# Adaptive Finite-Time Command Filtered Fault-Tolerant Control for Uncertain Spacecraft with Prescribed Performance

Authors: Zhongtian Chen; Qiang Chen; Xiongxiong He; Mingxuan Sun
Journal: Complexity
 (2018)

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## Abstract

In this paper, an adaptive finite-time fault-tolerant control scheme is proposed for the attitude stabilization of rigid spacecrafts. A first-order command filter is presented at the second step of the backstepping design to approximate the derivative of the virtual control, such that the singularity problem caused by the differentiation of the virtual control is avoided. Then, an adaptive fuzzy finite-time backstepping controller is developed to achieve the finite-time attitude stabilization subject to inertia uncertainty, external disturbance, actuator saturation, and faults. Through using an error transformation, the prescribed performance boundary is incorporated into the controller design to guarantee the prescribed performance of the system output. Numerical simulations demonstrate the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

Due to the significant role in guaranteeing the success of any spacecraft related mission, the attitude control of the spacecraft has obtained much attention, and numerous control schemes are proposed, such as adaptive control, sliding mode control, backstepping control,H∞ control, finite-time control, and so on [1–8]. Considering the specificity of the spacecraft’s working environment, the hardware of the spacecraft is unlikely repairable, and the faults or failures cannot be fixed with replacement parts after the spacecraft is launched. The existence of faults or failures can potentially cause all kinds of safety, economic, and environment problems, and they should be considered in the attitude control design.Different from the conventional control system without considering the possibility of fault occurrence, the fault-tolerant control (FTC) can guarantee desirable performance properties even the actuators are not healthy. In general, the existing FTC technique can be roughly classified into two categories: active FTC and passive FTC. The active FTC relies on the fault detection and diagnosis (FDD) algorithm to provide the real-time information of the system status and then reconfigure the controller to achieve the control objective. For instance, an active fault-tolerant control was proposed for the flexible spacecraft in [9], such that the attitude stabilization was achieved. Unlike active FTC, by using a single fixed controller, passive FTC dose not require any online fault information and reconfiguration mechanism. Therefore, the passive FTC is more succinct, easy to compute and suitable to the actual application. Most of FTC controllers designed for the spacecraft attitude control are passive [10–12]. In [10], an adaptive fuzzy fault-tolerant controller was proposed to achieve the spacecraft attitude tracking. A sliding mode control based attitude fault-tolerant controller was presented in [11], and the attitude was stabilized for the satellite with solar flaps. For those spacecrafts with redundancy actuators, the common method is using the actuator distribution matrix. In [12], a fault-tolerant control method based on distribution matrix was presented for the spacecraft attitude tracking, such that the finite-time convergence of the tracking error was achieved.Actuator saturation is another issue worthy of study, the actual actuators in the spacecraft have the nominal limit of the output, and saturated output definitely effaces the system performance. Although there exist works concerning the attitude control design with actuator saturation [13–15], the possibility of the emerging of actuator failure at the same time is ignored. When the actuator fault happens, in order to maintain the performance of the system, the need for large control torque leads to severe actuator saturation. Until now, numerous works have considered both issues and designed controllers for the spacecraft attitude control [16–18].Convergence speed is always significant in practically spacecraft system, and the finite-time control is able to provide faster convergence performance and higher tracking precision. Most of the finite-time control methods applied to the spacecraft have two kinds: homogeneous method [19, 20] and Lyapunov method [21–24]. In [19], a local continuous finite-time control scheme was proposed for the spacecraft system with an unknown inertia matrix and the homogeneous method based controller achieved attitude stabilization within a finite time. Homogeneous system theory was used to design a finite-time controller in [20], and the spacecraft attitude was stabilized within a finite time even with actuator saturation. In [5, 21], two adaptive terminal sliding mode controllers were proposed based on the Lyapunov method, such that the spacecraft attitude and angular velocity could converge to a small region of the origin within a finite time, respectively. Furthermore, the recent works [12, 16, 22] have achieved the finite-time spacecraft attitude stabilization or tracking in the presence of the actuator saturation and faults.The adaptive backstepping approach [25, 26], as a recursive Lyapunov-based scheme, has emerged as a powerful method to construct controllers for nonlinear systems since early 1990s. There are several works using the backstepping method to design controllers of the spacecraft [3, 27]; however, the closed-loop stability is achieved when time goes to infinity. When designing a finite-time backstepping controller, the differentiation of the virtual controller in recursive steps may lead to the singularity problem. Recently, the finite-time command filtered backstepping approach was proposed in [28, 29], where the first-order Levant differentiator was used to approximate the derivative of the virtual controller, such that the finite-time convergence can be achieved when the system model was known or partially known.All of the aforementioned works mainly focus on the steady-state behavior but ignore the transient performance such as convergence rate and overshot. To achieve the specific goals of spacecraft missions which need a specific rotating speed or a limitation of angular velocity, prescribed transient performance of the system output is very important. The widely used techniques to improve transient performance mainly include barrier Lyapunov function (BLF) [30–33], funnel control [34–36], prescribed performance control (PPC) [37–42], and so on. In order to guarantee the prescribed performance imposed on the transient and steady-state output error, the prescribed performance control (PPC) method was proposed by Bechlioulis and Rovithakis for uncertain nonlinear systems [37]. For the spacecraft attitude system with input saturation, a PPC based adaptive fault-tolerant control was presented in [43], such that the output of the system was constrained by the prescribed performance. Nevertheless, the designed controller in [43] could only guarantee the asymptotic uniform ultimate boundedness of the spacecraft system as the time goes to infinity.Motivated by the aforementioned discussions, the fuzzy finite-time attitude stabilization problem for spacecraft systems under the actuator saturation and faults is studied in this paper, and a fuzzy finite-time fault-tolerant controller is proposed to achieve prescribed transient and steady-state performance of the system output. However, it is a challenging work to design a controller to guarantee the prescribed transient performance when considering the input constraints including actuator saturation and faults. On the one hand, when the initial state is far away from the equilibrium state, the required control input is usually set relatively large to guarantee the fast transient response. But on the other hand, due to the effect of the input saturation and actuator fault, it is a hard work to keep the satisfactory transient response as usual. The main contributions of this paper are listed as follows.(1) A first-order command filter is presented at the second step of the backstepping design to approximate the derivative of the virtual control, such that the singularity problem caused by the differentiation of the virtual control is avoided.(2) An adaptive fuzzy finite-time backstepping controller is developed to achieve the finite-time attitude stabilization subject to inertia uncertainty, external disturbance, actuator saturation, and faults.(3) Through using an error transformation, the prescribed performance boundary is incorporated into the controller design to guarantee the prescribed performance of the system output.The rest of this paper is organized as follows. Section2 states the formulation of the spacecraft attitude stabilization problem. In Section 3, some preliminary knowledge and lemmas are given. In Section 4, the fuzzy finite-time fault-tolerant control scheme is proposed and followed by stability analysis. Simulation results are provided in Section 5, and the conclusion is summarized in Section 6.


