# Nonsingular Fast Terminal Adaptive Neuro-sliding Mode Control for Spacecraft Formation Flying Systems

Authors: Xiaohan Lin; Xiaoping Shi; Shilun Li; Sing Kiong Nguang; Liruo Zhang
Journal: Complexity
 (2020)

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## Abstract

In this paper, a nonsingular fast terminal adaptive neurosliding mode control for spacecraft formation flying systems is investigated. First, a supertwisting disturbance observer is employed to estimate external disturbances in the system. Second, a fast nonsingular terminal sliding mode control law is proposed to guarantee the tracking errors of the spacecraft formation converge to zero in finite time. Third, for the unknown parts in the spacecraft formation flying dynamics, we proposed an adaptive neurosliding mode control law to compensate them. The proposed sliding mode control laws not only achieve the formation but also alleviate the effect of the chattering. Finally, simulations are used to demonstrate the effectiveness of the proposed control laws.


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## Body

## 1. Introduction

With the increasing applications of the spacecraft formation flying system, the importance of improving system performance of complicated spacecraft with practical control design has gained much attention over recent years [1–5]. In the spacecraft formation flying system, stability is the most essential problem to be solved [6]. Many control methods have been applied to solve these problems in the literature, such as the adaptive control [7], sliding mode control [8], backstepping technique control [9], fuzzy control [10], optimal control [11], and H∞ control [12]. However, along with the goal of stabilization for the spacecraft, the disturbances and uncertainties should also be considered, which is often neglected in some research [13].Sliding mode control is widely used due to its advantages of robustness against external disturbances [14–16] and parameter uncertainties [17], especially in spacecraft formation flying system. In sliding mode control, chattering is a common phenomenon that often degrades system performance and even causes instability. To deal with this problem, the terminal sliding mode control is developed, in which some nonlinear terms are introduced in the sliding surface. In [18], the case of spacecraft final approach, combining with the continuously differential collision avoidance state constraint function is studied and an improved terminal sliding surface is presented. In [19], based on the conditions of external disturbances and inertial uncertain parameters, Ran et al. have proposed the adaptive fuzzy terminal sliding mode control for the second-order spacecraft attitude manoeuvre system and the convergence time of the sliding surface has been reduced. In [20], a new nonsingular terminal sliding mode (NTSM) surface has been introduced to eliminate singularity within the finite time. An adaptive nonsingular terminal sliding mode control also has been presented for an attitude tracking of spacecraft with actuator faults to avoid singularity [21]. Although several drawbacks of the sliding mode control have been compensated by the modified control laws, due to the high-frequency reaching control term in the control input [22], the chattering problem still needs to be solved. In [23], a fractional order control law based on the nonsingular terminal supertwisting sliding mode control is proposed to achieve system stability and accurately estimate the unknown model. Consequently, the nonsingular fast terminal sliding mode surface has the ability to reduce the chattering and solve the singularity problem.In practical spacecraft formation flying systems, internal and external disturbances are inevitable. Many control methods have been used to solve this problem [24, 25]. The disturbance observers have been widely considered to estimate the disturbances [26]. In [27], an extended state observer is introduced to estimate disturbances for the multiagent systems with unknown nonlinear dynamics and disturbances. Furthermore, by simultaneously taking the output measurement and delayed control input into account, the extended observer has been used to estimate the unmeasurable system states and the additive disturbances [28]. To have a smaller estimation error of the disturbance, high-order disturbance observers have been investigated in [29]. Among all types of high-order observer, the supertwisting disturbance observer stands out due to its ability of avoiding the excessively high order observer gains that can cause sensor noise within fixed bandwidth [30].Despite the external disturbances, uncertainties in practical formation flying systems are also inevitable; however, their impacts on the system’s performance have not been studied in most of the aforementioned results. In practice, parameter uncertainties may have direct impact on system performance [31]; hence, it is critical to deal with the problem of uncertainties. Due to the advantage of the neural networks in approximating nonlinear functions, it can be applied in dealing with the nonlinearity in the spacecraft formation systems as well [32–34]. Based on neural network and backstepping techniques, a distributed adaptive coordinated control algorithm has been proposed in [35], which effectively solves the chattering problem. To achieve robustness, in [36], a robust adaptive fuzzy PID-type sliding mode control has been designed with the neural network, with which the system stability is ensured and the transient performance is improved. In [37], the global adaptive neural control for a general class of nonlinear robot manipulators has been proposed, in which the transient performance and robustness are both improved. Compared with other neural networks [38–40], the Radial Basis Function (RBF) neural network stands out given its strong robustness and excellent performance of nonlinear approximation [41]. Therefore, it is desirable to use the RBF neural network to estimate and compensate for the nonlinear functions in the spacecraft formation flying system.Motivated by the above observations, in this paper, a novel nonsingular fast terminal adaptive neurosliding mode control for the spacecraft formation flying system is proposed. The proposed control law guarantees that the tracking errors converge to zero in finite time. By using an adaptive RBF neural network, the uncertainties in the spacecraft formation flying system is compensated. The contributions of the paper can be summarized as follows:(1)
External disturbances are estimated by a supertwisting disturbance observer in finite time. The supertwisting disturbance observer can avoid high observer gains that may amplify the noises of the sensors.(2)
Compared with [42], our novel fast nonsingular terminal sliding mode control law achieves the stability of spacecraft formation flying system within finite time. Moreover, the singularity is avoided and the chattering problem is also alleviated.(3)
If some of the nonlinear dynamics in the spacecraft formation system are unknown, an adaptive RBF neural network control law is proposed to approximate those nonlinear dynamics.The remaining parts of the paper are arranged as follows. The background of the spacecraft formation flying system is introduced in Section2. In Section 3, the proposed control laws and stability analysis are presented. Simulation results are given to verify the effectiveness of the proposed control laws in Section 4. Finally, some conclusions are given in Section 5.


### 1.1. Notations

The following notations are used throughout this paper. For a vectorx=x1,x2,…,xnT∈ℝn and xd≜diagx1,x2,…,xn, the standard sign function is defined as signx=signx1,signx2,…,signxnT. The Euclidean norm of vector x is denoted as x=xTx. For a matrix A∈ℝn×n, A1 denotes the maximum absolute column sum norm. A2=λmaxATA is the 2-norm, where λmax means the maximum eigenvalue. trA denotes the trace of A. In×n denotes n×n identity matrix. For a three-dimensional vector x=x1,x2,x3T, the skew-symmetric matrix x×∈ℝ3×3 is defined as follows:(1)x×=0−x3x2x30−x1−x2x10.


## 1.1. Notations

The following notations are used throughout this paper. For a vectorx=x1,x2,…,xnT∈ℝn and xd≜diagx1,x2,…,xn, the standard sign function is defined as signx=signx1,signx2,…,signxnT. The Euclidean norm of vector x is denoted as x=xTx. For a matrix A∈ℝn×n, A1 denotes the maximum absolute column sum norm. A2=λmaxATA is the 2-norm, where λmax means the maximum eigenvalue. trA denotes the trace of A. In×n denotes n×n identity matrix. For a three-dimensional vector x=x1,x2,x3T, the skew-symmetric matrix x×∈ℝ3×3 is defined as follows:(1)x×=0−x3x2x30−x1−x2x10.


## 2. Spacecraft Formation Systems and Preliminaries

In this section, the spacecraft formation flying system and some useful lemmas are presented.


