Published in IET Control Theory and ApplicationsReceived on 17th August 2011Revised on 30th December 2011doi: 10.1049/iet-cta.2011.0496
ISSN 1751-8644
Fault tolerant control with H∞ performance forattitude tracking of flexible spacecraftQ. Hu1 B. Xiao1 M.I. Friswell21Department of Control Science and Engineering, Harbin Institute ofTechnology, Harbin 150001,People’s Republic of China2College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UKE-mail: [email protected]
Abstract: A fault-tolerant control scheme with thruster redundancy is developed and applied to perform attitude trackingmanoeuvres for an orbiting flexible spacecraft. Based on the assumption of bounded elastic vibrations, an adaptive slidingmode controller is proposed to guarantee that all the signals of the resulting closed-loop attitude system are uniformly ultimatelybounded in the presence of an unknown inertia matrix, bounded disturbances and unknown faults. An H∞ performance indexis introduced to describe the disturbance attenuation performance of the closed-loop system. This approach is then extendedto the problem of elastic vibration without knowledge of the bounds a priori. The presented approach addresses thrustersaturation limits and the desired thruster force is guaranteed to stay within the limit of each thruster. Extensive simulationstudies have been conducted to demonstrate the closed-loop performance benefits compared to conventional control schemes.
1 Introduction
Spacecraft are generally designed with high reliabilitywithout any degradation in their attitude pointing accuracyand attitude stability. This high reliability is usually achievedby selecting highly reliable components, ensuring theirreliability by quality assurance and environmental testsconducted at component level, subsystem level and evenintegrated spacecraft level before launch. Despite theseefforts, faults do occur in orbit. The occurrence of a faultin any component of the spacecraft would degrade itsperformance, and consequently a fault tolerance capabilityis vital.
Actuators are vital components in a spacecraft attitudecontrol system, especially for high-performance spacecraft,that are required to undergo fast slewing or manoeuvring. Asa result, the consequences of an actuator fault are seriouscompared with other malfunctions during a mission. Forthis reason, almost all spacecraft systems are designed withactuator redundancy to increase system reliability. Boskovicet al. [1] applied the method of multiple models to detectand isolate actuator faults for spacecraft attitude controlsystems. Li et al. [2] used a dynamic recurrent neuralnetwork architecture, and proposed a fault detection andisolation (FDI) mechanism for thruster failures. Chen andSaif [3] presented a fault diagnosis approach to identifythruster faults by using an iterative learning observer. Wuand Saif [4] applied a robust FDI method based on neuralstate space models to a satellite attitude control subsystem,and investigated the robustness, sensitivity and stabilityproperties. Unfortunately, these methods require the FDIprocess to provide a correct decision on the actual status of
the component failure [5, 6]. Hence, there is great interestin fault-tolerant attitude control design for spacecraft byusing redundant actuators. Jin et al. [7] employed dynamicsinversion and time-delay theory to design a fault-tolerantcontroller (FTC) for a rigid satellite with four reactionwheels. For a spacecraft equipped with redundant thrusters,Cai et al. [8] proposed an indirect adaptive fault-tolerantattitude tracking control, in which a bound of the lumpedperturbations was introduced and updated online. Othermethods have been developed using actuator redundancy forreconfigurable air vehicle control to tackle actuator failures[9, 10].
System uncertainties and external disturbances are twofurther challenges to be addressed in the design of attitudecontrollers. Furthermore, the orbiting attitude slewingoperation introduces vibration in the flexible appendages,which may degrade the attitude pointing accuracy. Singh[11] presented an attitude control law design basedon linearisation and non-linear inversion, although thetechnique was not able to deal with system uncertainties.Since adaptive control can successfully address systemuncertainties of known structure, many investigations usingadaptive control to design spacecraft attitude control lawsare available [12–15]. Jasim et al. [15] developed anadaptive feedback control algorithm to provide asymptotictracking of commanded spacecraft motion without theknowledge of the spacecraft inertia parameters, althoughunmodelled dynamics and disturbances are not considered.Neural network control may be used as a removal techniquefor non-linear characteristics using relatively slow onlinefunction approximation [16, 17], providing the uncertaintiesin the plant do not change quickly. In recent years, sliding
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
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mode control (SMC) has become an effective technique inspacecraft control because of its inherent insensitivity androbustness to plant uncertainties and external disturbances[18–20]. However, these design methods require informationof the bounds on the uncertainties/disturbances for thecomputation of the control gains. Therefore to overcomethe drawbacks of each method, a combination of thesetechniques has been studied for flexible spacecraft withuncertain parameters and disturbances [21–24].
In this paper, an adaptive FTC strategy is suggested forflexible spacecraft with redundant thrusters. The proposedFTC scheme is based on SMC theory and applied toaddress the problems of unknown thruster faults, externaldisturbances and an unknown and time-varying inertiamatrix of the spacecraft. A key feature of the proposedapproach is that the design of the FTC is independent of thefaults. The resulting tracking performance is evaluated by anH∞ index from a torque level disturbance/elastic vibrationto a penalty signal for the tracking error. An adaptivemechanism is also adopted to update the unknown boundsof the system parameters. Furthermore, taking the practicalthruster limits into account, the desired thruster forcesare redesigned to remain within the limit constraints. Apulse-width pulse-frequency (PWPF) modulation techniqueis employed to transform the continuous control commandsinto equivalent discrete commands, that can be implementedwith on–off gas jet actuators.
