# Neural Network-Based Finite-Time Fault-Tolerant Control for Spacecraft without Unwinding

Authors: Chao Tan; Guodong Xu; Limin Dong; Han Zhao; Jun Li; Sai Zhang
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

In this paper, we focus on solving the problems of inertia-free attitude tracking control for spacecraft subject to external disturbance, unknown inertial parameters, and actuator faults. The robust control architecture is designed by using the rotation matrix and neural networks. In the presence of external disturbance and parametric uncertainties, a fault-tolerant control (FTC) scheme synthesized with the minimum-learning-parameter (MLP) algorithm is proposed to improve the reliability of the system when unknown actuator faults occur. These methods are developed based on backstepping to ensure that finite-time convergence is achievable for the entire closed-loop system states with low computational complexity. The validity and advantage of the designed controllers are highlighted by using Lyapunov-based analysis. Finally, the simulation results demonstrate the satisfactory performance of the developed controllers.


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## Body

## 1. Introduction

With the rapid development of space engineering, spacecraft attitude control has been studied extensively by researchers for its essential role in various space missions, involving deep space exploration, earth observation, Mars detection, etc. As an important part of spacecraft, control systems determine the success of these missions for the most part. However, designing controllers with satisfactory performance for spacecraft still remains a challenging task, which can be attributed to complex external disturbance, unknown inertial parameters, and actuator faults. Despite these obstacles, numerous methods have been designed to address the attitude tracking control problem, including backstepping control [1–4], sliding mode control [5–7], prescribed performance control [8–10], and event-triggered control [11, 12].Though fruitful results have been obtained for attitude control of spacecraft during the last decades, it still remains a challenge to design finite-time controllers under the circumstances of the external disturbance and unknown inertial matrix. On the one hand, modern space missions put forward high requirements in fast response and strong robustness for attitude control systems, which could be resorted to finite-time control strategies [13, 14]. On the other hand, inertial parameters are always time-varying with continuing fuel consumption during space missions, which makes it challenging to acquire accurate values of the inertial matrix. On account of the fast response requirement, finite-time control strategies [15–18] and appointed time approaches [19, 20] were adopted in the spacecraft system such that the control objective could be accomplished within a limited time. In [17, 18], terminal sliding mode-based methods were proposed for a single spacecraft and multiple spacecraft, respectively. Different from these two controllers, the prescribed performance control strategy was adopted in [19, 20], where the convergence time of the spacecraft system could be predefined by utilizing a nonlinear function. As for the unknown system parameters, NN-based control schemes [21, 22] were widely adopted for nonlinear systems, which have also been used to approximate unknown system dynamics in spacecraft systems [23, 24]. As for the stabilization of fractional-order systems with unknown parameters, the T-S fuzzy control theory synthesized with adaptive linear-like control rules shows its superior capability in dealing with uncertainties [25].Apart from the fast response and unknown system parameters, actuator faults are another important aspect that deserves special attention, especially for the need for high reliability in space activities, such as rendezvous and docking. Complex space environments and disturbances render the spacecraft vulnerable to actuator faults, which will deteriorate the control performance. To address this issue, researchers have introduced FTC schemes for various space applications [26–28]. Niu et al. proposed an observer-based FTC strategy for attitude stabilization of rigid spacecraft when time-varying faults occur [27]. Aiming at attitude synchronization control for multiple spacecraft, Zhou and Xia introduced a fault-tolerant method in the presence of modeling uncertainties [28]. With much more superiority in nonlinear fractional-order systems, Hashtarkhani and Khosrowjerdi proposed a fault-tolerant controller in conjunction with the backstepping terminal sliding mode technique for an uncertain faulty system [29]. Besides, fractional adaptive state feedback control laws based on backstepping were developed in [30] to help promote the stability of the system with actuator failures.Obviously, controllers in [26–28] have been constructed by using unit quaternion-based attitude descriptions, which have the inherent disadvantage of the unwinding problem. Due to this drawback, researchers focused on the rotation matrix-based attitude description, which can represent attitude motions globally and uniquely [31–33]. By resorting to the rotation matrix, Huang et al. proposed a fault-tolerant attitude tracking controller with finite-time stability [33]. Similarly, several control algorithms have been designed for rigid spacecraft with consideration of the unwinding problem [31, 32]. However, these controllers are not applicable for cases that inertial parameters are unknown. Therefore, further research is required to deal with the actuator fault, unknown system parameters, and unwinding problem.Inspired by the aforementioned obstacles in controller design for spacecraft, this paper focuses on providing two unwinding-free attitude tracking control architectures for spacecraft when there exist actuator faults and unknown inertial parameters. The main contributions of this paper are given as follows:(i)
A finite-time attitude tracking controller is constructed on the basis of the rotation matrix with unknown inertial parameters, so that finite-time tracking performance could be obtained for spacecraft attitude control. Compared with existing methods [16, 33], inertial parameters could remain unavailable, making the tracking controllers more applicable in aerospace engineering(ii)
Actuator faults and unknown system dynamics could be handled simultaneously with the aid of the FTC strategy proposed in this paper. Based on the MLP algorithm, unknown system dynamics can be approximated with low computational complexityThe rest of this paper is organized as follows. In Section2, the spacecraft model and preliminaries are presented. In Section 3, backstepping controllers are designed by utilizing MLP and FTC. In Section 4, the effectiveness of the proposed algorithms is validated through numerical simulations. Finally, the conclusions of this paper are presented in Section 5.


