# Finite-Time Orbit Control for Spacecraft Formation with External Disturbances and Limited Data Communication

Authors: Lei Xing; Dechao Ran; Jian Zhang; Li Huang
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

This work addresses the finite-time orbit control problem for spacecraft formation flying with external disturbances and limited data communication. A hysteretic quantizer is employed for data quantization in the controller-actuator channel to decrease the communication rate and prevent the chattering phenomenon caused by the logarithmic quantizer. Combined with the adding one power integrator method and backstepping technique, a new finite-time tracking control strategy with adaptation law is designed to ensure that the closed-loop system is practical finite-time stable, and that the tracking errors of relative position and velocity are bounded within finite-time despite with limited data communication and external disturbances. Finally, an example is shown to validate the effectiveness of the proposed finite-time tracking controller.


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## Body

## 1. Introduction

Spacecraft formation flying (SFF) is a concept that the functionality of traditional large spacecraft is replaced by a group of less-expensive, smaller, and cooperative spacecraft [1, 2]. In recent years, the study of spacecraft formation control has gradually became an active area of research owing to the reason that SFF is a primary technology for modern space missions, such as deep space exploration, spacecraft on-orbit servicing, and deep space imaging [3–6]. So far, a rich body of orbit control strategies for SFF has been presented, and the relative dynamic model of these strategies can be roughly categorized into two types, namely, linear dynamic model and nonlinear dynamic model [7, 8]. Considering the linearizing of relative dynamic model can induce model errors, various control schemes based on the nonlinear relative dynamic model for spacecraft formation flying have been proposed [9–12].It is to be noticed that all aforementioned control schemes can only guarantee the asymptotical stability of the controlled systems. A fast convergence rate is an essential demand for SFF. In the past few years, the finite-time control method has been widely adopted to design controllers for various nonlinear systems since it can guarantee the controlled systems have better disturbance rejection property, faster convergence rate, and higher precision control performance compared with the asymptotic control [13–16]. To date, the finite-time method has been found successfully applied to handle the spacecraft control problem [17, 18]. The design approaches for finite-time control for spacecraft main include three type approaches, namely, the adding one power integrator technique, the terminal sliding mode method, and the homogeneity theorem. However, the few studies have focused on finite-time orbit control based on the adding one power integrator technique for spacecraft formation.Despite the fact that the mentioned research results above can ensure the controlled system finite-time stability, most of the existing finite-time control strategies focused on traditional large spacecraft, in which the bandwidth of the communication channel is assumed to be not limited. However, in modern spacecraft formation control system, data communication between different modules is usually executed by wireless networks, which means that the bandwidth of the communication channel is limited [19, 20]. Although the use of wireless networks brings many advantages, such as lighter weight, implementation, and installation with less cost, some new challenges have inevitably been induced, for example, data missing, communication time delay, and quantization effect [21–24]. It is well known that when the data of the control module is transmitted to the actuator module by limited communication networks, the quantization errors caused by signal quantization have unavoidably emerged. If the effect of quantization errors is not compensated effectively, the control performance may be degraded or even let the system unstable. Therefore, it is desirable to design a new controller considering limited data communication for network-based spacecraft formation control systems. Recently, the attitude control problem with limited data communication for spacecraft has been studied in [25]. Unfortunately, the study concentrated on finite-time control for SFF with limited data communication is scarce.Motivated by the above discussion, the finite-time orbit control for SFF with limited data communication will be investigated. The main contribution of this work can be summarized as follows: (1) a novel finite-time tracking controller for SFF is proposed such that the tracking errors are bounded within finite time, in which the external disturbances are considered. (2) Compared with some existing finite-time controllers designed based on the terminal sliding mode method [26–28], the advantage of our method is that it can avoid noncontinuous and singular problem by using the backstepping approach and the adding one power integrator technique. (3) The needed communication rate is decreased by employing the hysteresis quantizer to quantify the control data, in which the chattering phenomenon induced by the logarithmic quantizer can be successfully prevented.The rest of the paper is organized as follows. The modeling and preliminaries are given in Section2. In Section 3, the main results are presented. In Section 4 and Section 5, the illustrative example and conclusions are presented, respectively.


