and 4 show thetime responses of the attitude quaternions and angular ve-locity under security mode. As we can see, the FTC +

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

14 Q. Hu et al.

Figure 16. Time response of the first three elastic modes under stuck failure mode.

RLSCA (solid line) achieves the best performance of thethree schemes The spacecraft attitude system reaches thecommanded target smoothly with a setting time less than50 s with a high accuracy of 10−4 in steady-state error. TheFTC + PI (dot–dashed line), achieving a similar responseto FTC + RLSCA, can be improved a lot in comparisonwith the PD + PI (dotted line) scheme. However, a longersetting time more than 70 s and a lower accuracy of 10−3 areobserved. Some slight oscillations also result which is dueto the fact that some control property such as the optimaldistribution of the demanded torques is not considered inthe CA scheme. Moreover, the PD + PI scheme leads to alongest setting time of about 100 s and serious oscillationsand steady-state error, because the robust property is not ex-plicitly considered in the PD + PI control design. Despitethe fact that there may exist further room for improvementwith fine-tuning of the control parameter sets, there is notmuch improvement in the attitude responses.The time re-sponse of thruster forces is shown in Figure 5. Note that ifthe restriction on the thruster control force magnitude 10 Nis considered explicitly, then the FTC + RLSCA producesless pulse frequency of thruster’s force, due to the proposedoptimal RLSCA method, mainly in comparison with theFTC + PI scheme. The PD + PI produces the largest fre-quency of the force pulse, which implies the energy optimalprinciple is working in the RLSCA law.

The time response of the first three elastic modes undersecurity mode is shown in Figure 6. In addition, from thepoint view of practice, the spacecraft attitude responsesusing the Euler angle

[ψ θ φ

]T(φ, θ and ψ denoting the

roll, pitch and yaw angles, respectively) are presented in

Figure 7 to clearly describe the attitude rotational process.It is clearly seen that fairly good control performance isachieved under FTC + RLSCA (solid line).

More specifically, the control performance comparisonof these schemes in security mode is given in Table 4.1in terms of the settling time and steady-state error whichfurther demonstrates the improved performance using theproposed control strategy under the multiple constraintsmentioned previously.

4.2 Response of outage failure mode

In this section, in order to demonstrate the fine performanceof the proposed control scheme, especially when an abrupt

Figure 17. Time responses of attitude angles under stuck failuremode.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

International Journal of Systems Science 15

Table 4.3. Comparison of control performance between two cases under stuck failure mode.

Control performance Settling time of q(s) Steady error of q Settling time of ω(s) Steady error of ω

FTC + RLSCA 110 5 × 10−4 110 6 × 10−4

PD + PI >>150 – >>150 –

thruster stuck failure, called outage, occurred in the fourththruster at the time instant t = 20 s , and the other failuresdefined as case (2) above. Figures 8 and 9 show the timeresponses of the attitude quaternions and angular velocityunder outage failure mode. It is clear that the attitudes areachieved to be stable before 50 s with a high accuracyof 10−4 in steady-state error when the proposed controlstrategies FTC + RLSCA (solid line) are used, while thePD + PI (dotted line) achieves a worse performance witha longer setting time of about 150 s with a lower accuracyof 10−3; even some slight oscillations exist.

Time responses of thruster forces under outage failuremode are shown in Figure 10. Figure 11 reveals the firstthree elastic modes and we can see the FTC + RLSCAscheme making a better sense. In addition, the spacecraftattitude responses using Euler angles are presented in Fig-ure 12 to clearly describe the attitude rotational process. Itis clearly seen that fairly better control performance witha shorter and more smooth trajectory is achieved underFTC + RLSCA.

Above all, as shown in Table 4.2, less setting time andhigher accuracy and a better transient process are still ob-tained for the proposed FTC + RLSCA under outage fail-ure mode, whereas for the PD + PI scheme, it shows adegradation of performance with a higher number of oscil-lations after a fault occurs.