## 2. Spacecraft Model and Preliminaries

### 2.1. Spacecraft Attitude Dynamics

The unit quaternion has been widely used for attitude description in most attitude controllers [9–11]. However, the unit quaternion-based algorithm results in the unwinding problem with high fuel consumption. To address this problem, the rotation matrix is adopted here to describe the attitude dynamics of the spacecraft, which can be defined as follows:
(1)Ṙ=Rω×,(2)Jω̇=−ω×Jω+δu+d,where R∈SO3 denotes the rotation matrix; ω∈R3×1 is the angular velocity; J∈R3×3 denotes the inertial matrix; u∈R3×1 and d∈R3×1 are the control torque and external disturbance of the spacecraft, respectively; and δ=diagδ1,δ2,δ3 is the fault coefficient, where 0<θ<δi≤1, i=1,2,3, with θ being a positive constant. For a vector a=a1,a2,a3T, a× is defined as follows:
(3)a×=0−a3a2a30−a1−a2a10.For the attitude controller design, attitude and angular velocity tracking errorsR~ and ω~ are defined in Equations (4) and (5), respectively, where Rd∈SO3 and ωd∈R3×1 are the commanded rotation matrix and angular velocity, respectively.
(4)R~=RdTR,(5)ω~=ω−R~Tωd.Thus, the attitude dynamics can be rewritten as follows:(6)R~̇=R~ω~×,(7)Jω~̇=F+δu+d,where F=−ω~+R~Tωd×Jω~+R~Tωd−JR~Tω̇d+Jω~×R~Tωd.Attitude error functions presented in Equations (8) and (9) were selected to facilitate the attitude controller design.
(8)eR~=121+trR~R~−R~T∨,(9)ψR~=2−1+trR~.Here,∨ denotes the inverse operation of ×. Combining Equations (1)–(9), error dynamics can be reconstructed as follows:
(10)ėR~=Eω~,(11)Jω~̇=F+δu+d,(12)E=121+trR~trR~I−R~T+2eR~eR~T,(13)F=−ω~+R~Tωd×Jω~+R~Tωd−JR~Tω̇d+Jω~×R~Tωd.Remark 1.
To ensure Equation (10) is valid, trR~≠−1 must be satisfied. As stated by Huang et al., trR~≠−1 can be derived if the condition eR~≤1 is true [31–33]. Consequently, eR~≤1 must be satisfied during space missions.Remark 2.
The inertial parametersJ are regarded as unknown variables in the controller design. Minimum-learning-parameter algorithm-based NNs are employed to manage the unknown inertial matrix. In order to facilitate the controller design, the inertial matrix J is assumed to be a positive definite diagonal matrix.