## 2. Problem Formulation

Considering the attitude stabilization problem for a rigid spacecraft, the modified Rodrigues parameter (MRP) based spacecraft system is described as [44](1)σ˙=141-σTσI3+2σ×+2σσTω=Gσω(2)Jω˙=-ω×Jω+u+dwhere σ=[σ1,σ2,σ3]T is the spacecraft attitude in body frame with respect to the inertial frame presented by MPRs, I3 is the identity matrix, and the operational symbol a× denotes the following skew-symmetric matrix for any vector a=[a1,a2,a3]T(3)a×=0-a3a2a30-a1-a2a10ω=[ω1,ω2,ω3]T is the body frame angular velocity with respect to inertial frame. J=J0+ΔJ is the inertia matrix of the spacecraft, where J0 denotes the nonsingular known nominal value of the inertia matrix and ΔJ is the bounded uncertainty. u and d in (2) denote the control torque and the bounded external disturbance torque, respectively. Considering the input saturation and actuator faults in the rigid spacecraft, the actual control torque u is further formulated as(4)u=Esat⁡vwhere E=diag{E1,E2,E3} with 0<Ei≤1(i=1,2,3) is the fault matrix and sat(v) is the saturated control input satisfying sat(v)=sat(v1),sat(v2),sat(v3), where v=[v1,v2,v3]T is the commanded control which needs to be designed later, sat(vi)=sgn(vi)×min⁡vi,vMi with sgn(·) denoting the sign function and vMi being the ith axis maximum torque.The nonlinear saturation functionsat(vi) can be approximated by the following smooth function g(vi):(5)gvi=vMi×tanh⁡vivMi=vMievi/vMi-e-vi/vMievi/vMi+e-vi/vMi.Then,sat(v) is rewritten as(6)sat⁡v=gv+dsvwhere g(v)=[g(v1),g(v2),g(v3)]T and ds(v)=[ds(v1),ds(v2),ds(v3)]T stands for the approximation error. It is noted that ds(vi) is a bounded function and its bound satisfying dsvi=satvi-gvi≤vMi1-tanh1.According to the mean value theorem, there exist constantsci(0<ci<1), i=1,2,3, such that the following inequality holds:(7)gvi=gvi′+gμvi-vi′,i=1,2,3where gμ=∂gvi/∂vivi=civi+1-civi′,vi′∈0,vi. By setting vi′=0, (6) is rewritten as(8)sat⁡v=Hv+dsvwhere H=diag{gμ(v1),gμ(v2),gμ(v3)}.The control torque (4) is rewritten as(9)u=Γv+Edsvwhere Γ=EH, and there exists an unknown positive constant b satisfying 0<b≤Γ<1.From (1), (2), and (9), the spacecraft system is further formulated as(10)σ˙=Gσωω˙=J0-1Γv+Ft+Dtwhere F(t)=J0-1(-ΔJω˙-ω×Jω) is the uncertainty and D(t)=J0-1(Eds(v)+d). Because ds(v) and d are bounded, there exists an unknown positive constant Dm satisfying D(t)≤Dm.The matrixG(σ) has the following properties [45]:(11)Gσ-1=161+σTσ2GσT(12)GσTGσ=1+σTσ42I3.From (11) and (12), it is obtained that 1/4≤G≤1/2.The control objective in this paper is to develop a fuzzy fault-tolerant finite-time control scheme for the spacecraft with inertia uncertainty, extra disturbance, input saturation, and actuator faults, such that the system outputσ converges into a small region of the origin within the prescribed bounds in a finite time.


## 3. Preliminaries

In this section, some preliminary knowledge critical for control design and satiability analysis is presented.


### 3.1. Finite-Time Differentiator

The first-order Levant differentiator [46] is formulated as(13)φ˙1=ιι=-β1φ1-αr1/2sgn⁡φ1-αr+φ2φ˙2=-β2sgn⁡φ2-φ˙1where αr is the input signal, β1,β2>0 are the parameters, and φ1 and ι are the estimations of αr and α˙r, respectively. Following lemma holds if the parameters β1 and β2 are chosen properly.Lemma 1 (see [46]).
If the inputαr is not affected by the noise, the following equalities are true within a finite time(14)φ1=αr,ι=α˙r.If the input is affected by noise and satisfying αr-αr′≤κ1, where αr′ is the original signal without interference, the following inequalities hold in a finite time:(15)φ1-αr′≤ς1κ1=ϖ1(16)ι-α˙r′≤ς2κ11/2=ϖ2where ς1 and ς2 are positive constants depended on the design parameters of the differentiator.


