a reliable fault tolerant attitude control system based on an...

Scientia Iranica D (2013) 20(6), 1999{2014

Sharif University of TechnologyScientia Iranica

Transactions D: Computer Science & Engineering and Electrical Engineeringwww.scientiairanica.com

A reliable fault tolerant attitude control system basedon an adaptive fault detection and diagnosis algorithmtogether with a backstepping fault recovery controller

H. Bolandi, M. Haghparast� and M. AbediDepartment of Electrical Engineering, Iran University of Science and Technology, P.O. Box 1684613114, Tehran, Iran.

Received 5 October 2011; received in revised form 2 October 2012; accepted 19 February 2013

KEYWORDSFault detection;Fault isolation;Fault recovery;Attitude control;Hardware in the looptest.

Abstract. This paper presents a practical solution to achieve a fault tolerant attitudecontrol system capable of Fault Detection and Diagnosis (FDD). The novelty of ourproposed strategy is in the accurate modeling of satellite dynamics by the Takagi-Sugenomethod. Based on this model, an adaptive observer has been utilized to achieve faultdiagnosis in reaction wheels of the Attitude Control System (ACS). For this, the occurredfaults in reaction wheels have been estimated using an adaptive algorithm which providesfault detection and identi�cation abilities. Moreover, in this paper, a recovery algorithmhas been utilized, combined with fault detection and identi�cation algorithm, to provide anadvanced decision support system. In this regard, for undertaking the remedial actions, abackstepping feedback linearization control law has been considered in which the estimatedfault has been utilized. Accordingly, the boundedness of the attitude control error isguaranteed, despite actuators fault. The developed algorithms provide a signi�cant degreeof autonomy to e�ectively handle satellite operation in the presence of ACS faults, withoutthe ground segment intervention. Through extensive simulation results, the designedalgorithms are shown to be robust and accurate. Also, designed algorithms are assessedthrough hardware in the loop test bed to evaluate their functions in an experimentalsituation.c 2013 Sharif University of Technology. All rights reserved.

1. Introduction

The increasing complexity of satellites, exposure tospace radiations and, more importantly, the limitedaccess of satellites cause that fault event in these spacevehicles be inevitable. In this regard, although themeasures, such as selecting highly reliable componentsand conducting quality assurance process, make itpossible to achieve high reliability, but mentionedstrategies only cause delay in fault occurrence [1].History of fault events in di�erent missions shows thats

*. Corresponding author. Tel.: +98 21 77204040;E-mail address: mehran [email protected] (M.Haghparast)

these events have always been challenging, even intoday's modern satellites. A statistical comparisonbased on data achieved from 1584 satellites between1984 and 2008 indicates that 36 percent of faults areattributed to the Attitude Determination and ControlSystem (ADCS) [2]. These events have been resultedin degradation of expected services, loss of vehiclecontrol or, in case of total failures, catastrophic loss ofmission. Accordingly, there is a need to develop faulttolerance tools in a safety critical subsystem, such asADCS, capable of detecting, identifying and isolatingany component fault.

Traditional Fault Detection and Isolation (FDI)mechanisms are based on hardware redundancy inwhich the required reliability is achieved by multiple

2000 H. Bolandi et al./Scientia Iranica, Transactions D: Computer Science & ... 20 (2013) 1999{2014

sensors and actuators accompanied by a voting sys-tem [3]. In these systems, the value that has themajority in outputs is considered as the desired outputvalue; thus prevents the spread of fault in the system.The Above systems, despite having extensive heritagein aerospace systems, are not suitable solutions for ap-plications in which mass, power and cost are restrictive.The rapid growth of digital implementation techniqueshave led to a new philosophy in FDI techniques,which are known as analytical redundancy [4]. Theseapproaches are used as a powerful alternative whichdo not su�er from the restrictions related to massand power. In fact, analytical methods provide thepossibility of replacing the hardware replication witha management of functional or analytical redundancyconstituted by the knowledge of system.

Analytical approaches which have been appliedto the attitude determination and control system arecategorized to model-based and data-based methods [4-7]. In this regard, by using neural networks andavailable data from a pulsed plasma thruster, thiselement has been modeled and the output di�erenceof the modeled element and real element has beenutilized as a measure for fault detection [8]. Also theauthors of [9-11] have used the above method for faultdetection. Note that the capabilities of the mentionedapproach are restricted only to the range of the storeddata, so utilizing this method for a system such as satel-lite, which may experience unpredicted operationalconditions, is risky and may lead to incorrect faultdeclaration. To deal with the above problem, model-based approaches have been proposed. Kalman �lter-based techniques are important model-based methodswhich have several applications in ADCS. In [12], twoKalman �lters have been used for fault detection andisolation which are based on the measured outputs fromgyros, sun sensor and magnetometer. Also, the prob-lems of fault detection and isolation based on ExtendedKalman Filter (EKF) and Unscented Kalman Filter(UKF) have been addressed for nonlinear dynamicsof satellites [13-16]. In [17], application of a bank ofinteracting multiple Kalman �lters for detection anddiagnosis of anticipated reaction wheels failures in ACShas been described and developed. Note that, theabove method imposes high computational loads, sincemost of the main modes (dynamics) of the systemshould be considered in the design process. Also,the modes which are not included cannot be isolated.Kalman �lter-based methods, although provide faultdetection and isolation capabilities, are not robustagainst disturbances and uncertainties in the satellitedynamics. To resolve the mentioned problem, anUnknown Input Observer (UIO) has been used in thethrusters of a satellite named MEX to develop faultdetection and isolation algorithms which are robustagainst disturbances [18,19]. Although these algo-