### 2.2. Assumptions

The following assumption is necessary for the attitude controller design:Assumption 1.
The external disturbanced and commanded angular velocity ωd are bounded and satisfy d≤D and ω̇d≤ω¯, where D and ω¯ are positive constants.


### 2.3. Mathematical Notation

In this paper, the notation⋅ denotes the Euclidean norm of a vector or the induced norm of a matrix. For any ξ∈R, sigξχ is defined as sigξχ=signξξχ. For a vector ξ=ξ1,ξ2,⋯,ξnT, sigξχ is defined as sigξχ=signξ1ξ1χ,⋯,signξnξnχT. Here, χ is a constant that satisfies 0<χ<1. Furthermore, tanhξ=tanhξ1,tanhξ2,⋯,tanhξnT. For a matrix A, the notation λmaxA and λminA represent the maximum eigenvalue and minimum eigenvalue of A, respectively.


### 2.4. Related Lemmas

For the satisfactory performance in approximating functions, RBF NNs have been widely used in controller design for nonlinear systems. Therefore, RBF NNs are employed herein to address the unknown system dynamics caused by the unavailable inertial matrix. The following lemmas are introduced to facilitate the design process.Lemma 3 [20].
For a random continuous functionfx, an ideal weight vector W exists that satisfies the following function:
(14)fx=WThx+ε,where W=w1,w2,⋯,wpT and x=x1,x2,⋯,xm are Ge-Lee vectors defining weight and input vectors, respectively; p and m are the node and input number, respectively; ε is the approximation error; and hx=h1x,⋯,h2xT is the Gaussian basis function vector with the following form:
(15)hix=exp−x−ci222bi2,i=1,⋯,p,where ci∈Rm and bi∈R are the center vector and Gaussian basis function vector of hix, respectively.Lemma 4 [33].
If a vectorx∈R3 satisfies RdTR=expx×, the inequality is always tenable. The term E defined in Equation (12) satisfies E=1/2. Additionally, if x≠π, E is always invertible.Lemma 5 [33].
Considering positive constantsρ and αi, i=1,2,⋯,n, if the inequality 1<ρ<2 holds, the following equation is obtained:
(16)α12+⋯+αn2ρ≤α1ρ+⋯+αnρ2.Lemma 6 [33].
For the system Equations (19) and (20), if a Lyapunov function V1 exists such that it satisfies the following equation:
(17)V̇1≤−aV1α,where a and α are positive constants satisfying 0<α<1, then the system converges to the origin within a finite time.Lemma 7 [34, 35].
Given positive scalersx∈R and σ>0, the following key inequation can be derived:
(18)0≤x−x2x2+σ2≤x−x2x+σ<σ.