## 2. Modeling and Preliminaries

### 2.1. Spacecraft Formation Flying Dynamic Model

The nonlinear relative motion dynamics of SFF can be described as follows [29]:
(1)x¨−2nẏ−ṅy−n2x=−μxR+q1t3−ul1ml+w1+u1mf,y¨+2nẋ+ṅx−n2y=u2mf−μy+rR+q1t3+μrR3−ul2ml+w2z¨+μzR+q1t3+ul3ml=w3+u3mf,,where q1t=x,y,zT is the relative position from the follower spacecraft to the leader spacecraft in the local coordinate frame, ut=u1,u2,u3T and ult=ul1,ul2,ul3T are the control inputs of the follower spacecraft and leader spacecraft, respectively, mf and ml denote the mass of the follower spacecraft and leader spacecraft, respectively, wt=w1,w2,w3T denotes the bounded external disturbance, μ denotes the Earth gravitational constant, n denotes the orbit angular velocity of the leader spacecraft, and R=0,r,0T is the position vector from the inertial coordinate attached to the center of Earth to the leader spacecraft described in the local coordinate frame.The position tracking errore1t and velocity tracking error e2t are defined as
(2)e1t=q1t−q1d=e11,e12,e13T,e2t=q2t−q2d=e21,e22,e23T,where q2t=ẋ,ẏ,żT, q1d=xd,yd,zdT m, and q2d=ẋd,ẏd,żdT m/s are the relative velocity vector, desired position state, and desired velocity state, respectively.Then, the error relative motion dynamics of SFF can be described as follows:(3)ė1t=e2t,ė2t=fq1,q2,q̇2d+1mut+wt,where
(4)fq1,q2,q̇2d=f1q11,q21,q̇2d1f2q12,q22,q̇2d2f3q13,q23,q̇2d3=2nẏ+ṅy+n2x−μxR+q1t3−ul1ml−x¨d−2nẋ−ṅx+n2y−μy+rR+q1t3+μrR3−ul2ml−y¨d−μzR+q1t3−ul3ml−z¨d.fq1,q2,q̇2d=f1q11,q21,q̇2d1,f2q12,q22,q̇2d2,f3q13,q23,q̇2d3T.Assumption 1.
The desired statesq1d and q2d are assumed to bounded, and their first two-order time derivatives are assumed to be bounded.Definition 2.
The solution of (3) can be regarded as finite-time bounded or practical finite-time stable if for all e0=e0, and there exists ν¯>0 and Trν¯,e0<∞, such that et<ν¯ for t≥Tr.Lemma 3 (see [30]).
Forxi∈R, i=1,⋯,m if q>1, we have
(5)∑i=1mxiq≤2q−1∑i=1mxiq,∑i=1mxi1/q≤∑i=1mxi1/q.Lemma 4 (see [31]).
Ifα and β be positive constants, then ςy,z>0 is a real-valued function. We have
(6)yαzβ≤αςy,zα+βyα+β+βςy,z−α/βα+βzα+β.


### 2.2. Fuzzy-Logic Systems

The IF-THEN rules of the FLSs are constructed:(7)Rl:ifx1isF1l,x2isF2l,⋯,andxnisFnl,thenyisGl,where x=x1,⋯,xnT∈ℝn is the input of FLSs, y∈ℝ is the output of FLSs, and Fil and Gl are the fuzzy sets associate with the membership functions μF2lxi and μGly, respectively. m=1,2,⋯,k and k are the number of rules. Then, the FLSs can be expressed as
(8)yx=∑m=1kΛm∏i=1nμFimxi∑m=1k∏i=1nμFimxi,where Λm=maxy∈ℝμGmy, and the fuzzy basis functions can be expressed by
(9)ξmx=∏i=1nμFimxi∑l=1k∏i=1nμFimxi.LetΛT=Λ1,Λ2,⋯,Λk and ξTx=ξ1,ξ2,⋯,ξk. Based on (9), we have ξTxξx<1. Then, considering (8) and (9), we can obtain
(10)yx=ΛTξx.Moreover, we can obtain the following property when the membership function is the Gaussian function.Lemma 5 (see [32]).
Iffx is a continuous function defined on a compact set Ω, then there exists an FLSs for any given positive constant ε∗>0 satisfying
(11)supx∈Ωfx−Λ∗Tξx≤ε∗,where Λ∗ is the optimal parameter vector.It is well known that FLSs have been used to deal with the uncertainties of nonlinear control systems owing to their universal approximation ability. In this paper, the FLSs will be used to approximateΩR2i.