rithms are considered robust techniques against distur-bances, they have been applied only for linear dynamicsof satellites. The residual generation technique basedon H1 theory which has been applied for Microscopesatellite is also designed for linear dynamics [20]. Incontrast, fault detection and isolation methods basedon sliding mode observers are an important class ofrobust techniques which have been applied for nonlin-ear dynamics of satellites [21-23]. Since, these methodsutilize a �xed structure (related parameters cannot beupdated online), a conservative upper-bound has beenassumed for existing faults and uncertainties in thesatellite dynamics. Another approach to compensatethe e�ects of uncertainties and disturbances are adap-tive observers, the parameters of which, in contrast tosliding mode, can be updated online, and hence theupper bound of faults and uncertainties is estimatedaccurately [24,25]. In this approach, the estimatedfault term is used as a measure for fault detectionand isolation. Adaptive observers applied to ACS,have been designed based on linear dynamics. Thescheme developed in this paper, resolve the problem ofapplication of adaptive observers to nonlinear dynamicsof satellites. To this end, satellite dynamics hasbeen modeled using Takagi-Sugeno approach whichis a new idea in the category of fault diagnosis inACS. This approach provides the satellite dynamicsapproximation based on combination of local linearmodels in di�erent operating points [26]. In this regard,by designing adaptive observer, using the obtainedmodel, the fault occurred in the actuators can beestimated, which in addition to ensuring boundedestimation error, fault detection and identi�cationfeatures are also achieved. On the other hand, forcompensating the fault e�ect and undertaking theappropriate subsequent actions, an active mechanismhas been suggested, which provides recon�gurationpossibility after fault event. To this end, for en-suring system stability before fault occurrence, back-stepping feedback linearization control law has beenapplied. After fault occurrence, controller structure isrecon�gured so that the fault term identi�ed by theadaptive observer is utilized as a compensating factor.Based on the suggested idea, attitude control error re-mains bounded, even after fault occurrence in reactionwheels. Therefore, the designed recovery algorithm,combined with the fault detection and identi�cationalgorithms, provides an advanced decision supportsystem which could expeditiously monitor the systemhealth, and subsequently, appropriate remedial actionsare undertaken to maintain the desired speci�cations.Developed algorithms are assessed through di�erentsimulated scenarios. In the simulation model, sensornoises, uncertainties and di�erent space disturbancesapplied to satellite are modeled. Also, hardware inthe loop facility has been developed to qualify the

H. Bolandi et al./Scientia Iranica, Transactions D: Computer Science & ... 20 (2013) 1999{2014 2001

designed algorithms in the presence of experimentallimitations.

The outline of this paper is as follows.: InSection 2, satellite dynamic model is obtained. InSection 3, satellite dynamics is modeled based onthe Takagi-Sugeno method. Design of fault detectionand identi�cation algorithms are introduced in Section4. Designed control law for the recovery purposesis included in Section 5. Numerical simulations fora number of faulty scenarios in the reaction wheelsare presented in Section 6. In Section 7, hardwarein the loop test results, conducted for the purpose ofevaluation of the developed algorithms, are described.Finally, conclusions are presented in Section 8.

2. Satellite dynamic model

Before designing the fault detection, identi�cationand recovery algorithms, mathematical model of theattitude control system should be obtained accurately.The satellite considered in this paper is a three axisstabilized satellite in which three reaction wheels areused as actuators. To analyze the satellite motion,three sets of coordinate systems are de�ned:

1. Earth-centered inertial frame with its origin atthe center of the earth, its Xi axis aligned withthe vernal equinox, its Zi axis aligned with thegeographic north and its Yi axis selected such as theabove coordinate system becomes right- handed;

2. Body-�xed frame which has its origin at the satel-lite's center of the mass and its axes (Xb, Yb andZb) aligned with the principal axes of the satelliteinertia;

3. Orbital frame which has its origin at the satellite'scenter of the mass and its Zo axis points toward thecenter of the mass of the earth, its Xo axis is in theplane of the orbit, perpendicular to the Zo axis andin the direction of the satellite velocity and its Yoaxis completes a three-axis right-handed orthogonalsystem.

The satellite is modeled as a rigid body having themoments of inertia matrix along the principal axes ofrotation, I = Diag3�3fIx; Iy; Izg. With the above con-siderations, satellite attitude dynamics which describesthe relations between angular velocities and the appliedtorques is obtained as follows [27]:

_~! = I�1��~! � I~! � ~! � Iw~!w � _hW + d� ; (1)

where ~!3�1 is the angular velocity vector of the satel-lite with reference to the inertial coordinate system,~!w3�1 =

�!wx !wy !wz

�T is the angular velocityvector of the reaction wheels with reference to thesatellite body frame, _hW3�1 =

� _hxw _hyw _hzw�T is

the control torque applied to the satellite by thereaction wheels, Iw3�3 = Diag3�3fIwx; Iwy; Iwzg is themoments of inertia matrix of the reaction wheels, d isthe disturbance torque which is applied to the satelliteand I3�3 is the moments of inertia matrix of thesatellite. Eq. (1) could be stated in the body frameas:

_!x = �x!y!z +Iwy!z!wy � Iwz!y!wz

Ix� _hxW

Ix;

_!y = �y!x!z +Iwz!x!wz � Iwx!z!wx

Iy� _hyW

Iy;

_!z = �z!y!z +Iwx!y!wx � Iwy!x!wy

Iz� _hzW

Iz; (2)

where:

�x = (Iy � Iz)=Ix;�y = (Iz � Ix)=Iy;�z = (Ix � Iy)=Iz: (3)

In this paper, detection and identi�cation of faultsoccurred in the reaction wheels have been considered.The main sources and causes of the mentioned faultsinjected to the reaction wheels are [17]:

� Viscous friction variations;� Unexpected changes in the satellite bus voltage;� Unexpected changes in the motor torque values.In Eq. (4), the fault model occurred in the reactionwheels, due to one or more of the above fault sources,have been proposed. As can be seen, the e�ect of thementioned fault sources has been modeled as Uf termin the satellite attitude dynamics.

_~! = I�1��~! � I~! � ~! � Iw~!w � _hW + Uf + d� :

(4)

As mentioned in Section 1, before designing thefault detection and identi�cation algorithms, nonlineardynamics of satellite should be modeled using theTakagi-Sugeno method, which is described in the nextsection.

3. Satellite dynamics modeling based on theTakagi-Sugeno method

The Takagi-Sugeno method is a powerful tool in accu-rate modeling of nonlinear systems. The useful featureof this method is to build a group of local linearmodels to describe the original nonlinear systems. Asa result, the fault diagnosis schemes for linear systems


can be extended to nonlinear systems. In this section,based on the above idea, satellite dynamics is modeledusing local linear dynamics combination. The obtainedmodel is used as a reference model for design of thefault detection and identi�cation algorithms in the nextsection.

Before describing the satellite attitude dynamicsmodeling based on the Takagi-Sugeno method, philoso-phy of this approach is presented for a general nonlinearsystem. To this end, a smooth time invariant systemis considered to be:

_x = f(x; u); y = g(x); (5)

where x 2 Rn is the system state variables, y 2 Rpis the system output and f : Rn � Rm ! Rn andg : Rn ! Rp are nonlinear smooth vector �elds whichsatisfy Lipchitz conditions.