## 2.1. Spacecraft Attitude Dynamics

The unit quaternion has been widely used for attitude description in most attitude controllers [9–11]. However, the unit quaternion-based algorithm results in the unwinding problem with high fuel consumption. To address this problem, the rotation matrix is adopted here to describe the attitude dynamics of the spacecraft, which can be defined as follows:
(1)Ṙ=Rω×,(2)Jω̇=−ω×Jω+δu+d,where R∈SO3 denotes the rotation matrix; ω∈R3×1 is the angular velocity; J∈R3×3 denotes the inertial matrix; u∈R3×1 and d∈R3×1 are the control torque and external disturbance of the spacecraft, respectively; and δ=diagδ1,δ2,δ3 is the fault coefficient, where 0<θ<δi≤1, i=1,2,3, with θ being a positive constant. For a vector a=a1,a2,a3T, a× is defined as follows:
(3)a×=0−a3a2a30−a1−a2a10.For the attitude controller design, attitude and angular velocity tracking errorsR~ and ω~ are defined in Equations (4) and (5), respectively, where Rd∈SO3 and ωd∈R3×1 are the commanded rotation matrix and angular velocity, respectively.
(4)R~=RdTR,(5)ω~=ω−R~Tωd.Thus, the attitude dynamics can be rewritten as follows:(6)R~̇=R~ω~×,(7)Jω~̇=F+δu+d,where F=−ω~+R~Tωd×Jω~+R~Tωd−JR~Tω̇d+Jω~×R~Tωd.Attitude error functions presented in Equations (8) and (9) were selected to facilitate the attitude controller design.
(8)eR~=121+trR~R~−R~T∨,(9)ψR~=2−1+trR~.Here,∨ denotes the inverse operation of ×. Combining Equations (1)–(9), error dynamics can be reconstructed as follows:
(10)ėR~=Eω~,(11)Jω~̇=F+δu+d,(12)E=121+trR~trR~I−R~T+2eR~eR~T,(13)F=−ω~+R~Tωd×Jω~+R~Tωd−JR~Tω̇d+Jω~×R~Tωd.Remark 1.
To ensure Equation (10) is valid, trR~≠−1 must be satisfied. As stated by Huang et al., trR~≠−1 can be derived if the condition eR~≤1 is true [31–33]. Consequently, eR~≤1 must be satisfied during space missions.Remark 2.
The inertial parametersJ are regarded as unknown variables in the controller design. Minimum-learning-parameter algorithm-based NNs are employed to manage the unknown inertial matrix. In order to facilitate the controller design, the inertial matrix J is assumed to be a positive definite diagonal matrix.


## 2.2. Assumptions

The following assumption is necessary for the attitude controller design:Assumption 1.
The external disturbanced and commanded angular velocity ωd are bounded and satisfy d≤D and ω̇d≤ω¯, where D and ω¯ are positive constants.


## 2.3. Mathematical Notation

In this paper, the notation⋅ denotes the Euclidean norm of a vector or the induced norm of a matrix. For any ξ∈R, sigξχ is defined as sigξχ=signξξχ. For a vector ξ=ξ1,ξ2,⋯,ξnT, sigξχ is defined as sigξχ=signξ1ξ1χ,⋯,signξnξnχT. Here, χ is a constant that satisfies 0<χ<1. Furthermore, tanhξ=tanhξ1,tanhξ2,⋯,tanhξnT. For a matrix A, the notation λmaxA and λminA represent the maximum eigenvalue and minimum eigenvalue of A, respectively.


## 2.4. Related Lemmas

For the satisfactory performance in approximating functions, RBF NNs have been widely used in controller design for nonlinear systems. Therefore, RBF NNs are employed herein to address the unknown system dynamics caused by the unavailable inertial matrix. The following lemmas are introduced to facilitate the design process.Lemma 3 [20].
For a random continuous functionfx, an ideal weight vector W exists that satisfies the following function:
(14)fx=WThx+ε,where W=w1,w2,⋯,wpT and x=x1,x2,⋯,xm are Ge-Lee vectors defining weight and input vectors, respectively; p and m are the node and input number, respectively; ε is the approximation error; and hx=h1x,⋯,h2xT is the Gaussian basis function vector with the following form:
(15)hix=exp−x−ci222bi2,i=1,⋯,p,where ci∈Rm and bi∈R are the center vector and Gaussian basis function vector of hix, respectively.Lemma 4 [33].
If a vectorx∈R3 satisfies RdTR=expx×, the inequality is always tenable. The term E defined in Equation (12) satisfies E=1/2. Additionally, if x≠π, E is always invertible.Lemma 5 [33].
Considering positive constantsρ and αi, i=1,2,⋯,n, if the inequality 1<ρ<2 holds, the following equation is obtained:
(16)α12+⋯+αn2ρ≤α1ρ+⋯+αnρ2.Lemma 6 [33].
For the system Equations (19) and (20), if a Lyapunov function V1 exists such that it satisfies the following equation:
(17)V̇1≤−aV1α,where a and α are positive constants satisfying 0<α<1, then the system converges to the origin within a finite time.Lemma 7 [34, 35].
Given positive scalersx∈R and σ>0, the following key inequation can be derived:
(18)0≤x−x2x2+σ2≤x−x2x+σ<σ.