4.3 Response of stuck failure mode

In this section, we assess that T1 gets stuck in −5 N and T4gets stuck in 2 N simultaneously at the time instant t = 20sto demonstrate the better performances achieved by theproposed strategies FTC + RLSCA (solid line) using thesame initial conditions and simulation parameters, andthe other failures, as defined in case (3) above. The effectof the stuck failure can be viewed as an additional constantdisturbance. Figures 13–14 present the time responses ofthe attitude quaternions and angular velocity under stuckfailure mode. It is seen that fairly better control perfor-mances are obtained under this severe scenario with twothrusters stuck especially meanwhile even though longersetting time (about 110 s) is required for the proposed con-trol strategies FTC + RLSCA (solid line). However, forthe PD + PI scheme (dotted line), the system is unstable inthe simulation time slot at least. Time responses of thrusterforces and the first three elastic modes under stuck failuremode are shown in Figures 15–16. In addition, the space-craft attitude responses using Euler angles are presented in

Figure 17 to clearly describe the attitude rotational process.It is clearly seen that fairly better control performanceswith a shorter and smoother trajectory are achieved un-der FTC + RLSCA. More specifically, the control per-formance comparison of these schemes under stuck failuremode is shown in Table 4.3, which further demonstrates theimproved performance using the proposed control strategyeven under actuator stuck failure mode.

Summarising all the cases (security mode and failuremode), the proposed control scheme FTC + RLSCA canimprove the normal performance more significantly thanthe PD and PI methods, whether the faults occur or not.It can also be observed that as more severe fault cases areconsidered in the design, the proposed controller can stillguarantee the system to be stable. Extensive simulationswere also done using different control parameters, distur-bance inputs and even combinations of the thrusters. Theseresults show that in the closed-loop system, attitude stabil-isation control and energy saving is accomplished as faras possible in spite of these undesired effects in the sys-tem. Moreover, the flexibility in the choice of velocity-freeFTC controller and RLSCA allocator parameters can beutilised to obtain the desirable performance while meetingthe constraints on the control magnitude and actuator fail-ures. These control approaches provide the theoretical basisfor the practical application of the advanced control theoryto spacecraft attitude control system design.

5. Conclusion

In this paper, a nonlinear velocity-free PI control schemehas been incorporated into a CA framework to solve the at-titude control problem of flexible spacecraft in the presenceof actuator faults/failures, external disturbances and actu-ator saturation, as well. The proposed scheme, which usesa nonlinear PI control law as the virtual feedback controlto converge to the desired attitude position under undesireddisturbances and, separately, employs the RLSCA schemeto suitably distribute the total virtual control effort into theactive actuators, including actuator faults/failures and satu-ration constraints, enables the overall scheme to cope withhigh attitude accuracy. Numerical implementation of thenew controller is presented to confirm the advantages andimprovements over existing controllers. While the influenceof the CA on the stability of the whole flexible spacecraftattitude control system is not considered, this is the subjectof future research.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

16 Q. Hu et al.

AcknowledgementsThe present work was supported by the National Natural Sci-ence Foundation of China (61004072, 61273175) and Programfor New Century Excellent Talents in University (NCET-11-0801), Heilongjiang Province Science Foundation for Youths(QC2012C024), Fundamental Research Funds for the Cen-tral Universities (HIT.NSRIF.2009003, HIT.BRETIII.201212) andDSO National Laboratories of Singapore through a StrategicProject Grant (Project No. DSOCL10004). The authors wouldalso like to thank the reviewers and the editor for their commentsand suggestions that helped to improve the paper significantly.

Notes on contributorsQinglei Hu received his BEng degree in2001 from the Department of Electrical andElectronic Engineering at the ZhengzhouUniversity, China, and his MEng. and PhDdegrees from the Department of ControlScience and Engineering at the Harbin In-stitute of Technology, China, with special-isation in controls, in 2003 and 2006, re-spectively. Since 2003, he has been with the

Department of Control Science and Engineering at the HarbinInstitute of Technology, and he was promoted to professor in2012. He worked as a postdoctoral research fellow in the Schoolof Electrical and Electronic Engineering, Nanyang TechnologicalUniversity, from 2006 to 2007, and from 2008 to 2009, he visitedthe University of Bristol as a Senior Research Fellow supportedby a Royal Society Fellowship. After that, he visited ConcordiaUniversity supported by the Natural Sciences and Engineering Re-search Council of Canada (NSERC) from 2010 to 2011, and againthe Nanyang Technological University, Singapore, in 2012. Hisresearch interests include variable structure control and applica-tions, spacecraft fault tolerant control and applications, spacecraftformation flying, etc. In these areas, he has published more than80 technical papers. He was an Associate Editor for the Journalof the Franklin Institute.