Takagi-Sugeno method is based on obtaining sev-eral local linear models of system and combination ofthem according to the current operating point. Thesystem model in any of these operating points is statedas [26]:

if z1(t) is M i1, z2(t) is Mi2;

and � � � zq(t) is M iq then:_x(t) = Aix(t) +Biu(t) + �i;

y(t) = Cix(t); (6)

where [z1(t); z2(t); � � � ; zq(t)] is the vector of premisevariables, M i1; � � � ;M iq represent fuzzy sets (referto [28]) and Ai, Bi, Ci and �i are constant matri-ces which are obtained by the Jacobean linearizationmethod as [28]:

Ai =@f@x��(xi;ui) ;

Bi =@f@u��(xi;ui) ;

Ci =@g@x��(xi;ui) ; (7)

in which (xi; ui) is the system operating point. Sincethe points, in which the linearization is done, may notbe the system equilibrium points, �i is utilized to dealwith the e�ect of the system modeling uncertainty:�i = f(xi; ui)�Aixi �Biui: (8)

The total Takagi-Sugeno fuzzy system is then writtenas:

_x(t) =lXi=1

hi(z)(Aix(t) +Biu(t) + �i) + �fx;

y(t) =lXi=1

hi(z)(Cix(t) + ci) + �fy; (9)

where �fx and �fy denote respectively the modelinguncertainty and the local linearization uncertainty inthe state and output equations, l is the number of fuzzyrules and hi(z) denotes the weighting coe�cient of anyof the linear models computed as:

hi(z) =�i(z)lPi=1

�i(z); �i(z) =

qYj=1

M ij(z): (10)

In the above equation, the following relation holds forhi(z):

lXi=1

hi(z) = 1; hi(z) > 0: (11)

In nonlinear dynamics modeling of satellite, based onEq. (2) and (7), the Ai, Bi and Ci matrices are obtainedas:

Ai =

26666640 �x!zi� Iwz!wziIxx

�y!zi+Iwz!wziIyy 0

�z!yi� Iwy!wyiIzz �z!xi+ Iwx!wxiIzz�x!yi+

Iwy!wyiIxx

�y!xi� Iwx!wxiIyy0

3777775 ;

Bi = B = I�1 =

241=Ixx 0 00 1=Iyy 00 0 1=Izz

35 ;Ci = C = I3�3: (12)

Therefore, the vector of z(t) is selected as [!x(t) !y(t)!z(t) !wx(t) !wy(t) !wz(t)]. Also �i is considered asEq. (8) for any of the operating points. In this regard,operating point selection for local linearization of satel-lite is done in a way that covers operational areas of thesatellite. So, the satellite dynamics together with themodeling uncertainties can be described as:

_!(t) =lXi=1

hi(!)(Ai!(t) + �i)�B _hw(t) + �fx:(13)

Based on Eqs. (4) and (13), attitude control systemdynamics, after fault occurrence in actuator(s), isobtained as:

_!(t) =lXi=1

hi(!)(Ai!(t) + �i)�B _hw(t)

+BUF (t) + �fx: (14)


Clearly, based on Eq. (12), the output variables arethe same as the system state variables, i.e. the angularvelocity vector of the satellite.

4. Design of the fault detection andidenti�cation algorithms

In this section, design of the fault detection and iden-ti�cation algorithms, based on the obtained Takagi-Sugeno model, is presented. For this, an adaptivefault diagnosis observer is investigated to estimate theangular velocities and accurately reconstruct the faultterms occurred in the reaction wheels. So, using thementioned idea, it is possible to detect the faults due toactuators, identify them and determine their behaviorwith reference to time at ACS. The above featureswhich have been achieved by the suggested approachare very critical to provide the required level of satelliteautonomy. Especially, it is important to note that thementioned methodology, proposes a practical solutionwhich can be implemented onboard.

The suggested adaptive observer constructedbased on the Takagi-Sugeno model is described by:

_~! =lXi=1

hi(Ai~! + �i + LiC(! � ~!))

�B _hw(t) +BÛF (t); (15)where

_Uf (t) is the fault term estimation and Li is the

observer gain. This gain should be selected such thatthe convergence of the state estimation errors to zerois guaranteed. Accordingly, the state estimation errordynamics, using the observer presented in Eq. (15) andthe system dynamics presented in Eq. (14) is obtainedas:

_e! =lXi=1

hi(Ai � LiC)e! +Bef (t); (16)where e!(t) is the state estimation error of (! � ~!)and ef (t) is the actuators fault estimation error of(Uf (t) � ~Uf (t)). Here, it is assumed that the fault isslowly varying. This assumption is due to the fact thatin practical conditions, the fault occurred in actuatorsdoes not have drastic variations [29]. Therefore, it isobtained that:

_ef (t) = � _̂UF (t): (17)Now, we are ready to present our result which is anadaptive diagnostic algorithm for estimating the fault.

Theory. If there exists a positive de�nite matrix �such that:

(Ai� LiC)T� +�(Ai� LiC) < 0; i=1; 2; � � � ; l;(18)

then the observer designed in Eq. (15) and the followingadaptive fault estimation algorithm:

_̂UF (t) = BT�e!; (19)

can realize limt!1 e!(t) = 0 and limt!1 ef (t) = 0.

Proof. Consider the Lyapunov function V (t) =eT!�e! + eTf ef ; its derivative with reference to time is:

_V (t) = _eT!�e! + eT!� _e! + _e

Tf ef + e

Tf _ef

=eT!lXi=1

hi(Ai � LiC)T�e!

+ eTf BT�e! + eT!�

lXi=1

hi(Ai � LiC)e!

+ e!�Bef + _eTf ef + eTf _ef

=lXi=1

hieT! [(Ai � LiC)T� + �(Ai � LiC)]e!

+ eTf BT�e! + eT!�Bef + _e

Tf ef + e

Tf _ef

=lXi=1

hieT! [(Ai � LiC)T� + �(Ai � LiC)]e!

+ eTf BT�e! + eT!�Bef � _̂UTf ef + eTf _̂Uf : (20)

From Eqs. (19) and (20) we obtain:

_V (t) =lXi=1

hieT! [(Ai � LiC)T� + �(Ai � LiC)]e!:(21)

Therefore, according to hi(!) � 0 (Eq. (11)), asu�cient condition for _V (t) � 0 is that Eq. (18) issatis�ed. So, _V (t) � 0 guarantees limt!1 e!(t) =0 and limt!1 ef (t) = 0. As a result, the factlimt!1 ef (t) = 0 makes Ûf (t) converge to Uf (t)without any estimation error.