## 3. Design of the Controller

In this section, a fault-tolerant attitude tracking controller is presented to ensure the stability of the closed-loop system with finite-time convergence. For better applicability, MLP and adaptive laws are proposed to estimate unknown system information. Besides, a backstepping-based FTC scheme is proposed with an auxiliary system, thus improving system reliability when an unknown actuator fault occurs.Previous to designing the attitude controller, it is necessary to definex1=eR~=x11,x12,x13T and x2=Eω~−ω~p,1=x21,x22,x23T, so that the error dynamics in Equations (10) and (11) can be expressed as follows:
(19)ẋ1=Eω~,(20)ẋ2=Ėω~+EJ−1F+δu+d−ω~̇p,1.Step 1.
To prove thatx1 converges to a region containing the origin within a finite time, the virtual angular velocity ω~p,1 is expressed as follows:
(21)ω~p,1=−k1x1x12+σ2−k2x1,where k1 and k2 are positive constants and k2 should satisfy k2>1/2.
The Lyapunov function is defined as follows:(22)V1=12x1Tx1.
By the utilization of Equations (19)–(21) as well as x2=Eω~−ω~p,1, the derivative of V1 can be derived as follows:
(23)V̇1=x1Tẋ1=x1TEω~=x1Tx2+ω~p,1=−k1x1Tx1x12+σ2−k2x1Tx1+x1Tx2.
Using Lemma7, the following equation is derived:
(24)V̇1=−k1x1Tx1x12+σ2−k2x1Tx1+x1Tx2≤−k1x1−k2−12x12+x222+k1σ.Step 2.
With Remark2 and the result in Equation (24), consider the Lyapunov function as follows:
(25)V2∗=V1+12x2TJx2.
Then, the derivative ofV2∗ can be calculated as follows:
(26)V̇2∗=V̇1+x2TJẋ2=−k1x1−k2−12x12+x222+k1σ+x2TJĖω~+x2TE−ω~+R~Tωd×Jω~+R~Tωd−JR~Tω̇d+Jω~×R~Tωd+d+u−E−1Jω~̇p,1.
The notation is selected as follows:(27)H=JĖω~−Jω~̇p,1+E−ω~+R~Tωd×Jω~+R~Tωd−JR~Tω̇d+Jω~×R~Tωd.
From Assumption1 and Lemma 4, the term x2TEd is known to satisfy the following condition:
(28)x2TEd≤D2x2.
Thus, Equation (27) is rewritten as follows:
(29)V̇2∗≤−2k1V11/2−2k2−1V1+x222+k1σ+x2TH+D2x2+x2TEu.
Observing Lemma3, H can be expressed as follows:
(30)H=W∗Th+ε,where W∗=W1∗,W2∗,W3∗T is a Ge-Lee matrix defining the weight matrix, h=h1,h2,h3T is a Ge-Lee matrix defining the radial basis function matrix, and ε is the approximation error satisfying ε≤εd with εd>0. Wi∗, i=1,2,3, and hi are three-dimensional Ge-Lee vectors. Combining Equations (29)–(31) results in
(31)V̇2∗≤−2k1V11/2−2k2−1V1+x222+k1σ+x2TW∗Th+ε+D2x2+x2TEu.
Here, the termx2TW∗Th can be defined as follows:
(32)x2TW∗h=∑i=13x2iWi∗Thi.
Equation (31) is further equivalent to the following:
(33)V̇2∗≤−2k1V11/2−2k2−1V1+x222+k1σ+∑i=13βx2i2hi22η+3η2+x2D2+εd+x2TEu,where β=max1≤i≤3Wi∗2 and η is a positive constant. Thus, β is easier to estimate than W∗. To simplify the design, we introduce a constant D⌢ satisfying D⌢=D/2+εd.Step 3.