### 2.3. Hysteretic Quantizer

In this paper, the hysteresis quantizerQ· is introduced to quantify the control signal, which can be expressed as follows [33]:
(12)Qui≜uiksgnuiifuik1+η<ui≤uik1−η,u̇ik<0,uik<ui≤uik1−η,u̇ik>0,uik1+δsgnuiifuik<ui≤uik1−η,u̇ik<0,uik1−η≤ui≤uik1+η1−η,u̇ik>0,0,if0≤ui<umin1+η,u̇ik<0,umin1+η≤ui≤umin,u̇ik>0,Quit−othercasewhere
(13)uik=ρ1−kumin,k=1,2,⋯n,and umin>0, 0<ρ<1, and η=1−ρ/1+ρ. Qui is in the set of U=0,±uik,±uik1+δ. The size of the dead-zone for Qui is determined by umin. The map of the hysteresis quantizer Qui for ui>0 is given in Figure 1.Remark 6.
The parameterρ can be considered as a measure of quantization density. From (13), it is well known that the smaller ρ is, the coarser the quantizer becomes. Furthermore, η approaches 1 while ρ approaches zero, and then Qui will have fewer quantization levels since ui ranges over that interval [34]. Thus, the needed communication rate between the actuator module and control module is lower.Remark 7.
Unlike the logarithmic quantizer, additional quantization levels have been added in (12) to prevent oscillations. Besides, the parameter ρ chosen should be based on a principle that the controlled system is stable can be ensured.Figure 1
Map ofQui for ui>0.To facilitate the next controller design, the hysteretic quantizerQui is decomposed into the following form:
(14)Qui=ϕitui+dit,where ϕit and dit are nonlinear functions.Lemma 8.
The nonlinear blue functionsϕit and dit, respectively, satisfy
(15)1−η≤ϕit≤1+η,dit≤umin.Proof.
From Figure2 and according to the sector-bound properties, for ui≥umin, we have
(16)1−ηui≤Qui≤1+ηuit.
Forui≤umin, when Qui=0, one has
(17)0=ϕitui+dit.Figure 2
Block diagram of the finite-time control scheme.Define(18)ϕit=Quiui,ui>umin1,ui≤umindit=0ui>umin−uit,ui≤umin.Then,Qui=ϕitui+dit holds, where ϕit and dit satisfy (15).


### 2.4. Control Objective

The objective of this paper is to develop a finite-time adaptive tracking controllerut such that the position state q1t and velocity state q2t can track the desired state q1d and q2d in finite-time, and that the tracking error states e1t and e2t are finite-time bounded with limited data communication and external disturbances.