### 3.2. Fuzzy Logic System

A typical fuzzy logic system (FLS) consists of four parts: the fuzzy rules, the fuzzifier, the fuzzy inference engine, and the defuzzifier. The foundation of the FLS is a group of fuzzy If-Then rules as follows:Rulel:Ifx1isF1landx2isF2land…andxnisFnl,ThenyisMl,l=1,2,…,N.wherex=[x1,x2,…,xn]T is the input of the FLS, y is the FLS output, Fil and Ml denote the fuzzy sets relating to the membership functions μGl(y) and μFil(xi), respectively, and N is the rules number.Combining the singleton fuzzifier, center average defuzzification, and product inference, the output of the FLS is obtained as(17)yx=∑l=1Ny¯l∏i=1nμFilxi∑l=1N∏i=1nμFilxiwherey¯l=maxy∈RμGl(y) and μFilxi is the membership function value of the fuzzy variable. The fuzzy basis function is defined as(18)Φl=∏i=1nμFilxi∑l=1N∏i=1nμFilxi.DefineW=[y¯1,y¯2,…,y¯N]T=[w1,w2,…,w3]T as the ideal constant weight vector, and the (17) is expressed as(19)yx=WTΦxwhere Φ(x)=[Φ1(x),Φ2(x),…,ΦN(x)]T is the basis function vector. The relationship between the FLS and the unknown nonlinear function in the system is given in the following lemma.Lemma 2.
For any continuous functionf(x) defined on a compact Ω, then for any constant μ>0, there exists an FLS WTΦ(x) such that(20)supx∈Ω⁡fx-WTΦx≤μ.


### 3.3. Prescribed Performance Function

As a priori guaranteeing prescribed behavioral bounds on the output of the system, the prescribed performance function (PPF) is designed as follows:(21)ρt=ρ0-ρ∞e-κt+ρ∞where ρ0, ρ∞, and κ are positive parameters, where κ is the prescribed minimum exponential convergence rate and ρ∞ stands for the maximum steady-state error, respectively, and it guarantees the following inequality:(22)-δ_itρt<σi<δ¯itρt,i=1,2,3.In order to relax the assumption that the initial condition should be precisely known to guarantee the prescribed transient in the classic PPF, e.g., [37], the functions δ_i and δ¯i are satisfying the following properties [47].(1)δ_i and δ¯i are positive and strictly decreasing; (2) limt→0δ_i=+∞, limt→∞δ_i=C1, C1∈R+, limt→0δ¯i=+∞, and limt→∞δ¯i=C2, C2∈R+,An example of suchδ_i and δ¯i is given by(23)δ¯˙i=-piδ¯i+qi,pi,qi∈R+δ_˙i=-aiδ_i+bi,ai,bi∈R+where ai,bi,pi,qi,i=1,2,3 are positive constants.For the purpose of designing the control law to guarantee the prescribed performance bounds (22), the error transformations are presented as(24)εi=12ln⁡δ_it+σit/ρtδ¯it-σit/ρt,i=1,2,3.Through employing the error transformation (24), the output of the original system σ can be guaranteed within the prescribed bound provided that the transformed error ε=[ε1,ε2,ε3]T is stable.Lemma 3 (see [37, 48]).
System (10) is invariant under the error transformation (24), and the stabilization of the transformed error ε can guarantee the output σ converge with the prescribed performance described by (22).The derivative of theεi is given as(25)ε˙i=riσ˙i+ϑi,i=1,2,3where ri,i=1,2,3, is(26)ri=12ρt1σit/ρt+δ_it-1σit/ρt-δ¯itand the ϑi,i=1,2,3, is(27)ϑi=δ_˙it-ρ˙tσit/ρ2t2σit/ρt+δ_it+δ¯˙it+ρ˙tσit/ρ2t2σit/ρt-δ¯it.Substituting (1) into (25) yields(28)ε˙=rGω+ϑwhere r=diagr1,r2,r3 and ϑ=[ϑ1,ϑ2,ϑ3]T.


## 3.1. Finite-Time Differentiator

The first-order Levant differentiator [46] is formulated as(13)φ˙1=ιι=-β1φ1-αr1/2sgn⁡φ1-αr+φ2φ˙2=-β2sgn⁡φ2-φ˙1where αr is the input signal, β1,β2>0 are the parameters, and φ1 and ι are the estimations of αr and α˙r, respectively. Following lemma holds if the parameters β1 and β2 are chosen properly.Lemma 1 (see [46]).
If the inputαr is not affected by the noise, the following equalities are true within a finite time(14)φ1=αr,ι=α˙r.If the input is affected by noise and satisfying αr-αr′≤κ1, where αr′ is the original signal without interference, the following inequalities hold in a finite time:(15)φ1-αr′≤ς1κ1=ϖ1(16)ι-α˙r′≤ς2κ11/2=ϖ2where ς1 and ς2 are positive constants depended on the design parameters of the differentiator.