Bo Li received his BEng degree in 2010from the Department of Automation at theHenan University, China, and his MEng. de-gree from the Department of Control Sci-ence and Engineering at the Civil AviationUniversity of China, China, with specialisa-tion in controls, in 2013. Since 2011, he hasbeen studying in the Department of ControlScience and Engineering at the Harbin In-

stitute of Technology (HIT) as an exchange student. Currently,he is pursuing a PhD degree in HIT. His research interests in-clude spacecraft attitude control and control allocation, space-craft fault tolerant control and applications, etc. He has publishedsignificantly on the above subjects with several technical paperswhile enjoying the application of the theory through astronauticconsulting.

Danwei Wang received his PhD and MSEdegrees from the University of Michigan,Ann Arbor, in 1989 and 1984, respectively.He received his BE degree from the SouthChina University of Technology, China, in1982. Since 1989, he has been with theSchool of Electrical and Electronic Engi-neering, Nanyang Technological University(NTU), Singapore. Currently, he is a profes-

sor and deputy director of the Robotics Research Centre, NTU.

He has served as general chairman, technical chairman and at var-ious positions in international conferences, such as the Interna-tional Conference on Control, Automation, Robotics and Vision(ICARCV), IEEE/RSJ Intelligent Robots and Systems (IROS),and IEEE International Conference on Robotics, Automation andMechatronics (RAM). He has served as an associate editor of theConference Editorial Board, IEEE Control Systems Society, from1998 to 2005. He is presently an associate editor of the Interna-tional Journal of Humanoid Robotics and an invited guest editorof various international journals. He is a recipient of Alexan-der von Humboldt Fellowship, Germany. His research interestsinclude robotics, satellite dynamics and control, control theoryand applications. He has published widely in the areas of itera-tive learning control, repetitive control, fault diagnosis and failureprognosis, satellite formation dynamics and control, as well asmanipulator/mobile robot dynamics, path planning and control.

Eng Kee Poh obtained his BEng (electri-cal engineering), First Class Honours, in1986 from the National University of Sin-gapore. Subsequently, he received his MSc(electrical engineering: systems) and PhD(electrical engineering: systems) from theUniversity of Michigan in 1990 and 1993,respectively. He is presently a distinguishedmember of technical staff (DMTS) cum lab-

oratory head in DSO. Concurrently, he is also an adjunct associateprofessor at the Nanyang Technological University and an As-sociate Fellow of AIAA. His current research is in the areas ofcontrol design for unmanned platforms and navigation systemsignal processing.

ReferencesAkella, M.R. (2001), ‘Rigid Body Attitude Tracking Without An-

gular Velocity Feedback’, Systems and Control Letters, 42 (4),321–326.

Akella, M.R., Valdivia, A., and Kotamraju, G.R. (2005), ‘Velocity-Free Attitude Controllers Subject to Actuator Magnitude andRate Saturations’, Journal of Guidance, Control, and Dynam-ics, 28 (5), 659–666.

Ali, I., Radice, G., and Kim, J. (2010), ‘Backstepping ControlDesign with Actuator Torque Bound for Spacecraft AttitudeManeuver’, Journal of Guidance, Control, and Dynamics,33 (1), 254–259.

Alwi, H., and Edwards, C. (2008), ‘Fault Tolerant Control UsingSliding Modes with Online Control Allocation’, Automatica,44, 1859–1866.

Boskovic, J.D., Li, S.M., and Mehra, R.K. (2004), ‘Robust Track-ing Design for Spacecraft Under Control Input Saturation’,Journal of Guidance, Control, and Dynamics, 27 (4), 627–633.

Buffington, J.M., and Enns, D.F. (1996), ‘Lyapunov Stability Anal-ysis of Daisy Chain Control Allocation’, Journal of Guidance,Control, and Dynamics, 19 (6), 1226–1230.

Cai, W.C., Liao, X.H., and Song, Y.D. (2008), ‘Indirect Ro-bust Adaptive Fault-Tolerant Control for Attitude Trackingof Spacecraft’, Journal of Guidance Control and Dynamics,31 (5), 1456–1463.

Casavola, A., and Garone, E. (2010), ‘Fault Tolerant AdaptiveControl Allocation Schemes for Over-Actuated Systems’, In-ternational Journal of Robust and Nonlinear control, 20 (17),1958–1980.