This completes the proof. �A nice feature of the adaptive method proposed

in this paper is that the estimation of_Uf (t), not only

enables fault detection, but also provides the shape offault which is used for fault accommodation. In anideal condition in which modeling is considered withoutuncertainties,

_Uf (t) term is zero when no faults are

present, and this term changes suddenly when thefaults occurs. However, since the model has beenutilized for attitude control dynamics of the satellitea�ected by the uncertainties and disturbances, these


factors are also estimated by the adaptive observertogether with faults. In other words, the fault estima-tion accompanies errors due to the mentioned factors.To resolve the above problem, a threshold has beenselected, i.e. a fault is declared only if

_Uf (t) exceeds the

selected threshold. Eq. (22) shows the decision makingprocess for the fault declaration:8 Tr Fault occurrence in actuators��Ûf �� Tr No fault occurrence in actuators (22)where Tr is the threshold for fault declaration, and itsvalue is selected based on: the uncertainties due to thelocal linear models (�fx), the disturbances which areapplied to satellite and the accuracy of sensors whichmeasures the angular velocities of satellite. Therefore,based on the designed observer in Eq. (15), it is possibleto detect and identify the faults that occur in theactuators.

5. Design of the recovery algorithm

As mentioned before, design of a fault tolerant attitudecontrol system has been considered in this paper tomaintain high reliability for satellite control systemagainst possible reaction of wheels faults. This con-troller provides a solution to meet the desired stabilityand performance despite the occurrence of faults. So, itis necessary to design a recovery algorithm, in additionto the fault detection and identi�cation algorithms.For this, a control strategy based on the backstep-ping feedback linearization approach is investigatedto ensure the three axis stability of satellite, andthen the compensating control law which utilizes thefault estimation is designed to compensate the faulte�ect. Backstepping method is a recursive scheme,and the idea of it is to design a controller recursivelyby considering some of the state variables as virtualcontrols and also to design intermediate control laws forthem. To give a clear idea of the above controller, thesatellite kinematics relations, considering the orbitalframe as reference, are obtained as [27]:24 _�_�

_

35 =24 sec � sin !�cos !�sin� sec � sin !�

35+

241 tan � sin� tan � cos�0 cos� � sin�0 sin� sec � cos� sec �

3524!x!y!z

35 : (23)Eq. (23) together with the satellite dynamic Eqs. (1)can be rewritten as:

_E = S(E;!o) + g(E)!;

_! = D(!; !w)�B _hW +BUf ; (24)

where E3�1 =��

�is the vector of Euler angles,

!0 is the orbital angular velocity of satellite, S(E;!o)and g(E) are vector �elds which can be easily derivedfrom Eq. (23) and D is a nonlinear smooth function as:

D = I�1 (�~! � I~! � ~! � Iw~!w) : (25)The backstepping feedback linearization control lawshould be designed so that control of the Euler anglesvector E3�1, i.e. attitude angles of satellite, is achieved.For this, �rst, with the assumption of no fault existencein the actuators, two separate steps are considered:

- Step one: In this step, due to Eq. (23), ! isconsidered as input of Eqs. (24), and based on thefeedback linearization theory [30] is selected accord-ing to Eq. (26). Using this control law provides theconvergence of the satellite attitude angles to desiredvalues.!d = g(E)�1(�S(E)� �E(E � Ed) + _Ed): (26)

In the above equation, E3�1 =��d �d d

�is

the desired attitude angle of satellite, _Ed3�1 is thedesired variation rate of the attitude angles, and�E3�3 is the design matrix with constant coe�cients.If angular velocities of satellite are selected accordingto Eq. (26), we will have:

_eE + �EeE = 0; (27)

in which eE3�1 = E � Ed is the attitude trackingerror. Eq. (27) reveals that if �E3�3 is positivede�nite, then E3�1 tends to the desired attitude,Ed3�1 , asymptotically.

- Step two: In this step the applied torque to thesatellite is designed so that the ! calculated in the�rst step is achieved. For this purpose, the desiredcontrol torque is calculated as follows:

_hWd = �B�1(�D(!; !w)� �(! � !d) + _!d);(28)

where !d is the desired angular velocity of satellitethat was obtained in the �rst step according toEq. (26), and �3�3 is the design constant. If thecontrol torque Eq. (28) is applied to Eqs. (24), weobtain:

_et + �et = 0; (29)

in which et3�1 = ! � !d is the angular velocitytracking error, and shows that when no faults existin the reaction wheels, if �3�3 is positive de�nite,then the angular velocities of the satellite tend to thedesired angular velocities (Eq. (26)) in a �nite time.Considering Eqs. (24), we can rewrite kinematicsequations as follows:

_E = S(E;!o) + g(E)! � g(E)!d + g(E)!d= S(E;!o) + g(E)!d + g(E)et: (30)


Also, according to Eqs. (26) and (27), the attitudetracking error dynamics in the presence of angularvelocity tracking error can be written as:

_eE = ��EeE + g(E)et: (31)As Eq. (29) reveals, angular velocity tracking errorconverges to zero in a �nite time. So, de�ning theaugmented error as ea =

�eTt eTE

�T , we obtain:_ea =

� ��3�3 03�3g(E(t))3�3 ��E3�3

�ea: (32)

Based on Theorem 8-13 of [31], since �E and � arestable and diagonal matrices, the augmented errordynamics is stable if and only if the time varying matrixof g(E(t)) is bounded. So the problem of convergenceof attitude tracking is turned to evaluating if g(E(t)) isbounded or not. Due to Eqs. (26), (31) and (32), threeconditions must be considered to guarantee successfultracking:

1. With attention to Eq. (20), G(E) must be bounded.In this regard, Eq. (23) reveals that this conditiondoes not hold just in the singularity point ofkinematics at � = 90�. So the stability condition ofsystem holds in the whole region of system excepta single point. The single e�ect of this singularityis that �d = 90� isn't tractable.

2. According to Eq. (26), g(E) must be invertible.Inverse of this matrix is equivalent to Eq. (33), soit exists and this assumption applies no constrainson the design.

g(E)�1 =

241 0 � sin �0 cos� sin� cos �0 � sin� cos� cos �

35 : (33)3. Due to Eq. (31), for a successful attitude tracking,

the rate of convergence of the angular velocitytracking error must be much more than the rateof convergence of the attitude tracking error. Thisrequirement is achieved by selecting the properdesign coe�cients, �E and �. For this purpose,the following condition must be held:

eigen value(�)i >> eigen value(�E)i;

i = x; y; z: (34)

This condition guarantees faster tracking of an-gular velocity rather than the attitude tracking, whichleads to a successful attitude tracking.