The paper is organised as follows. Flexible spacecraftmodelling and control problem formulations are summarisedin Section 2. Attitude tracking FTC laws based on SMC,and the adaptive case with H∞ performance, are derived inSection 3. Numerical simulations are presented in Section4 to demonstrate various features of the proposed controllaw. Finally, the paper is completed in Section 5 with someconcluding comments.
Notation: The notation ‖ · ‖ denotes the Euclidean normof vectors and the induced norm of matrix. tr(A) denotesthe trace of the matrix A. For ∀χ , y ∈ R, we define theprojection mapping operator Projχ (y) as
Projχ (y) =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
0, if χ = χmax and y < 0−y, if χmin < χ < χmax
−y, if χ = χmax and y ≥ 0−y, if χ = χmin and y ≤ 00, if χ = χmin and y > 0
where χmin and χmax are the known constants. For ∀b =[b1 b2 b3]T ∈ R
3, we define the linear operator [15] L(b) :R
3 → R3×6 by
L(b) =[
b1 0 0 0 b3 b2
0 b2 0 b3 0 b1
0 0 b3 b2 b1 0
]
2 Flexible spacecraft dynamics and thecontrol problem
2.1 Kinematic equation
The unit quaternion is adopted to describe the attitude of thespacecraft using a global representation without singularities[25]. The unit quaternion q is defined by
q =[
cos(�/2)n sin(�/2)
]=
[q0
q
](1)
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
where n is the Euler axis, � is the Euler angle, q0 and q =[q1 q2 q3]T are the scalar and vector components of theunit quaternion, respectively. Moreover, the equation qTq +q2
0 = 1 is satisfied. Then, the kinematics equation is
[q0
q
]= 1
2
[ −qT
q0I + S(q)
]ω (2)
where ω ∈ R3 is the angular velocity of the body-fixed
reference frame of the spacecraft with respect to an inertialreference frame and expressed in the body-fixed referenceframe, I ∈ R3×3 represents the identity matrix, and for ∀x =[x1 x2 x3]T ∈ R
3 we define a skew-symmetric matrixS(x) as
S(x) =[
0 −x3 x2
x3 0 −x1
−x2 x1 0
](3)
2.2 Relative attitude error kinematics
Let qe = [q0e qTe ]T denote the relative attitude error from a
desired reference frame to the body-fixed reference frame.Then
qe = q ⊗ q−1d = [q0e qT
e ]T (4)
where q−1d denotes the inverse of the desired quaternion
qd with the definition q−1d = [q0d − qT
d ]T, and ⊗ is theoperator for quaternion multiplication, defined as
qa ⊗ qb =[
q0aq0b − qTa qb
q0aqb + q0bqa − S(qa)qb
](5)
for any given two quaternions qa and qb. The relativespacecraft attitude error dynamics is then
[q0e
qe
]= 1
2
[ −qTe
q0eI + S(qe)
]ωe (6)
with the angular velocity error ωe given by
ωe = ω(t) − Rdωd(t) (7)
where Rd is the rotation matrix from the desired referenceframe to the body-fixed reference frame, and ωd is theangular velocity of the desired reference frame with respectto the inertial reference frame.
2.3 Flexible spacecraft dynamics
Under the assumption of small elastic displacements, thedynamic equations of a spacecraft with flexible appendagesare given by Gennaro [26]
J ω + δTη = −S(ω)(Jω + δTη) + u(t) + d(t) (8)
η + Cη + Kη + δω = 0 (9)
where J ∈ R3 is positive definite symmetric inertia matrix
of the spacecraft, u(t) ∈ R3 is the control torque acting on
the main body and generated by gas jets and d(t) ∈ R3 is
the external disturbance (environmental disturbances, solarradiation etc.). In addition, C ∈ R
the damping and stiffness matrices, respectively, and aredefined as
C = diag{2ξi�1/2i , i = 1, 2, . . . , N }
K = diag{�i, i = 1, 2, . . . , N } (10)
where N ∈ R is the number of elastic modes considered,�
1/2i is the natural frequency and ξi is the corresponding
damping ratio. δ ∈ RN×3 is the coupling matrix between
the elastic and rigid structures and η ∈ RN is the modal
coordinate vector.To facilitate the subsequent control formulation, the
following two assumptions are made.
Assumption 1: The external disturbance d(t) is unknown buthas the property that d(t) ∈ L2(0, ∞). Thus the disturbanceenergy, ∫∞
0 |d(t)| dt, is bounded.
Assumption 2: The elastic oscillation and its velocity areassumed to be bounded. Thus ‖η(t)‖ and ‖η(t)|| are boundedduring the whole attitude control process.
Remark 1: The net disturbance force, d(t), acting on thespacecraft consists mainly of gravitational perturbations,atmospheric drag and solar radiation pressure preservativeforces, and these disturbance torques are bounded. It istherefore reasonable to make Assumption 1 for spacecraft.