## 2.1. Spacecraft Formation Flying Dynamic Model

The nonlinear relative motion dynamics of SFF can be described as follows [29]:
(1)x¨−2nẏ−ṅy−n2x=−μxR+q1t3−ul1ml+w1+u1mf,y¨+2nẋ+ṅx−n2y=u2mf−μy+rR+q1t3+μrR3−ul2ml+w2z¨+μzR+q1t3+ul3ml=w3+u3mf,,where q1t=x,y,zT is the relative position from the follower spacecraft to the leader spacecraft in the local coordinate frame, ut=u1,u2,u3T and ult=ul1,ul2,ul3T are the control inputs of the follower spacecraft and leader spacecraft, respectively, mf and ml denote the mass of the follower spacecraft and leader spacecraft, respectively, wt=w1,w2,w3T denotes the bounded external disturbance, μ denotes the Earth gravitational constant, n denotes the orbit angular velocity of the leader spacecraft, and R=0,r,0T is the position vector from the inertial coordinate attached to the center of Earth to the leader spacecraft described in the local coordinate frame.The position tracking errore1t and velocity tracking error e2t are defined as
(2)e1t=q1t−q1d=e11,e12,e13T,e2t=q2t−q2d=e21,e22,e23T,where q2t=ẋ,ẏ,żT, q1d=xd,yd,zdT m, and q2d=ẋd,ẏd,żdT m/s are the relative velocity vector, desired position state, and desired velocity state, respectively.Then, the error relative motion dynamics of SFF can be described as follows:(3)ė1t=e2t,ė2t=fq1,q2,q̇2d+1mut+wt,where
(4)fq1,q2,q̇2d=f1q11,q21,q̇2d1f2q12,q22,q̇2d2f3q13,q23,q̇2d3=2nẏ+ṅy+n2x−μxR+q1t3−ul1ml−x¨d−2nẋ−ṅx+n2y−μy+rR+q1t3+μrR3−ul2ml−y¨d−μzR+q1t3−ul3ml−z¨d.fq1,q2,q̇2d=f1q11,q21,q̇2d1,f2q12,q22,q̇2d2,f3q13,q23,q̇2d3T.Assumption 1.
The desired statesq1d and q2d are assumed to bounded, and their first two-order time derivatives are assumed to be bounded.Definition 2.
The solution of (3) can be regarded as finite-time bounded or practical finite-time stable if for all e0=e0, and there exists ν¯>0 and Trν¯,e0<∞, such that et<ν¯ for t≥Tr.Lemma 3 (see [30]).
Forxi∈R, i=1,⋯,m if q>1, we have
(5)∑i=1mxiq≤2q−1∑i=1mxiq,∑i=1mxi1/q≤∑i=1mxi1/q.Lemma 4 (see [31]).
Ifα and β be positive constants, then ςy,z>0 is a real-valued function. We have
(6)yαzβ≤αςy,zα+βyα+β+βςy,z−α/βα+βzα+β.


## 2.2. Fuzzy-Logic Systems

The IF-THEN rules of the FLSs are constructed:(7)Rl:ifx1isF1l,x2isF2l,⋯,andxnisFnl,thenyisGl,where x=x1,⋯,xnT∈ℝn is the input of FLSs, y∈ℝ is the output of FLSs, and Fil and Gl are the fuzzy sets associate with the membership functions μF2lxi and μGly, respectively. m=1,2,⋯,k and k are the number of rules. Then, the FLSs can be expressed as
(8)yx=∑m=1kΛm∏i=1nμFimxi∑m=1k∏i=1nμFimxi,where Λm=maxy∈ℝμGmy, and the fuzzy basis functions can be expressed by
(9)ξmx=∏i=1nμFimxi∑l=1k∏i=1nμFimxi.LetΛT=Λ1,Λ2,⋯,Λk and ξTx=ξ1,ξ2,⋯,ξk. Based on (9), we have ξTxξx<1. Then, considering (8) and (9), we can obtain
(10)yx=ΛTξx.Moreover, we can obtain the following property when the membership function is the Gaussian function.Lemma 5 (see [32]).
Iffx is a continuous function defined on a compact set Ω, then there exists an FLSs for any given positive constant ε∗>0 satisfying
(11)supx∈Ωfx−Λ∗Tξx≤ε∗,where Λ∗ is the optimal parameter vector.It is well known that FLSs have been used to deal with the uncertainties of nonlinear control systems owing to their universal approximation ability. In this paper, the FLSs will be used to approximateΩR2i.