### 2.1. Spacecraft Formation Flying System

The relative orbit motion under the chaser spacecraft coordinate frame is shown in Figure1. FI denotes the equatorial inertial coordinate frame (OIXIYIZI), Ft is located at the centroid of the target spacecraft, and the coordinate axis of the target spacecraft coincides with the main inertia axis FI, the xt-axis points to the solar panel, the yt-axis is perpendicular to the longitudinal plane of xt-axis, and zt-axis satisfies the rule of the right hand. The chaser spacecraft coordinate frame Fc is similar to the coordinate frame Ft.Figure 1
The relative orbit motion model of the spacecraft formation flying system.Assume that the active control force of the target spacecraft is zero, and the dynamic models of the target spacecraft and the chaser spacecraft are given as follows:(2)r¨t=−μert3rt+dctmt,(3)r¨c=−μerc3rc+dccmc+fccmc,where mt and mc denote the masses of the target spacecraft and the chaser spacecraft and dct∈ℝ3 and dcc∈ℝ3 denote the external perturbation forces of the target spacecraft and the chaser spacecraft, respectively. fcc∈ℝ3 is the control force of the chaser spacecraft. rt=rt2 and rc=rc2, where rt and rc denote the position vectors of the target spacecraft and chaser spacecraft to the geocentric, respectively.Defineρ=rc−rt; then, from (2) and (3) the relative acceleration vector is obtained as(4)ρ¨=−μerc3rc+μert3rt+dccmc−dctmt+fccmc.Furthermore, letρc denote the projection vector in the chaser spacecraft body coordinate frame Fc, which denotes(5)fc=−μerc3rc+μert3rt,dc=dcc−mcdctmt,uc=fcc.Then,(6)ρ¨c+ωc×ωc×ρc+2ωc×ρ˙c+ω˙c×ρc=fc+dcmc+ucmc,where ωc=ωc1,ωc2,ωc3T is the angular velocity of Fc relative to the inertial coordinate frame FI under Fc frame.In this paper, the desired relative position and velocity underFc frame are assumed to be ρcd and 0, respectively. Then, the error vector can be expressed as ec=ρc−ρcd; then, (6) can be written as the Euler–Lagrange equation form to express the orbit relative motion:(7)mce¨c+Ace˙c+Bcec+gc=uc+dc,where(8)Ac=2mcωc×,Bc=mc−ωc22−ωc32ωc1ωc2−ω˙c3ωc1ωc3+ω˙c2ωc1ωc2+ω˙c3−ωc12−ωc32ωc2ωc3−ω˙c1ωc1ωc3−ω˙c2ωc2ωc3+ω˙c1−ωc12−ωc22,gc=−mcfc+mcr¨dc+Acr˙dc+Bcrdc.The Modified Rodrigues Parameters (MRPs) are used to describe the attitude dynamics of the chaser spacecraft which can be defined as(9)σ=q^1+q¯=ketanφ4,where q=q¯,q^TT denotes attitude quaternion and ke and φ∈−2π,2π denote Euler axis and Euler angular of the attitude rotation, respectively.Then, the attitude kinematics and the dynamics of the chaser spacecraft are given as(10)σ˙=Gσωc,Gσ=141−σTσI3×3+2σ×+2σσT,Jω˙c=−ωc×Jωc+uτ+dτ,where J∈ℝ3×3 denotes the inertia matrix of the chaser spacecraft and uτ∈ℝ3 and dτ∈ℝ3 denote the control torque and the external disturbance torque of the chaser spacecraft, respectively.Defining the error attitudeσe and the error angular velocity ωe as(11)σe=1−σdTσdσ−1−σTσσd+2σ×σd1+σTσσdTσd+2σdTσ,ωe=ωc−Rσeωd,Rσ=I3×3−41−σTσ1+σTσ2σ×+8σ×1+σTσ2σ×,where σd∈ℝ3 denotes the desired attitude and ωd∈ℝ3 denotes the desired angular velocity. Furthermore, the attitude model of the chaser spacecraft can be expressed as the Euler–Lagrange form:(12)Mτσ¨e+Aτσ˙e+gτ=G−Tσedτ+G−Tσeuτ,where(13)Mτ=G−TσeJG−1σe,Gσe=141−σeTσeI3×3+2σe×+2σeσeT,Aτ=−G−TσeJG−1σeσ˙e×G−1σe+JG˙−1σe+G−TσeJRσeωd×+Rσeωd×J−JRσeωd×G−1σe,gτ=G−TσeRσeωd×JRσeωd+JRσeω˙d.In order to describe the attitude-orbit coupling dynamics motion of the spacecraft formation uniformly, define(14)xe=σeec,u=uτuc,d=dτdc.Then, based on (5) and (22), the attitude-orbit coupling model is given as(15)x˙e=xde,(16)Mx˙de=−Axde−g+C1u+C2d,where(17)M=Mτ03×303×3mcI3×3,(18)A=Aτ03×303×3Ac,(19)g=gτBcec+gc,(20)C1=C2=G−Tσe03×303×3I3×3.Remark 1.
It can be seen in (12), the attitude and orbit in the spacecraft formation flying system are coupled. Therefore, it is necessary to build the spacecraft coupling motion model (15) and (16) to enhance the control accuracy and efficiency. From (15) and (16), it can be seen that the coupling of attitude and orbit is reflected by the matrix Bc and the vector dτ simultaneously. Bc represents the attitude effect on orbit. In addition, the gravity gradient torque contains orbit information, and the disturbance torque can reflect the influence of the orbit on the attitude, thus dτ can reflect the influence of the orbit on the attitude.Assumption 1.
The external disturbance vectord=d1,…,d6 in the spacecraft formation flying system (15) and (16) is assumed to satisfy d≤dm and di.≤ςi, where dm>0 and ςi>0.Assumption 2.
All the state variables of the spacecraft formation flying system are assumed to be measurable.


### 2.2. Preliminary Knowledge

In this section, some related lemmas and definitions in this paper are given below.Lemma 1 (see [43]).
Forx˙=fx,t, f0,t=0 and x∈U0⊂Rn, where f:U0×R+⟶Rn is continuous with respect to x on an open neighborhood U0 of the origin x=0. Suppose that there are a positive-definite function Vx,t (defined on U^×R+, where U^∈U0∈Rn is a neighborhood of the origin), real numbers c>0 and 0<α<1, such that V˙x,t+cVαx,t is negative semidefinite on U^. Then, Vx,t is locally finite-time convergent, or equivalently, and becomes 0 locally in finite time, with its setting time T≤Vxt0,t01−α/c1−α for a given initial condition xt0 in a neighborhood of the origin in U^.Lemma 2 (see [44]).
Suppose Lyapunov functionVz^ is defined on a neighborhood U⊂Rn of the origin, and V˙z^+λ0Vℓ≤0 in which λ0>0, 0<ℓ<1. Then, there exists an area U0⊂Rn such that any Vz^ that start from U0⊂Rn can reach Vz^≡0 within finite time Tf≤V1−ℓ0/λ01−ℓ in which V0 is the initial time of Vz^.Definition 1.
(see [45]). For the autonomous system,(21)z^˙=fz^,where z^∈RN and f0=0, if the Lyapunov function Vz^ satisfies the following conditions:(a)
Vz^ is a positive continuous differentiable function(b)
There existα1>0, α2>0, p∈0,1, and an open neighborhood U⊂U0 containing the origin, such that the inequality V˙z^≤−α1Vz^−αVpz^ holds; then, it can be concluded that the origin is the finite time stable equilibrium of system (20)