Remark 2: Assumption 2 is also satisfied for flexiblespacecraft systems since damping always exists in flexiblestructures, even if it is small, such that the magnitudes ofelastic vibration and its velocity are bounded.
From the definition of ωe in (7), and the spacecraftdynamics given by (8), the attitude tracking error equationcan be derived as [8]
where R is the rotation matrix related to the errorquaternion vector qe and given by R = (q2
0e− qT
e qe)I +2qeqT
e − 2q0e S(qe). For the attitude tracking model givenby (11), consider the situation when an actuator fault occurs,especially when the actuator loses total or partial controlpower. A simple model of the thruster faults is incorporatedinto the non-linear spacecraft attitude error dynamics modeland is described by
where τ ∈ Rl denotes the propulsion force vector producedby l thrusters, and D ∈ R
3×l is the thruster distributionmatrix. For a given spacecraft, D is available and canbe made full-row rank by properly placing the thrustersat certain locations and directions on the spacecraft. Thediagonal matrix E ∈ R
l×l characterises the health of thethrusters, and is defined as
E = diag{e1, e2, . . . , el} (13)
with 0 ≤ ei ≤ 1 (i = 1, 2, . . . , l) indicating the actuatorhealth for the ith thruster.
Remark 3: In the actuation effectiveness matrix, E, the caseei = 1 represents no fault in the ith thruster, while ei = 0represents the complete failure of the ith thruster from whichno force is generated. The case 0 < ei < 1 corresponds tothe ith thruster partially losing actuation power. Hence, thematrix E is an unknown or possibly even time-varying,diagonal matrix.
2.4 Control problem statements
The signal z = [ρ1qTe ρ2σ
T]T is used to evaluate theattitude control performance, where σ defines a slidingsurface and ρi > 0 (i = 1, 2) are weighting coefficients.Suppose the level of disturbance attenuation is given asγ > 0. The control objective is to determine a control lawτ(t), from the flexible spacecraft attitude tracking systemgiven by (12), such that:
1. All the signals of the resulting closed-loop attitudetracking system are uniformly ultimately bounded;2. Torque level disturbances and/or elastic vibrationattenuation with respect to the attitude quaternion error,along with the angular velocity penalty, is ensured usingthe H∞ performance index.
Further, these two objectives should be met despiteunknown external disturbances and system uncertainties,severe thruster faults and force limits on each thruster.
3 Derivation of adaptive sliding mode FTCsystem with H∞ performance
3.1 Adaptive sliding mode FTC design
A robust adaptive SMC strategy is proposed for theattitude tracking control problem. By considering the errorquaternion and the angular velocity vector, the followingsliding surface is proposed
σ = ωe + βqe (14)
where β > 0 is a design parameter.
Remark 4: To facilitate the controller design, let d � d(t) −δTη − S(ω)δTη be considered as the lumped disturbances.From Assumptions 1 and 2, it can be shown that the energyof d is bounded, that is, there exist a constant (unknown)μ > 0 such that
The elements of the unknown symmetric inertia matrix J =[Jij] are represented by
�1 � [J11 J22 J33 J23 J13 J12]T (17)
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
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Then, (16) can be rewritten as
J σ = M1�1 + DEτ(t) + d (18)
where
M1(ωe, Rωd, Rωd, qe) = −S(ωe + Rωd)L(ωe + Rωd)
+ L(S(ωe)Rωd − Rωd)
+ L
{β
2[q0eI + S(qe)] ωe
}(19)
For the controller derivation, consider the Lyapunov-likefunction defined by
V = 1
2σ TJσ + β[qT
e qe + (1 − qe0)2] + 1
2αtr(ϒTϒ)
+ 1
2�T
1�−11 �1 + 1
2�T
2�−12 �2 (20)
where �i ∈ R6×6 (i = 1, 2) are positive definite symmetric
constant matrices, α is a positive constant and the elementsof the unknown symmetric matrix (dJ/dt) are representedby
�2 �[
dJ11
dt
dJ22
dt
dJ33
dt
dJ23
dt
dJ13
dt
dJ12
dt
]T
(21)
ϒ and �i denote the estimates of the uncertain matrix ϒ �DEDT and the uncertain vector �i (i = 1, 2), respectively,and the estimate errors are defined as ϒ � ϒ − ϒ and �i ��i − �i (i = 1, 2).
Differentiating V with respect to time and inserting (18)gives
V = σ TM1�1 + βσ Tqe + σ TDEτ(t) + σ Td + σ TM2�2
− β2qTe qe + 1
αtr(ϒT ˙
ϒ) + �T1�
−11
˙�1 + �T
2�−12
˙�2 (22)
wheredJ
dtσ = L(σ )�2 = 2M2�2 (23)
We are now ready to summarise the first result for theattitude tracking problem.
Theorem 1: Consider the flexible spacecraft attitude trackingsystem given by (6) and (15), with Assumptions 1 and 2.The following control law is applied
τ(t) = −DTϒ−1{M1�1 + M2�2 + βqe + κσ } (24)
and updated by
˙ϒ = αστ T (25)
˙�1 = �1M1σ (26)
˙�2 = �2M2σ (27)
Suppose that the control gains are chosen to satisfy
κ − 1
4γ 2− ρ2
2 > 0 (28)
β > ρ1 > 0 (29)
where κ is a positive control gain to be determined. Then,the control objectives (a) and (b) as stated in Section 2.4 areachieved.