## 2.3. Hysteretic Quantizer

In this paper, the hysteresis quantizerQ· is introduced to quantify the control signal, which can be expressed as follows [33]:
(12)Qui≜uiksgnuiifuik1+η<ui≤uik1−η,u̇ik<0,uik<ui≤uik1−η,u̇ik>0,uik1+δsgnuiifuik<ui≤uik1−η,u̇ik<0,uik1−η≤ui≤uik1+η1−η,u̇ik>0,0,if0≤ui<umin1+η,u̇ik<0,umin1+η≤ui≤umin,u̇ik>0,Quit−othercasewhere
(13)uik=ρ1−kumin,k=1,2,⋯n,and umin>0, 0<ρ<1, and η=1−ρ/1+ρ. Qui is in the set of U=0,±uik,±uik1+δ. The size of the dead-zone for Qui is determined by umin. The map of the hysteresis quantizer Qui for ui>0 is given in Figure 1.Remark 6.
The parameterρ can be considered as a measure of quantization density. From (13), it is well known that the smaller ρ is, the coarser the quantizer becomes. Furthermore, η approaches 1 while ρ approaches zero, and then Qui will have fewer quantization levels since ui ranges over that interval [34]. Thus, the needed communication rate between the actuator module and control module is lower.Remark 7.
Unlike the logarithmic quantizer, additional quantization levels have been added in (12) to prevent oscillations. Besides, the parameter ρ chosen should be based on a principle that the controlled system is stable can be ensured.Figure 1
Map ofQui for ui>0.To facilitate the next controller design, the hysteretic quantizerQui is decomposed into the following form:
(14)Qui=ϕitui+dit,where ϕit and dit are nonlinear functions.Lemma 8.
The nonlinear blue functionsϕit and dit, respectively, satisfy
(15)1−η≤ϕit≤1+η,dit≤umin.Proof.
From Figure2 and according to the sector-bound properties, for ui≥umin, we have
(16)1−ηui≤Qui≤1+ηuit.
Forui≤umin, when Qui=0, one has
(17)0=ϕitui+dit.Figure 2
Block diagram of the finite-time control scheme.Define(18)ϕit=Quiui,ui>umin1,ui≤umindit=0ui>umin−uit,ui≤umin.Then,Qui=ϕitui+dit holds, where ϕit and dit satisfy (15).


## 2.4. Control Objective

The objective of this paper is to develop a finite-time adaptive tracking controllerut such that the position state q1t and velocity state q2t can track the desired state q1d and q2d in finite-time, and that the tracking error states e1t and e2t are finite-time bounded with limited data communication and external disturbances.