## 2.1. Spacecraft Formation Flying System

The relative orbit motion under the chaser spacecraft coordinate frame is shown in Figure1. FI denotes the equatorial inertial coordinate frame (OIXIYIZI), Ft is located at the centroid of the target spacecraft, and the coordinate axis of the target spacecraft coincides with the main inertia axis FI, the xt-axis points to the solar panel, the yt-axis is perpendicular to the longitudinal plane of xt-axis, and zt-axis satisfies the rule of the right hand. The chaser spacecraft coordinate frame Fc is similar to the coordinate frame Ft.Figure 1
The relative orbit motion model of the spacecraft formation flying system.Assume that the active control force of the target spacecraft is zero, and the dynamic models of the target spacecraft and the chaser spacecraft are given as follows:(2)r¨t=−μert3rt+dctmt,(3)r¨c=−μerc3rc+dccmc+fccmc,where mt and mc denote the masses of the target spacecraft and the chaser spacecraft and dct∈ℝ3 and dcc∈ℝ3 denote the external perturbation forces of the target spacecraft and the chaser spacecraft, respectively. fcc∈ℝ3 is the control force of the chaser spacecraft. rt=rt2 and rc=rc2, where rt and rc denote the position vectors of the target spacecraft and chaser spacecraft to the geocentric, respectively.Defineρ=rc−rt; then, from (2) and (3) the relative acceleration vector is obtained as(4)ρ¨=−μerc3rc+μert3rt+dccmc−dctmt+fccmc.Furthermore, letρc denote the projection vector in the chaser spacecraft body coordinate frame Fc, which denotes(5)fc=−μerc3rc+μert3rt,dc=dcc−mcdctmt,uc=fcc.Then,(6)ρ¨c+ωc×ωc×ρc+2ωc×ρ˙c+ω˙c×ρc=fc+dcmc+ucmc,where ωc=ωc1,ωc2,ωc3T is the angular velocity of Fc relative to the inertial coordinate frame FI under Fc frame.In this paper, the desired relative position and velocity underFc frame are assumed to be ρcd and 0, respectively. Then, the error vector can be expressed as ec=ρc−ρcd; then, (6) can be written as the Euler–Lagrange equation form to express the orbit relative motion:(7)mce¨c+Ace˙c+Bcec+gc=uc+dc,where(8)Ac=2mcωc×,Bc=mc−ωc22−ωc32ωc1ωc2−ω˙c3ωc1ωc3+ω˙c2ωc1ωc2+ω˙c3−ωc12−ωc32ωc2ωc3−ω˙c1ωc1ωc3−ω˙c2ωc2ωc3+ω˙c1−ωc12−ωc22,gc=−mcfc+mcr¨dc+Acr˙dc+Bcrdc.The Modified Rodrigues Parameters (MRPs) are used to describe the attitude dynamics of the chaser spacecraft which can be defined as(9)σ=q^1+q¯=ketanφ4,where q=q¯,q^TT denotes attitude quaternion and ke and φ∈−2π,2π denote Euler axis and Euler angular of the attitude rotation, respectively.Then, the attitude kinematics and the dynamics of the chaser spacecraft are given as(10)σ˙=Gσωc,Gσ=141−σTσI3×3+2σ×+2σσT,Jω˙c=−ωc×Jωc+uτ+dτ,where J∈ℝ3×3 denotes the inertia matrix of the chaser spacecraft and uτ∈ℝ3 and dτ∈ℝ3 denote the control torque and the external disturbance torque of the chaser spacecraft, respectively.Defining the error attitudeσe and the error angular velocity ωe as(11)σe=1−σdTσdσ−1−σTσσd+2σ×σd1+σTσσdTσd+2σdTσ,ωe=ωc−Rσeωd,Rσ=I3×3−41−σTσ1+σTσ2σ×+8σ×1+σTσ2σ×,where σd∈ℝ3 denotes the desired attitude and ωd∈ℝ3 denotes the desired angular velocity. Furthermore, the attitude model of the chaser spacecraft can be expressed as the Euler–Lagrange form:(12)Mτσ¨e+Aτσ˙e+gτ=G−Tσedτ+G−Tσeuτ,where(13)Mτ=G−TσeJG−1σe,Gσe=141−σeTσeI3×3+2σe×+2σeσeT,Aτ=−G−TσeJG−1σeσ˙e×G−1σe+JG˙−1σe+G−TσeJRσeωd×+Rσeωd×J−JRσeωd×G−1σe,gτ=G−TσeRσeωd×JRσeωd+JRσeω˙d.In order to describe the attitude-orbit coupling dynamics motion of the spacecraft formation uniformly, define(14)xe=σeec,u=uτuc,d=dτdc.Then, based on (5) and (22), the attitude-orbit coupling model is given as(15)x˙e=xde,(16)Mx˙de=−Axde−g+C1u+C2d,where(17)M=Mτ03×303×3mcI3×3,(18)A=Aτ03×303×3Ac,(19)g=gτBcec+gc,(20)C1=C2=G−Tσe03×303×3I3×3.Remark 1.
It can be seen in (12), the attitude and orbit in the spacecraft formation flying system are coupled. Therefore, it is necessary to build the spacecraft coupling motion model (15) and (16) to enhance the control accuracy and efficiency. From (15) and (16), it can be seen that the coupling of attitude and orbit is reflected by the matrix Bc and the vector dτ simultaneously. Bc represents the attitude effect on orbit. In addition, the gravity gradient torque contains orbit information, and the disturbance torque can reflect the influence of the orbit on the attitude, thus dτ can reflect the influence of the orbit on the attitude.Assumption 1.
The external disturbance vectord=d1,…,d6 in the spacecraft formation flying system (15) and (16) is assumed to satisfy d≤dm and di.≤ςi, where dm>0 and ςi>0.Assumption 2.
All the state variables of the spacecraft formation flying system are assumed to be measurable.


## 2.2. Preliminary Knowledge

In this section, some related lemmas and definitions in this paper are given below.Lemma 1 (see [43]).
Forx˙=fx,t, f0,t=0 and x∈U0⊂Rn, where f:U0×R+⟶Rn is continuous with respect to x on an open neighborhood U0 of the origin x=0. Suppose that there are a positive-definite function Vx,t (defined on U^×R+, where U^∈U0∈Rn is a neighborhood of the origin), real numbers c>0 and 0<α<1, such that V˙x,t+cVαx,t is negative semidefinite on U^. Then, Vx,t is locally finite-time convergent, or equivalently, and becomes 0 locally in finite time, with its setting time T≤Vxt0,t01−α/c1−α for a given initial condition xt0 in a neighborhood of the origin in U^.Lemma 2 (see [44]).
Suppose Lyapunov functionVz^ is defined on a neighborhood U⊂Rn of the origin, and V˙z^+λ0Vℓ≤0 in which λ0>0, 0<ℓ<1. Then, there exists an area U0⊂Rn such that any Vz^ that start from U0⊂Rn can reach Vz^≡0 within finite time Tf≤V1−ℓ0/λ01−ℓ in which V0 is the initial time of Vz^.Definition 1.
(see [45]). For the autonomous system,(21)z^˙=fz^,where z^∈RN and f0=0, if the Lyapunov function Vz^ satisfies the following conditions:(a)
Vz^ is a positive continuous differentiable function(b)
There existα1>0, α2>0, p∈0,1, and an open neighborhood U⊂U0 containing the origin, such that the inequality V˙z^≤−α1Vz^−αVpz^ holds; then, it can be concluded that the origin is the finite time stable equilibrium of system (20)


## 3. Control Law Design

In this section, first, we employ a supertwisting observer to estimate the external disturbances in the system. Second, we propose a fast nonsingular terminal sliding mode control law along with the supertwisting disturbance observer for the spacecraft formation flying system. In the last section, the nonlinear matricesA and g in (18) and (19), respectively, are assumed to be unknown, and an adaptive neurosliding model control law for the spacecraft formation flying system is proposed.