The occurrence of fault in actuators causes satel-lite to deviate from the desired attitude. In thiscondition, the fault term, Uf , appears in the satellitedynamics. So, in the above faulty situations, a

compensating algorithm should be activated to toleratethe fault e�ect and restore the satellite to the desiredattitude. In this regard, to handle the fault e�ect inthe actuator(s), a compensating term is added to thecontrol law as [29]:

_hW = _hWd + _hcompensation: (35)

Here, this compensating term is selected as the faultterm estimated by the adaptive observer, so we willhave:

_hcompensation = Ûf : (36)

Therefore, using the above compensating control lawin faulty conditions, the tracking error dynamics ofangular velocities with the consideration of Eqs. (24),(28) and (36) is obtained as:

_et + �et +Bef = 0: (37)

De�ning the new augmented error as eaug =�eTf eTt eTE

�T , we can write:_eaug =

24�BT� 03�3 03�3�B ��3�3 03�30 g(E(t))3�3 ��3�3

35 eaug: (38)Again, based on Theorem 8-13 of [31], since �E and �are stable and diagonal matrices, the augmented errordynamics is stable if and only if the time varying matrixof g(E(t)) is bounded, which was discussed before. Thisequation also applies another condition in the designof � that BT� must be positive de�nite. Attention tothe structure of B reveals that this condition is alwayssatis�ed. So even in the fault occurrence conditionsin the reaction wheels, the satellite tracks its desiredattitude.

6. Simulation results

In this section, simulation results of the developedfault detection, identi�cation and recovery algorithmsare presented to demonstrate the capabilities of thedeveloped algorithms. These simulations are carriedout for a Low Earth Orbit (LEO) satellite, withthe altitude 700 kilometers, which has the imagingmission. Satellite moments of inertia are consideredIxx = 4:92 kgm2, Iyy = 5 kgm2 and Izz = 1:55kgm2. Also, three reaction wheels aligned with theprincipal axes of the satellite, and each with themoment of inertia 0.003 kgm2, are utilized as actuators.For simulation purposes, the e�ect of disturbancesincluding geomagnetic disturbance torques, gravitygradient disturbances, aerodynamic disturbances andsolar radiation disturbances are considered. Afteranalysis and simulation of the above disturbances, a


maximum value, 10�5 N.m, has been obtained as theirresultant e�ect, which has been applied in the simula-tion model. Accordingly, the threshold value is selectedas �ve times of the magnitude of the disturbance value,i.e. 5 � 10�5 N.m. Also, the output error of thesensors for measuring the satellite angular velocities aremodeled as a white Gaussian noise with the standarddeviation 10�5 rad/sec. For modeling the satellitedynamics based on the Takagi-Sugeno method, twooperating points of

��!max !max� for angular velocityand

��!wheelmax !wheelmax� for rotational velocity ofreaction wheels are considered. The maximum velocityof satellite after separation from launcher !max ischosen as 7 deg/sec which is consistent with the dataavailable from di�erent launchers. Maximum value ofthe reaction wheels velocities depends on their charac-teristics which here is considered 6000 r.p.m. Also, foreach of the components of the angular velocities andreaction wheels velocities, two fuzzy triangular mem-berships functions k = x; y; z;M!kNegative;M

!kPositive and

k = x; y; z;M!wheelkNegative;M!wh eelkPositive have been considered.

So, the satellite nonlinear dynamics is described with64 rules. Ai, Bi and Ci matrices are obtained accordingto Eqs. (12), based on the satellite moments of inertiaand assumed operating points. Design matrices Li; i =1; � � � ; 64 have been obtained using MATLAB, so thatConditions (18) and (30) are satis�ed.

For this, after performing various simulations,the above coe�cients are tuned so that the suitableresponses are achieved. Details of these results arenot included here, due to space limitations. Also,design matrices in the controller part are selectedas �E3�3 = Diag3�3f0:1 0:1 0:1g and �3�3 =Diag3�3f1 1 1g. To investigate and study theperformance capabilities of the fault detection andidenti�cation algorithms, due to the reaction wheelsfaults, the following three scenarios are considerednext:

- Scenario 1: In the �rst scenario, fault does not occurin the reaction wheels. In this scenario, the faultestimation has been presented in Figure 1, and theestimation error of satellite angular velocities hasbeen depicted in Figure 2. These �gures reveal thatestimation error of the angular velocities is negli-gible, and the fault estimation in reaction wheelsis less than the selected threshold, which has beenshown by dash lines. So, as expected, no fault hasbeen declared. As can be observed in Figure 1,the fault estimation is not exactly zero, which maybe attributed to the disturbances and noise e�ects.Also, it is evident that the threshold value has beenappropriately and �nely selected to avoid false faultdeclaration.

- Scenario 2: In the second scenario, an abrupt faulthas been occurred in the reaction wheel, aligned

Figure 1. Fault estimation of actuators in the �rstscenario.

Figure 2. Estimation error of the satellite angularvelocities in the �rst scenario.

with the x axis, at a time between 200 sec and250 sec, and with the magnitude 10�3 N.m. In theseconditions, Figure 3 presents the fault estimation inreaction wheels, and Figure 4 shows the estimationerror of angular velocities. The results depictedin Figure 3 imply that only the fault estimationvalue, aligned with the x axis, has been exceededthe selected threshold, and so, fault is only declaredin this axis. Therefore, the designed algorithmhas provided the nice feature of isolating the faultycomponent. Moreover, the fault estimation value,compared to the real value, illustrates that theadaptive observer has the capability to �nely andaccurately estimate the fault terms. This feature


Figure 3. Fault estimation of actuators in the secondscenario.

Figure 4. Estimation error of the satellite angularvelocities in the second scenario.

leads to the boundedness of the estimation errorof angular velocities, despite the fault occurrencein the reaction wheels. This error deviates only inthe fault occurrence times which damps immediately.Note that in Figure 3, performance of the developedmethod has been compared with a fault estimatorbased on the Sliding Mode Observer (SMO) [21,23].As shown in this �gure, the fault estimated bythe Fuzzy-Adaptive Observer (FAO) represents anovershoot of 1.7%, settling time of 13 sec andestimation error of 0.1%; however, the SMO methodprovides an overshoot of 55%, the settling time of 28sec and the estimation error of 2%.

- Scenario 3: To further substantiate the FDD algo-rithm robustness, in this scenario, incipient faults

with the slope 10�5 N.m have been applied inall three reaction wheels. The Above faults haveoccurred at 200 sec. In this regard, Figure 5shows the fault estimation in reaction wheels, andthe estimation error of angular velocities has beendepicted in Figure 6. As can be seen in Figure 5,the fault estimation in all three axes violates theselected threshold at a considered time interval,which veri�es the performance of the fault detectionalgorithm. Also, it reveals that the fault magnitudesare accurately estimated. Figure 6 shows that theestimation error of angular velocities is bounded inspite of fault occurrence, which attributed to thefault estimation capability of the adaptive observer.Note that in Figure 5, performance of the devel-

Figure 5. Fault estimation of actuators in the thirdscenario.

