Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance

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Research article

Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance ∗

Xiuwei Huang a , , Guangren Duan a,b a b

Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, PR China State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, PR China

article

info

Article history: Received 1 November 2018 Received in revised form 28 August 2019 Accepted 28 August 2019 Available online xxxx Keywords: Attitude tracking control Prescribed performance Nonlinear extended state observer Dynamic surface control Fault-tolerant control Actuator saturation

a b s t r a c t Capture and control of a failed spacecraft can be achieved by a space manipulator installed in a service spacecraft. After this target has been captured, the combined spacecraft must be controlled in a prescribed way. The attitude attacking control of the combined spacecraft system is one major challenge since the mass properties of the whole spacecraft system and configuration matrix of the reaction wheels change, especially when actuator fault occurs. In this paper, a nonlinear disturbanceobserver-based fault-tolerant attitude control scheme is developed for the combined spacecraft with prescribed performance. Firstly, an approach is given to reconstruct the attitude tracking dynamics of the combined spacecraft with reaction wheels. Then, a fault-tolerant controller, based on dynamic surface method and nonlinear extended state observer, is developed whereby performance in the light of convergence time, stability and accuracy with inertia uncertainty, actuator saturation and external disturbance can be prescribed. Finally, comparative simulations in both actuator faults and actuator fault-free cases are conducted to show the superiority of the developed attitude tracking control method. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Nowadays, on-orbit failures are increasing and account for losses of billions of dollars [1]. Among these failed spacecraft, some are still functional but also need to be decommissioned for lacking fuel or momentum exchange device. For the purpose of prolonging the operational lifetime of these spacecraft, capturing the spacecraft through a space robotic arm mounted on a base spacecraft is an effective way [2,3]. After the nonfunctioning targets have been captured, the attitude takeover control of the combined spacecraft becomes an urgent and important problem. Once the target is captured, the dynamics of the combined spacecraft will change largely because of the increase in mass and variation in the center-of-mass [4]. In addition, the configuration matrix of reaction wheels suffering a large change will also influence the new attitude dynamics. The development of a new model of the combined spacecraft with reaction wheels has been addressed in [4], but in that case the position vectors of the reaction wheel was assumed to be known in the new body frame. In this paper, a new attitude dynamic model of the combined spacecraft considering the new configuration matrix of ∗ Corresponding author. E-mail addresses: [email protected] (X. Huang), [email protected] (G. Duan).

reaction wheels, will be determined in the new body frame of the combined system. For the combined spacecraft, actuator fault or failure is more likely to occur because the attitude control system only exists on the service spacecraft. Therefore, fault-tolerant control (FTC) is essential to improve the safety and reliability of the combined spacecraft. The existing approaches for FTC system design contains two main types: passive and active [5,6]. Comparing with the active FTC, the passive not only can endure several actuator faults simultaneously, but also reduce the computational burden [7]. During the past decades, extensive studies about passive FTC method have been explored for attitude control system of spacecraft [8–12]. Among these methods, faulttolerant control method with extended state observer (ESO) is more widely used, where high accuracy for estimating the mismatched disturbance and robustness of the closed-loop system are ensured [13]. In [8], a terminal-sliding-mode-based observer (TSMO) to rebuild all the states in the attitude system with actuator failures within finite time was presented. To improve the accuracy and weaken the chattering effect in the TSMO, a secondorder nonlinear-extended-state observer (NESO) is proposed for coupled spacecraft tracking maneuver [14]. Besides fault-tolerant ability, prescribed performances in terms of convergence time, stability and accuracy are also needed to be considered in the attitude tracking system. The ability

https://doi.org/10.1016/j.isatra.2019.08.041 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: X. Huang and G. Duan, Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.041.

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Fig. 1. Before and after capture of the target with space manipulator.

for an attitude control to induce known and prescribed performances that can be analytically validated, which means it can be implemented with certainty. The idea of the prescribed performance was firstly developed in [15] and [16] for the feedback linearizable nonlinear system. For this system, the control designer can prescribe an arbitrarily small residual set for the tracking error, the convergence rate of the closed-loop states is equal or more than a predefined value and prescribed maximum overshoot though more control torque may be needed. Though paper [2] also investigated the attitude takeover control problem with prescribed performance, actuator fault was not considered. In [7], spacecraft attitude tracking control problem with prescribed performance was addressed, where backstepping control method was used during the adaptive fault-tolerant controller design process. To avoid ‘explosion of complexity’ in backstepping method, [17] proposed a dynamic surface control method for a nonlinear MIMO strict-feedback systems. In the paper, a NSEO-based fault-tolerant controller is developed to ensure that attitude tracking error converges to zero within prescribed bound for combined spacecraft by dynamic surface method. The main contributions are presented in the following: 1. The attitude tracking dynamic equation of the combined spacecraft with reaction wheels is constructed in the new body frame, based only on knowledge of the position vector of the mass center of the combined spacecraft, which in practice can be estimated in-situ. 2. A fault-tolerant controller based on NESO and dynamic surface method is developed, to ensure that the attitude tracking error converges to one small neighborhood of zero satisfying prescribed performance with inertia uncertainty, actuator saturation and external disturbance. This paper is presented in the following structure. The attitude tracking dynamics model of combined spacecraft with reaction wheels is constructed considering inertia uncertainties, actuator faults/failures, actuator saturation and external disturbance in Section 2. In Section 3, dynamic surface control (DSC) method combining NESO will be proposed for the attitude tracking control system of combined spacecraft. Finally, Section 4 presents comparative simulation results of combined spacecraft with reaction wheels which illustrates the effective of the proposed method. Notation: Following notations are defined for the text. Rn×m and Rn are used to represent the n × m-dimensional real matrices space and n-dimensional Euclidean space, respectively. For any matrix A ∈ Rn×m , AT represents the transposition of matrix A. For [ ]T any vector g = g1 g2 g3 ∈ R3 , g × ∈ R3×3 represents its skew-symmetric matrix, in which

⎡

0

g × = ⎣ g3

−g2

−g3 0 g1

g2

⎤

−g 1 ⎦ 0

and satisfies g × f = g × f for any f ∈ R3 . I n represents For any γ ∈ R and a vector c = ]T matrix. [the n × n identity m c1 c2 · · · cm ∈ R , we define sigγ (c) = |c1 |γ sgn(c1 )

[

|c2 |γ sgn(c2 )