### 3.1. The Fast Nonsingular Terminal Sliding Mode Control Law with a Supertwisting Disturbance Observer

The block diagram of the fast nonsingular terminal sliding mode control law with the supertwisting disturbance observer is shown in Figure2. In Figure 2, it can be seen that the overall control law consists of two parts: a fast nonsingular terminal sliding mode control law along with the supertwisting disturbance observer. The supertwisting disturbance observer is applied to estimate the disturbances for the spacecraft formation flying system if the system dynamics are known.Figure 2
The block diagram of the control law based on the supertwisting disturbance observer.


#### 3.1.1. Supertwisting Disturbance Observer

In practical applications, the disturbances in the spacecraft formation flying system are fast time-varying disturbances. Hence, it is difficult for conventional observers to estimate those disturbances. In this section, a supertwisting disturbance observer is employed to estimate those disturbances. The supertwisting disturbance observer can estimate the disturbance in finite time and also it avoids high-order observer gains with fixed bandwidth.According to the attitude-orbit coupling system (15) and (16), the disturbance vector d can be rewritten as follows:(22)d=C2−1Mx..e+Ax.e+g−C1u.Define the state variables as follows:(23)x1=∫0tddt=C2−1Mx.e−∫0tddtC2−1Mx.e+C2−1Ax.e+g−C1udt,x2=d.Then, the disturbance dynamic system for the spacecraft formation flying is given as(24)x˙1=x2,x˙2=d˙.For each disturbancedi in the spacecraft formation flying system, the disturbance’s model is given as follows:(25)x˙1i=x2i,x˙2i=d˙i,i=1,2,…,6.The supertwisting disturbance observer is given as follows:(26)x˜˙1i=−λ1e1i1/2signe1i+x˜2i,x˜˙2i=−λ2signe1i,where λ1 and λ2 denote the positive observer gains, x˜1i and x˜2i are the estimations of the state vectors x1i and x2i, and e1i=x˜1i−x1i and e2i=x˜2i−x2i are the estimation errors, respectively.According to the attitude-orbit coupling system (15) and (16) and the supertwisting state observer (26), the estimation errors can be rewritten as(27)e˙1i=−λ1e1i1/2signe1i+e2i,e˙2i=−λ2signe1i−d.i.Define the state vectorsκ=κ1,κ2T, where κ1=e1i1/2signe1i, κ2=e2i.Then, the derivatives of the state vectorsκ1 and κ2 can be written as(28)κ˙1=−λ12e1i−1e1i+12e1i−1/2e2i,κ˙2=−λ2e1i−1e1i−d.i.Similar to [46], the following Lyapunov function for (28) is constructed:(29)Vst=κTθ1κ=12λ12+4λ2κ12−λ1κ1κ2+κ22,where θ1=1/2λ12+4λ2−λ1−λ12.Then, the derivative of Lyapunov functionVst with respect to the time is given as(30)V˙st=λ12+4λ2κ1κ.1−λ1κ.1κ2−λ1κ1κ.2+2κ2κ.2=λ12+4λ2e1i−1/2e1i−λ12e1i−1e1i+12e1i−1/2e2i+λ122e1i−1e1ie2i−λ12e1i−1/2e2i2−λ1e1i−1/2e1i−λ2e1i−1e1i−d.i−2λ2e1i−1e1ie2i−2e2id.i=λ12+4λ2−λ12e1i−1/2+12e1i−1e1ie2i+λ122e1i−1e1ie2i−λ12e1i−1/2e2i2+λ1λ2e1i−1/2+λ1e1i−1/2e1id.i−2λ2e1i−1e1ie2i−2e2id.i.Furthermore, (30) can be written as follows:(31)V.st=−λ132+λ1λ2e1i1/2+λ12e2isigne1i−λ12e2i2e1i1/2+λ1e1i1/2signe1id.i−2e2id.i=−λ12e1i1/2λ12+2λ2e1i−2λ1e2ie1i1/2signe1i+e2i2+λ12e1i1/22e1isigne1i−4λ1e1i1/2e2id.i≤−λ12e1i1/2λ12+2λ2e1i−2λ1e2ie1i1/2signe1i+e2i2+λ12e1i1/22e1i+4λ1e1i1/2e2id.i.Sinceκ12−2κ1κ2+κ22≥0, (31) can be rewritten as(32)V.st≤−λ12e1i1/2λ12+2λ2κ12−2λ1κ1κ2+κ22+λ12e1i1/22κ12+4λ1κ1κ2ςi≤−λ12e1i1/2λ12+2λ2κ12−2λ1κ1κ2+κ22+λ12e1i1/22κ12+2λ1κ12+2λ1κ22ςi.Define the following matrix as(33)θ2=λ12+2λ2−2λ1+1ςiλ1−λ1−λ11−2ςiλ1.Then, (32) can be concluded as(34)V.st≤−λ12e1i1/2κTθ2κ.From (33), by selecting appropriate observer gains, the matrix θ2 can be a positive definite matrix. Then, (34) becomes(35)V.st≤−λ1λminθ22e1i1/2κ2,where λminθ2 is the minimum eigenvalue of θ2.θ1 must to be a positive definite matrix to be Lyapunov matrix; hence, the observer gains λ1 and λ2 need to be selected appropriately so that θ1>0. Let us define the minimal and maximal eigenvalues of θ1 as λminθ1>0 and λmaxθ1>0, respectively. Then, we have(36)λminθ1κ2≤Vst≤λmaxθ1κ2.Furthermore, by observing from the state vectorsκ and κ1, the following can be concluded:(37)e1i1/2=κ12≤κ.Finally, combining (36) with (37), (35) can be derived as(38)V.st+λ0Vst1/2≤0,λ0=λ1λminθ11/2λminθ22λmaxθ1>0.Based on (37) and Lemma 1, we prove that limt⟶tde1i=0 and limt⟶tde2i=0 with i=1,2,…,6. Furthermore, the estimations errors of the state vector x1i and x2i can converge in finite time, and the convergence time is td=2Vst1/2/λ0.Remark 2.
To guarantee that the trajectory errors converge to zero within finite time, the supertwisting disturbance observer gainsλ1 and λ2 should be selected appropriately so that θ1 and θ2 are positive definite matrices.