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
Before we proceed with the Proof of Theorem 1, thefollowing remarks are in order.
Remark 5: The conclusion of Theorem 1 is independent ofthe initial conditions q(0), ω(0) and the reference trajectoriesqd and ωd. Therefore the proposed control law (24) canguarantee the spacecraft attitude to follow any desiredattitude from any initial attitude with any angular velocity.In addition, the initial conditions for the parameter updatinglaws in (25)–(27) are selected as ϒ(0) = I , �1(0) = 0 and�2(0) = 0 for convenience.
Remark 6: The actuation effectiveness matrix E is not usedin the control scheme (24), and so there is no need toinclude a health monitoring unit to identify or estimatewhich actuator is unhealthy, and fault isolation is notrequired. The thruster fault accommodation/compensationis performed automatically and adaptively by the proposedcontrol algorithm. There are l thrusters (l > 3) properlymounted on the spacecraft and the remaining active thrustersare assumed to be able to produce a combined forcesufficient to perform the required attitude manoeuvres.Moreover, the number of completely failed thrusters is nomore than l − 3 to guarantee the attitude manoeuvres may beaccomplished. If the number of failed thrusters is more thanl − 3, then the system will become under-actuated, and thedesigned controller in (24) will not guarantee the stability ofthe resulting closed-loop attitude system. The under-actuatedsystem is not considered further in this paper.
This feature is necessary to build affordable and effectivefault-tolerant spacecraft control schemes. To stabilise thesystem, the rank of matrix D should be equal to 3, thatis, the remaining active thrusters are able to produce anefficient actuating torque vector for the spacecraft to performthe given mission.
Remark 7: In the updating laws (25)–(27), it is known thateven a small disturbance may lead to the divergence of theestimates of the parameters. To avoid this rare but possiblecase, a discontinuous projection proposed in [27] is used forthe adaptive laws in (25)–(27), and the adaptation laws canbe rewritten as follows
˙ϒ = Projϒ (αστ T) (30)
˙�1 = Proj�1
(�1M1σ) (31)
˙�2 = Proj�2
(�2M2σ) (32)
Here, we have slightly abused the notation by using Projto stand for both scalar-valued and vector-valued projectionoperators.
Proof of Theorem 1: In view of (22) and the proposedcontrol law (24), one has
2‖σ‖2 is employed in thederivation of (33). Then, integrating inequality (33) fromt = 0 to ∞ yields
V (∞) +∫∞
0
‖z‖2 dt ≤ V (0) + γ 2
∫∞
0
‖d‖2 dt (34)
Clearly, we see in (34) that the L2 gain from the unknownbounded disturbance, caused by the external disturbancesand elastic vibrations, to the penalty signal z is prescribed bythe constant γ . (34) shows that the H∞ control performanceis achieved according to the definition in [28]. Thus theproof is completed.
3.2 Modified adaptive sliding mode FTC design
Section 3.1 designed a controller for the attitude trackingmanoeuvre for flexible spacecraft by an adaptive SMC lawwith an unknown inertia matrix and disturbance torques.However, the assumption of boundedness of the lumpedperturbation must be satisfied in advance. To relax theassumption, a modified adaptive SMC is proposed, in whichthe effect of elastic vibrations is also considered in therobustness design. Introduce the state
Theorem 2: Consider the flexible spacecraft attitude trackingsystem governed by (6) and (37) with Assumption 1.Suppose that the adaptive SMC input τ(t) is determinedby (41) (see below) and the updating law
˙ϒ = αστ T (42)
˙�2 = �2M2σ (43)
˙�3 = �3M3σ (44)
where �i+1 (i = 1, 2) are positive definite symmetricmatrices. The control parameters are chosen such that
κ − ρ22 > 0 (45)
β > ρ1 > 0 (46)
Then, for all possible initial conditions q(0), ω(0) andreference trajectories qd and ωd , the control objectives (a)and (b) as stated in Section 2.4 are achieved.
Proof: The starting point is again the choice of anappropriate Lyapunov function, which is here taken as
V = 1
2σ TJ0σ + β[qT
e qe + (1 − qe0)2]
+ 1
2αtr(ϒTϒ) + 1
2�T
3�−13 �3 + 1
2�T
2�−12 �2 (47)
By taking the time derivative of V along trajectoriesgenerated from (38) to (46), one has
˙V ≤ −κ‖σ‖2 + σ T(M3�3 + M2�2)
+ [(‖δT[K C]‖ + ‖ω‖‖[0 δT]‖)2 + 1]‖σ‖2
4γ 2
+ σ Tϒτ (t) + γ 2
{∥∥∥∥[
ηψ
]∥∥∥∥2
+ ‖d‖2
}− β2qT
e qe
+ 1
αtr(ϒT ˙
ϒ) + �T3�
−13
˙�3 + �T
2�−12
˙�2
= −(κ − ρ22)‖σ‖2 − (β2 − ρ2
1)‖qe‖2
+ γ 2
{∥∥∥∥[
ηψ
]∥∥∥∥2
+ ‖d‖2
}− ‖z‖2 (48)
If we choose k2 � min{(κ − ρ22), (β
2 − ρ21)}, then
˙V ≤ −k2(‖σ‖2 + ‖qe‖2) + γ 2
{∥∥∥∥[
ηψ
]∥∥∥∥2
+ ‖d‖2
}− ‖z‖2
(49)
The validation of the specifications of cases (a) and (b)follows the same argument developed in Section 3.1.Note that to keep the estimated parameters bounded, thediscontinuous projection is also applied to the adaptive lawin (42)–(44). This completes the proof. �
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
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Remark 8: In Theorems 1 and 2, there are many parametersthat need to be determined, such as γ , ρ1, ρ2, β, κ and �i.Note that here parameters ρ1 and ρ2 are the weightingcoefficients of the penalty signal z and are usually selectedas 1. Parameters β and �i are easily determined accordingto the requirements of the closed-loop system.