## 3.2. Fuzzy Logic System

A typical fuzzy logic system (FLS) consists of four parts: the fuzzy rules, the fuzzifier, the fuzzy inference engine, and the defuzzifier. The foundation of the FLS is a group of fuzzy If-Then rules as follows:Rulel:Ifx1isF1landx2isF2land…andxnisFnl,ThenyisMl,l=1,2,…,N.wherex=[x1,x2,…,xn]T is the input of the FLS, y is the FLS output, Fil and Ml denote the fuzzy sets relating to the membership functions μGl(y) and μFil(xi), respectively, and N is the rules number.Combining the singleton fuzzifier, center average defuzzification, and product inference, the output of the FLS is obtained as(17)yx=∑l=1Ny¯l∏i=1nμFilxi∑l=1N∏i=1nμFilxiwherey¯l=maxy∈RμGl(y) and μFilxi is the membership function value of the fuzzy variable. The fuzzy basis function is defined as(18)Φl=∏i=1nμFilxi∑l=1N∏i=1nμFilxi.DefineW=[y¯1,y¯2,…,y¯N]T=[w1,w2,…,w3]T as the ideal constant weight vector, and the (17) is expressed as(19)yx=WTΦxwhere Φ(x)=[Φ1(x),Φ2(x),…,ΦN(x)]T is the basis function vector. The relationship between the FLS and the unknown nonlinear function in the system is given in the following lemma.Lemma 2.
For any continuous functionf(x) defined on a compact Ω, then for any constant μ>0, there exists an FLS WTΦ(x) such that(20)supx∈Ω⁡fx-WTΦx≤μ.


## 3.3. Prescribed Performance Function

As a priori guaranteeing prescribed behavioral bounds on the output of the system, the prescribed performance function (PPF) is designed as follows:(21)ρt=ρ0-ρ∞e-κt+ρ∞where ρ0, ρ∞, and κ are positive parameters, where κ is the prescribed minimum exponential convergence rate and ρ∞ stands for the maximum steady-state error, respectively, and it guarantees the following inequality:(22)-δ_itρt<σi<δ¯itρt,i=1,2,3.In order to relax the assumption that the initial condition should be precisely known to guarantee the prescribed transient in the classic PPF, e.g., [37], the functions δ_i and δ¯i are satisfying the following properties [47].(1)δ_i and δ¯i are positive and strictly decreasing; (2) limt→0δ_i=+∞, limt→∞δ_i=C1, C1∈R+, limt→0δ¯i=+∞, and limt→∞δ¯i=C2, C2∈R+,An example of suchδ_i and δ¯i is given by(23)δ¯˙i=-piδ¯i+qi,pi,qi∈R+δ_˙i=-aiδ_i+bi,ai,bi∈R+where ai,bi,pi,qi,i=1,2,3 are positive constants.For the purpose of designing the control law to guarantee the prescribed performance bounds (22), the error transformations are presented as(24)εi=12ln⁡δ_it+σit/ρtδ¯it-σit/ρt,i=1,2,3.Through employing the error transformation (24), the output of the original system σ can be guaranteed within the prescribed bound provided that the transformed error ε=[ε1,ε2,ε3]T is stable.Lemma 3 (see [37, 48]).
System (10) is invariant under the error transformation (24), and the stabilization of the transformed error ε can guarantee the output σ converge with the prescribed performance described by (22).The derivative of theεi is given as(25)ε˙i=riσ˙i+ϑi,i=1,2,3where ri,i=1,2,3, is(26)ri=12ρt1σit/ρt+δ_it-1σit/ρt-δ¯itand the ϑi,i=1,2,3, is(27)ϑi=δ_˙it-ρ˙tσit/ρ2t2σit/ρt+δ_it+δ¯˙it+ρ˙tσit/ρ2t2σit/ρt-δ¯it.Substituting (1) into (25) yields(28)ε˙=rGω+ϑwhere r=diagr1,r2,r3 and ϑ=[ϑ1,ϑ2,ϑ3]T.


