Improved Repetitive Control for High-Precision Satellite Rendezvous

，

College of Automation Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，P.R.China

Abstract: This paper is focused on control design for high-precision satellite rendezvous systems.A relative motion model of leader-follower satellites described by relative orbit elements（ROE）is adopted，which has clear geometric meaning and high accuracy.An improved repetitive control（IRC）scheme is proposed to achieve high-precision position and velocity tracking，which utilizes the advantage of repetitive control to track the signal precisely and conquers the effects of aperiodic disturbances by adding a nonsingular terminal sliding mode（NSTSM）controller.In addition，the nonlinear state error feedback（NLSEF）is used to improve the dynamic performance of repetitive controller and the radial basis function （RBF） neural networks are employed to approximate the unknown nonlinearities.From rigorous Lyapunov analysis，the stability of the whole closed-loop control system is guaranteed.Finally，numerical simulations are carried out to assess the efficiency and demonstrate the advantages of the proposed control scheme.

Keywords：satellite rendezvous；relative orbit elements；repetitive control；sliding mode control；radial basis function（RBF）neural network

Nomenclature

Ω/rad Right ascension of ascending node ω/rad rad Argument of perigee i/Orbit inclination a/km Semi-major axis e Eccentricity θ/rad True anomaly u/rad Argument of attitude D/(rad·s-1) Relative average drift rate Δe Relative eccentricity vector Δi Relative inclination vector ΔM/rad Difference of mean argument of latitude nˉ/(rad·s-1) Average angular velocity n/(rad·s-1) Orbit angular velocity μ/(m3·s-2) Earth gravitational constant r/km Distance between satellite and the Earth

Δr/km Distance between two satellites x p1 Relative position xp2 Relative velocity xp1d Expected relative position xp2d Expected relative velocity

0 Introduction

Nowadays，people expect satellites to have longer life spans，for example，refueling a satellite when it runs out of fuel，repairing a satellite when failure happens，or removing components when a satellite assignment is completed.These are called on-orbit services（OOS）for satellites.For these operations，the satellite rendezvous technology is essential.

To gain better satellite rendezvous control performance，an accurate model of satellite formation flying dynamics is essential.Considerable effortshave been devoted to relative motion dynamical equations.One of the most famous and widespread models is Clohessy-Wiltshire（C-W）equations of motion［1］，which is used to describe the relative position and velocity between leader and follower satellites in close rendezvous phase［2］.The C-W equations set a rotating Cartesian coordinate system with the origin on the leader satellite，resembling Hill equations in the form［3］.Thus，the equations are also referred to as Hill-Clohessy-Wiltshire（HCW）equations.In order to have a simple and linear expression，the C-W equations make some assumptions，such as the leader satellite orbit being circular，and the Earth being symmetrically sphere.These assumptions not only affect the model precision，but also have effects on fuel consumption.

Among work aiming at promoting the application of C-W equations［4-10］，Tschauner-Hempel（TH）equations［9］describe another widely used model for formation flying，taking the eccentricity of the leader satellite orbit into consideration.

Some work followed another idea to obtain a precise model.A set of six constants can determine an arbitrary satellite orbit， including the orbit shape，position，and satellite motion information.The six constants are named Kepler elements，or absolute orbit elements（AOE）in this paper.Hill［3］mentioned another set of six constants，named relative orbit elements （ROE） in his lunar theory，which determines the relative motion between leader and follower satellites.There is correspondence and association between absolute and relative orbit elements.The relative motion model is established and analyzed based on ROE［11-12］.The ROE has clear geometrical meanings，making the model easy to understand，and the model based on ROE unifies the expression of both elliptical orbit and circular orbit without singularities，which C-W equations and T-H equations fail to do.Most importantly，the model based on ROE is proved to achieve high precision［13］.Despite the abovementioned advantages of ROE，some studies［11-13］focus on trajectory analysis，or only design simple control schemes.Common controllers are usually described in Cartesian coordinates.The rendezvous models described by ROE need transformation before they are suitable for common controllers.

Based on the relative position and relative velocity described by ROE［11-12］，our study transfers the relative motion model into the form of Cartesian equations.In this way，the relative motion model in this paper is more suitable for design process control.