Figure 6. Estimation error of the satellite angularvelocities in the third scenario.


Table 1. Performance of the proposed method in comparison to the sliding mode observer approach.

Fault detectiontime

Overshoot Settlingtime

Faultestimationaccuracy

Sensitivityto sample

rate

Fault type SMO FAO SMO FAO SMO FAO SMO FAO SMO FAO

Abrupt 2.1 sec 2.5 sec 55% 1.7% 28 sec 13 sec 2% 0.1% High Normal

Incipient 10 sec 12 sec | | | | 4.6% 1.9% High Normal

oped method has been compared with the SMO.It can be seen that the estimation error of 1.9% isattained using the developed method; however, anestimation error of 4.6% is provided by the SMOmethod. Similarly, by carrying out simulations fordi�erent scenarios, e�ectiveness of the proposed faultdetection, isolation and identi�cation algorithms isdemonstrated. The associated graphs are not shownhere for brevity.

Table 1 summarizes the results obtained by the FAOin comparison to the SMO. According to this table,although the fault detection times, provided by theoutlined methods, are almost close to each other(sliding mode estimator yields a somewhat betterperformance), the estimation accuracy achieved by theFAO is signi�cantly better than the SMO. Also, ourinvestigations show that the SMO approach is moresensitive to the sampling time variations. Accordingly,the estimation ability attained, using the FAO, pro-vides a better match, compared to the real fault value.The outlined ability is especially important about thecompensators which use the estimated fault term tomaintain the system stability. In fact, a more accuratefault estimation can eventually lead to a more robustremedial action.

Next step is veri�cation of the recovery algorithm.For this purpose, satellite attitude error is shown inFigure 7. As can be observed from this simulation,the value 0.0062 deg has been obtained as the pointingaccuracy in presence of noise and disturbance torques.So, it reveals that an acceptable orientation responsehas been achieved by the backstepping feedback lin-earization control law, and the satellite has reachedthe desired angle. Now, performance of the recoveryalgorithm is investigated against abrupt faults withthe magnitude 10�3 N.m, which are introduced in allthree reaction wheels at 120 sec. For this, Figure 8depicts the satellite attitude errors about the roll, pitchand yaw axes, in which the compensator controllerhas not been utilized. Based on the obtained results,tracking errors 147 deg about the yaw axis, 0.44 degabout the pitch axis and 0.47 deg about the roll axishave been gained. So, the system performance issigni�cantly degraded after actuator faults, which maylead to undesired e�ects on mission imaging resolution.

Figure 7. Attitude errors about the roll, pitch and yawaxes in case of no fault occurrence at the satellite reactionwheels and without fault compensation.

Figure 8. Attitude errors about the roll, pitch and yawaxes in case of fault occurrence at the satellite reactionwheels and without fault compensation.

Moreover, occurrence of an abrupt fault with a largermagnitude or an intermittent fault may lead to a moresevere e�ect or even loss of the satellite control.

Therefore, the necessity of utilizing a compensatorcontroller, together with the designed algorithm isevident. With activation of the compensator controller,


Figure 9. Attitude errors about the t roll, pitch and yawaxes in case of fault occurrence at the satellite reactionwheels and together with fault compensation.

the satellite attitude errors have been illustrated againin Figure 9. It can be seen that high control precisionand a good tracking process are still obtained, despitefault existence in actuators and deviation of theirgenerated control torques from the desired values.So, the satellite maintains the desired attitude, ande�ectiveness and applicability of the recovery algorithmare veri�ed. Summarizing all the cases, it is notedthat the proposed adaptive observer provides fault

identi�cation with an acceptable accuracy. Utilizingthis capability combined with the threshold selectionprocess leads to a high reliability attributed to the faultdetection and identi�cation processes.

Another important merit of the designed algo-rithm is that in case of fault occurrence in an actuator,only the fault term associated with that actuatorviolates the selected threshold, which typically providesthe fault isolation capability. Simulation results alsoshow that the recovery algorithm can signi�cantlyimprove the normal performance mode, and so thisfeature together with the outlined fault detection,isolation and identi�cation capabilities achieves an au-tomatic and independent fault tolerant attitude controlsystem.

7. Hardware In the Loop (HIL) test

To evaluate the potentialities of utilizing the proposedalgorithms in real applications, and investigate theircapability to onboard implementation, in this section,the results of the Hardware In the Loop (HIL) testsare presented. These experimental results validatethe desired requirements attributed to the designedalgorithms considering the real conditions, which thesatellite may experience in the orbital situations. Forthis, hardware in the loop test bed is planned accordingto Figure 10, which provides the ability to test FaultDetection, Isolation and Recovery (FDIR) algorithms

Figure 10. Di�erent elements of the hardware in the loop test bed.


in close to a real-time condition. This facility includesthe following main elements:

� Simulator computer: this simulator which has beencreated by the lab view software provides the pos-sibility to accurately model the satellite dynam-ics, considering the factors such as uncertainties,space disturbances, sensor noise and reaction wheelmodels. In this environment, we also have thefault simulator to monitor the parameters relatedto the FDIR algorithms, and analyze their perfor-mance. The simulator computer also provides aspecial monitoring environment to track the controlalgorithms, their generated moments which apply tothe satellite, and the angular rates which have beenachieved by them.

� On-board electronics: the fault detection, identi�ca-tion and recovery algorithms have been implementedin this element. The processor utilized in thisproduct is selected from the 8051 family which, inaddition to satisfy our technical speci�cations, hasa history of use in di�erent space missions. Thiselement also provides the exibility to implementand evaluate di�erent attitude control and FDIRalgorithms. As seen from Figure 10, the onboardelectronics receives the satellite attitude, satelliteangular rates and reaction wheel angular rates as itsinput states, and after process and execution of theimplemented FDIR algorithms, generates the torquecommands and FDI output signals, which should betransmitted to the simulator computer. The torquecommand is generated by the recovery algorithmafter fault declaration by the FDI algorithm, whichshould be applied to the satellite simulator model toguarantee its stability.

� Interface circuits: Interface circuits consist of theappropriately selected I=O cards which have the roleof receiving the output states of the computer simu-lator and transmitting the output signals generatedby the FDIR algorithm.

Figure 11 shows a real view of the mentioned elementsand their connections. connected to each other.