## 4. Main Results

### 4.1. Control Design

Define virtual statesz1 and z2 as(29)z1=εz2=ω-xcwhere xc=[xc1,xc2,xc3]T is the output of the following finite-time command filter(30)φ˙1i=-β1φ1i-αi1/2sgn⁡φ1i-αi+φ2iφ˙2i=-β2sgn⁡φ2i-φ˙1i,i=1,2,3where the input α=α1,α2,α3T is the virtual control to be designed later and the output xci=φ1i,i=1,2,3.Based on the command filtered backstepping control approach, the compensated stabilization errors are given bys1=z1-ξ1, s2=z2-ξ2, where ξ1 and ξ2 are the error compensating signals to reduce the influence of xc-α.Choose the Lyapunov function asV1=1/2s1Ts1. Using (28), the time derivative of V1 is(31)V˙1=s1Ts˙1=s1Tz˙1-ξ˙1=s1TrGω+ϑ-ξ˙1=s1TrGz2+xc+ϑ-ξ˙1=s1TrGz2+rGα+rGxc-α+ϑ-ξ˙1.The virtual controlα and error compensation ξ1 are designed as(32)α=G-1r-1-k1z1-τ1sigγ⁡s1-ϑand(33)ξ˙1=-k1ξ1+rGxc-α+rGξ2-l1sig⁡ξ1where k1,τ1,l1>0, 0<γ<1 are design parameters, sigγ(s1)=[s11γsgn(s11), s12γsgn(s12),s13γsgn(s13)]T, and sig(ξ1)=sgn(ξ11),sgn(ξ12),sgn(ξ13)T with ξ1i(0)=0, i=1,2,3.Substituting (32) and (33) into (31) yields(34)V˙1=s1T-k1s1+rGs2-τ1sigγ⁡s1+l1sig⁡ξ1.Construct the second Lyapunov function as(35)V2=V1+12s2Ts2.Taking the derivative ofV2 along with (10) and (29) yields(36)V˙2=V˙1+s2Ts˙2=V˙1+s2Tω˙-x˙c-ξ˙2=V˙1+s2TJ0-1Γv+F+D-x˙c-ξ˙2.The error compensation is designed asξ2=ξ˙2=[0,0,0]T. Let F¯=F-x˙c+rGs1+s2. Substituting (34) into (36) and using the fact that G is symmetric lead to(37)V˙2=V˙1+s2TF¯-rGs1-s2+J0-1Γv+D=-k1s1Ts1-τ1∑13s1iγ+1+s1Tl1sig⁡ξ1-s2Ts2+s2TF¯+J0-1Γv+D.The fuzzy logic systems (19) are utilized to approximate the unknown nonlinear F¯. From Lemma 2, for any given constant μ>0, there always exists an FLS such that(38)F¯i=WiTΦiZn+δi,i=1,2,3where Φi=[Φi1,Φi2,…,ΦiN]T is the basis function vector, the approximation error δi satisfied δi≤μ, and Zn=[ωT,xcT,x˙cT]T.By Young’s inequality and (38), the following inequalities hold:(39)s2TF¯≤∑i=13s2iWiTΦi+∑i=13s2iμ≤θ∑i=13bs2iΦiTΦi2h2+3h22+s222+3μ22(40)s2TD≤s222+Dm22(41)s1Tl1sig⁡ξ1≤l1s122+l12where θ=1/bmax⁡W12,W22,W32.The commanded controller is designed as(42)v=J0-k2s2-θ^ηs22h2-τ2sigγ⁡s2where k2,τ2,h are positive design parameters, η=diagΦ1TΦ1,Φ2TΦ2,Φ3TΦ3, adn θ^ is the estimation of θ.It is obtained for the commanded controller (42) that(43)s2TJ0-1Γv=s2T-k2Γs2-Γθ^ηs22h2-τ2Γsigγ⁡v2≤-k2bs2Ts2-θ^∑i=13bs2i2ΦiTΦi2h2-τ2b∑i=13s2iγ+1.Using (39)-(43), the derivative of the V2 is simplified as(44)V˙2≤-k1-0.5l1s1Ts1-τ1∑i=13s1iγ+1-k2bs2Ts2-τ2b∑i=13s2iγ+1+θ-θ^∑i=13bs2i2ΦiTΦi2h2+3h22+3μ22+Dm22+l12.The update law ofθ^ is designed as(45)θ^˙=λ∑i=13s2i2ΦiTΦi2h2-m1θ^where λ and m1 are positive design parameters.


### 4.2. Stability Analysis

Before providing the stability analysis, the following two lemmas are given.Lemma 4 (see [49]).
For0<a<1 and xi∈R, i=1,2,3…, the following inequality holds:(46)∑i=13xia+1≥∑i=13xi2a+1/2.Lemma 5 (see [50]).
For any real number0<γ<1,λ1,λ2>0, an continuous positive-definite Lyapunov function V(x) satisfied the from as V˙(x)+λ1V(x)+λ2Vγ(x)≤0, then V≡0 can be achieved in a finite time and the setting time can be estimated by(47)Treach≤1λ11-γln⁡λ1V1-γ0+λ2λ2where V(0) is the initial value of V(x).Theorem 6.
Consider the spacecraft stabilization system described in (1) and (2) subject to input saturation and actuator fault (4) with finite-time command filter (30), the virtual control (32), the error compensation (33), the controller (42), and the update law (45), and then
(i) the transformed errorε converges into a small region of the origin in a finite time, and ω is bounded in a finite time;
(ii) the prescribed control performance ofσ (22) is preserved.Proof.
Construct the Lyapunov function as(48)V=V2+b2λθ~2where θ~=θ-θ^.
According to (44), the time derivative of V is(49)V˙≤-k1-0.5l1s1Ts1-τ1∑i=13s1iγ+1-k2bs2Ts2-τ2b∑i=13s2iγ+1+∑i=13s2i2ΦiTΦi2h2bθ~+3h22+3μ22+Dm22+l12-bθ~θ^˙λ.Substituting update law (45) into (49) yields(50)V˙≤-k1-0.5l1s1Ts1-τ1∑i=13s1iγ+1-k2bs2Ts2-τ2b∑i=13s2iγ+1+3h22+3μ22+Dm22+l12+m1bθ~θ^λ.According to Young’s inequality, the following inequality is hold:(51)m1bθ~θ^λ≤-3m1bθ~24λ+m1bθ2λ.Considering 0<γ<1, it is concluded that(52)m1bθ~22λγ+1/2-m1bθ~22λ≤14.Using (51), (52), and Lemma 4, the time derivative of V is expressed as(53)V˙≤-k1-0.5l1s1Ts1-τ1∑13s1i2γ+1/2-k2bs2Ts2-τ2b∑13s2i2γ+1/2-m1bθ~24λ-m1bθ~22λγ+1/2+3h22+3μ22+Dm22+l12+m1bθ2λ+14which leads to(54)V˙≤-λ1V-λ2Vγ+1/2+μ1where λ1=min⁡2k1-l1,2k2b,0.5m1, λ2=min⁡2γ+1/2τ1,2γ+1/2τ2b,m1γ+1/2, and μ1=3h2/2+3μ2/2+Dm2/2+l1/2+m1bθ2/λ+1/4. According to Lemma 5, it is concluded that s1 and s2 converge to the small region si≤max⁡2μ1/λ1,2μ1/λ22/γ+1, i=1,2, in a finite time T1≤1/λ11-γln⁡λ1V1-γ(0)+λ2/λ2. From the definition s1=ε-ξ1, s2=z2-ξ2, if the finite-time convergence of the ξi is guaranteed, then it is confirmed that the states z1 and z2 can converge into a small neighbourhood of the origin within a finite time.
Sinceξ2=ξ˙2=[0,0,0]T is given in the control design, in order to show that ξ1 is bounded in a finite time, the following Lyapunov function is chosen:(55)V3=12ξ1Tξ1.
DifferentiatingV3 along with (33) yields(56)V˙3=ξ1Tξ˙1=ξ1T-k1ξ1+Gxc-α+Gξ2-l1sig⁡ξ1=-k1ξ1Tξ1-l1∑13ξi+ξ1TGξ2+ξ1TGxc-α.
According to the Lemma1, xc-α≤ϖ can be achieved in a finite time T2, and combining G≤1/2, the following inequality is obtained:(57)ξ1TGxc-α≤12ϖ∑13ξi.
Substituting (57) and ξ2=[0,0,0]T into (56) leads to(58)V˙3≤-k1ξ1Tξ1-l1-12ϖ∑13ξi21/2≤-2k1V3-2l1-12ϖV31/2.
According to Lemma5, it is illustrated that ξ1 can converge to the origin in the finite time T3 by choosing suitable parameter satisfying l1>1/2ϖ. Since z1=s1+ξ1 and z2=s2+ξ2, the transformed error ε=z1 converges to a small region of the origin within the finite time TF=T1+T2+T3, and from the definition ω=z2+xc, ω is also bounded. According to Lemma 3, the stabilization of transformed error ε is sufficient to guarantee σ converge with prescribed performance described by (22). This completes the proof.Remark 7.
From (54) and (58), it is seen that the increase of k1,k2,τ1,τ2,l1 and decrease of γ lead to better convergence speed, but large τ1,l1 and small γ will result in chattering problem. Consequently, the choice of the parameters τ1,l1, and γ should be considered with a trade between the convergence speed and chattering reduction.