Studies have been done on controling a follower satellite to maintain a certain position against a leader satellite.At the early stage， open-loop control schemes were attempted in satellite rendezvous［14-15］，which could hardly satisfy the precision and robustness standard of satellite operations.To deal with the outer disturbances and model uncertainties，most methods［16-29］took them as an external unknown term， and designed close-loop control schemes to reduce the impact of the disturbances on the models.Sliding mode control is a popular control scheme in various circumstances［23-29］.In particular，terminal sliding mode （TSM） controller is proved to have finite convergence time and good anti-interference performance［29］.However，the control precision can be further enhanced if the perturbations causing model uncertainties and external disturbances are analyzed separately［15，23］.As an important indicator in satellite rendezvous mission，tracking precision was not the priority in some studies［24-29］，and the precision level achieved by them can be further improved.

Sliding mode control provides stability for tracking.However，in order to meet the precision requirement of satellite rendezvous mission，the control scheme needs modification.Our study combines the advantages of the radial basis function（RBF）of sliding mode control and the repetitive control，consequently improving the precision of the control system，when the stability of the system is guaranteed.

The repetitive controller is commonly seen in high-precision periodic control tasks.The repetitive control loop contains a dynamic model of the track-ing signal，and the control variable of the last period is added into that of the current period.In time，the repetitive controller is able to track any periodic signals with no steady state error.The impact of the repetitive controller adds up for each period，and a proper control output can remain even when tracking error comes to zero.Hence，the repetitive controller is applied widely in various areas.Nevertheless，there are few examples of satellite rendezvous mission using the repetitive control，because aperiodic disturbances in outer space are amplified by the repetitive controller［30］.

Considering model complexity increased by the repetitive controller and the nonlinearity of the control system， the nonlinear state error feedback（NLSEF） containing an extended state observer（ESO）is utilized to estimate the model states and improve the dynamic performance of the repetitive controller.ESO requires very little information and observes well.The error feedback law made by NLSEF is more effective than the linear control law，like traditional PID control.However，it is difficult to analyze the convergence characteristics of NLSEF with ESO.In our study，the repetitive controller improved by NLSEF is regarded as an augmented system，and sliding mode control is used to strengthen the robustness of the system while RBF neural networks are used to estimate model uncertainties.

Compared with the previously work on satellite rendezvous control，the main contributions of this paper are as follows：First，state-space equations of the leader-follower satellites are deduced by using relative position and velocity based on ROE.The model has clear geometric meaning and high accuracy，and most closed-loop control methods can be applied to this model directly.Second，to achieve highprecision satellite rendezvous with external disturbances and model uncertainties，our study takes advantage of the repetitive controller，and tracks expected position and velocity signals accurately.Third，a sliding mode controller is employed to improve the robustness of the repetitive controller，and guarantees that the system is converged in finite time.Further，NLSEF is added to the repetitive controller to adjust the dynamic performance，and the model uncertainties caused by various perturbations are approximated by RBF neural networks.

The remainder of our paper is organized as follows.In Section 1，the dynamic model of leader-follower satellites is proposed.In Section 2，the controller combining the repetitive control and sliding mode control is developed.In Section 3，the stability and convergence property of the control scheme is analyzed strictly.In Section 4，a numeric simulation is presented to verity of the control scheme.Finally，conclusions are drawn in Section 5.

Notations：A specification of symbols is explained in advance.

For an arbitrary vectorv=[v1v2v3]T∈R3and an arbitrary real constantc＞0，the exponent of the vectorvcis defined as

where for any real numberξ

1 Problem Formulation

In this paper，the leader satellite is in an elliptical orbit，and orbits without any extra control.The control scheme is aimed at the follower satellite.

Our model uses ROE to construct dynamical equations.The model has clear geometrical meanings，unifies the expression of both the elliptical orbit and the circular orbit without singularities，and achieves high precision［14］.

1.1 Relative orbit elements

The traditional AOE is the right ascension of ascending node（RAAN）Ω，argument of perigeeω，orbit inclinationi，semi-major axisa，the eccentricity of the orbite，and initial mean anomalyM0.

Six ROEs are the relative average drift rateD，relative eccentricity vector Δe=[ΔexΔey]T，rela-tive inclination vector Δi=[ΔixΔiy]T，and difference of mean argument of latitude ΔM'［31］.Δidescribes the orientation of the follower satellite orbit plane.Δedescribes the shape of the orbit plane.Ddescribes the size of the orbit and ΔM'describes the position of the orbit.