Remark 9: We can see from (34) and (49) that thehigh robustness to external disturbance/elastic vibrations isguaranteed by a prescribed level γ and the smaller γ is, thebetter the resulting attitude control accuracy. However, smallvalues of γ require large control inputs. Hence, the valueof γ will be determined according to the attitude pointingaccuracy requirement of the considered spacecraft mission.
From the above analysis, controller (41) enables thespacecraft to track a desired trajectory in the presence ofparameter uncertainties, disturbances and even unknownthruster faults. From a practical perspective, one of the majorissues in the attitude control system design is that the signalτ(t) in (41) generated by the control law might not beimplemented because of physical constraints. A commonexample of such a constraint is actuator saturation, whichimposes limitations on the magnitude of the achievablecontrol input. When the actuator saturation is considered, theactual adaptive SMC being implemented is different from(41) as follows.
τ(t) = −τmaxSat(DTτt) (50)
where τt is given by (41), and Sat(DTτt) is defined as
Sat(DTτt) =
⎧⎪⎪⎨⎪⎪⎩
DTσ
‖D‖‖σ‖ , τmax ≤ ‖DTτt‖DTτt
τmax, 0 ≤ ‖DTτt‖ < τmax
(51)
|τi| ≤ τ imax and τmax = min{τ 1
max, τ 2max, . . . , τ l
max} (i = 1, 2, . . . , l)denotes the minimal value of the maximal allowable thrusterforce τ i
max of each thruster. Note that here τi (i = 1, 2, . . . , l)denotes the components of the thruster control forces. Thenthe stability analysis result is given in the following.
Theorem 3: Consider the flexible spacecraft attitude trackingsystem given by (6) and (37) with Assumption 1. Supposethat the functional thrusters are able to produce a combinedforce sufficient to allow the spacecraft to follow a giventarget in the sense that there exists a constant ε such that theτmax strictly dominates the unknown uncertainty, disturbanceand possible elastic vibration. That is
τmax ≥ ‖D‖λ
(∥∥∥∥M3�3 + M2�2 + H + δT[K C][
ηψ
]
−S(ω)δTψ + d
∥∥∥∥ + ε
)(52)
where λ is a design parameter and less than the minimumeigenvalue of DEDT. The control law in (50) with (51) isthen applied. For all possible initial conditions q(0) andω(0) and the reference trajectories qd and ωd, the controlobjectives (a) and (b) as stated in Section 2.4 are guaranteed.
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
Proof: From (50), two cases are needed to prove the stabilityof the closed-loop attitude system.
Case 1: For ‖DTτt‖ ≥ τmax, we consider a new Lyapunovfunction V = (1/2)σ TJ0σ + β[qT
e qe + (1 − qe0)2], and then
the time derivative of V can be calculated as
˙V ≤ ‖σ‖{−λτmax
‖ D‖ +∥∥∥∥M3�3 + M2�2 + H
+ δT[K C][
ηψ
]− S(ωe + Rωd)δ
Tψ + d
∥∥∥∥}
− β2qTe qe
≤ −ε‖σ‖ − β2qTe qe ≤ 0 (53)
Case 2: For ‖DTτt‖ < τmax, we consider the same Lyapunovfunction V as defined in (47), and the Lyapunov derivativecan be algebraically rearranged in steps identical to thoseemployed in deriving (48), and the same argument as inTheorem 2 can be applied. This completes the proof. �Remark 10: Based on the stability analysis in the previoussection, the designed FTC law can be summarised asfollows:
Step 1: Check the amplitude value of thruster limit such thatτmax = min{τ 1
max, τ 2max, . . . , τ l
max}.Step 2: Choose the control gain β such that the slidingsurface σ = 0 is stable and the system has the specifiedconvergence speed.
Step 3: Choose the scalar γ according to the attitude pointingaccuracy of the considered spacecraft mission.
Step 4: Based on Remarks 8 and 9, design the parametersκ , ρ1, ρ2, ε, α and �i (i = 1, 2, 3) such that the conditions in(28)–(29) or (45)–(46) are satisfied.
Step 5: Compute variables Mi (i = 1, 2, 3) by using (19),(23) and (39), respectively.
Step 6: Construct the designed control laws (24), (41) and(50) with the computed gain matrices and variables.