## 4.1. Control Design

Define virtual statesz1 and z2 as(29)z1=εz2=ω-xcwhere xc=[xc1,xc2,xc3]T is the output of the following finite-time command filter(30)φ˙1i=-β1φ1i-αi1/2sgn⁡φ1i-αi+φ2iφ˙2i=-β2sgn⁡φ2i-φ˙1i,i=1,2,3where the input α=α1,α2,α3T is the virtual control to be designed later and the output xci=φ1i,i=1,2,3.Based on the command filtered backstepping control approach, the compensated stabilization errors are given bys1=z1-ξ1, s2=z2-ξ2, where ξ1 and ξ2 are the error compensating signals to reduce the influence of xc-α.Choose the Lyapunov function asV1=1/2s1Ts1. Using (28), the time derivative of V1 is(31)V˙1=s1Ts˙1=s1Tz˙1-ξ˙1=s1TrGω+ϑ-ξ˙1=s1TrGz2+xc+ϑ-ξ˙1=s1TrGz2+rGα+rGxc-α+ϑ-ξ˙1.The virtual controlα and error compensation ξ1 are designed as(32)α=G-1r-1-k1z1-τ1sigγ⁡s1-ϑand(33)ξ˙1=-k1ξ1+rGxc-α+rGξ2-l1sig⁡ξ1where k1,τ1,l1>0, 0<γ<1 are design parameters, sigγ(s1)=[s11γsgn(s11), s12γsgn(s12),s13γsgn(s13)]T, and sig(ξ1)=sgn(ξ11),sgn(ξ12),sgn(ξ13)T with ξ1i(0)=0, i=1,2,3.Substituting (32) and (33) into (31) yields(34)V˙1=s1T-k1s1+rGs2-τ1sigγ⁡s1+l1sig⁡ξ1.Construct the second Lyapunov function as(35)V2=V1+12s2Ts2.Taking the derivative ofV2 along with (10) and (29) yields(36)V˙2=V˙1+s2Ts˙2=V˙1+s2Tω˙-x˙c-ξ˙2=V˙1+s2TJ0-1Γv+F+D-x˙c-ξ˙2.The error compensation is designed asξ2=ξ˙2=[0,0,0]T. Let F¯=F-x˙c+rGs1+s2. Substituting (34) into (36) and using the fact that G is symmetric lead to(37)V˙2=V˙1+s2TF¯-rGs1-s2+J0-1Γv+D=-k1s1Ts1-τ1∑13s1iγ+1+s1Tl1sig⁡ξ1-s2Ts2+s2TF¯+J0-1Γv+D.The fuzzy logic systems (19) are utilized to approximate the unknown nonlinear F¯. From Lemma 2, for any given constant μ>0, there always exists an FLS such that(38)F¯i=WiTΦiZn+δi,i=1,2,3where Φi=[Φi1,Φi2,…,ΦiN]T is the basis function vector, the approximation error δi satisfied δi≤μ, and Zn=[ωT,xcT,x˙cT]T.By Young’s inequality and (38), the following inequalities hold:(39)s2TF¯≤∑i=13s2iWiTΦi+∑i=13s2iμ≤θ∑i=13bs2iΦiTΦi2h2+3h22+s222+3μ22(40)s2TD≤s222+Dm22(41)s1Tl1sig⁡ξ1≤l1s122+l12where θ=1/bmax⁡W12,W22,W32.The commanded controller is designed as(42)v=J0-k2s2-θ^ηs22h2-τ2sigγ⁡s2where k2,τ2,h are positive design parameters, η=diagΦ1TΦ1,Φ2TΦ2,Φ3TΦ3, adn θ^ is the estimation of θ.It is obtained for the commanded controller (42) that(43)s2TJ0-1Γv=s2T-k2Γs2-Γθ^ηs22h2-τ2Γsigγ⁡v2≤-k2bs2Ts2-θ^∑i=13bs2i2ΦiTΦi2h2-τ2b∑i=13s2iγ+1.Using (39)-(43), the derivative of the V2 is simplified as(44)V˙2≤-k1-0.5l1s1Ts1-τ1∑i=13s1iγ+1-k2bs2Ts2-τ2b∑i=13s2iγ+1+θ-θ^∑i=13bs2i2ΦiTΦi2h2+3h22+3μ22+Dm22+l12.The update law ofθ^ is designed as(45)θ^˙=λ∑i=13s2i2ΦiTΦi2h2-m1θ^where λ and m1 are positive design parameters.