## 3. Main Results

In this section, a novel finite-time orbit tracking control strategy for formation spacecraft will be proposed. To facilitate the control strategy propose, the intermediate variablesR1i and R2i is defined as follows:
(19)R1i=e1i1/α1−σ1i1/α1,R2i=e2i1/α2−σ2i1/α2,i=1,2,3,where α1=1, 0<α2=1+τ<1, and τ=−m/n<0 be the ration of an even integer m and an odd integer n. σ1i and σ2i are the virtual controller. Under the backstepping method framework, the control strategy develop procedure is described as follows:Step 1.
Propose of virtual control lawσ1i and σ2i.
Consider a Lyapunov function candidate asV1=∑i=13∫σ1ie1is1/α1−σ1i1/α12−α2ds.
Design the virtual controllerσ1i as follows:
(20)σ1i=0.
The derivative ofV1 is obtained as follows:
(21)V̇1=∑i=13R1i2−α2ė1i=∑i=13R1i2−α2e2i−σ2i+∑i=13R1i2−α2σ2i,and then applying Lemmas 3 to 5 yields
(22)R1i2−α2e2i−σ2i≤R1i2−α2e2i−σ2i≤2R1i2−α2R2iα2≤R1i2+ςR2i2,where ς=α22−α22−α2/α2>0 is a positive constant. The virtual control scheme is proposed as
(23)σ2i=−2R1iα2.
Then substituting (22) and (23) into above (21)
(24)V̇1≤∑i=13R1i2+∑i=13ςR2i2+∑i=13R1i2−α2σ2i≤∑i=13R1i2+∑i=13ςR2i2−2∑i=13R1i2≤∑i=13ςR2i2−∑i=13R1i2.Step 2.
Propose of controllerui.
Consider the Lyapunov function as follows:(25)V=V1+ϖ+1/2∑i=13ψ~i2,
where0<α2+τ<1(26)ϖ=∑i=13∫σ2ie2is1/α2−σ2i1/α22−α3ds.
The time derivative ofϖ is derived as
(27)ϖ̇=∑i=13R2i2−α3ė2i+∑i=132−α3−dσ2i1/α2dt∫σ2ie2is1/α2−σ2i1/α21−α3ds.
Note that(28)∫σ2ie2is1/α2−σ2i1/α21−α3ds≤R2i1−α3e2i−σ2i≤21−α2R2i1−τ≤2R2i1−τ.
Based on (28), we have
(29)2−α3−dσ2i1/α2dt∫σ2ie2is1/α2−σ2i1/α21−α3ds≤2−α321+α2/α2e2iR2i1−τ≜χe2i,σ2i.
Substituting (29) into the time derivative of V, it yields
(30)V̇=V̇1+ϖ̇+∑i=13ψ~iψ~i≤∑i=13ςR2i2−∑i=13R1i2+∑i=13R2i2−α3ė2i+χe2i,σ2i+∑i=13ψ~iψ~i≤∑i=13ςR2i2−∑i=13R1i2+∑i=13R2i2−α3R2ifiq1i,q2i,q̇2di+1mQui+wi+χe2i,σ2i−∑i=13ψ~iψ^̇i.The controllerui is designed as
(31)ui=−m1−ηR2iα3ℓiψ^i+1+fiq1i,q2i,q̇2di.The adaptive law of ψ^i is designed as
(32)ψ^̇i=ℓiR2i2−μiψ^i,where ℓi>0 and μi>0 are designed parameters.Substituting (31) into (30) yields
(33)V̇≤∑i=13ςR2i2−∑i=13R1i2+∑i=13R2i2−α3fiq1i,q2i,q̇2di+1mϕitui+dit+wi+∑i=13χe2i,σ2i−∑i=13ψ~iψ^̇i≤∑i=13ςR2i2−∑i=13R1i2+R2i2−α31mdit+wi−∑i=13R2i2ℓiψ^i+1+∑i=13χe2i,σ2i−∑i=13ψ~iψ^̇i.LetΩR2i=R2iτ−1ςR2i2+R2i2−α31/mdit+wi+χe2i,σ2i. Then, (33) can be rewritten as
(34)V̇≤−∑i=13R1i2−∑i=13R2i2ℓiψ^i+1−∑i=13ψ~iψ^̇i+∑i=13R2i1−τΩR2i.By using the approximation ability of FLSs in Lemma5, we have
(35)ΩR2i=φiR2i∗Tξi+ε1.For convenience, let(36)Ω=φi∗Tξi+ε1≤φ¯i∗Tξ¯i,where φi∗T is the optimal fuzzy weight vector, ε1≤ε1∗, with ε1∗ being a positive constant. φ¯i∗T=φi∗T,ε1∗ and ξ¯iT=ξiT,1. According to Lemmas 3 to 4, and noting that 0<ξiTξi≤1, one has
(37)R2i1−τΩ≤R2i1−τψi1−τ2≤ℓiR2i2ψi+Γi,where ψ~i=ψi−ψ^i, ψ=2φ¯i∗2/1−τ, and ℓi>0 are positive design parameters, Γi=1+τ/22/1−τℓi−1−τ/1+τ, and ψ^i is the estimate of the parameter ψi. Substituting (32) and (37) into (34) yields
(38)V̇≤−∑i=13R1i2−∑i=13R2i2ℓiψ^i+1−∑i=13ψ~iψ^̇i+∑i=13ℓiR2i2ψi+∑i=13Γi≤−∑i=13R1i2−∑i=13R2i2ℓiψ^i+1−∑i=13ψ~iℓiR2i2−μiψ^i+∑i=13ℓiR2i2ψi+∑i=13Γi≤−∑i=13R1i2−∑i=13R2i2+∑i=13μiψ~iψ^i+∑i=13Γi.The main theorem of this paper is presented as follows.Theorem 9.
Consider the dynamic model of SFF described by (1) with the hysteretic quantizer given in (14). If the controller is designed as (31), and the adaptation law is given in (32), then the tracking errors e1i and e2i,i=1,2,3 will converge into a region of the origin within finite-time.Proof.
DenotingV¯=V1+ϖ, one has
(39)V¯=∑j=12∑i=13∫σjiejis1/αj−σji1/αj2−αj+1ds≤2∑i=13R1i2−τ+2∑i=13R2i2−τ.
Choosing0<γ=2/2−τ<1, and according to Lemma 3, one then has
(40)V¯γ≤2∑i=13R1i2−τ+2∑i=13R2i2−τγ≤2∑i=13R1i2+2∑i=13R2i2.
According to the definition ofV and V¯, we can obtain
(41)V=V¯+12∑i=13ψ~i2
According to (38) and (39), one has
(42)V̇≤−12Vγ+12Vγ−∑i=13R1i2−∑i=13R2i2+∑i=13μiψ~iψ^i+∑i=13Γi≤−12Vγ+12V¯γ+1212∑i=13ψ~i2γ−∑i=13R1i2−∑i=13R2i2+∑i=13μiψ~iψ^i+∑i=13Γi≤−12Vγ+12∑i=13ψ~i2γ+∑i=13μiψ~iψ^i+∑i=13Γi.
Moreover, applying Lemma4(43)μiψ~iψ^i≤−μi2ψ~i2+μi2ψi2,12∑i=13ψ~i2γ≤∑i=13μi2ψ~i2+∑i=13νi,where νi=1−γγ/μiγ/1−γ.Then,(44)V̇≤−12Vγ+∑i=13νi+μi2ψi2+∑i=13Γi≤−12κVγ−121−κVγ+∑i=13νi+μi2ψi2+∑i=13Γi≤−12κVγ−121−κVγ+ϑ,where ϑ=∑i=13νi+μi/2ψi2+∑i=13Γi. From (44), we can obtain that V̇≤−12κVγ if and only if Vγ>2ϑ/1−κ, which means V<0 out the set Vγ≤2ϑ/1−κ. The required time of reach the bounded set is Tr≤2V1−γe0/κ1−γ. Thus, as the analysis in [16], we can conclude that V can reach the set within finite time based on Definition 2. Furthermore, we can obtain that the tracking errors e1i and e2i,i=1,2,3 converge into a region of the origin with finite-time.The design procedure of the controller could be visualized from the bloc diagram shown in Figure2.Remark 10.
By employing the property0<ξiTξi≤1 of FLSs [16], only one adaptation law for ψ^̇i is designed to proposed the controller. Furthermore, the effects of quantization errors and external disturbances can be eliminated by the designed adaptation law.Remark 11.
Compared with the finite-time controllers using the terminal sliding mode method, the finite-time controller (31) is proposed by using the adding one power integrator technique and backstepping approach to guarantee the finite-time convergence and overcome the noncontinuous and singular problem.