···

|cm |γ sgn(cm )

]T

where sgn(·) denotes the sign function. Meanwhile, ∥·∥ represents the standard Euclidean norm and |·| represents the absolute value. 2. Problem description 2.1. Attitude tracking model building of combined spacecraft with reaction wheels As shown in Fig. 1 [4], before capture, there are two objects in the space: a rigid service spacecraft with one rigid space manipulator and a rigid target spacecraft. After capture, these objects are connected as a combined spacecraft system. Then, the following assumptions are made: (1) Once the target spacecraft has been captured, all joints of space manipulators are locked and the combined spacecraft is seen as one rigid body. (2) The whole attitude control system of combined spacecraft is taken over by the service spacecraft. (3) The position of the mass center of the combined spacecraft is known. That assumption is reasonable, as it is demonstrated in [18], the knowledge of the mass center for the spacecraft body after capturing the debris can be determined, in-situ, by using a least-squares method to a high precision. Modified Rodrigues Parameters (MRPs) vector σ = e tan(ϑ/4) ∈ R3 is used to denote the combined spacecraft’s attitude in the body frame of combined spacecraft Fc (Oc xc yc zc ), where ϑ is Euler’s angle and e is Euler’s principal rotation axis [3]. Then the kinematics of the combined spacecraft is denoted by:

σ˙ = G(σ )ω

(1)

]T

[ , ω = ω1

]T

in which σ = σ1 σ2 σ3 ω2 ω3 is the angular velocity of the combined spacecraft expressed in frame Fc (Oc xc yc zc ), moreover, G(σ ) is kinematic matrix expressed as

[

] 1[ (1 − σ T σ )I3 + 2σ × + 2σσ T (2) 4 Now considering that the service spacecraft are mounted with four reaction wheels. Three of them are mounted where their spin axes are parallel to the body frame Fs (Os xs ys zs ), respectively, and the other one is mounted where its spin axis points to some fixed direction. After the target spacecraft has been captured, the mass center will shift. The configuration of reaction wheels before and after capturing target spacecraft are shown in Fig. 2. According to the results in [10] and [19], the angular momentum of the whole reaction wheels respect to service spacecraft’s G(σ ) =

Please cite this article as: X. Huang and G. Duan, Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.041.

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3

According to the above analysis, the entire angular momentum of combined spacecraft considering reaction wheels assembled on the service spacecraft is denoted by

¯ w = J ω + C¯ J w Ωw H = Jω + H

(3)

where J is the inertia matrix of the combined spacecraft, which can be computed by [4]. Then the dynamic equation of the attitude control system expressed in the frame F c (Oc xc yc zc ) can be denoted as

˙ w + ω × H = T ext ˙ + C¯ J w Ω Jω

Fig. 2. Reaction wheels configuration before and after capturing target spacecraft.

mass center Os expressed in the frame Fs (Os xs ys zs ) can be denoted as

where C is the reaction wheels’ configuration matrix expressed in the frame Fs (Os xs ys zs ), defined as [20] C = c1

where T ext is the whole external torque imposing on the combined spacecraft. In this paper, this total external torque can be written as T ext = ug + d with the gravity gradient torque ug ∈ R3 and the disturbance torque d ∈ R3 which satisfies the following one. Assumption 1. The external disturbance d and its time derivative are also bounded with unknown positive constants. And ug is written as [21]

H w = C hw

[

(4)

c2

c3

c4

]

ug = 3ω02 R 3 (σ ) × J R 3 (σ )

(5)

in which ω0 is the value of orbit angular velocity and R 3 (σ ) is expressed as 8σ3 σ3 − 4σ2 (1 − σ T σ )

⎡ 1

⎤

with the position of ith reaction wheel c i in frame Fs (Os xs ys zs ). And

R 3 (σ ) =

hw = Ωw J w

Then the kinematics and dynamics of the combined spacecraft expressed in Fc (Oc xc yc zc ) is

]T

[ = Ωw 1

where Ωw Ωw2 Ωw3 Ωw4 ∈ R is the angular velocity of reaction wheels respect to service spacecraft expressed in the reaction wheels frame and J w = diag{Jw1 , Jw2 , Jw3 , Jw4 } ∈ R4×4 is the whole reaction wheels’ the inertia tensor. Based on the above assumptions, since the combined spacecraft is seen as a rigid body, the angular rate of reaction wheels which respect to the combined spacecraft is the same to the angular velocity of reaction wheels respect to the service spacecraft. And both J w and Ωw are expressed in the reaction wheels frame, the change of inertia tensor and mass center of the spacecraft will not change that value. Thus, the value of hw will not change even in the new body frame. Now let us denote the angular momentum of the whole reaction wheels respect to combined spacecraft’s ¯ w , then mass center Oc expressed in frame Fc (Oc xc yc zc ) as H 4

¯ w = C¯ hw H where C¯ is the reaction wheels’ configuration matrix expressed in the frame Fc (Oc xc yc zc ), defined as

[ C¯ = c¯ 1

c¯ 2

c¯ 3

c¯ 4

]

with the position of ith reaction wheel c¯ i expressed in frame Fc (Oc xc yc zc ) and can be written as c¯ i = R sc (c i − r c ) where r c is the vector from Os to Oc expressed in the frame Fs (Os xs ys zs ) and R sc is the rotation matrix from Fs (Os xs ys zs ) to Fc (Oc xc yc zc ). Remark 1. Comparing to [4] where the configuration matrix C¯ was assumed to be known, this paper first reconstructs the configuration matrix C¯ only based on the mass center of combined spacecraft r¯ c and the relationship between C¯ and r¯ c is first established.