Cui, L., and Yang, Y. (2011), ‘Disturbance Rejection and RobustLeast-Squares Control Allocation in Flight Control System’,

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014

International Journal of Systems Science 17

Journal of Guidance, Control, and Dynamics, 34 (6), 1632–1643.

De, R.A. (2010), ‘Adaptive Spacecraft Attitude Tracking Controlwith Actuator Saturation’, Journal of Guidance, Control, andDynamics, 33 (5), 1692–1695.

Durham, W.C. (1993), ‘Constrained Control Allocation’, Journalof Guidance, Control, and Dynamics, 16 (4), 717–725.

Fu, Y.P., Cheng, Y.H., Jiang, B., Yang, M.-K. (July 25–28, 2011),‘Fault Tolerant Control with On-line Control Allocation forFlexible Satellite Attitude Control System’, in The 2nd Inter-national Conference on Intelligent Control and InformationProcessing, Harbin, China, pp. 42–46.

Ghaoui, L.E., and Lebret, H. (1997), ‘Robust Solutions to LeastSquares Problems with Uncertain Data’, SIAM Journal onMatrix Analysis and Applications, 18 (4), 1035–1064.

Ghaoui, L.E., Oustry, F., and Lebret, H. (1998), ‘Robust Solu-tions to Uncertain Semi-Definite Programs’, SIAM Journalon Optimization, 9 (1), 33–52.

Godard, and Kumar, K.D. (2011), ‘Robust Attitude Stabilizationof Spacecraft Subject to Actuator Failures’, Acta Astronautica,68 (7–8), 1242–1259.

Harkegard, O. (2003), ‘Backstepping and Control Allocation withApplications to Flight Control’, Ph.D. Thesis, Linkoping Uni-versity, Linkoping, Sweden.

Harkegard, O. (2004), ‘Dynamic Control Allocation Using Con-strained Quadratic Programming’, Journal of Guidance, Con-trol, and Dynamics, 27 (6), 1028–1034.

Hu, Q.L. (2009), ‘Robust Adaptive Sliding Mode Attitude Ma-neuvering and Vibration Damping of Three-Axis-StabilizedFlexible Spacecraft with Actuator Saturation Limits’, Nonlin-ear Dynamics, 55 (4), 301–321.

Hu, Q.L., and Ma, G.F. (2005), ‘Variable Structure Control andActive Vibration Suppression of Flexible Spacecraft Dur-ing Attitude Maneuver’, Aerospace Science and Technology,9 (4), 307–317.

Jin, J., Ko, S., and Ryoo, C.K. (2008), ‘Fault Tolerant Control forSatellites with Four Reaction Wheels’, Control EngineeringPractice, 16 (10), 1250–1258.

Jin, X.Z., Yang, G.H., and Peng, L. (2012), ‘Robust AdaptiveTracking Control of Distributed Delay Systems with Actuatorand Communication Failures’, Asian Journal of Control, 14(5), 1282–1298.

Junkins, J.L., Akella, M.R., and Robinett, R.D. (1997), ‘Nonlin-ear Adaptive Control of Spacecraft Maneuvers’, Journal ofGuidance, Control, and Dynamics, 20 (6), 1104–1110.

Khalil, H. (2002), Nonlinear Systems (3rd ed.), Upper SaddleRiver, NJ: Prentice-Hall, Chapter 4.

Li, X.J., and Yang, G.H. (2012), ‘Adaptive Actuator Failure Ac-commodation for Linear Systems with Parameter Uncertain-ties’, IET Control Theory and Applications, 6 (2), 274–285.

Li, X.J., and Yang, G.H. (2012), ‘Robust Adaptive Fault-TolerantControl for Uncertain Linear Systems with Actuator Failures’,IET Control Theory and Applications, 6 (10), 1544–1551.

Liang, Y.W., Xu, S.D., and Tsai, C.L. (2007), ‘Study of VSC Re-liable Designs with Application to Spacecraft Attitude Stabi-lization’, IEEE Transactions on Control Systems Technology,15 (2), 332–338.

Liu, K.F., Xia, Y.Q., Zhu, Z., and Basin, M.V. (2012), ‘SlidingMode Attitude Tracking of Rigid Spacecraft with Distur-bance’, Journal of the Franklin Institute, 349 (2), 413–440.