## 4. Illustrative Example

In this section, an illustrative example is shown to illustrate the efficiency of the proposed finite-time control scheme. The leader spacecraft and follower spacecraft mass areml=1 kg and mf=1 kg, respectively. For simplicity, the leader spacecraft is assumed in a circular reference orbit of radius 6728 km. The input control force is limited as ui≤umax=1N, i=1,2,3.The initial relative position and velocity areq10=250,−50,0T m and q20=0,0,0T m/s, respectively. The desired relative position and relative are q1d=0,0,0T m and q2d=0,0,0T m/s, respectively. The external disturbance is wt=0.01cos0.5t,0.01cos0.5t,0.01cos0.5t. The parameters of the finite time control scheme (31) and the adaptation law (33) are chosen as ℓi=0.001, μi=0.001, ψ^i0=0.01, i=1,2,3α1=1, α2=3/5, and α3=1/5, and the quantization parameters are designed as ρ=0.4, umin=0.000001.The tracking errorse1i and e2i,i=1,2,3 of the finite-time controller (31) are shown Figures 3 and 4, respectively. As observed from Figure 2, the position tracking errors can converge to almost zero with 150 s, and the position tracking errors is within e1i<0.01 m at 500 s. The control force is given in Figure 5, which force magnitude is limited to 1 N. Figure 6 shows the quantized control force for the finite-time controller (31). The response curves of ψ^i are shown in Figure 7. Clearly, the simulation results verify the validity of the hysteresis quantizer and illustrate the efficiency of the proposed finite-time control scheme.Figure 3
Response curves of position tracking errors.Figure 4
Response curves of velocity tracking errors.Figure 5
Response curves of control force.Figure 6
Response curves of quantized control force.Figure 7
Response curves of the adaptive parameterψ^i,i=1,2,3.


## 5. Conclusion

In this paper, the problem of finite-time orbit control for SFF with limited data communication and external disturbances was studied. We have applied the hysteretic quantizer to quantize the data of the controller-actuator channel to decrease the communication rate. By combining with the adding one power integrator method and the backstepping technique, a finite-time adaptive controller has been developed. Under the proposed control strategy, the effects of quantization errors and external disturbances have been eliminated. Moreover, the finite-time stability of the controlled system and bounded of tracking errors of position and velocity within finite time are guaranteed by the developed controller. Finally, an example has been shown to illustrate the effectiveness of the proposed control strategy. Future work will focus on dealing with the finite-time tracking control problem for SFF with actuator faults.



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