In Fig.1，Ois the center of Earth，PlandPfare perigees of the leader and the follower satellite orbits；BlandBfare ascending nodes of the leader and the follower satellite orbits；elandefare eccentricity vectors of the leader and the follower satellite the orbits，and Δeis relative eccentricity vector；ilandifare inclination of the leader and the follower satellite orbits；P'fande'fare projections ofPfandefin the leader satellite orbit plane.

Fig.1 Geometric description of ROE

A strict equivalence relation between AOE and ROE is presented in Ref.［31］.However，in satellite rendezvous situation，two satellites are close to each other.That is Δr al?1，where Δris the distance between two satellites，andalis the semi-major axis of the leader satellite.Then the relationship between ROEs by AOE can be expressed as［31］

where subscript l represents the leader satellite，and subscript f the follower satellite；is the average angular speed，andμthe Earth gravitational constant.

1.2 Relative dynamical equations

In the local vertical local horizontal（LVLH）frameL-XYZwith the leader satellite centroid as the origin，as shown in Fig.2，whereXis along the radial direction of the leader satellite，Yis normal to the orbital plane of the leader satellite，andZcompletes the right-handed Cartesian frame.

Fig.2 Schematic diagram of LVLH frame

The relative dynamical equations of two close satellites are as follows［31］

where

andr=a( 1-e2)( 1+ecosθ)is the distance be-tween the leader satellite and the Earth；u=θ+ωis the argument of the latitude of leader satellite；θis the true anomaly of the leader satellite；Δ1is an additional translation term alongXdirection；andΔ2is a pendulum term ofΔ1on the basis of satellite’s elliptical orbit.

Assumption 1［31］Relative average drift rateDand difference of mean argument of latitude ΔM'are bounded.The first and the second derivatives ofDand ΔM'are bounded.

Remark 1Since two satellites are close to each other，the differences of their RAANsΩand orbit inclinationsican be considered small.It is rational to suppose that the differences between their orbit sizes and orbit positions are small，which are described by relative average drift rateDand the difference of mean argument of latitude ΔM'.

Assumption 2［31］Additional termsΔ1andΔ2are bounded.The first and the second derivatives of the additional terms are bounded.

Remark 2Except in some special cases where orbit eccentricityeis close to 1，the numerical magnitudes ofΔ1andΔ2are small.

andBased on Assumptions 1 and 2，σ1，σ2and their first and the second derivatives are bounded.Then Eq.（2）turns into

The variables with respect to time in Eq.（3）are argument of latitudeu=θ+ωand distance between the leader satellite and the Earthr.Their derivatives with respect to time are

Taking the derivate of Eq.（3）with respect to time leads to

According to Eqs.（3，4），the state space equations can be obtained as follows

Eq.（5）is a state space expression in Cartesian coordinate system，which is suitable for control design，and the adaptability and high precision of ROE are still preserved.

1.3 Transformation

The true anomalyθpresents the angular distance of a satellite past the point of periapsis measured in degrees，and increases monotonically with time.Then the state variables changing in time can be transformed，and the true anomalyθis chosen as an independent variable.It is more convenient for the control design process.

Variablexis taken as example［32］.The relationship between the derivative with respect to time dxdt=and the derivative with respect to true anomaly dxdθ=x'can be expressed as

wherenis the orbit angular velocity of the leader satellite，and

Then Eq.（5） can be transformed into state space equations with true anomalyθas independent variable

Eq.（6）can be rewritten as

where

and

As the satellite orbits are in the outer space，they are confronted with various perturbations and disturbances.It is impossible to establish a perfect model.Perturbations lead to measuring errors and modeling inaccuracy， and are presented by ΔA(x，y，z，θ)∈R3.External disturbances are presented byd0(θ)∈R3.

Definexp1=[x y z]Tandxp2=[vx vy vz]Tas the relative position and relative velocity of the follower satellite， respectively.Based on Eq.（7），the satellite rendezvous model is expressed as

whered=d0+σ∈R3，andu∈R3is the control force.

Assumption 3The unknown disturbancesdand the derivativesd'are bounded，i.e.|di|≤whereandare known constants.