{

(1 + σ T σ )

8σ3 σ2 + 4σ1 (1 − σ T σ )

⎣

4(σ − σ − σ 2 3

2 2

2 1)

+ (1 − σ σ ) T

⎦

(6)

2

σ˙ = G(σ )ω ˙ w + ug + d ˙ = −ω × (J ω + C¯ J w Ωw ) − C¯ J w Ω Jω

(7)

Furthermore, the desired attitude and angular velocity σ d and ωd expressed in the frame Fd are satisfying the following assumption. Assumption 2. Desired signals ωd , σ d and σ˙ d are assumed to be known and bounded. Same to [22], the attitude and angular velocity tracking error can be denoted by

⎧ × T T ⎨σ = σ d (σ σ − 1) + σ (1 − σ d σ d ) − 2σ d σ e T T T 1 + σ σσ d σ d + 2σ d σ ⎩ ωe = ω − R(σ e )ωd

(8)

with rotation matrix R(σ e ) from Fd to Fc denoted by R(σ e ) = R(σ )R(σ d )T

(9)

and can be expressed as R(σ e ) =

1 (1 + σ σ e T e

T × 2 (1 + σ Te σ e )2 I 3 − 4σ × e (1 − σ e σ e ) + 8(σ e )

[ )2

]

(10) Thereby, the attitude tracking kinematics and dynamics of combined spacecraft with reaction wheels is denoted by

⎧ ⎨σ˙ e = G(σ e )ωe × ¯ ˙ = −W × Jω e (J W e + C J w Ωw ) + J ω e R(σ e )ω d ⎩ e ¯ ˙ w + ug + d −J R(σ e ω˙ d ) − C J w Ω where W e = ωe + R(σ e )ωd and G(σ e ) = 2σ e σ Te ].

1 4

(11)

[(1 − σ Te σ e )I 3 + 2σ × e +

Please cite this article as: X. Huang and G. Duan, Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.041.

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As stated in [23], inertia property of the combined spacecraft is difficult to obtain from ground or be estimated by on-line identification. Thus for the actual inertia matrix, there is a certain deviation comparing with the estimated one. Now define J = J 0 + ∆J

(12)

where J 0 represents the estimated inertia matrix of the combined spacecraft, ∆J represents error matrix between the actual inertia matrix of the combined spacecraft and the estimated one, and the following assumption is satisfied. Assumption 3. J 0 is nonsingular and ∆J is bounded.

T3 ), Υ4 (t − T4 )}, where Ti is the time when faults occur, and Υi (t − Ti ) is the time schedule when ith wheel fault occurs, i.e. { 1 − e−ai (t −Ti ) , if t ≥ Ti (16) Υi (t − Ti ) = 0, if t < Ti in which ai > 0 represents the fault evolution rate. Then, attitude tracking control system for combined spacecraft with reaction wheels containing a general actuator faults model is stated by

⎧ ⎨σ˙ e = G(σ e )ωe 1 × × ¯ ω˙ = −J − 0 W e (J 0 W e + C J w Ωw ) + ω e R(σ e )ω d ⎩ e −1 ¯ −1 −R(σ e )ω˙ d − J 0 C J w sat(uc ) + J 0 ug0 + δ21 + δ22

(17)

where

Furthermore 1 ˜ J −1 = J − 0 + ∆J

(13)

1 −1 −1 −1 where ∆J˜ = −J − J 0 . Taking (12) and (13) into 0 ∆J (I 3 + J 0 ∆J ) the system (11) and defining the reaction wheels’ angular accel˙ w as the control input, we can get the dynamics eration vector Ω and kinematics of the combined spacecraft with reaction wheels under inertia uncertainty as follows

⎧ ⎨σ˙ e = G(σ e )ωe 1 × × ¯ ω˙ = −J − 0 W e (J 0 W e + C J w Ωw ) + ω e R(σ e )ω d ⎩ e −1 ¯ − 1 ˙ w + J 0 ug0 + δ21 −R(σ e )ω˙ d − J 0 C J w Ω

(14)

1¯ ¯ δ22 = −J − 0 C J w Υ(t , T fault )[(Λ(t) − I 4 )sat(uc ) + uc ]

Furthermore, commanded angular acceleration vector of reaction wheels uc with saturation can be rewritten as sat(uc ) = uc + ∆u

(18)

[ ]T with ∆u = ∆u1 ∆u2 ∆u3 ∆u4 and ⎧ ⎨uci max − uci , if uci > uci max ∆ui = 0, if uci min ≤ uci ≤ uci max , i = 1, 2, 3, 4 ⎩ uci min − uci , if uci < uci min

(19)

ug0 = 3ω

where uci max and uci min are the upper bound and lower bound of uci , respectively. Then the attitude tracking dynamics and kinematics of the combined spacecraft with reaction wheels under actuator faults and saturation is rewritten by

with

⎧ ⎪ σ˙ = G(σ e )ωe ⎪ ⎪ e ⎨ 1 × × ¯ ω˙ e = −J − 0 W e (J 0 W e + C J w Ωw ) + ω e R(σ e )ω d − 1 − 1 − ⎪−R(σ e )ω˙ d − J 0 C¯ J w uc + J 0 ug0 − J 0 1 C¯ J w ∆u + δ2 ⎪ ⎪ ⎩ ˙ w = uc + δ3 Ω

where

σ ) × J 0 R 3 (σ ) −1 ¯ δ21 = −∆J˜ W × W× e (J 0 W e + C J w Ωw ) − J e ∆J W e − 1 2 ˜ ¯ ˜ ˙ w + ∆J ug + 3ω0 J 0 R 3 (σ ) × ∆J R 3 (σ ) + J −1 d − ∆J C J w Ω 2 0 R3(

⎡ ⎤ 0

R 3 (σ ) = R(σ e )R(σ d ) ⎣0⎦ 1 Remark 2. In this paper, we directly use the angular acceleration ˙ w as the control input, which is also vector of reaction wheels Ω the actuator input in reality. It means that the designed control signals can apply to the attitude tracking control system directly without any control allocation methods. 2.2. Attitude model with actuator fault and constraints According to the analysis in [24] and [25], four different faults will come to reaction wheel: failure to respond to control signals, rased bias torque, reduced reaction torque, and successive generation of reaction torque. Taking both the constraints on the angular acceleration vector of reaction wheels and the fault occurrence rate into account, the relationship between the actual angular acceleration vector and the commanded angular acceleration vector of reaction wheels is written as

˙ w = sat(uc ) + Υ(t , T fault )[(Λ(t) − I 4 )sat(uc ) + u¯ c ] Ω

(15)

]T

[

where uc = uc1 uc2 uc3 uc4 denotes the commanded angular acceleration vector of reaction wheels, u¯ c = [ ]T u¯ c1 u¯ c2 u¯ c3 u¯ c4 is the acceleration vector bias, Λ(t) = diag{Λ1 , Λ2 , Λ3 , Λ4 } denotes the effectiveness matrix of the reaction wheels. The matrix Υ(t , T fault ) ∈ R4×4 with T fault =