Schaub, H., Akella, M.R., and Junkins, J.L. (2001), ‘Adaptive Con-trol of Nonlinear Attitude Motions Realizing Linear Closed

Loop Dynamics’, Journal of Guidance, Control, and Dynam-ics, 24 (1), 95–100.

Sharma, R., and Tewari, A. (2004), ‘Optimal Nonlinear Trackingof Spacecraft Attitude Maneuvers’, IEEE Transactions onControl Systems Technology, 12 (5), 677–682.

Shi, Z., Li, B., Hu, Q., and Ma, G. (2012), ‘Dynamic Control Al-location Based-on Robust Adaptive Backstepping Control forAttitude Maneuver of Spacecraft with Redundant ReactionWheels’, in The 24th Chinese Control and Decision Confer-ence, May 23–25, Taiyuan, China: IEEE, pp. 1354–1359.

Sidi, M.J. (1997), Spacecraft Dynamics and Control, Cambridge:Cambridge University Press.

Subbarao, K. (2004), ‘Nonlinear PID-like Controllers for Rigid-Body Attitude Stabilization’, The Journal of the AstronauticalSciences, 52 (1–2), 61–74.

Subbarao, K., and Akella, M.R. (2004), ‘Differentiator-Free Non-linear Proportional-Integral Controllers for Rigid-Body At-titude Stabilization’, Journal of Guidance, Control, and Dy-namics, 27 (6), 1092–1096.

Tafazoli, S., and Khorasani, K. (2006), ‘Nonlinear Control andStability Analysis of Spacecraft Attitude Recovery’, IEEETransactions on Aerospace and Electronic Systems, 42 (3),825–845.

Tang, S.Y., Zhang, S.J., and Zhang, Y.L. (2011), ‘A ModifiedDirect Allocation Algorithm with Application to RedundantActuators’, Chinese Journal of Aeronautics, 24 (3), 299–308.

Tsiotras, P. (1996), ‘Stabilization and Optimality Results for theAttitude Control Problem’, Journal of Guidance, Control, andDynamics, 19 (4), 772–779.

Tsiotras, P. (1998), ‘Further Passivity Results for the AttitudeControl Problem’, IEEE Transactions on Automatic Control,43 (11), 1597–1600.

Tsiotras, P. (1996), ‘Stabilization and Optimality Results for theAttitude Control Problem’, Journal of Guidance, Control, andDynamics, 19 (4), 772–779.

Wallsgrove, R.J., and Akella, M.R. (2005), ‘Globally Stabiliz-ing Saturated Attitude Control in the Presence of BoundedUnknown Disturbances’, Journal of Guidance, Control, andDynamics, 28 (5), 957–963.

Xiao, B., Hu, Q., and Ma, G. (2011), ‘Adaptive Quaternion-BasedOutput Feedback Control for Flexible Spacecraft AttitudeTracking with Input Constraints’, Proceedings of the Insti-tution of Mechanical Engineers, Part I: Journal of Systemsand Control Engineering, 225 (2), 226–240.

Xiao, B., Hu, Q.L., and Zhang, Y.M. (2011), ‘Fault-Tolerant Atti-tude Control for Flexible Spacecraft Without Angular VelocityMagnitude Measurement’, Journal of Guidance Control andDynamics, 34 (5), 1556–1561.

Xin, M., and Pan, H. (2010), ‘Integrated Nonlinear Optimal Con-trol of Spacecraft in Proximity Operations’, InternationalJournal of Control, 83 (2), 347–363.

Zhao, D.Y., and Zou, T. (2012), ‘A Finite-Time Approach to For-mation Control of Multiple Mobile Robots with TerminalSliding Mode’, International Journal of Systems Science, 43(11), 1998–2014.

Zhu, Z., Xia, Y.Q., and Fu, M.Y. (2011), ‘Attitude Stabilizationof Rigid Spacecraft with Finite-Time Convergence’, Inter-national Journal of Robust and Nonlinear Control, 21 (6),686–702.

Zou, A.M., and Kumar, K.D. (2010), ‘Adaptive Attitude Control ofSpacecraft Without Velocity Measurements using ChebyshevNeural Network’, Acta Astronautica, 66, 769–779.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

9:56

11

Aug

ust 2

014