4 Simulated example and comparison ofmethods
The numerical application of the proposed control schemeto the attitude tracking control of a flexible spacecraftis presented using MATLAB/SIMULINK. The spacecraftparameters are chosen from [26] with the first fourelastic modes considered at �
1/21 = 0.7681, �
1/22 = 1.1038,
�1/23 = 1.8733, �
1/24 = 2.5496 rad/s and damping ratios ξ1 =
0.0056, ξ2 = 0.0086, ξ3 = 0.013 and ξ4 = 0.025. There aresix thrusters to be distributed symmetrically on three axisof the spacecraft body frame, and every two thrusters aremounted as a pair. For instance, thrusters 1 and 2 arearranged to produce a torque about the X -axis, thrusters 3and 4 are arranged to generate a torque about the Y -axis andthrusters 5 and 6 are mounted to produce a torque about theZ-axis. In this case, the propulsion force is perpendicularto the corresponding axis such that distribution matrix canbe simply determined by the distance ri (i = 1, 2, . . ., 6)between the centre of mass of spacecraft and the positionof the thruster. In the following simulation, r1 = r2 = 0.7,r3 = r4 = 0.6 and r5 = r6 = 0.5, such that the distributionmatrix D is given by
To validate the controller performance with and withoutvarious combinations of thruster faults, attitude trackingis considered in the simulation, and suppose the desiredattitude quaternion and angular velocity are selected asshown in (55) and (56) (see below) with the initial conditionsset at q0(0) = 0.173648, q1(0) = −0.263201, q2(0) =0.789603, q3(0) = −0.526402 and, ωd(0) = [0 0 0]T.In addition, the initial modal variables and their timederivatives, ηi(0) and ηi(0) (i = 1, 2, 3, 4), are assumed tobe ηi(0) = ηi(0) = 0, that is, the flexible appendages areinitially undeformed. The simulated periodic disturbancetorque is (see (57))
and the time-varying part of the moment inertia matrix isassumed to be given by
where rand (·) is a random number generator between −1and 1. For the purpose of illustration, the following faultscenario is considered: the first thruster totally fails after 6 s,the second thruster loses 60% of its control power after 10 s,the third thruster totally fails after 10 s, the fourth thrusterloses 60% of its control power after 12 s, the fifth thrustertotally fails after 15 s and the sixth thruster loses 40% of itscontrol power after 16 s.
In the context of the simulation, the gas jets (thrusters)produce on-off control actions, while the control signalscommanded by the SMC in (24), (41) or (50) are continuous(the discontinuous switching only occurred on the slidingsurface). Thus the control signals need to be implementedin conjunction with the on–off actuators. For discrete-typeactuators, continuous signals are converted into equivalentdiscrete signals by PWPF modulation [25]. The PWPFmodulator produces a pulse command sequence to the
Fig. 1 Time histories of the gas jet forces using the proposed control methods
Fig. 2 Time histories for the attitude tracking control using the proposed methods
a Error quaternionb Angular velocity errorc Vibration displacements
Case 1: proposed ASMFTC with faults (solid line); Case 2: proposed ASMFTC without faults (dashed line)
Case 3: proposed MASMFTC with faults (dotted line); Case 4: proposed MASMFTC without faults (dot-dash line)
thruster by adjusting the pulse width and pulse frequency.In its linear range, the average torque produced equalsthe demanded torque input. In this paper, details of the
characteristics and implementation of PWPF modulation arenot given (see [25]). Furthermore, the simulations have beenrendered more realistic by considering thruster limits, and it
is assumed that the maximum value of the control force foreach thruster (gas jet) is 10 N, that is, τmax = 10 N.
4.1 Proposed control without explicitlyconsidering thruster limits
To show the effect of the proposed adaptive sliding modefault-tolerant controller (ASMFTC) in (24), simulations wereperformed with the given initial conditions and the faultscenario in (58). Fig. 1a shows the time histories of the sixgas jets. Once a thruster has failed the force produced bythat thruster is zero. The time histories of error quaternion,error angular velocity and modal displacements of spacecraftare shown in Figs. 2a–c (solid line). In comparison with thenominal response with functioning actuators in Figs. 2a–c(dashed line), the tracking performance degrades to somedegree once the actuator failure is introduced, and thesettling time increases. However, the system ultimatelyregulates the tracking error to near zero within 30 s. Inaddition, the elastic vibrations are passively suppressedand the oscillations settle within 30 s. The last plot inFig. 2c shows the vibration energy response, which isdescribed by E = ηTη + ηTKη, and this energy showsalmost zero oscillations after 30 s. This illustrates thatthe designed controller is capable of reducing the systemvibration while maintaining the tracking capability of thespacecraft.
The attitude tracking system is also controlled by usingthe proposed modified adaptive sliding mode fault tolerantcontrol (MASMFTC) law in (41), where the other controllerparameters are fixed for a fair composition. The samesimulation case is repeated, and the results are shown inFigs. 1b and 2a–c (dotted line). The simulation of both the
nominal system and the case with failed gas jets also showthe ability of the controller to follow the desired referencesignals. Moreover, the oscillations are further suppressedbecause the elastic vibrations are considered in the designduring tracking. These results support the theoretical resultthat the performance of the controller can be achieved withthe parameter updating law even in the presence of anunknown inertia matrix.