[

T1

T2

T3

T4

]T

denotes the time when reaction wheels faults

occur, i.e. Υ(t , T fault ) = diag{Υ1 (t − T1 ), Υ2 (t − T2 ), Υ3 (t −

(20)

where δ2 = δ21 + δ22 and δ3 = ∆u + Υ(t , T fault )[(Λ(t) − I 4 )(uc + ∆u) + u¯ c ]. In this paper, our target is to develop a NESO-based faulttolerant control scheme for the attitude tracking system (20) achieving two goals: (1) the whole states of attitude tracking closed-loop control system are uniformly ultimately bounded. (2) attitude tracking error σ e in system (20) converges to a small residual set with prescribed performance. 3. Control system design This section will propose a NESO-based fault-tolerant controller to make the attitude converge to the desired command signal and guarantee the closed-loop attitude tracking system satisfies prescribed performance. And the structure of control scheme of attitude tracking problem for combined spacecraft with reaction wheels is displayed in Fig. 3. 3.1. Nonlinear extended state observer design In this subsection, a NESO is constructed to estimate the lumped disturbance within finite time with high accuracy. The core idea of the NESO is to regard lumped disturbance of the sys[ ]T tem as a new extended state [26]. Define x1 = σ e = x11 x12 x13 ,

]T

x2 = ωe = x21 x22 x23 , and a new extended state variable x3 = δ2 with x˙ 3 = δ˙ 2 = g (t), and the following assumption is satisfied.

[

Please cite this article as: X. Huang and G. Duan, Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.041.

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Fig. 3. Folw chart of the developed control scheme for combined spacecraft with reaction wheels.

Assumption 4 ([14,27]). The lumped disturbance δ2 in system (20) is differentiable and bounded by an unknown positive constant, and its derivative g (t) is also bounded by ∥g (t)∥ ≤ g¯ with ¯ a positive constant g.

(21)

1 ˙ d + J− + x× 2 R(x1 )ω d − R(x1 )ω 0 ug0 B = −J −1 C¯ J w 0

In order to decrease chattering effect and enhance the dynamical performance, we use the second-order NESO developed in [14], where both linear correction terms and nonlinear terms are contained and its form is established in the following form z˙ 1 = A + Buc + B∆uc + z 2 − κ1 sigα (ev 1 ) − κ2 ev 1

(22)

z˙ 2 = −κ3 sig2α−1 (ev 1 ) − κ4 ev 1

where z 1 , z 2 are the observer outputs representing the estimation [ ]T values of x2 , x3 ; ev 1 = z 1 − x2 = ev 11 ev 12 ev 13 is the estimation error, α ∈ [0.5, 1], and κi = diag{κi1 , κi2 , κi3 } ∈ R3×3 with κij > 0 for all i = 1, 2, 3, 4, j = 1, 2, 3.

]T

[

Now define ev 2 = z 2 − x3 = ev 21 ev 22 ev 23 , then the derivative of ev 1 and ev 2 can be obtained from (21) and (22) e˙ v 1 = −κ1 sigα (ev 1 ) − κ2 ev 1 + ev 2

{

(23)

e˙ v 2 = −κ3 sig2α−1 (ev 1 ) − κ4 ev 1 − g (t)

Lemma 1 ([14]). Consider the system (21) combined with the NESO in (22), the observer errors in (23) will converge to the region

 

{

}

(24)

in finite time with

⎧(   ) α  ⎫ ⎨ g¯ ζ  2α−1 g¯    ⎬ ζ j j mj = min , ⎩ λmin (Ξ1 ) λmin (Ξ2 ) ⎭

(25)

where λmin (Ξi ) denotes the minimum eigenvalue of matrix Ξi (i = [ ]T [ ] 1, 2), ξ j = sigα (ev 1j ) ev 1j ev 2j , ζ j = κ1j κ2j −2 and positive matrices

⎡ Ξ1 = κ1j ⎣

⎢

κ3j + ακ1j2

0

0

κ4j + (2 + α )κ2j2 −(α + 1)κ2j

−κ1j α

⎤

⎥ −κ2j ⎦ 1

κ3j κ4j >

κ2j2 κ3j α

+ (α 2 + 2α + 1)κ1j2 κ2j2

(26)

⎧(   ) α  ⎫ ⎨ g¯ ζ  2α−1 g¯    ⎬ ζ j j ≪1 min , ⎩ λmin (Ξ1 ) λmin (Ξ2 ) ⎭

1 × ¯ A = −J − 0 (x2 + R(x1 )ω d ) (J 0 (x2 + R(x1 )ω d ) + C J w Ωω )

H = ξ j ∈ R3 |ξ j  ≤ mj

κ4j + κ2j2 −κ2j

0

Remark 3. By choosing suitable κij and α such that

where

{

0

0

if the positive observer gains κij and α satisfy

Then, the system (20) can be rewritten and extended as

⎧ ⎨x˙ 1 = G(x1 )x2 x˙ = A + Buc + B∆u + x3 ⎩ 2 x˙ 3 = g (t)

⎡ κ3j + (2α + 1)κ1j2 ⎢ Ξ2 = κ2j ⎣ 0

⎤ −κ1j α ⎥ −(α + 1)κ2j ⎦ α

α and if 2α− is selected to be large enough, the elements in set H 1 will become sufficiently small, which shows that high accuracy performance of the proposed observer can be ensured.

Remark 4. The observer used not only keeps the main merits of terminal sliding mode control, but decreases the chattering effect and increases the accuracy of estimation. According to (23), it can be found out that when ev 1 , ev 2 are far away from origin, linear terms κ2 ev 1 , κ4 ev 1 dominate over the nonlinear ones κ1 sigα (ev1 ), κ3 sigα (ev1 ) because the linear system converges to zero with an exponential convergence speed; when ev 1 , ev 2 are near origin, a finite-time convergence rate is ensured for the nonlinear terms dominate the observer. In short, observer in (22) has high precision with a faster convergence speed no matter comparing to pure linear or nonlinear observer. 3.2. Fault-tolerant control design A NESO-based fault-tolerant controller with prescribed performance is designed for the attitude tracking system of combined spacecraft (21) in this part. Firstly, the attitude tracking error σ e of system (21) is transformed into a bounded vector by prescribed performance control (PPC) technique [15,16]. Then, DSC method is used to achieve a desired tracking performance for the attitude control system (21) based on the transformed vector. Meanwhile, the estimated lumped disturbance based on the NESO (22) is contained in the controller to compensate the effect of the lumped disturbance δ2 . Define the following state variables e2 = x2 − x2d

(27)

y 2 = x2d − x¯ 2

(28)

where e2 is the virtual surface error, x2d and x¯ 2 are state variable and intermediate control function defined later, and y 2 denotes the boundary-layer error between x2d and x¯ 2 .