In the healthy condition of the actuator,δi=1 is ensured. However, in practical applications, external disturbance and actuator faults always occur simultaneously. Thus, the design of a fault-tolerant controller is necessary.
The fault-tolerant control scheme is designed as follows:(34)u=E−1u1+u2,(35)u1=−k3x2−k4sigx2λ−β^x3∘h∘h2η−D^tanhx2μ1,(36)u2=−γ^u1tanhx2u1μ2,(37)β^̇=c1∑i=13x2i2hi22η−c4β^,(38)γ^̇=c2u1x2tanhu1x23μ2−c5γ^,(39)D^̇=c3x2tanhx23μ1−c6D^,where β^, D^, and γ^ are estimations of β, D, and γ, respectively, with γ=1−θ/θ, k3>1/2, k4>0, μ1>0, μ2>0, and ci>0, i=1,2,⋯,6.Remark 8.
Inertial parameters are totally unknown to designers in this paper. To deal with the unmodeled dynamics, the MLP method is adopted to ensure finite-time convergence for the spacecraft system. In Equation (35), the term k3x2+k4sigx2λ is introduced for finite-time convergence while β^x2∘h∘h/2η and D^tanhx2/μ1 are designed to estimate unknown dynamics and the upper bound of external disturbances.Remark 9.
When0<δi<1 holds, the spacecraft encounters the partial actuator effectiveness loss fault, which will cause considerable performance degradation. To this end, the fault-tolerant controller is proposed as Equation (34). Here, u1 implies the norm part of this controller dedicating to forcing spacecraft towards the desired attitude. u2 defines the compensation part for the actuator fault. With this control strategy, finite-time stability is achievable for tracking errors even when actuator faults occur.Theorem 10.
For the attitude error dynamics expressed in Equations (10), (11), (19), and (20) with Assumption 1, then, the virtual angular velocity ω~p,1 is derived as Equation (21). If the virtual control law and actual control law are designed as in Equations (21) and (34)–(39), the following conclusions can be obtained:
(i)
The region of attraction is represented as follows:(40)x1T0x10+x2T0Jx20+1c1β~20+θc2γ~20+1c3D~20<1.(ii)
x2 will be stabilized in the region max2/λminJΔ4/ρ4,2/λminJΔ4/ρ41/λ+1 within a finite time. Here, ρ4=min2k1,k42/λminJ1+λ/2,c4c12k5−1/k51/2,c5c22k6−1/θk61/2,c6c32k7−1/k71/2 and Δ4=c4k5/2β2+c6k7/2D⌢2+c5k6/2γ2+3κDμ1+μ2γθ+k1σ+3η/2. ki, i=5,6,7, are positive constants satisfying ki>0.5Proof.
The Lyapunov function can be established as follows:(41)V3=V2∗+12c1β~2+θ2c2γ~2+12c3D~2.
Considering Equations (33) and (34), the derivation can be calculated as follows:
(42)V̇3=V̇2∗+1c1β~β~̇+θc2γ~γ~̇+1c3D~D~̇≤−2k1V11/2−2k2−1V1+x222+k1σ+∑i=13βx2i2hi22η+3η2+x2TI−δu1+x2Tu1+x2Tδu2+x2D2+ε−1c1β~β^̇−θc2γ~γ^̇−1c3D~D^̇.
Then, learning from Equations (35) and (36), one has