Figure 12 depicts the test results for a scenarioin which abrupt faults have occurred in each of thethree reaction wheels. As this �gure reveals, in spite ofthe faults in the reaction wheels, estimation errors ofthe implemented adaptive observer remains bounded.Also, the fault estimations track the real fault pro�les,and so validate the fault detection and identi�cationperformances in an onboard implementation test. Asseen in this �gure, the fault estimations are corruptedby noise, due to the measuring sensors. To copewith this event and prevent false fault declaration, thefault estimation signals have been �ltered by onboardelectronics, as it was represented in Figure 12. This

Figure 11. Hardware in the loop test facility.

�gure shows that the �ltered signals provide smoothpro�les which are more compatible with the real faultpro�les, and hence it is possible to achieve a moreaccurate and �ne FDD process.

To evaluate the FDD performance in a case inwhich the FDD algorithms are exposed to a highseverity of sensor noises, Figure 13 presents the faultsimulator outputs in a condition where the measuredsignals are corrupted by a noise with the standarddeviation 10�3 rad/sec. As the �ltered signals depictedin this �gure reveal, even in these harsh conditions, thefault declaration process works correctly, and the faultestimation pro�le tracks the real fault pro�le appliedto the reaction wheels.

Figures 14 and 15 represent the satellite attitudeangles and angular velocities achieved by the recoveryalgorithm implemented in the onboard electronics.Figure 14 shows the variables of interest in conditionswhere no compensation has been accomplished. Ascan be observed, an o�set has appeared in the yawaxis angle (). Figure 15 depicts the results whenthe compensation algorithm has been activated. It isevident from this �gure that despite fault existence inthe actuators, the satellite maintains its stability, andno o�set has appeared in the satellite states.

Table 2 indicates a comparison between simula-tion and experimental results of the developed recoveryalgorithm. As this table illustrates, there are somedeviations between the results depicted. According toour expectation, the obtained results in the simulationcase is somewhat better than the experimental casewhich is reasonable due to reasons detailed below:

1. Accuracy of the oat calculations in the on-boardelectronics processor is more restrictive than theCPU of the simulator computer, so some deviationsbetween the related results occur.

2. The major deviation between these two resultsoriginates from the analog to digital and digital toanalog conversions.


Figure 12. Fault detection and identi�cation test results after fault occurrence in the reaction wheels.

Figure 13. Fault detection and isolation test results after intermittent fault occurrence in the reaction wheel, aligned withthe x axis, considering a measuring noise with the standard deviation of 10�3 rad/sec.

Table 2. Simulation and experimental results of the recovery algorithm.

No faultFault application in allthree reaction wheels

with no recovery activation

Fault application in allthree reaction wheels and

utilizing the recovery algorithmSimulation

resultsExperimental

resultsSimulation

resultsExperimental

resultsSimulation

resultsExperimental

resultsRoll error 0:0062� 0:09� 0:4778� 0:48� 0:0086� 0:07�

Pitch error 0:0017� 0:15� 0:4433� 0:36� 0:0023� 0:14�

Yaw error 0:0019� 0:08� 1:475� 3:20� 0:0023� 0:08�


Figure 14. Satellite attitude angles after fault occurrence in each of the three reaction wheels and without faultcompensation.

Figure 15. Satellite attitude angles after fault occurrence in each of the three reaction wheels and with activation of thefault compensating algorithm.

Since the data must be transferred between the on-board electronics and the simulator computer, a 16bit conversion is inevitable, so this conversion errormay lead to deterioration of the experimental results.Note that since in real situations there is no need totransfer data between the onboard electronics andthe computer, this error is not a problem. In fact, therequested attitude determination data received directlyfrom the attitude sensors and the generated torque bythe reaction wheels is directly applied on the satellitebody. As a result, although there are some deviationsbetween simulation results and experimental results,

due to the reasons stated, these deviations are notcritical, and the results obtained are accurate enoughfor application in orbit.

8. Conclusions

This paper described both theoretical and experimentalresults focused on a study of fault tolerant attitudecontrol system. The proposed FDD algorithms pro-vide the ability of actuators fault detection, accurateidenti�cation of their magnitudes and also the isolationpossibility between them. The above capabilities


were achieved under the conditions where no priorinformation about the fault nature was used in theFDD design process. The FDD design was evaluatedusing several di�erent types of reaction wheel failurescenarios, through which the desired performance werevalidated taking into account the disturbances, uncer-tainties and unknown faults.

In this paper, also the design steps of the re-covery algorithm, based on a backstepping feedbacklinearization controller, were presented, in which thefault magnitude estimation accomplished by the FDDalgorithm was utilized. As the simulation resultsrevealed, this algorithm provided the compensationability of the fault e�ect such that no deviation of thesystem from the desired pointing attitude occurred.These analytical investigations demonstrated the ro-bustness of the control design to the disturbances,measurement noise and unknown faults, and so theexpected performance was achieved. To highlightthe potentialities of the proposed algorithms in realapplications, hardware in the loop test facility wasplanned to study the digital implementation of thedesigned algorithms, and provide more accurate andrealistic results. As observed from the test results,the developed algorithms maintained their desiredperformances, which validated their feasibility in real-time implementations. Future work is planned tostudy reliability and dependability analyses which areof paramount importance for aerospace applications.

References

1. Venkateswaran, N., Siva, M.S. and Goel, P.S. \Ana-lytical redundancy based fault detection of gyroscopesin spacecraft applications", Acta Astronautica, 50(9),pp. 535-545 (2002).

2. Castet, J.F. and Saleh, J.H. \Satellite and satellitesubsystems reliability: Statistical data analysis andmodeling, reliability engineering and system safety",Reliability Engineering & System Safety, 94, pp. 1718-1728 (2009).

3. Patton, R.J. \Fault detection and diagnosis inaerospace systems using analytical redundancy", Com-puting and Control Engineering Journal, pp. 127-136(1991).

4. Hwang, I. and Kim, S. \A survey of fault detection,isolation and recon�guration methods", IEEE Trans-actions on Control Systems Technology, 18(3), pp. 636-653 (2010).

5. Frank, P.M. \Fault diagnosis in dynamic systems usinganalytical and knowledge-based redundancy-a surveyand some new results", Automatica, 26(3), pp. 459-474 (1990).

6. Iserman, R. \Model-based fault detection anddiagnosis-status and applications", Annual Reviews inControl, 29, pp. 71-85 (2005).

7. Venkatasubramanian, V., Rengaswamy, R. andKavuri, S.N. \A review of process fault detection anddiagnosis. Part I: Quantitative model-based methods",Computers & Chemical Engineering, 27, pp. 293-311(2003).