#### 3.1.2. Fast Nonsingular Terminal Sliding Mode Control Law with Supertwisting Disturbance Observer

To design a fast nonsingular terminal sliding mode control law, the following sliding mode surface is chosen:(39)s=xe+∈1xedc1signxe+∈2xdedn/msignxde,where ∈1,∈2>0, c1>n/m, and 1<n/m<2 are all positive constants.If the spacecraft formation flying system state errors reach the sliding modes=0, then(40)xe+∈1xedc1signxe+∈2xdedn/msignxde=0.From (15), substituting the equation x.e=xde into (40), we have(41)x.e=−1∈2m/nxe+∈1xedc1signxem/n=−1∈2m/nxem/n1+∈1xedc1−1m/n.In order to analyze the fast convergence of the nonsingular fast terminal sliding mode surface, assume that the integration time of the state variablexe denotes as xeti and xeti+ts, respectively. Then, integrating the time along both sides of (41),(42)∫xetixeti+ts1xem/ndxe=−∫titi+ts1∈2m/n1+∈1xec1−1m/ndτ≤−∫titi+ts1∈2m/ndτ.Furthermore, it can be seen from (42) that the tracking errors of the spacecraft formation flying system can converge to zero in a limited time ts≤∈2m/n/n−mxeti1−m/n.Remark 3.
The proposed nonsingular fast terminal sliding mode surface (39) indicates that when the system state variable xe is far from the equilibrium point, the term xe plays the major role in making the system trajectory converge at a fast rate. When xe approaches to the equilibrium point, the term ∈1xedc1signxe plays the dominant role to achieve rapid convergence of the state trajectory.A design procedure for a fast nonsingular terminal sliding mode control law with a supertwisting disturbance observer is given in the following theorem.Theorem 1.
Consider the spacecraft formation flying system described by (15) and (16) with(43)u=C1−1Axde+g−C2x˜2−C1−1Mu0+u1+u2,where(44)u0=1∈2mnxded2−n/msignxde,(45)u1=c1∈1∈2mnxded2−n/mxedc1−1signxde,(46)u2=α1signs+α2sηsigns,and −α1signs−α2sηsigns is the reaching law with α1>0, α2>0, and η>1. Then, the proposed control law along with the supertwisting disturbance observer (26) can guarantee the tracking errors of the spacecraft formation flying system converge to zero in finite time.Proof.
From (39), the derivative of the sliding surface s is given as follows:(47)s.=xe.+∈1c1xedc1−1xde+∈2nmxdedn/m−1xde.=xe.+∈1c1xedc1−1xde+∈2nmxdedn/m−1M−1−Axde−g+C1u+C2d.
Consider the following Lyapunov function:(48)V1t=12sTs+∑i=16V1i.
Taking the time derivative of (48), we have(49)V˙1t=sTs.+∑i=16V˙1i=sTxde+∈1c1xedc1−1xde+∈2nmxdedn/m−1−M−1C2x2−d−u0+u1+u2−α0∑i=16V1i1/2.
Substituting the proposed control law (44)–(46) into (49),(50)V˙1t=−sT∈2nmxdedn/m−1α1signs+α2sηsigns+sT∈2nmxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2.
Definexdemin=minxde1n/m−1,…,xde6n/m−1, α¯1=xdemin∈2n/mα1, and α¯2=xdemin∈2n/mα2, then(51)V˙1t≤−α¯1s−α¯2sη+1+∈2nmsTxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2≤−α¯1s−α¯2sη+1+∈2nmsxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2.
Defineαe=∈2n/mxdedn/m−1M−1C2x2−d and select α¯e=α¯1−αe≥0; we have(52)V˙1t≤−α¯es−α0∑i=16V1i1/2.
Finally, defineαm=min2α¯e,α0, then (52) can be rewritten as(53)V˙1t≤−αm12s+∑i=16V1i1/2≤−αm12s2∑i=16V1i1/2=−αmV1t1/2.Remark 4.
When the system state variable is far from the fast nonsingular terminal sliding mode surface, the termα2sηsigns in the reaching law plays the major role. On the contrary, the term α1signs in the reaching law dominates. By choosing different α1 and α2 and selecting appropriate parameters in the fast terminal sliding mode surface, the convergence speed can be controlled. Moreover, compared with the conventional reaching law in [47], the proposed reaching law has faster convergence and better sliding performance. When s=0 and s˙=0, the speed of the state variable approaching the sliding surface is reduced to zero, which can alleviate the chattering phenomenon.


### 3.2. Fast Nonsingular Terminal Sliding Mode Control Law Based on the Adaptive Neural Network

In this section, an adaptive RBF neural network is employed to approximate the unknown nonlinearA and g in the spacecraft formation flying system (15) and (16), which is not easy to be precisely estimated in practical situations. The block diagram of a fast nonsingular terminal sliding mode control law based on the adaptive neural network is depicted in Figure 3.Figure 3
The block diagram of the control law based on the adaptive neural network.The structure of RBF neural network with multiple inputs and multiple outputs is shown in Figure4.Figure 4
The structure of RBF neural network.It can be seen that the RBF neural network has three layers including input layer, output layer, and hidden layer. Denotex=x1,x2,…,xnT as the input, and ϕix is the Gaussian function of the ith neuron in the hidden layer. Then, the Gaussian function can be expressed as below:(54)ϕix=exp−x−ci22gi2,i=1,2,…,6,where ci is the ith neural radial basis function and gi is the width of the ith neural radial basis function. In this section, we choose fnn to approximate and compensate the unknown nonlinear parts in the spacecraft formation flying system, where fnn=−Axde−g is the unknown nonlinear term. We rewrite fnn as follows:(55)fnn=WiTϕixi+ξi,where Wi denotes the ideal weighting matrix and ξi denotes the approximation error, i=1,2,…,6.The estimation offnn can be expressed as follows:(56)f^nn=W^iTϕixi,where W^i denotes the estimation of Wi.Theorem 2.
Consider the spacecraft formation flying system described by (15) and (16), with the following control law and the adaptive updating law of the RBF neural network:(57)u=C1−1f^nn−u3−C1−1Mu0+u1+u2,where(58)u0=1∈2mnxded2−n/msignxde,(59)u1=c1∈1∈2mnxded2−n/mxedc1−1signxde,(60)u2=α1signs+α2sηsigns,(61)u3=MM−1C11signsdm,(62)W^.i=Πwϕixiϱi,where ϱi is ith element of the term sT∈2n/mxdedn/m−1M−1, i=1,2,…,6, and Πw is a positive symmetrical matrix. The tracking errors of the spacecraft formation flying system will converge to zero.Proof.
Define the error of the weighting vector asW˜i=Wi−W^i. Consider the following Lyapunov function:(63)V2t=12sTs+12∑i=1ntrW˜iTΠw−1W˜i.
Taking the derivative of (63) with respect to time and substituting the proposed control law (57)–(62) and the adaptive updating law (62), then (63) can be converted as(64)V˙2t=sTs.−∑i=1ntrW˜iTΠw−1W˜˙i=sT−∈2nmxdedn/m−1M−1C21signsdm−M−1C2d+∈2nmxdedn/m−1M−1f˜nn+∈2nmxdedn/m−1α1signs+α2sηsigns−∑i=1ntrW˜iTΠw−1W˜˙i.
Since(65)sT∈2nmxdedn/m−1−M−1C21signsdm+M−1C2d=−∑i=16si∈2nmxdedn/m−1M−1C2signsidm+∑i=16si∈2nmxdedn/m−1M−1C2di≤−∑i=16si∈2nmxdedn/m−1M−1C2dm+∑i=16si∈2nmxdedn/m−1M−1C2di≤0.
Therefore,(66)V˙2t≤sT∈2nmxdedn/m−1M−1f˜nn+sT∈2nmxdedn/m−1−α1signs−α2sηsigns−∑i=1ntrW˜iTΠw−1W˜˙i≤sT∈2nmxdedn/m−1M−1∑i=1nW˜iTϕixi+ξi+sT∈2nmxdedn/m−1−α1signs−α2sηsigns−∑i=1ntrW˜iTϕixisT∈2nmxdedn/m−1M−1.
Since(67)sT∈2nmxdedn/m−1M−1∑i=1nW˜iTϕixi=−∑i=1ntrW˜iTϕixisT∈2nmxdedn/m−1M−1,(66) can be rewritten as follows:(68)V˙2t≤−sT∈2nmxdedn/m−1α1signs+α2sηsigns+∈2nmxdedn/m−1sTM−1∑i=1nξi≤−α¯1s−α¯2sη+1+∈2nmsTxdedn/m−1xdedn/m−1M−1∑i=1nξi≤−α¯1s−α¯2sη+1+∈2nmsxdedn/mn/m−1−1M−1∑i=1nξi.
Defineξh=∈2n/mxdedn/m−1M−1∑i=1nξi, then we have(69)V˙2t≤−ξ¯s,where ξ¯=α¯1−ξh.
Therefore, it can be concluded that the tracking errors of the spacecraft formation flying system converge to zero and the proof is completed.