Please cite this article as: X. Huang and G. Duan, Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.041.

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To ensure the tracking error x1 within the predefined bounded, following the idea of [15,16,28,29], the prescribed performance can be described as:

−δi,min χi (t) < x1i (t) < δi,max χi (t), i = 1, 2, 3, ∀t > 0

(29)

where χi (t) is a smooth decreasing function formulated as χi (t) = (χi0 − χi∞ )e−γi t + χi∞ with χi0 > χi∞ > 0, γi > 0, i = 1, 2, 3. χi0 can be set to fulfill −δi,min χi0 < x1i (0) < δi,max χi0 , and δi max , δi min are chosen positive constants. Here, χi0 , χi∞ represent the initial error bound and the maximum predefined steady error, respectively, and γi is the decreasing rate of χi (t). In order to accomplish the performance in (29), the constrained tracking error is transformed to one equivalent unconstrained vector, specifically, we define x1i (t) = χi (t)h ¯ [εi (t)], i = 1, 2, 3

(30)

is a smooth and strictly increasing function satisfying these two properties: (1) −δi,min < h ¯ (εi ) < δi,max ; (2) limεi →+∞ h¯ (εi ) = δi,max and limεi →−∞ h¯ (εi ) = −δi,min . It is easy to find out that function h ¯ (·) is strictly monotonically increasing, thus its inverse function exists, then the vector εi (t) can be derived as 1

εi (t) = h¯ −1 (λi (t)) =

2

ln

δi,min + λi δi,max − λi

, i = 1, 2, 3

(31)

in which λi (t) = x1i (t)/χi (t). Furthermore, the time derivative of λi (t) is

λ˙ i (t) = =

d(x1i /χi ) dt 1

χi

=

1

χi

(x˙ 1i − λi χ˙ i )

(σ˙ ei − λi χ˙ i ) =

[

1

χi

(I 3i G(x1 )x2 − λi χ˙ i )

[

(32)

]

[

where I 31 = 1 0 0 , I 32 = 0 1 0 and I 33 = 0 Then the derivative of εi (t) can be deduced as

ε˙ i =

1 ∂ h¯ −1 (λi ) λ˙ i = ∂λi 2

(

1

λi + δi,min

−

)

1

λi − δi,max

0

]

1 .

2

ln

(33)

δi,min , i = 1, 2, 3 δi,max

(34)

Remark 5. We will prove that the e1 (t) converges to the neighborhood of zero and is bounded. According to definition (31) and transformation (34), λ will also converge to the neighborhood of zero. For λi (t) = x1i (t)/χi (t), then x1i (t) will converge to zero with faster convergence rate compared with χi (t), i = 1, 2, 3.

]T

Based on (33) and (34), define χ = χ1 χ2 χ3 , r = diag{r1 , r2 , r3 } and λ = diag{λ1 , λ2 , λ3 }, we can obtain that

[

˙) e˙ 1 (t) = r(G(x1 )x2 − λχ

(35)

Now the controller design procedure is stated in the followings. Step 1: For (35), we define the virtual control law x¯ 2 as −1

x¯ 2 = −(rG(x1 ))

V˙ 1 = eT1 e˙ 1

= eT1 r(G(x1 )(e2 + x2d ) − λχ˙ ) = eT1 r(G(x1 )(e2 + y 2 + x¯ 2 ) − λχ˙ ) = −k1 eT1 e1 + eT1 rG(x1 )(e2 + y 2 )

= A + Buc + B∆u + x3 − x˙ 2d

˙) (k1 e1 − r λχ

(39)

(40)

Then the actuator command for the system (40) is designed as uc = B† [−A − k2 e2 − G T (x1 )re1 − k3 Bς − z 2 + y 2 /τ ]

(41)

where k2 and k3 are positive constants and B† = BT (BBT )−1 and ς is the output of the antiwindup saturation compensator

ς˙ = −k4 ς + ∆u

(42)

where k4 is a positive constant. Now consider the Lyapunov function candidate: 1

(eT e2 + ς T ς ) 2 2 then its derivative along the system (40) and (42) satisfies

(43)

V˙ 2 = eT2 e˙ 2 + ς T ς˙

+ eT2 B∆u − k4 ς T ς + ς T ∆u

λ˙ i

in which ri = 21χ ( λ +δ1 − λ −δ1 ) can be calculated in terms i i i i,max i,min of x1i and χi (t) for i = 1, 2, 3. [ ]T [ ]T Define e1 = e11 e12 e13 and ε = ε1 ε2 ε3 , and the following state transformation 1

1

(38) eT e1 2 1 then along the system (35), the derivative of V1 is given by V1 =

= eT2 (−k2 e2 − G T (x1 )re1 − k3 Bς − ev2 )

= ri (I 3i G(x1 )x2 − λi χ˙ i )

e1i (t) = εi (t) −

where its initial value is set as x2d (0) = x¯ 2 (0) Now let us consider the augmented Lyapunov function:

V2 =

]

(37)

e˙ 2 = x˙ 2 − x˙ 2d

ε −ε ε −ε h ¯ [εi (t)] = (δi max e i − δi min e i )/(e i + e i )

)

τ x˙ 2d + x2d = x¯ 2

Step 2: Taking the derivative of e2 , we can obtain that

where εi (t) is the transformed vector, and

(

To avoid repeatedly differentiating x¯ 2 , based on dynamic surface control method, let x¯ 2 passes through one first-order filter:

(36)

(44)

3.3. Stability analysis Based on Eqs. (35), (36), (37) and (42), the closed-loop attitude tracking dynamics control system of the combined spacecraft with actuator saturation is expressed as

⎧ ˙) e˙ 1 (t) = r(G(x1 )x2 − λχ ⎪ ⎪ ⎨ e˙ 2 (t) = A + Bsat(uc ) + x3 − x˙ 2d ⎪y˙ 2 = −y 2 /τ − x˙¯ 2 ⎪ ⎩ ς˙ = −k4 ς + ∆u

(45)

with the NESO (22). Then based on the dynamic surface control theory proposed in [30], the main result of this paper is obtained. Theorem 1. Consider the combined spacecraft attitude tracking model with reaction wheels (20), If Eq. (26) is satisfied and the 2 parameter of controller (41) satisfies 16k1 > 3rM τ , 4k2 k6 > 3 and 2 2 8k2 k4 > 9k3 k5 with k5 = ∥B∥, k6 > 0 and rM will be defined later, then the whole states of resulting closed-loop system are uniformly ultimately bounded, and attitude tracking error σ e is retained within the prescribed bound (29) for all t ≥ 0. Proof. Similar to the analysis in [30], we have

  x˙¯ 2  ≤ µ(e1 , e2 , y 2 )

(46)

Please cite this article as: X. Huang and G. Duan, Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.041.