1.4 Control objective

In our research，the main difficulties in designing a satellite rendezvous controller are as follows：

（1）Though ROE has specific geometric meaning and high precision，it is necessary to make some assumptions and simplifications to obtain a succinct rendezvous model described by ROE［11-12］.When it is transformed into the Cartesian form，the precisionhas to be assured.

（2）After employing the improved repetitive controller，with the existence of ESO，it is difficult to provide the theoretic proof of the stability of the augmented system［31］.Subsequent control schemes are required to handle the increased uncertainties.

（3）The key purpose of our study is to design a control scheme to achieve high-precision satellite rendezvous.Our study outperforms other studies［23-29］with an accuracy level of 5×10-4m.

Based on the relative motion model described in Eq.（8），our study aims at designing an improved repetitive control（IRC）scheme，making relative positionxp1and relative velocityxp2track given signalsxp1d=[xdydzd]Tandxp2d=[vxdvydvzd]Tprecisely，even with the existence of model uncertainties and unknown disturbances，and the ultimate thrust the engine can generate is prescribed.The relative position errorxp1-xp1dis bounded，and the relative speed errorxp2-xp2dconverges to zero in finite time.

2 Control Design

First，as shown in Fig.3，a repetitive controller is designed to track expected position and velocity signals precisely.An NLSEF is used to improve the dynamic performance of the repetitive controller.

Fig.3 Configuration of repetitive controller

Second，as shown in Fig.4，the original rendezvous model and the repetitive controller are regarded as an augmented system.An adaptive sliding mode controller is designed for the augmented system，and ensures the system has good stability and convergence property with disturbances.RBF neural networks are designed to approximate the model uncertainties，and sliding mode surface is designed as nonsingular terminal sliding mode （NSTSM）surface.

Fig.4 Configuration of repetitive controller and adaptive sliding mode controller

2.1 Repetitive controller

Essentially，the repetitive controller is a highprecision feedback control system，implanting the dynamic model of the external signal in the controller.The repetitive controller can track period signal with no steady state error.

A proper feedback loop can improve the dynamic performance of the repetitive controller.

NLSEF is robust and adaptable，and does not rely on a specific model.This avoids the derivation of the repetitive controller.In NLSEF structure，an extended state observer is utilized to achieve state feedback and model compensation.

As shown in Fig.4，the repetitive controller includes a delay elemente-sTand dynamic model of the external signalq(s)，whereTis the delay constant.εc∈R6is the output of the repetitive controller，andut∈R3is the output of the repetitive controller with NLSEF.

According to Lemma 1 in Ref.［33］（see the Appendix），to track the states of the rendezvous model controlled by the repetitive controller，the observer in NLSEF is designed as

wherez1∈R3，z2∈R3andz3∈R3are state variables of the observer.The functionsg1(·)，g2(·) andg3(·)in Eq.（9）are chosen as［31］

whereb0i(i=1，2，3)，α1andδ1are adjustable parameters，and function

Especially，denoteap=z3∈R3，which compensates the unknown part of the model，and denotewhich tracks the state variables of the model controlled by the repetitive controller.

Denote the output of the repetitive controllerwhereεc1∈R3andεc2∈R3.The NLSEF control lawu0∈R3is expressed as

whereb1，b2，αandδare adjustable parameters.

The satellite rendezvous model and the repetitive controller with NLSEF are regarded as an augmented system.Based on the rendezvous model described in Eq.（8），observer described in Eq.（9），and control law described in Eq.（10），the augmented system can be described as

whereF2=F1+u0-ap.

Although the augmented system described in Eq.（11）is able to track expected relative position and velocity precisely without additional control forceu，it is poor at dealing with unknown disturbances.An adaptive sliding mode controller is designed to address this problem.

2.2 Sliding mode controller

Sliding mode control belongs to variable structure control.The design process is independent from system structure and external disturbances，which makes the sliding mode controller robust.

RBF neural networks are feedforward networks.Theoretically，the output of the neural networks can approximate any function with any accuracy.The networks can be divided into three parts：Input layer，hidden layer，and output layer.

The hidden layer contains some nodes，which are also known as neural cells.The number of hidden layer nodes areN.h=[hx hy hz]∈RN×3is the RBF.Variablexis taken as an example.The output of thejth neural node is chosen as a Gaussian function

andhx=[hx1hx2…hxj…hxN]T∈RN，wherej=1，2，…，Npresents the serial number of the node；cxjis the central vector of each node，andbxjis the breadth of Gaussian function of each node.