(43)V̇3≤−2k2−1V1+x222+k1σ+∑i=13βx2i2hi22η+3η2+x2TI−δu1−k3x22−k4x21+λ−x2Tβ^x2∘h∘h2η−D^x2Ttanhx2μ1−γ^θx2Tu1tanhx2u1μ2+D⌢x2−1c1β~β^̇−θc2γ~γ^̇−1c3D~D^̇≤−2k2−1V1−k3−12x22+k1σ+∑i=13β~x2i2hi22η+3η2+1−θx2u1−D^x2+3D^μ1κ−γ^θx2u1+3μ2γ^θκ+D⌢x2−1c1β~β^̇−θc2γ~γ^̇−1c3D~D^̇=−2k2−1V1−k3−12x22+∑i=13β~x2i2hi22η+θγ~x2u1+D~x2−1c1β~β^̇−θc2γ~γ^̇−1c3D~D^̇+3κD^μ1+μ2γ^θ+k1σ+3η2.
Combining adaptive laws in Equations (37)–(39), one can obtain the following result:
(44)V̇3≤−2k2−1V1−k3−12x22+∑i=13β~x2i2hi22η+θγ~x2u1+D~x2−β~∑i=13x2i2hi22η−c4β^−θγ~u1x2tanhu1x23μ2−c5γ^−D~x2tanhx23μ1−c6D^+3κD^μ1+μ2γ^θ+k1σ+3η2≤−2k2−1V1−k3−12x22+c4β~β^+3κμ2γ~θ+c5θγ~γ^+3κμ1D~+c6D~D^+3κD^μ1+μ2γ^θ+k1σ+3η2≤−2k2−1V1−k3−12x22−c42β~2−c5θ2γ~2−c62D~2+c42β2+c5θ2γ2+c62D2+3κDμ1+μ2γθ+k1σ+3η2≤−ρ3V3+Δ3,where ρ3=min2k2−1,2k3−1/λmaxJ,c1c4,c2c5,c3c6 and Δ3=c4/2β2+c6/2D2+c5θ/2γ2+k1σ+3η/23κDμ1+μ2γθ.
Consequently,V3 is asymptotically stable and converges to a region containing the origin. To proceed, the derivation of V3 in Equation (42) can be rewritten as follows:
(45)V̇4≤−2k1V11/2+x222+k1σ+∑i=13βx2i2hi22η+3η2+x2TI−δu1+x2Tu1+x2Tδu2+x2D2+ε−1c1β~β^̇−1c3D~D^̇−θc2γ~γ^̇.
Combining the control laws Equations (34)–(39), the following expression is obtained:
(46)V̇4≤−2k1V11/2−k4x21+λ+c4β~β^+c5θγ~γ^+c6D~D^+3κDμ1+μ2γθ+k1σ+3η2=−2k112x1Tx11/2−k42λminJ1+λ/212x2TJx21+λ/2+c4β~β^+c5θγ~γ^+c6D~D^+3κDμ1+μ2γθ+k1σ+3η2.
For the termc4β~β^, the following condition always exists:
(47)c4β~β^=c4−β~2+β~β≤c4−β~2+12k5β~2+k52β2≤−c42k5−12k5β~2+c4k52β2,where k5 is a positive constant. If it satisfies c42k5−1/2k5β~2≥1, the following equation is obtained:
(48)c42k5−12k5β~21/2+c4β~β^≤c42k5−12k5β~2+c4β~β^≤c4k52β2.
Ifc42k5−1/2k5β~2<1 exists, the following equation is obtained:
(49)c42k5−12k5β~21/2c42k5−1/2k5β~2<1<c42k5−12k5β~21/2c42k5−1/2k5β~2≥1.
Then, it follows(50)c42k5−12k5β~21/2+c4β~β^≤c4k52β2.
The same results forc5θγ~γ^ and c6D~D^ are obtained.
Consequently, Equation (46) can be rewritten as follows:
(51)V̇4≤−2k112x1Tx11/2−k42λminJ1+λ/212x2TJx21+λ/2−c4c12k5−1k51/212c1β~21/2−c5c22k6−1θk61/2θ2c2γ~21/2−c6c32k7−1k71/212c3D~21/2+c42k5−12k5β~21/2+c52k6−12k6γ~21/2+c62k7−12k7D~21/2+c4β~β^+c5θγ~γ^+c6D~D^+3κDμ1+μ2γθ+k1σ+3η2≤−ρ4minV41/2,V41+λ/2+c42k5−12k5β~21/2+c52k6−12k6γ~21/2+c62k7−12k7D~21/2+c4β~β^+c5θγ~γ^+c6D~D^+3κDμ1+μ2γθ+k1σ+3η2.
In terms of Equation (50), we have
(52)V̇4≤−ρ4minV41/2,V41+λ/2+c4k52β2+c6k72D⌢2+c5k62γ2+3κDμ1+μ2γθ+k1σ+3η2=−ρ4minV41/2,V41+λ/2+Δ4.
From Equation (52), it can be concluded that the result V̇4≤0 always holds. Consequently, x1, x2, β~, γ~, and D~ are all bounded. To keep x12<1, one can obtain
(53)V4≤V40≤12x1T0x10+12x2T0Jx20+12c1β~20+12c2D~20<12.
Therefore, (i) has been proved.
Meanwhile, according to Lemma6, x2 will be stabilized to the region max2/λminJΔ4/ρ4,2/λminJΔ4/ρ41/λ+1 within a finite time.
Thus, Theorem10 is proven.