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where µ is a continuous function. The Lyapunov function candidate of system (45) is selected as V = V1 + V2 + V3

(47)

1 T y y . 2 2 2

where V3 = Then along closed-loop system (45), the derivative of V is V˙ = V˙ 1 + V˙ 2 + V˙ 3

= −k1 eT1 e1 + eT1 rG(x1 )(e2 + y 2 ) + eT2 B∆u + eT2 (−k2 e2 − G T (x1 )re1 − k3 Bς − ev2 ) + −k4 ς T ς + ς T ∆u + y T (−y 2 /τ − x˙¯ 2 ) 2

2

2

≤ −k1 ∥e1 ∥ − k2 ∥e2 ∥ − k4 ∥ς∥2 −

1

∥y ∥2 τ 2 + ∥e1 ∥ ∥r ∥ ∥G(x1 )∥ ∥y 2 ∥ + ∥e2 ∥ ∥ev2 ∥ + k3 k5 ∥e2 ∥ ∥ς∥ + k5 ∥e2 ∥ ∥∆u∥ + ∥ς∥ ∥∆u∥ + |µ| ∥y 2 ∥

(48)

Now consider the set Ξ = {e1 , e2 , y 2 : V ≤ ϱ} with a positive constant ϱ, thus Ξ is a compact set. Since r is a continuous matrix function, there must be a positive constant rM satisfying ∥r ∥ ≤ rM within  the compact set Ξ . And for 1/4 ≤ ∥G(x1 )∥ = (1 + σ T σ )/4 ≤ 1/2, (48) can be rewritten as V˙ ≤ −k1 ∥e1 ∥2 − k2 ∥e2 ∥2 − k4 ∥ς∥2 −

1

τ

∥y 2 ∥2

7 ∑

Θi

(49)

i=1

Θ4 = Θ5 = Θ6 = Θ7 =

2 k1

∥e1 ∥2 −

k2

2k4

∥e2 ∥2 −

3 rM

∥ς∥2 −

3 2 rM

2 3τ

2 k2

(∥e1 ∥ −

k1 2

2k4

∥e1 ∥2 − ( 3k23 k25

2k2 3

−

1

) ∥e2 ∥2

4k6

3 3k25

−

) ∥ς∥2 − (

2

2 rM

) ∥y 2 ∥2 4k2 3τ 8k1 3 3τ +( + ) ∥∆u∥2 + |µ|2 + k6 ∥ev2 ∥2 4k2 4k4 4

−(

−

4. Simulation

≤ −γ V + ϵ

(50)

where

γ = 2 min{ ϵ=(

3k25 4k2

k1 k2

+

2

,

3

3 4k4

−

1 4k6

,

2k4

) ∥∆u∥2 +

3 3τ 4

−

3k23 k25 4k2

,

2 3τ

−

Remark 6. Compared with the backstepping method in [27], dynamic surface method is proposed in this paper. We note that the proposed control law does not involve the differentiation of G(x1 )−1 and thus can prevent the explosion of terms.

Remark 8. Comparing to [7], although fault-tolerant attitude tracking problems are studied in both papers, this work uses MRPs to represent attitude rather than quaternion and the faulttolerant model considered in this paper is more generalized, and also a NESO-based dynamic surface control method was used in this paper rather than the adaptive method. Comparing to [14] where an adaptive controller with an ESO was proposed to deal with spacecraft attitude tracking problem, this paper not only investigates a new fault-tolerant attitude tracking problem for combined system, but also proposes a control law ensuring the attitude tracking error converges to one small neighborhood of zero with prescribed performance.

then we can get V˙ ≤ −

■

Remark 7. It can be seen from the above proof process that the convergence rate of closed-loop states is mainly determined by γ , larger γ causes faster convergence rate. And γ is associated with the parameters of the controller, saturation compensator and observer k1 , k2 , k3 , k4 and τ , the larger k1 , k2 , k4 and smaller k3 , τ will cause the increase of γ , then a faster convergence rate of states in closed-loop system will be attained.

∥y 2 ∥2 ,

∥y 2 ∥) 2 + ∥y 2 ∥2 , 2k1 8k1 3k2 k2 3k3 k5 ∥ς∥)2 + 3 5 ∥ς∥2 , − (∥e2 ∥ − 3 2k2 4k2 2 3k 3k5 k2 ∥∆u∥)2 + 5 ∥∆u∥2 , − (∥e2 ∥ − 3 2k2 4k2 k4 3 3 2 ∥∆u∥) + ∥∆u∥2 , − (∥ς∥ − 3 2k4 4k4 1 3τ 3τ − (∥ y 2 ∥ − |µ|)2 + |µ|2 , 3τ 2 4 1 ∥e2 ∥2 + k6 ∥ev2 ∥2 4k6

Θ2 = − Θ3 =

k1

m21 + m22 + m23 . Then according to the above analysis and Eq. (52), ϵ ≤ η can be obtained with an unknown scalar η. If we select a positive scalar ϱ satisfying V˙ ≤ −γ ϱ + η < 0, then for any V (0) ≤ ϱ, V˙ ≤ −γ ϱ + η < 0 is guaranteed on V = ϱ. Thus V ≤ ϱ is an invariant set, which means that V (t) ≤ ϱ can be ensured for all t > 0 if V (0) ≤ ϱ is satisfied. Additionally, it is reasonable to assume that γ ϱ = η/υ with 0 < υ < 1. Implied by (50), V˙ < 0 for V ≥ ϱ∗ inside the invariant set, where ϱ∗ = η/γ = υϱ. Then ϱ∗ can be small enough by adjusting the value of υ . According to the definition of V (t), the error surface ∥e1 ∥ and ∥e2 ∥, saturation compensator ∥ς∥ and the boundary-layer error ∥y 2 ∥ are uniformly ultimately bounded, and moreover, could also become arbitrarily small by appropriately adjusting designed parameters of controller (41), filter (37) and compensator (42) as t → ∞. Furthermore, if the transformed vector e1 is bounded, then the attitude tracking error x1 (also σ e ) will be retained within a set, that is,

is true based on Remark 5.