The same simulation of the attitude tracking manoeuvreare repeated with a traditional PD controller and theresults are shown in Fig. 3 (solid line). The trackingperformance of the PD controller is significantly degradedafter the thruster faults are introduced; severe oscillationsalso occur after the thruster failures, as demonstrated inthe modal displacement and vibration energy responsesshown in Fig. 3b (solid line). Furthermore, the closed-loop system becomes unstable. Although some improvementmay be possible with different design control parametersets, there is little improvement in the attitude and velocityresponses.
For further comparison, the simulation is repeated usingthe conventional adaptive sliding mode controller (CASMC)designed in [15] for the system. The results of the simulationare shown in Fig. 4 (solid line). The CASMC shows sometracking ability because of its robustness, although thetracking performance is degraded after the thrusters fail.Fig. 4 shows that the attitude responses can be improvedsignificantly compared to the PD case, but result in severevibration compared with the proposed methods. Figs. 1–4 show that the two proposed designs perform betterthan the existing controller designs, even when existingdesigns adapt to the system parameters under externaldisturbances.
Fig. 3 Time histories for the attitude tracking control using PD control
a Error quaternionb Vibration displacements
Case 1: fault case (solid line); Case 2: fault-free case (dotted line)
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
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Fig. 4 Time histories for the attitude tracking control using CASMC
a Error quaternionb Vibration displacements
Case 1: fault case (solid line); Case 2: fault-free case (dotted line)
Fig. 5 Time histories for the attitude tracking control using the proposed methods with saturation limits
4.2 Proposed control explicitly consideringthruster limits
The proposed methods can achieve the desired trackingperformance with different thruster faults, when no thrusterlimits are explicitly considered. However, the controllerparameters must be selected carefully and in practice thereexists control output constraints. Here the limit of thethruster force is assumed to be 10 N. In order to overcomethese constraints, the modified controller design in (50)is employed for the system considered. To demonstratethe effectiveness of this controller, the simulations arerepeated with the previous two fault scenarios and the sameinitial conditions. The results are shown in Fig. 5, anddemonstrate that the proposed controller given in (50) workseffectively within the operational control limit, even whenthe thrusters fail.
Summarising all of the cases (normal and fault cases),it is noted that the proposed controllers can significantlyimprove the tracking performance compared to the PD andCASMC methods, in both theory and simulation. Also,in the fault case, the proposed controllers have betterperformance than the conventional controllers. In addition,extensive simulations were performed using different controlparameters, disturbance inputs and even combinations ofthruster faults. These results show that closed-loop systemattitude control and vibration stabilisation are accomplishedin spite of these undesired effects in the system. Moreover,the flexibility in the choice of the control parameterscan be utilised to obtain desirable performance whilemeeting the constraints on the control magnitude andelastic deflection. These control approaches provide thetheoretical basis for the practical application of the methodsof advanced control theory to flexible spacecraft attitudecontrol systems.
5 Acknowledgments
This work was partially supported by the NationalNatural Science Foundation of China (project numbers61004072 and 61174200) and the program for New CenturyExcellent Talents in University (project number NCET-11-0801).
6 Conclusions
A fault-tolerant adaptive SMC scheme has been developedfor flexible spacecraft attitude tracking manoeuvres withredundant thrusters in the presence of parametricuncertainties, disturbance and even unknown faults. Theproposed control design methods do not require anyfault detection, isolation or identification process toidentify the faults. The control formulation is based onLyapunov’s direct stability theorem by incorporating theH∞ performance criterion into the controller design. Thecontroller designs are evaluated using numerical simulationand compared with other control schemes from the literatureusing different types of thruster failure scenarios. The resultsshow that the robust FTC is able to recover from actuatorfailure and to guarantee that the reference attitude can befollowed, while the conventional schemes are unable toperform the attitude tracking manoeuvre. Moreover, thecontrol objective can be achieved even with thrust saturation.The approach assumed that the system is controllable withthe remaining actuators, and that the sufficient control torque
is available to counteract the undesirable effects producedwhen the failed actuator settles to an arbitrary value. Theattitude control problem studied assumed full state feedback;however, in practice angular velocity measurements may notbe available because of cost limitations or implementationconstraints. Therefore attitude FTC without angular velocitymeasurements should be investigated, and is the subject offurther investigation.