The ideal output is

whereWi*∈RN(i=x，y，z) are the ideal weights，andεin(i=x，y，z) are bonded approximate errors，is a constant［34］.

The estimated output is

whereare estimated weights.The estimated weight vector is defined as

The estimated weight errors of each axis

and the estimated weight error vector is defined as

A backstepping algorithm［35］is used to design the sliding mode control law.The backstepping algorithm is divided into two steps.

Step 1Define the relative position error asδ1=xp1-x1d∈R3.Taking derivative ofδ1leads to

The estimated weightsare updated by

whereΓiandσiare adjustable parameters.

Define the virtual control variable asr2d∈R3and design the virtual control as

wherek=diag(kx，ky，kz)∈R3×3is an adjustable parameter.

Step 2Define relative speed error asδ2=xp2-r2d∈R3.Taking derivative ofδ2leads to

Usually，the sliding mode surface is designed linear.To obtain a more robust performance，and make the satellite come and stay at the designated position in limited time， this paper employs NSTSM surfaces∈R3as［34］

whereβandγare adjustable parameters，andγis a quotient of two odd numbers，satisfying 1 ＜γ＜2.

The sliding mode control variableuf∈R3is designed as

where

wherep1andp2are adjustable parameters.

3 Analysis of Convergence and Boundedness

The satellite rendezvous model described in Eq.（8）is established in Section 1.In Section 2.1，the model is augmented by a repetitive controller with NLSEF to gain precise tracking ability，described in Eq.（11）.And in Section 2.2，an adaptive sliding mode controller is used to assure the stability and robustness of the augmented system with unknown disturbances and model uncertainties.

This section uses Lyapunov-like composite energy functions（CEF）to analyze the convergence and boundedness of the system［36］.

Theorem 1Based on Assumptions 1，2 and 3，for dynamic system described in Eq.（11），the adaptive law is designed as Eq.（16），the NSTSM surface is employed as Eq.（19），and the controller is adopted as Eq.（20），then relative position errorδ1is bounded，the relative speed errorδ2converges to zero in finite time，and the relative stations of follower satellitexp1andxp2are able to converge to designate rendezvous states.

Proof

（1）Convergence of relative speed errorδ2Define a Lyapunov function as

Taking derivative with respect to the true anomalyθof Eq.（21）leads to

Taking derivation of Eq.（19）leads to

According to Eq.（23），Eq.（22）is written as

According to Eqs.（18，10，20），we have

Taking derivative of Eq.（25）leads to

Substituting Eqs.（26，20）into Eq.（24）leads to

Expand Eq.（27）into the form of addition of triaxial components

According to the limitation of the parameters thatβ＞0 and 1 ＜γ＜2，thenβγδ'2

(γ-1)≥0.If parameterp1＞ands≠0，thenV'1≤0is promised，and equality only holds whenδ'2=0.The specific situation whenδ'2=0ands≠0is discussed below.

Taking derivative of Eq.（18）leads to

Substituting Eqs.（10，20）into Eq.（29）leads to

Sinceδ2'=0andp1＞，Eq（.30）turns into

It can be observed that whens＞0，δ″2＜0，δ'2decreases；whens＜0，δ″2＞0，δ'2increases.That meansδ'2=0andδ2≠0is not a stable state，so is the stateV'1=0.Therefore，the system states will tend towards sliding mode surfaces=0 and the relative speed errorδ2will converge as well［37］.

（2）Boundedness of relative position errorδ1and estimated weight error vectorW?

Define a Lyapunov function as

Taking derivative of Eq.（31）leads to

Substituting Eqs.（15，10）into Eq.（32）leads to

According to Eq.（12），Eq.（33）turns into Substituting Eqs.（17，14）into Eq.（34）leads to

Expanding Eq.（35）leads to

Substituting the adaptive law described in Eq.（16）into Eq.（36）leads to

With 2ab≤a2+b2，the following inequality holds

It is proved thatδ2converges within a finite time in subsection（1），Eq.（37）turns into

According to Eq.（14），Eq.（38）turns into

Eq.（39）can be rewritten as

As long as parameterskandσare designed properly，μ＞0 can be satisfied，and a conclusion can be drawn that the relative position errorδ1is stable.