## 3.1. The Fast Nonsingular Terminal Sliding Mode Control Law with a Supertwisting Disturbance Observer

The block diagram of the fast nonsingular terminal sliding mode control law with the supertwisting disturbance observer is shown in Figure2. In Figure 2, it can be seen that the overall control law consists of two parts: a fast nonsingular terminal sliding mode control law along with the supertwisting disturbance observer. The supertwisting disturbance observer is applied to estimate the disturbances for the spacecraft formation flying system if the system dynamics are known.Figure 2
The block diagram of the control law based on the supertwisting disturbance observer.


### 3.1.1. Supertwisting Disturbance Observer

In practical applications, the disturbances in the spacecraft formation flying system are fast time-varying disturbances. Hence, it is difficult for conventional observers to estimate those disturbances. In this section, a supertwisting disturbance observer is employed to estimate those disturbances. The supertwisting disturbance observer can estimate the disturbance in finite time and also it avoids high-order observer gains with fixed bandwidth.According to the attitude-orbit coupling system (15) and (16), the disturbance vector d can be rewritten as follows:(22)d=C2−1Mx..e+Ax.e+g−C1u.Define the state variables as follows:(23)x1=∫0tddt=C2−1Mx.e−∫0tddtC2−1Mx.e+C2−1Ax.e+g−C1udt,x2=d.Then, the disturbance dynamic system for the spacecraft formation flying is given as(24)x˙1=x2,x˙2=d˙.For each disturbancedi in the spacecraft formation flying system, the disturbance’s model is given as follows:(25)x˙1i=x2i,x˙2i=d˙i,i=1,2,…,6.The supertwisting disturbance observer is given as follows:(26)x˜˙1i=−λ1e1i1/2signe1i+x˜2i,x˜˙2i=−λ2signe1i,where λ1 and λ2 denote the positive observer gains, x˜1i and x˜2i are the estimations of the state vectors x1i and x2i, and e1i=x˜1i−x1i and e2i=x˜2i−x2i are the estimation errors, respectively.According to the attitude-orbit coupling system (15) and (16) and the supertwisting state observer (26), the estimation errors can be rewritten as(27)e˙1i=−λ1e1i1/2signe1i+e2i,e˙2i=−λ2signe1i−d.i.Define the state vectorsκ=κ1,κ2T, where κ1=e1i1/2signe1i, κ2=e2i.Then, the derivatives of the state vectorsκ1 and κ2 can be written as(28)κ˙1=−λ12e1i−1e1i+12e1i−1/2e2i,κ˙2=−λ2e1i−1e1i−d.i.Similar to [46], the following Lyapunov function for (28) is constructed:(29)Vst=κTθ1κ=12λ12+4λ2κ12−λ1κ1κ2+κ22,where θ1=1/2λ12+4λ2−λ1−λ12.Then, the derivative of Lyapunov functionVst with respect to the time is given as(30)V˙st=λ12+4λ2κ1κ.1−λ1κ.1κ2−λ1κ1κ.2+2κ2κ.2=λ12+4λ2e1i−1/2e1i−λ12e1i−1e1i+12e1i−1/2e2i+λ122e1i−1e1ie2i−λ12e1i−1/2e2i2−λ1e1i−1/2e1i−λ2e1i−1e1i−d.i−2λ2e1i−1e1ie2i−2e2id.i=λ12+4λ2−λ12e1i−1/2+12e1i−1e1ie2i+λ122e1i−1e1ie2i−λ12e1i−1/2e2i2+λ1λ2e1i−1/2+λ1e1i−1/2e1id.i−2λ2e1i−1e1ie2i−2e2id.i.Furthermore, (30) can be written as follows:(31)V.st=−λ132+λ1λ2e1i1/2+λ12e2isigne1i−λ12e2i2e1i1/2+λ1e1i1/2signe1id.i−2e2id.i=−λ12e1i1/2λ12+2λ2e1i−2λ1e2ie1i1/2signe1i+e2i2+λ12e1i1/22e1isigne1i−4λ1e1i1/2e2id.i≤−λ12e1i1/2λ12+2λ2e1i−2λ1e2ie1i1/2signe1i+e2i2+λ12e1i1/22e1i+4λ1e1i1/2e2id.i.Sinceκ12−2κ1κ2+κ22≥0, (31) can be rewritten as(32)V.st≤−λ12e1i1/2λ12+2λ2κ12−2λ1κ1κ2+κ22+λ12e1i1/22κ12+4λ1κ1κ2ςi≤−λ12e1i1/2λ12+2λ2κ12−2λ1κ1κ2+κ22+λ12e1i1/22κ12+2λ1κ12+2λ1κ22ςi.Define the following matrix as(33)θ2=λ12+2λ2−2λ1+1ςiλ1−λ1−λ11−2ςiλ1.Then, (32) can be concluded as(34)V.st≤−λ12e1i1/2κTθ2κ.From (33), by selecting appropriate observer gains, the matrix θ2 can be a positive definite matrix. Then, (34) becomes(35)V.st≤−λ1λminθ22e1i1/2κ2,where λminθ2 is the minimum eigenvalue of θ2.θ1 must to be a positive definite matrix to be Lyapunov matrix; hence, the observer gains λ1 and λ2 need to be selected appropriately so that θ1>0. Let us define the minimal and maximal eigenvalues of θ1 as λminθ1>0 and λmaxθ1>0, respectively. Then, we have(36)λminθ1κ2≤Vst≤λmaxθ1κ2.Furthermore, by observing from the state vectorsκ and κ1, the following can be concluded:(37)e1i1/2=κ12≤κ.Finally, combining (36) with (37), (35) can be derived as(38)V.st+λ0Vst1/2≤0,λ0=λ1λminθ11/2λminθ22λmaxθ1>0.Based on (37) and Lemma 1, we prove that limt⟶tde1i=0 and limt⟶tde2i=0 with i=1,2,…,6. Furthermore, the estimations errors of the state vector x1i and x2i can converge in finite time, and the convergence time is td=2Vst1/2/λ0.Remark 2.
To guarantee that the trajectory errors converge to zero within finite time, the supertwisting disturbance observer gainsλ1 and λ2 should be selected appropriately so that θ1 and θ2 are positive definite matrices.