7 References
1 Boskovic, J.D., Li, S.M., Mehra, R.K.: ‘Intelligent control of spacecraftin the presence of actuator failures’. Proc. 38th IEEE Conf. onDecision and Control, Phoenix AZ, USA, 1999, pp. 4472–4477
2 Li, L., Ma, L.Y., Khorasani, K.: ‘A dynamic recurrent neuralnetwork fault diagnosis and isolation architecture for satellite’sactuator/thruster failures’, Lecture Notes Comput. Sci., 2005, 3498, (3),pp. 574–583
3 Chen, W., Saif, M.: ‘Observer-based fault diagnosis of satellite systemssubject to time varying thruster faults’, J. Dyn. Syst. Meas. Control,2007, 129, (3), pp. 352–356
4 Wu, Q., Saif, M.: ‘Robust fault diagnosis for a satellite largeangle attitude system using an iterative neuron PID observer’.Proc. American Control Conf., Minneapolis, USA, June 2006,pp. 5710–5715
5 Mao, Z., Jiang, B.: ‘Fault identification and fault tolerant control fora class of networked control systems’, Int. J. Innov. Comput. Inf.Control, 2007, 3, (5), pp. 1121–1130
6 Li, C.W., Tong, S.C., Wang, Y.T.: ‘Fuzzy adaptive fault tolerantsliding mode control for SISO nonlinear systems’, Int. J. Innov.Comput. Inf. Control, 2008, 4, (12), pp. 3375–3384
7 Jin, J.H., Ko, S.H., Ryoo, C.K.: ‘Fault tolerant control for satelliteswith four reaction wheels’, Control Eng. Pract., 2008, 16, (10),pp. 1250–1258
8 Cai, W.C., Xiao, X.L., Song, Y.D.: ‘Indirect robust adaptive fault-tolerant control for attitude tracking of spacecraft’, J. Guid. ControlDyn., 2008, 31, (5), pp. 1456–1463
9 Hess, R., Wells, S.R.: ‘Sliding mode control applied to reconfigurableflight control design’, J. Guid. Control Dyn., 2003, 26, (3),pp. 452–462
10 Kim, D., Kim, Y.: ‘Robust variable structure controller design forfault tolerant flight control’, J. Guid. Control Dyn., 2000, 23, (3),pp. 430–437
11 Singh, S.N.: ‘Rotational maneuvers of nonlinear uncertain spacecraft’,IEEE Trans. Aerosp. Electron. Syst., 1988, 24, (2), pp. 114–123
12 Singh, S.N., Araujo, A.D.: ‘Adaptive control and stabilization ofelastic spacecraft’, IEEE Trans. Aerosp. Electron. Syst., 1999, 35, (1),pp. 115–122
13 Maganti, G.B., Singh, S.N.: ‘Simplified adaptive control of an orbitingflexible spacecraft’, Acta Astronaut., 2007, 61, (7–8), pp. 575–589
14 Singh, S.N., Zhang, R.: ‘Adaptive output feedback control ofspacecraft with flexible appendages by modeling error compensation’,Acta Astronaut., 2004, 54, (4), pp. 229–243
15 Jasim, A., Vincent, T.C., Dennis, S.B.: ‘Adaptive asymptotic trackingof spacecraft attitude motion with inertia matrix identification’, J.Guid. Control Dyn., 1998, 21, (5), pp. 684–691
16 Nayeri, M.R.D., Alasty, A., Daneshjou, K.: ‘Neural optimal control offlexible spacecraft slew maneuver’, Acta Astronaut., 2004, 55, (10),pp. 817–827
18 Chen, H.M., Chen, Z.Y., Su, J.P.: ‘Design of a sliding mode controllerfor a water tank liquid level control system’, Int. J. Innov. Comput.Inf. Control, 2008, 4, (12), pp. 3149–3160
19 Hu, Q.L., Ma, G.F., Xie, L.H.: ‘Robust and adaptive variable structureoutput feedback control of uncertain systems with input nonlinearity’,Automatica, 2008, 44, (3), pp. 552–559
20 Iyer, A., Singh, S.N.: ‘Variable structure slewing control and vibrationdamping of flexible spacecraft’, Acta Astronaut., 1991, 25, (1),pp. 1–9
21 Hu, Q.L., Shi, P., Gao, H.J.: ‘Adaptive variable structure andcommanding shaped vibration control of flexible spacecraft’, J. Guid.Control Dyn., 2007, 30, (3), pp. 804–815
22 Zeng, Y., Araujo, A.D., Singh, S.N.: ‘Output feedback variablestructure adaptive control of a flexible spacecraft’, Acta Astronaut.,1999, 44, (1), pp. 11–22
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
www.ietdl.org
23 Hu, Q.L., Ma, G.F.: ‘Adaptive variable structure maneuvering controland vibration reduction of three-axis stabilized flexible spacecraft’,Eur. J. Control, 2006, 12, (6), pp. 654–668
24 Lee, K.W., Nambisan, P.R., Singh, S.N.: ‘Adaptive variable structurecontrol of aircraft with an unknown high-frequency gain matrix’,J. Guid. Control Dyn., 2008, 31, (1), pp. 194–203
25 Sidi, M.J.: ‘Spacecraft dynamics and control’ (Cambridge UniversityPress, 1997)
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1388–1399doi: 10.1049/iet-cta.2011.0496
26 Gennaro, S.D.: ‘Output stabilization of flexible spacecraft with activevibration suppression’, IEEE Trans. Aerosp. Electron. Syst., 2003, 39,(3), pp. 747–759
27 Yao, B., Tomizuka, M.: ‘Smooth robust adaptive sliding mode controlof robot manipulators with guaranteed transient performance’, Trans.ASME J. Dyn. Syst. Meas. Control, 1996, 118, (4), pp. 764–775
28 Schaft, A.J.: ‘L2 gain and passivity techniques in nonlinear control’(Springer-Verlag Press, 2000)
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