where

Θ1 = −

Then we can obtain V (t) ≤ V (t0 )e−γ t + γ1 ϵ based on comparison principle. For Ξ is a compact set, there must be a maximum value of µ(e1 , e2 , y 2 ) on Ξ , and ∥∆u∥ can be bounded by a scalar [31]. Based on Lemma 1, the NESO error satisfies ∥ev 2 ∥ ≤ √

−δi,min χi (t) < x1i (t) < δi,max χi (t), i = 1, 2, 3, ∀t > 0

+ (rM /2) ∥e1 ∥ ∥y 2 ∥ + ∥e2 ∥ ∥ev2 ∥ + k3 k5 ∥e2 ∥ ∥ς∥ + k5 ∥e2 ∥ ∥∆u∥ + ∥ς∥ ∥∆u∥ + |µ| ∥y 2 ∥ =

7

2 rM

8k1

|µ|2 + k6 ∥ev2 ∥2 ,

} > 0,

(51) (52)

In this part, comprehensive simulations performed on a combined spacecraft were applied in the MATLAB/Simulink environment. In this combined spacecraft, a 3-DOF space manipulator connect the service spacecraft and service spacecraft. The target spacecraft carrying a push-broom-type remote sensing camera, with unfinished mission reorienting the satellite itself in some desired areas to take a sequence of high-definition pictures [32], has failed attitude control system. After the target spacecraft has been captured, the combined spacecraft is required to track the desired attitude by attitude maneuver. The involved differential

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Table 1 Performance function parameters.

4.1. The case of actuator fault free

ith

χi0

χi∞

γi

δi min

δi max

1 2 3

0.11 0.14 0.2

4 × 10−6 2 × 10−6 1.4 × 10−6

0.12 0.12 0.08

0.7 1 1

1 0.7 0.3

equations were simulated with a fixed-step Runge–Kutta solver (0.1 s). During the whole simulation, the model used is equation (13) and the relevant parameters is subsequently defined. The nominal inertia matrix of the combined spacecraft is J 0 = diag{10, 15, 20} kg m2 , and the uncertain one is ∆J = diag{2, −1, 2} kg m2 [27]. For each reaction wheel, the inertia is J wi = 0.338 kg m2 , i = 1, 2, 3, 4, the maximum and minimum value on the angular acceleration vector of reaction wheels are uc max = [10, 10, 10, 10]T and uc min = [−7, −7, −7, −7]T rad/s2 . The reaction wheels’ configuration matrix of the service spacecraft expressed in the frame Fs (Os xs ys zs ) is set as

⎡

1

0

0

C = ⎣0

1

0

0

0

1

√ ⎤ √ 1/ 3⎦ √ 1/ 3 1/ 3

and we assume that the reaction wheels configuration matrix C¯ expressed in the frame Fc (Oc xc yc zc ) can be expressed as

⎡ −0.9992 C¯ = ⎣−0.0394 0.0009

−0.0300 0.7455 −0.6658

−0.0256 0.6653 0.7461

⎤ −0.6086 0.7913 ⎦ 0.0469

Furthermore, the external disturbance model is 10 times the one in [33], which is 5 cos(0.02t)

⎡

⎤

d = 10−3 × ⎣−5 cos(0.025t)⎦ N m 6 cos(0.04t) The combined spacecraft in the simulation is assumed to be tumbling, which means its initial angular velocity is nonzero. Same to [33], the initial attitude is σ (0) = [0.323, −0.194, −0.388]T , the initial angular velocity is ω(0) = [−0.01, 0.02, −0.03]T rad/s. Besides, other initial variables are given as ς = z 1 = z 2 = [0, 0, 0]T . Without loss of generality, the same to [33], the desired signal is governed by the initial attitude σ d (0) = [0.217, −0.109, −0.163]T and the corresponding time-varying angular velocity is ωd = [0.02 cos(t /100), −0.01 sin(t /100), −0.03 cos(t /100)]T rad/s. And the parameters of prescribed-performance bound (PPB) are presented in Table 1. The parameters of nonlinear ESO are set as 0.2

⎡

κ1 = ⎣ 0

0 0.08

⎡

κ3 = ⎣ 0 0

0

0

0.2

0.03

⎤

⎡

0 ⎦ , κ2 = ⎣ 0 0 .2

0 0 0.08 0

0 0.03

0 0

⎤

0.06

⎡

0

⎤

0 ⎦ 0.03

0

0 ⎦ , κ4 = ⎣ 0 0.08

0

0 0.06 0

0

⎤

0 ⎦ 0.06

and α = 0.7. The controller parameters are set as k1 = 0.1, k2 = 4, k3 = 0.01, k4 = 22, k6 = 1 and τ = 0.1, such that 2 16k1 > 3rM τ , 8k2 k6 > 3 and 8k2 k4 > 9k23 k25 with k25 = 1.72 ∗ 10−5 are satisfied.