## 4.2. Stability Analysis

Before providing the stability analysis, the following two lemmas are given.Lemma 4 (see [49]).
For0<a<1 and xi∈R, i=1,2,3…, the following inequality holds:(46)∑i=13xia+1≥∑i=13xi2a+1/2.Lemma 5 (see [50]).
For any real number0<γ<1,λ1,λ2>0, an continuous positive-definite Lyapunov function V(x) satisfied the from as V˙(x)+λ1V(x)+λ2Vγ(x)≤0, then V≡0 can be achieved in a finite time and the setting time can be estimated by(47)Treach≤1λ11-γln⁡λ1V1-γ0+λ2λ2where V(0) is the initial value of V(x).Theorem 6.
Consider the spacecraft stabilization system described in (1) and (2) subject to input saturation and actuator fault (4) with finite-time command filter (30), the virtual control (32), the error compensation (33), the controller (42), and the update law (45), and then
(i) the transformed errorε converges into a small region of the origin in a finite time, and ω is bounded in a finite time;
(ii) the prescribed control performance ofσ (22) is preserved.Proof.
Construct the Lyapunov function as(48)V=V2+b2λθ~2where θ~=θ-θ^.
According to (44), the time derivative of V is(49)V˙≤-k1-0.5l1s1Ts1-τ1∑i=13s1iγ+1-k2bs2Ts2-τ2b∑i=13s2iγ+1+∑i=13s2i2ΦiTΦi2h2bθ~+3h22+3μ22+Dm22+l12-bθ~θ^˙λ.Substituting update law (45) into (49) yields(50)V˙≤-k1-0.5l1s1Ts1-τ1∑i=13s1iγ+1-k2bs2Ts2-τ2b∑i=13s2iγ+1+3h22+3μ22+Dm22+l12+m1bθ~θ^λ.According to Young’s inequality, the following inequality is hold:(51)m1bθ~θ^λ≤-3m1bθ~24λ+m1bθ2λ.Considering 0<γ<1, it is concluded that(52)m1bθ~22λγ+1/2-m1bθ~22λ≤14.Using (51), (52), and Lemma 4, the time derivative of V is expressed as(53)V˙≤-k1-0.5l1s1Ts1-τ1∑13s1i2γ+1/2-k2bs2Ts2-τ2b∑13s2i2γ+1/2-m1bθ~24λ-m1bθ~22λγ+1/2+3h22+3μ22+Dm22+l12+m1bθ2λ+14which leads to(54)V˙≤-λ1V-λ2Vγ+1/2+μ1where λ1=min⁡2k1-l1,2k2b,0.5m1, λ2=min⁡2γ+1/2τ1,2γ+1/2τ2b,m1γ+1/2, and μ1=3h2/2+3μ2/2+Dm2/2+l1/2+m1bθ2/λ+1/4. According to Lemma 5, it is concluded that s1 and s2 converge to the small region si≤max⁡2μ1/λ1,2μ1/λ22/γ+1, i=1,2, in a finite time T1≤1/λ11-γln⁡λ1V1-γ(0)+λ2/λ2. From the definition s1=ε-ξ1, s2=z2-ξ2, if the finite-time convergence of the ξi is guaranteed, then it is confirmed that the states z1 and z2 can converge into a small neighbourhood of the origin within a finite time.
Sinceξ2=ξ˙2=[0,0,0]T is given in the control design, in order to show that ξ1 is bounded in a finite time, the following Lyapunov function is chosen:(55)V3=12ξ1Tξ1.
DifferentiatingV3 along with (33) yields(56)V˙3=ξ1Tξ˙1=ξ1T-k1ξ1+Gxc-α+Gξ2-l1sig⁡ξ1=-k1ξ1Tξ1-l1∑13ξi+ξ1TGξ2+ξ1TGxc-α.
According to the Lemma1, xc-α≤ϖ can be achieved in a finite time T2, and combining G≤1/2, the following inequality is obtained:(57)ξ1TGxc-α≤12ϖ∑13ξi.
Substituting (57) and ξ2=[0,0,0]T into (56) leads to(58)V˙3≤-k1ξ1Tξ1-l1-12ϖ∑13ξi21/2≤-2k1V3-2l1-12ϖV31/2.
According to Lemma5, it is illustrated that ξ1 can converge to the origin in the finite time T3 by choosing suitable parameter satisfying l1>1/2ϖ. Since z1=s1+ξ1 and z2=s2+ξ2, the transformed error ε=z1 converges to a small region of the origin within the finite time TF=T1+T2+T3, and from the definition ω=z2+xc, ω is also bounded. According to Lemma 3, the stabilization of transformed error ε is sufficient to guarantee σ converge with prescribed performance described by (22). This completes the proof.Remark 7.
From (54) and (58), it is seen that the increase of k1,k2,τ1,τ2,l1 and decrease of γ lead to better convergence speed, but large τ1,l1 and small γ will result in chattering problem. Consequently, the choice of the parameters τ1,l1, and γ should be considered with a trade between the convergence speed and chattering reduction.