Furthermore，multiplying both sides of Eq.（40）byeμθleads to

Integrating Eq.（41）leads to

According to Eq.（31），there is

For ?θ≥0，the following result can be obtained

Then the Lyapunov function described in Eq.（31）can be written as

where

There holds the following inequality

According to Eq.（42），Eq.（43）can be transformed into

where

and a compact set is defined as

According to Lemma 2（see the Appendix），the relative position errorδ1and estimate weight error vectorare bounded in finite time.

（3）Finiteness of convergence time ofδ2

There are two convergence processes.Suppose that the system state reaches the sliding mode surface at a timeθ1.Afterwards，δ2andδ'2converge at a timeθ2.Then the convergence time of the system isθs=θ1+θ2.

When the sliding mode surface converges to zero，i.e.s=δ2+βδ'2

γ=0，then

Define a Lyapunov function as

Taking derivative of Eq.（45）leads to

Substitute Eq.（44）into Eq.（46）and rewrite Eq.（46）in the form of addition of triaxial components

Eq.（47）can also be rewritten as

whereα=，p=( 1+γ)2γ.

According to Lemma 3（see the Appendix），the settling time of each axis can be expressed as

wherei=x，y，z.Thenθ2≤θ2i.The relative speed errorδ2converges within finite timeθs=θ1+θ2.

4 Simulation

In this section，to validate that the control scheme designed is effective，the application of the proposed control scheme to a satellite rendezvous is demonstrated［35］.

The AOE of the leader satellite is shown in Table 1.Suppose a satellite weighting 50 kg can provide maximum output of 2 N using a chemical acceleration module，and is equipped with electric propulsion offering a thrust of 10—100 mN［38-40］.Suppose the initial relative position of two satellites isxp1(θ0)=[-800 600 500]Tm，the initial relative velocity isxp2(θ0)=[2 3 2]Tm/s.The expected relative position of two satellites isxp1d=[-20 20 20]Tm，and the expected relative velocity isxp2d=[0 0 0]Tm/s.

Table 1 Initial orbit elements of the leader satellite

The satellite orbit is affected by various pertur-bations and disturbances in outer space，leading to model uncertainties.Major perturbations include nonspherical perturbations，the third body attraction perturbations，sunlight pressure perturbations and so on.The numerical values of these perturbations are small compared to Earth gravity.Nevertheless，their long term influences cannot be ignored.

ΔApresents model uncertainties，which are related to system states and true anomaly，and the external disturbancesdare only related to true anomaly.ΔAanddare considered as

（1）Case 1：Repetitive control scheme without disturbances

By implanting a dynamic model of the tracking signal into the controller，the repetitive controller can track any periodic signal with no steady state error.An ideal situation is set，where no disturbances exist，and the position tracking ability of repetitive control is analyzed.Fig.5 shows the position tracking results by the repetitive controller.Fig.5 indicates that repetitive control can make position states track the expected signals precisely after the true anomaly of 10 rad.

Fig.5 Position tracking by repetitive controller

The steady state error is illustrated in Fig.6.It can be observed that the position tracking erroreis bounded and converges to zero domain.The precision of relative position tracking is within 2 m.

Fig.6 Relative position error by repetitive controller

NLSEF is able to improve the dynamic performance of the repetitive controller.Parametersb1andb2of NLSEF control lawu0contribut the most to the regulating process.It can be observed from Fig.7 that the NLSEF helps repetitive controller gain less overshoot and faster respond speed.

Fig.7 Regulating process by NLSEF

（2）Case 2：Repetitive control scheme with disturbances

In reality，various disturbances in outer space are inevitable.In this case，the model uncertainties ΔAand unknown disturbancesdare considered in the simulation，and the system is still controlled by a repetitive control scheme only.In Fig.8，the position tracking errors are demonstrated and compared with that in Case 1.

Fig.8 Precision comparison in two situations

From Fig.8，it is obvious that if the system is controlled by the repetitive control scheme only，the position tracking errors tend to diverge with the impact of disturbances.The results prove that the repetitive control does little work dealing with the disturbances，and improvement is necessary for satellite rendezvous.