### 3.1.2. Fast Nonsingular Terminal Sliding Mode Control Law with Supertwisting Disturbance Observer

To design a fast nonsingular terminal sliding mode control law, the following sliding mode surface is chosen:(39)s=xe+∈1xedc1signxe+∈2xdedn/msignxde,where ∈1,∈2>0, c1>n/m, and 1<n/m<2 are all positive constants.If the spacecraft formation flying system state errors reach the sliding modes=0, then(40)xe+∈1xedc1signxe+∈2xdedn/msignxde=0.From (15), substituting the equation x.e=xde into (40), we have(41)x.e=−1∈2m/nxe+∈1xedc1signxem/n=−1∈2m/nxem/n1+∈1xedc1−1m/n.In order to analyze the fast convergence of the nonsingular fast terminal sliding mode surface, assume that the integration time of the state variablexe denotes as xeti and xeti+ts, respectively. Then, integrating the time along both sides of (41),(42)∫xetixeti+ts1xem/ndxe=−∫titi+ts1∈2m/n1+∈1xec1−1m/ndτ≤−∫titi+ts1∈2m/ndτ.Furthermore, it can be seen from (42) that the tracking errors of the spacecraft formation flying system can converge to zero in a limited time ts≤∈2m/n/n−mxeti1−m/n.Remark 3.
The proposed nonsingular fast terminal sliding mode surface (39) indicates that when the system state variable xe is far from the equilibrium point, the term xe plays the major role in making the system trajectory converge at a fast rate. When xe approaches to the equilibrium point, the term ∈1xedc1signxe plays the dominant role to achieve rapid convergence of the state trajectory.A design procedure for a fast nonsingular terminal sliding mode control law with a supertwisting disturbance observer is given in the following theorem.Theorem 1.
Consider the spacecraft formation flying system described by (15) and (16) with(43)u=C1−1Axde+g−C2x˜2−C1−1Mu0+u1+u2,where(44)u0=1∈2mnxded2−n/msignxde,(45)u1=c1∈1∈2mnxded2−n/mxedc1−1signxde,(46)u2=α1signs+α2sηsigns,and −α1signs−α2sηsigns is the reaching law with α1>0, α2>0, and η>1. Then, the proposed control law along with the supertwisting disturbance observer (26) can guarantee the tracking errors of the spacecraft formation flying system converge to zero in finite time.Proof.
From (39), the derivative of the sliding surface s is given as follows:(47)s.=xe.+∈1c1xedc1−1xde+∈2nmxdedn/m−1xde.=xe.+∈1c1xedc1−1xde+∈2nmxdedn/m−1M−1−Axde−g+C1u+C2d.
Consider the following Lyapunov function:(48)V1t=12sTs+∑i=16V1i.
Taking the time derivative of (48), we have(49)V˙1t=sTs.+∑i=16V˙1i=sTxde+∈1c1xedc1−1xde+∈2nmxdedn/m−1−M−1C2x2−d−u0+u1+u2−α0∑i=16V1i1/2.
Substituting the proposed control law (44)–(46) into (49),(50)V˙1t=−sT∈2nmxdedn/m−1α1signs+α2sηsigns+sT∈2nmxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2.
Definexdemin=minxde1n/m−1,…,xde6n/m−1, α¯1=xdemin∈2n/mα1, and α¯2=xdemin∈2n/mα2, then(51)V˙1t≤−α¯1s−α¯2sη+1+∈2nmsTxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2≤−α¯1s−α¯2sη+1+∈2nmsxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2.
Defineαe=∈2n/mxdedn/m−1M−1C2x2−d and select α¯e=α¯1−αe≥0; we have(52)V˙1t≤−α¯es−α0∑i=16V1i1/2.
Finally, defineαm=min2α¯e,α0, then (52) can be rewritten as(53)V˙1t≤−αm12s+∑i=16V1i1/2≤−αm12s2∑i=16V1i1/2=−αmV1t1/2.Remark 4.
When the system state variable is far from the fast nonsingular terminal sliding mode surface, the termα2sηsigns in the reaching law plays the major role. On the contrary, the term α1signs in the reaching law dominates. By choosing different α1 and α2 and selecting appropriate parameters in the fast terminal sliding mode surface, the convergence speed can be controlled. Moreover, compared with the conventional reaching law in [47], the proposed reaching law has faster convergence and better sliding performance. When s=0 and s˙=0, the speed of the state variable approaching the sliding surface is reduced to zero, which can alleviate the chattering phenomenon.


## 3.1.1. Supertwisting Disturbance Observer

In practical applications, the disturbances in the spacecraft formation flying system are fast time-varying disturbances. Hence, it is difficult for conventional observers to estimate those disturbances. In this section, a supertwisting disturbance observer is employed to estimate those disturbances. The supertwisting disturbance observer can estimate the disturbance in finite time and also it avoids high-order observer gains with fixed bandwidth.According to the attitude-orbit coupling system (15) and (16), the disturbance vector d can be rewritten as follows:(22)d=C2−1Mx..e+Ax.e+g−C1u.Define the state variables as follows:(23)x1=∫0tddt=C2−1Mx.e−∫0tddtC2−1Mx.e+C2−1Ax.e+g−C1udt,x2=d.Then, the disturbance dynamic system for the spacecraft formation flying is given as(24)x˙1=x2,x˙2=d˙.For each disturbancedi in the spacecraft formation flying system, the disturbance’s model is given as follows:(25)x˙1i=x2i,x˙2i=d˙i,i=1,2,…,6.The supertwisting disturbance observer is given as follows:(26)x˜˙1i=−λ1e1i1/2signe1i+x˜2i,x˜˙2i=−λ2signe1i,where λ1 and λ2 denote the positive observer gains, x˜1i and x˜2i are the estimations of the state vectors x1i and x2i, and e1i=x˜1i−x1i and e2i=x˜2i−x2i are the estimation errors, respectively.According to the attitude-orbit coupling system (15) and (16) and the supertwisting state observer (26), the estimation errors can be rewritten as(27)e˙1i=−λ1e1i1/2signe1i+e2i,e˙2i=−λ2signe1i−d.i.Define the state vectorsκ=κ1,κ2T, where κ1=e1i1/2signe1i, κ2=e2i.Then, the derivatives of the state vectorsκ1 and κ2 can be written as(28)κ˙1=−λ12e1i−1e1i+12e1i−1/2e2i,κ˙2=−λ2e1i−1e1i−d.i.Similar to [46], the following Lyapunov function for (28) is constructed:(29)Vst=κTθ1κ=12λ12+4λ2κ12−λ1κ1κ2+κ22,where θ1=1/2λ12+4λ2−λ1−λ12.Then, the derivative of Lyapunov functionVst with respect to the time is given as(30)V˙st=λ12+4λ2κ1κ.1−λ1κ.1κ2−λ1κ1κ.2+2κ2κ.2=λ12+4λ2e1i−1/2e1i−λ12e1i−1e1i+12e1i−1/2e2i+λ122e1i−1e1ie2i−λ12e1i−1/2e2i2−λ1e1i−1/2e1i−λ2e1i−1e1i−d.i−2λ2e1i−1e1ie2i−2e2id.i=λ12+4λ2−λ12e1i−1/2+12e1i−1e1ie2i+λ122e1i−1e1ie2i−λ12e1i−1/2e2i2+λ1λ2e1i−1/2+λ1e1i−1/2e1id.i−2λ2e1i−1e1ie2i−2e2id.i.Furthermore, (30) can be written as follows:(31)V.st=−λ132+λ1λ2e1i1/2+λ12e2isigne1i−λ12e2i2e1i1/2+λ1e1i1/2signe1id.i−2e2id.i=−λ12e1i1/2λ12+2λ2e1i−2λ1e2ie1i1/2signe1i+e2i2+λ12e1i1/22e1isigne1i−4λ1e1i1/2e2id.i≤−λ12e1i1/2λ12+2λ2e1i−2λ1e2ie1i1/2signe1i+e2i2+λ12e1i1/22e1i+4λ1e1i1/2e2id.i.Sinceκ12−2κ1κ2+κ22≥0, (31) can be rewritten as(32)V.st≤−λ12e1i1/2λ12+2λ2κ12−2λ1κ1κ2+κ22+λ12e1i1/22κ12+4λ1κ1κ2ςi≤−λ12e1i1/2λ12+2λ2κ12−2λ1κ1κ2+κ22+λ12e1i1/22κ12+2λ1κ12+2λ1κ22ςi.Define the following matrix as(33)θ2=λ12+2λ2−2λ1+1ςiλ1−λ1−λ11−2ςiλ1.Then, (32) can be concluded as(34)V.st≤−λ12e1i1/2κTθ2κ.From (33), by selecting appropriate observer gains, the matrix θ2 can be a positive definite matrix. Then, (34) becomes(35)V.st≤−λ1λminθ22e1i1/2κ2,where λminθ2 is the minimum eigenvalue of θ2.θ1 must to be a positive definite matrix to be Lyapunov matrix; hence, the observer gains λ1 and λ2 need to be selected appropriately so that θ1>0. Let us define the minimal and maximal eigenvalues of θ1 as λminθ1>0 and λmaxθ1>0, respectively. Then, we have(36)λminθ1κ2≤Vst≤λmaxθ1κ2.Furthermore, by observing from the state vectorsκ and κ1, the following can be concluded:(37)e1i1/2=κ12≤κ.Finally, combining (36) with (37), (35) can be derived as(38)V.st+λ0Vst1/2≤0,λ0=λ1λminθ11/2λminθ22λmaxθ1>0.Based on (37) and Lemma 1, we prove that limt⟶tde1i=0 and limt⟶tde2i=0 with i=1,2,…,6. Furthermore, the estimations errors of the state vector x1i and x2i can converge in finite time, and the convergence time is td=2Vst1/2/λ0.Remark 2.
To guarantee that the trajectory errors converge to zero within finite time, the supertwisting disturbance observer gainsλ1 and λ2 should be selected appropriately so that θ1 and θ2 are positive definite matrices.