## 5. Simulation

In order to illustrate the effectiveness of the proposed control scheme, the simulation results and discussions are presented in this section. The spacecraft model is expressed as (1) and (2) where the initial values of the state parameters are set as (59)σ0=-0.3,-0.4,0.2Tω0=0,0,0Trad/s.The nominal inertia matrix is(60)J0=35034327010410192and the uncertainty ΔJ is (61)ΔJ=diag⁡5sin⁡0.1t,7sin⁡0.2t,9sin⁡0.3tkg·m2.The external disturbance is(62)d=ω2+0.05sin⁡0.8t,cos⁡0.5t,sin⁡0.3tTN·m.The maximum torque of the actuators isvMi=8N·m, i=1,2,3. In order to reflect the fault condition of real actuators such as flywheels, a time-varying loss of actuator effectiveness fault E=diag⁡E1,E2,E3 is given as [9] (63)E1=0.2+0.1sin⁡0.2πt,t≥15sE2=0.2+0.1sin⁡0.3πt,t≥12sE3=0.2+0.1sin⁡0.4πt,t≥10s.In practice, to perform a high-precision stabilization and safety during the maneuvers, the stateσi should be stabilize to a small region, i.e., σi≤1×10-3, i=1,2,3, and no overshoot is allowed.The comparative simulations are given to verify the effectiveness of the proposed control scheme. For the notation convenience, the three compared control schemes are given as follows.M1: the proposed control scheme including the finite-time command filter (30), virtual control (32), error compensation (33), commanded controller (42), and the update law (45). The control parameter settings are shown in Table 1.Table 1
Parameters of the control scheme.
Parameter Value Parameter Value Parameter Value k1 0.1 β 1 3 p1,q1 5,1 k2 0.8 β 2 4 a1,b1 5,4 l1 0.01 h 0.5 p2,q2 5,1 τ1 0.01 m 1 0.001 a2,b2 5,5 τ2 0.5 ρ 0 0.5 p3,q3 5,4 γ 0.3 ρ ∞ 0.001 a3,b3 5,1 λ 2 κ 0.1M2: the conventional backstepping control scheme [26]. The virtual control α, commanded controller v, and fuzzy adaptive update law are given by(64)α=-k1G-1z1(65)v=J0-k2z2-θ^ηs22h2(66)θ^˙=λ∑i=13s2i2ΦiTΦi2h2-m1θ^where z1=σ, z2=ω-α, and the control parameters k1,k2,λ,h are set the same as M1 scheme.M3: the adaptive fast terminal sliding mode control scheme proposed in [22]. The sliding function S is given by(67)S=ω+K1σ+K2Sauwhere K1, K2 are design parameters and Sau=[Sau1,Sau2,Sau3]T is(68)Saui=sgn⁡σiδir,ifσi≥ε2-rεr-1σi+2-rεr-2sgn⁡σiσi2,ifσi<εwhere ε>0 is a small constant scalar; 0<r<1 is a design parameter.The control law is(69)v=-τS-ρsig0.5⁡S-vsstwhere τ,ρ are design parameters, sig0.5⁡S=[S10.5sgn⁡S1,S20.5sgn⁡S2, S30.5sgn⁡S3]T, and vss(t) is(70)vsst=SSF^t+SS2∑i=13oi4pic^it,ifSF^t>ϵSϵF^2t,ifSF^t≤ϵwhere ϵ>0 is a small constant scalar, F^(t)=c^1(t)+c^2(t)ω+c^3(t)ω2, and the update laws are given by(71)c^˙1=-o1c^1+p1Sc^˙2=-o2c^2+p2Sωc^˙3=-o3c^3+p3Sω2where oi,pi,i=1,2,3, are design parameters. The parameters are chosen as r=9/11, K1=0.4I3, K2=0.2I3, τ=10I3, ρ=1, oi=0.1, pi=2, i=1,2,3, ϵ=0.01, and ε=0.0001.The simulation results are shown in Figures1–5. The attitude described by MRPs σi(t), i=1,2,3, along with the corresponding performance bounds is depicted in Figure 1. As shown in Figure 1, although all the three schemes could achieve the attitude stabilization, the M2 and M3 fail to meet the prescribed transient and steady-state error bound, while the proposed M1 scheme remains in the prescribed bound all the time. The angular velocities ω of the three schemes are depicted in Figure 2, which shows that M1 can provide better angular velocity performance than M2 and M3. From Figures 1 and 2, it is concluded that the attitude stabilization with prescribed performance and high precision is achieved in a finite time with the proposed control scheme. Figure 3 shows the actual control signals u, which are quite similar for different three schemes. The three-dimensional trajectory of the transformed error ε(t) is shown in Figure 4; it can be seen that the transformed error can converge to the neighbourhood of the origin in a finite time (around 40 s) subject to the actuator fault and input saturation. The convergence performance of the estimated parameter θ^ is shown in Figure 5, and it is clear that the parameter θ^ converges to a positive constant. From Figures 1–5, it is concluded that the proposed control scheme can achieve the prescribed performance within a finite time in the presence of the inertia uncertainty, external disturbance, actuator saturation, and faults.Figure 1
Attitudeσ along with corresponding performance bounds.Figure 2
Angular velocityω.Figure 3
Actual Control torqueu.Figure 4
Three-dimensional trajectory of the transformed errorε(t).Figure 5
Estimated parameterθ^.


## 6. Conclusion

The problem of attitude stabilization with guaranteed transient and steady-state performance has been investigated in this paper for the spacecraft systems with inertia uncertainty, external disturbance, actuator saturation, and faults. The singularity problem caused by the differentiation of the virtual control is avoided by the proposed first-order command filter. Then, incorporate the prescribed performance boundary into the controller design by using the error transformation to guarantee the prescribed performance of the system output. The control scheme stabilizes the system within a finite time with the proposed adaptive finite-time fault-tolerant control scheme. Finally, simulation results have been provided to verify the effectiveness of the proposed control algorithm.



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*Source: 4912483-2018-11-06.xml*