（3）Case 3：IRC scheme with disturbances

Given unknown disturbances and model uncertainties，the response curves of the relative position maintained by the IRC scheme designed in our study are shown in Fig.9，and the response curves of the relative velocity are shown in Fig.10.

Fig.9 Position tracking by IRC

Fig.10 Velocity tracking by IRC

In Figs.9，10，relative position and velocity response curves converge after 5 rad.The results prove that the controller is stable and convergent despite the disturbances.

Note that the velocities peak at first 0.05 rad.According to Fig.11，the maximum control forcesneeded are less than 2 N and meet the requirement.

Fig.11 Control variable by IRC in transferring state

Figs.12，13 demonstrate the performance of the IRC scheme at a steady state，emphasizing the period of 16 rad to 25 rad，where the follower satellite keeps hovering against the leader satellite.

When the satellites come to a steady state，the precision of the relative position is better than 5×10-4m，and the control force is less than 100 mN.The accuracy achieves millimeter level，and the maintenance forces can be afforded by an electric propulsion［39］.Moreover， the output curves are quite smooth without obvious buffeting.

To demonstrate the advantages of the IRC scheme，the contribution of the repetitive controller is eliminated in Fig.14.

Fig.12 Relative position error by IRC

Fig.13 Control variable by IRC in steady state

Fig.14 Precision comparison between two control schemes

Apparently，the precision of the system deteriorates when the contribution of the repetitive con-troller is eliminated.The results prove that the IRC scheme improves the accuracy of the sliding mode controller by more than one order of magnitude.

Further，to demonstrate that the IRC scheme designed in this paper has the advantage of converging in finite time，a comparative simulation is shown in Fig.15，comparing IRC with the traditional backstepping control scheme.

Fig.15 Convergence comparison between two control schemes

As analyzed in Section 3，and according to Lemma 3，the IRC ensures that after the system states reach the sliding mode surface，the relative position errorδ1converges to a small neighborhood of the originwithin finite time

In Case 3，we can find in Fig.15 that IRC makes the system states reach the neighborhood[ -0.077，0.077]m withinθs=5.23 rad，while the traditional backstepping control is unable to converge to neighborhood in time.

Fig.15 proves that the IRC converges in finite time.Nevertheless，the rapidity of the IRC is not satisfying，and worth further investigation.

5 Conclusions

Our study aims at high-precision satellite rendezvous control，and proposes an improved repetitive control（IRC）scheme.The controlled model described by ROE in the form of state space equations has both advantages of high accuracy and great adaptability.The repetitive controller makes the follower satellite track the expected signals precisely，while the siding mode controller solves the divergence problem caused by the aperiodic disturbances and makes the repetitive controller robust.Moreover，NLSEF is effective to improve the dynamic performance of the controller.

The rapidity of the controller is not studied in this paper，and could be further improved in future work.Also，it is worth considering the situation when the rendezvous object is uncooperative in the future.

Appendix

Lemmal 1［41］Consider a dynamic system

An extended state observer is constructed instead of identifying the unknown functionf(x1，x2) in the system

The state variables of the observerz1andz2track the state variables of the system described in Eq.（A1）x1andx2.Especially，a(t)estimatesf(x1，x2) in real time.

Lemma 2［42］LetD?Rndenote a domain containing the origin，there is a continuously differentiable and positivedefinite functionL(t，x)∈[0，∞)×D，which satisfies the following conditions

where?is a positive real number；and for ?t≥0，?x∈D，it is promised thatW1(x)，W2(x) andW3(x) are all continuous and positive definite functions onD，and there is a positive constantc，so that Ξ={W1(x)≤candW2(x)＜c}is a compact subset ofD.Then the functionL(t，x) is able to converge within a bounded compact set in finite time.

Lemma 3［43］Consider a system of differential equations

wherex=[x1，x2，…，xn]T∈Rn，f(x)：Rn→Rnis continuous on Rnandf(0)=0.

For the system described in Eq.（A2），if there exists a continuous positive definite functionV(x)：Rn→R，and so does the continuous first-order partial derivative with respect to anyx，such that there exist real numbersαandp∈(0，1)，and they satisfy that

Then the origin is a globally finite time stable equilibrium of Eq.（A2）.That means the system can converge to the neighborhood of designate point within timeT.The upper bound of the settling time is