In that part, the case that the whole reaction wheels operate normally is considered. Beside the proposed algorithm (DSC + PPC), traditional PD-controller uc = −B† (β1 σ e + β2 ωe ) [34](PD), the robust fault-tolerant control method (RFT) in [35] and proposed algorithm without PPC (DSC) are used to show the advantages of the developed control method. The design parameters of PD and RTF controllers are selected through trial and error to make three controllers have nearly identical convergence rates. The time trajectories of closed-loop system are shown in Figs. 4–9. Figs. 4–6 present the time history of attitude tracking error trajectories by DSC+PPC, DSC, RFT, and PD methods, Fig. 7 presents the time history of angular error trajectory by DSC+PPC method, the time history of control input by DSC+PPC, DSC, RFT, PD methods is shown in Fig. 8, and the time history of the lumped disturbance δ2 estimated error ev 2 of the lumped disturbance by DSC+PPC method is shown in Fig. 9. From Figs. 4–6, it is shown that satisfied attitude tracking errors have been accomplished with the developed DSC+PPC, which means that the developed controller can make sure all the attitude tracking errors converge to a predefined tiny neighborhood of zero at a faster speed compared with PD, RFT or DSC method. Furthermore, the proposed controller can guarantee the prescribed control performance, while all the attitude tracking errors by DSC, RFT and PD fail though DSC method achieve better convergence performance than PD method after 100 s. Fig. 7 shows that the trajectory of the angular tracking error ωe is less than 1 × 10−7 after 70 s. Together with the attitude tracking errors in Figs. 4 and 6, it could be found that attitude tracking errors σ e with prescribed performance and angular tracking errors ωe with high precise are accomplished by the developed control algorithm under no actuator fault case. From Fig. 8, the control saturation effect can be clearly seen, where the control torque is constrained in the prescribed limitation. Both RFT and PD methods produce much more control torques than the other two methods. Comparing to DSC, a bit more control torque is needed in the proposed method. This is reasonable and acceptable for a better state performance. Finally, the lumped disturbance and estimated parameter errors are provided in Fig. 9. It is obvious to observe that inertia uncertainties, actuator saturation and external disturbance on the attitude tracking system is restrained by comprising the estimated value in actuator input and the estimated error will be less than 2 × 10−6 after 30 s, which thinks to the precise estimation of the lumped disturbance by the NESO. In short, the effectiveness and the superiority of the developed PPC-based attitude tracking controller has been demonstrated under no actuator fault case. 4.2. The case of actuator faults In that part, the case that the reaction wheels suffer from actuator faults is considered. Beside the proposed algorithm (DSC + PPC-FTC), both traditional PD-controller uc = −B† (k1 σ e + k2 ωe ) (PD), the robust fault-tolerant control method (RFT) in [35] (RFT-FTC) and proposed algorithm without PPC (DSC-FTC) under actuator fault are used to show the advantages of the developed control method. The design parameters of PD and RTF controllers are the same to the ones in actuator fault-free case. The parameters of actuator faults are stated as follows. The reaction wheel RW1 loses 50% power (i.e. , e1 = 0.5) and with a bias of u¯ 2 = 0.3 after 5 s. The reaction wheel RW2 also undergoes 50% power off (i.e., e2 = 0.5) after 10 s without a bias. The reaction wheel RW3 experiences a bias with the value of u¯ 3 = −0.3 without power off (i.e., e3 = 1) during the time interval [5, 15] s. The fourth wheel RW4 is always healthy (i.e., e4 = 1, u¯ 3 = 0). The fault evolution

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Fig. 4. Time histories of spacecraft attitude without actuator fault.

Fig. 6. Time histories of spacecraft attitude without actuator fault.

Fig. 5. Time histories of spacecraft attitude without actuator fault.

Fig. 7. Time histories of spacecraft attitude angular velocity without actuator fault.

rate ai = 1, i = 1, 2, 3, 4. The time trajectories of closed-loop system are shown in Figs. 10–15. Figs. 10–12 present the time history of attitude tracking error trajectories with actuator faults by DSC+PPC−FTC, DSC−FTC, RFT−FTC and PD methods, Fig. 13 presents the time history of angular error trajectory with actuator faults by DSC+PPC−FTC method, Fig. 14 shows the time history of control input by DSC+PPC−FTC, DSC−FTC, RFT−FTC and PD. Fig. 15 shows the time history of the lumped disturbance δ2 and estimated error ev 2 of the lumped disturbance with actuator faults by DSC + PPC-FTC method. As we can see from Figs. 10–13 by the method of DSC+PPC− FTC, the attitude tracking errors can also converge to zero within prescribed bounds with smaller convergence rate compared to the actuator fault-free case. Similarly, the attitude tracking errors by DSC−FTC, RFT−FTC and PD violate the given bound. Furthermore, Fig. 14 shows that the actuator torque is also saturated in the prescribed constraints, but it its magnitude is larger comparing to actuator fault-free case, which means more control torque is needed to keep the system performance when actuators

faults occur. In Fig. 15, the lumped disturbance and estimated parameter errors with actuator faults are presented. Although the lumped disturbance is much larger comparing to actuator faultfree case, it is also well estimated by the NESO and the estimated error is a little bit larger than the no-fault case. As a result, an acceptable system performance can still be maintained though actuator faults exist, which demonstrates that the developed attitude tracking controller has prominent robustness even under actuator faults. 5. Conclusion In this paper, a NESO-based FTC control method with prescribed performance has been designed for the rebuilt attitude tracking control system. Firstly, compared to the existing works, the attitude tracking dynamics for the combined spacecraft with reaction wheels was firstly clearly and completely stated. Then,

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Fig. 10. Time histories of spacecraft attitude with actuator faults. Fig. 8. Time histories of spacecraft control torque without fault.

Fig. 11. Time histories of spacecraft attitude with actuator faults. Fig. 9. Time histories of the lumped disturbance δ2 and estimated error ev 2 without fault.

based on the established equation, a NESO was applied to evaluate the integrated disturbance comprised of inertia uncertainties, actuator fault, actuator uncertainties and external disturbance. Furthermore, a prescribed performance function and an associated error transformation method were suggested to transform the attitude tracking error to the transformed vector, and then a dynamic surface based FTC method avoiding the problem of “explosion of complexity” was presented for the attitude tracking system of the combined spacecraft. Besides, tuning conditions of controller parameters were obtained according to Lyapunov analysis. Moreover, the prescribed performance of the attitude tracking is ensured and the other closed-loop states is uniformly ultimately bounded. Finally, comparable numerical examples were shown to illustrate the performance superiority of the developed control scheme comparing to other methods. In the future, output

feedback control of the combined spacecraft will be possible areas of further research. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported by the Major Program of National Natural Science Foundation of China (Nos. 61690210, 61690212), Self-Planned Task (No. SKLRS201716A) of State Key Laboratory of Robotics and System, China (HIT) and the National Natural Science Foundation of China (No. 61333003).

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Fig. 12. Time histories of spacecraft attitude with actuator faults. Fig. 14. Time histories of spacecraft control torque with faults.

Fig. 13. Time histories of spacecraft attitude angular velocity with actuator faults.

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Please cite this article as: X. Huang and G. Duan, Fault-tolerant attitude tracking control of combined spacecraft with reaction wheels under prescribed performance. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.041.