8. Valdes, A. and Khorasani, K. \A pulsed plasmathruster fault detection and isolation strategy forformation ying of satellites", Applied Soft Computing,10, pp. 746-758 (2010).

9. Li, Z.Q., Ma, L. and Khorasani, K. \A dynamicneural network-based reaction wheel fault diagnosis forsatellites", International Joint Conference on NeuralNetworks Sheraton Vancouver Wall Centre Hotel, Van-couver, BC, Canada, pp. 3714-3721 (2006).

10. Fan, C., Jin, Z., Zhang, J. and Tian, W. \Applicationof multi sensor data fusion based on RBF neuralnetworks for fault diagnosis of SAMs", Seventh Inter-national Conference on Control, Robotics and Vision,3, pp. 1557-1562 (2002).

11. Zhao, S. and Khorasani, K. \A recurrent neuralnetwork based fault diagnosis scheme for a satellite",The 33rd Annual Conference of the IEEE IndustrialElectronics Society, Nov. 5-8, Taipei, Taiwan, pp. 2660-2665 (2007).

12. Pirmoradi, F.N., Sassani, F. and Silva, C.W.D. \Faultdetection and diagnosis in a spacecraft attitude deter-mination system", Acta Astronautica, 65, pp. 710-729(2009).

13. Okatan, A., Hajyev, C. and Hajiyeva, U. \Kalman�lter innovation sequence based fault detection in Leosatellite attitude determination and control system",3rd International Conference on Recent Advances inSpace Technologies RAST `07, Istanbul, pp. 411-416(2007).

14. Xiong, K., Chan, C.W. and Zhang, H.Y. \Detection ofsatellites attitude sensor faults using the UKF", IEEETransactions on Aerospace and Electronics Systems,43(2), pp. 480-491 (2007).

15. Soken, H.E. and Hajiyev, C. \Pico satellite attitudeestimation via robust unscented Kalman �lter in thepresence of measurement faults", ISA Transactions,49, pp. 249-256 (2010).

16. Bae, J. and Kim, Y. \Attitude estimation for satel-lite fault tolerant system using federated unscentedKalman �lter", International Journal of Aeronautical& Space Science, 11(2), pp. 80-86 (2010).

17. Tudorou, N. and Khorasani, K. \Satellite fault diagno-sis using a bank of interacting Kalman �lters", IEEETransactions on Aerospace and Electronics Systems,43(4), pp. 1334-1350 (2007).

18. Patton, R.J., Uppal, F.J., Simani, S. and Polle, B.\Robust FDI applied to thruster faults of a satellitesystem", Control Engineering Practice, 18, pp. 1093-1109 (2010).

19. Patton, R.J., Uppal, F.J., Simani, S. and Polle,B. \Reliable fault diagnosis scheme for a spacecraftattitude control system", Proc. IMechE, Part O: J.Risk and Reliability, 222, pp. 139-152 (2008).


20. Henry, D. \Robust fault diagnosis of the microscopesatellite micro-thrusters", IFAC Fault Detection, Su-pervision and Safety of Technical Processes, Beijing,pp. 342-347 (2006).

21. Jiang, T. and Khorasani, K. \A fault detection,isolation and reconstruction strategy for a satellite'sattitude control subsystem with redundant reactionwheels", IEEE International Conference on Systems,Man and Cybernetics, Montreal, Que, pp. 3146-3152(2007).

22. Wu, L., Zhang, Y. and Li, H. \Research on faultdetection for satellite attitude control systems basedon sliding mode observers", IEEE International Con-ference on Mechatronics and Automation, Changchun,China, pp. 4408-4413 (2009).

23. Wu, Q. and Saif, M. \Robust fault diagnosis of asatellite system using a learning strategy and secondorder sliding mode observer", IEEE Systems Journal,4(1), pp. 112-121 (2010).

24. Zhang, K., Jiang, B. and Shi, P. \Adaptive observer-based fault diagnosis with application to satelliteattitude control systems", Second International Con-ference on Innovative Computing, Information andControl, Kumamoto, pp. 508-508 (2007).

25. Wang, J. Jiang, B. and Shi, P. \Adaptive observerbased fault diagnosis for satellite attitude control sys-tems", International Journal of Innovative Computing,Information and Control, ICIC International, 4(8),pp. 1921-1929 (2008).

26. Tanaka, K. and Wang, H.O., Fuzzy Control SystemDesign and Analysis, John Wiley & Sons, pp. 5-10(2001).

27. Sidi, M.J., Spacecraft Dynamics and Control, Cam-bridge University Press, pp. 88-95 (1997).

28. Jiang, B., Gao, Z., Shi, P. and Xu, Y. \Adaptive fault-tolerant tracking control of near-space vehicle usingTakagi-Sugeno fuzzy models", IEEE Trans. Fuzzy Sys-tems, 18(5), pp. 1000-1007 (2010).

29. Ichalal, D., Marx, B., Ragot, J. and Maquin, D.\Fault tolerant control for Takagi-Sugeno systems withunmeasurable premise variables by trajectory track-ing", IEEE International Symposium on IndustrialElectronics, pp. 2097-2102 (2010)

30. Khalil, H., Nonlinear Systems, 2nd Ed. Upper SaddleRiver, NJ, Prentice-Hall, pp. 588-601 (1996).

31. Chen, C.T., Linear System Theory and Design, Holt,Rinehart and Winston, pp. 400-404 (1970).

Biographies

Hossein Bolandi received his DSc degree in Electri-cal Engineering from George Washington University,Washington, D.C., in 1990. Since 1990 he has beenwith the College of Electrical Engineering, Iran Uni-versity of Science and Technology, Tehran, Iran, wherehe is an associate professor. His research interests arein attitude determination and control subsystems ofsatellites, adaptive control, automation and guidance,and navigation and control.

Mehran Haghparast received his BSc and MScdegrees in Electrical Engineering from Iran Universityof Science and Technology, Tehran, Iran in 2008 and2011, respectively. His research interests are in attitudecontrol subsystems of satellites, fault detection andisolation system, robust control and applied controltheory.

Mostafa Abedi received his BSc degree in ElectricalEngineering from the Chamran University and hisMSc degree from the Iran University of Science andTechnology in 2004 and 2007, respectively. He is nowPhD candidate in the electrical department, Iran Uni-versity of Science and Technology, Tehran. His researchinterests are in attitude determination of satellites,satellite sensors, fault detection and isolation.

a reliable fault tolerant attitude control system based on an...

Documents