## 4. Simulation Results

This section is dedicated to authenticating the stability and validity of the controllers presented in the theoretical analysis. The main parameters, inertial matrix, disturbance torque, initial state, desired rotation matrix, and angular velocity are selected as follows:J=22.700023.300024.5kg⋅m2, R0=0.4977−0.8674−0.00070−0.00081.0000−0.8674−0.4977−0.0004, ω0=0.1,0.1,0.1Trad/s, Rd0=I3×3, ωd=0.1×sint/40,−cost/50,−sint/60Trad/s, and d=0.02×sin0.1t,cos0.2t,sin0.2tT, respectively.To show the adverse effect of the actuator faults, the control performance of the controller in [22] is shown under two scenarios δ=diag0.3,0.3,0.3,ift>80sdiag1,1,1,else and δ=diag0.2,0.2,0.2,ift>80sdiag1,1,1,else, where d=0.05×sin0.1t,cos0.2t,sin0.2tTN⋅m. Attitude tracking errors are presented in Figure 1, which implies that system performance degradation will be caused as δ increases. Thus, it concludes that fault-tolerant control strategies are necessary for spacecraft during space missions in case of actuator faults.Figure 1
Attitude tracking errors under differentδ.Then, the actuator faults are set as follows:δ=diag0.5,0.4,0.3,ift>100sdiag1,1,1,else. The design parameters of the fault-tolerant controller are determined as follows: k1=0.05, k2=0.5, k3=2, k4=1, c1=0.001, c2=0.001, c3=0.001, c4=2,, c5=2, c6=2, μ1=1, μ2=1, and λ=0.9. The following parameters are selected for the RBF NN: bi=2, p=20, and the input vectors of the RBF NN satisfy ωi∈−0.5,0.5rad/s, ω̇i∈−0.5,0.5rad/s, x1i∈−0.3,0.3, x2i∈−0.5,0.5rad/s, and x2di∈−0.5,0.5rad/s. Simulation results for this method are presented in Figures 2–8. From the simulation results, we can conclude that the control objective can be achieved within 60 s in the case of actuator faults. Additionally, the chattering problem does not appear in the control torque. Thus, the points mentioned in Theorem 10 are illustrated by the simulation results.Figure 2
Attitude tracking error using the finite-time controller.Figure 3
Derivative ofeR~ using the finite-time controller.Figure 4
Angular velocity tracking error using the finite-time controller.Figure 5
Estimation ofβ.Figure 6
Estimation ofD⌢.Figure 7
Estimation ofγ.Figure 8
Control torque using the finite-time controller.


## 5. Conclusions

In this paper, the unwinding-free problem is solved for spacecraft attitude tracking control missions. The proposed control schemes not only exhibit excellent robustness against actuator faults and disturbance but also address the unknown inertial parameters. Additionally, the application of the MLP method results in a controller with less computational complexity. The simulation results prove the effectiveness of the proposed methods.



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*Source: 9269438-2021-02-24.xml*