## 3.1.2. Fast Nonsingular Terminal Sliding Mode Control Law with Supertwisting Disturbance Observer

To design a fast nonsingular terminal sliding mode control law, the following sliding mode surface is chosen:(39)s=xe+∈1xedc1signxe+∈2xdedn/msignxde,where ∈1,∈2>0, c1>n/m, and 1<n/m<2 are all positive constants.If the spacecraft formation flying system state errors reach the sliding modes=0, then(40)xe+∈1xedc1signxe+∈2xdedn/msignxde=0.From (15), substituting the equation x.e=xde into (40), we have(41)x.e=−1∈2m/nxe+∈1xedc1signxem/n=−1∈2m/nxem/n1+∈1xedc1−1m/n.In order to analyze the fast convergence of the nonsingular fast terminal sliding mode surface, assume that the integration time of the state variablexe denotes as xeti and xeti+ts, respectively. Then, integrating the time along both sides of (41),(42)∫xetixeti+ts1xem/ndxe=−∫titi+ts1∈2m/n1+∈1xec1−1m/ndτ≤−∫titi+ts1∈2m/ndτ.Furthermore, it can be seen from (42) that the tracking errors of the spacecraft formation flying system can converge to zero in a limited time ts≤∈2m/n/n−mxeti1−m/n.Remark 3.
The proposed nonsingular fast terminal sliding mode surface (39) indicates that when the system state variable xe is far from the equilibrium point, the term xe plays the major role in making the system trajectory converge at a fast rate. When xe approaches to the equilibrium point, the term ∈1xedc1signxe plays the dominant role to achieve rapid convergence of the state trajectory.A design procedure for a fast nonsingular terminal sliding mode control law with a supertwisting disturbance observer is given in the following theorem.Theorem 1.
Consider the spacecraft formation flying system described by (15) and (16) with(43)u=C1−1Axde+g−C2x˜2−C1−1Mu0+u1+u2,where(44)u0=1∈2mnxded2−n/msignxde,(45)u1=c1∈1∈2mnxded2−n/mxedc1−1signxde,(46)u2=α1signs+α2sηsigns,and −α1signs−α2sηsigns is the reaching law with α1>0, α2>0, and η>1. Then, the proposed control law along with the supertwisting disturbance observer (26) can guarantee the tracking errors of the spacecraft formation flying system converge to zero in finite time.Proof.
From (39), the derivative of the sliding surface s is given as follows:(47)s.=xe.+∈1c1xedc1−1xde+∈2nmxdedn/m−1xde.=xe.+∈1c1xedc1−1xde+∈2nmxdedn/m−1M−1−Axde−g+C1u+C2d.
Consider the following Lyapunov function:(48)V1t=12sTs+∑i=16V1i.
Taking the time derivative of (48), we have(49)V˙1t=sTs.+∑i=16V˙1i=sTxde+∈1c1xedc1−1xde+∈2nmxdedn/m−1−M−1C2x2−d−u0+u1+u2−α0∑i=16V1i1/2.
Substituting the proposed control law (44)–(46) into (49),(50)V˙1t=−sT∈2nmxdedn/m−1α1signs+α2sηsigns+sT∈2nmxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2.
Definexdemin=minxde1n/m−1,…,xde6n/m−1, α¯1=xdemin∈2n/mα1, and α¯2=xdemin∈2n/mα2, then(51)V˙1t≤−α¯1s−α¯2sη+1+∈2nmsTxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2≤−α¯1s−α¯2sη+1+∈2nmsxdedn/m−1M−1C2x2−d−α0∑i=16V1i1/2.
Defineαe=∈2n/mxdedn/m−1M−1C2x2−d and select α¯e=α¯1−αe≥0; we have(52)V˙1t≤−α¯es−α0∑i=16V1i1/2.
Finally, defineαm=min2α¯e,α0, then (52) can be rewritten as(53)V˙1t≤−αm12s+∑i=16V1i1/2≤−αm12s2∑i=16V1i1/2=−αmV1t1/2.Remark 4.
When the system state variable is far from the fast nonsingular terminal sliding mode surface, the termα2sηsigns in the reaching law plays the major role. On the contrary, the term α1signs in the reaching law dominates. By choosing different α1 and α2 and selecting appropriate parameters in the fast terminal sliding mode surface, the convergence speed can be controlled. Moreover, compared with the conventional reaching law in [47], the proposed reaching law has faster convergence and better sliding performance. When s=0 and s˙=0, the speed of the state variable approaching the sliding surface is reduced to zero, which can alleviate the chattering phenomenon.


## 3.2. Fast Nonsingular Terminal Sliding Mode Control Law Based on the Adaptive Neural Network

In this section, an adaptive RBF neural network is employed to approximate the unknown nonlinearA and g in the spacecraft formation flying system (15) and (16), which is not easy to be precisely estimated in practical situations. The block diagram of a fast nonsingular terminal sliding mode control law based on the adaptive neural network is depicted in Figure 3.Figure 3
The block diagram of the control law based on the adaptive neural network.The structure of RBF neural network with multiple inputs and multiple outputs is shown in Figure4.Figure 4
The structure of RBF neural network.It can be seen that the RBF neural network has three layers including input layer, output layer, and hidden layer. Denotex=x1,x2,…,xnT as the input, and ϕix is the Gaussian function of the ith neuron in the hidden layer. Then, the Gaussian function can be expressed as below:(54)ϕix=exp−x−ci22gi2,i=1,2,…,6,where ci is the ith neural radial basis function and gi is the width of the ith neural radial basis function. In this section, we choose fnn to approximate and compensate the unknown nonlinear parts in the spacecraft formation flying system, where fnn=−Axde−g is the unknown nonlinear term. We rewrite fnn as follows:(55)fnn=WiTϕixi+ξi,where Wi denotes the ideal weighting matrix and ξi denotes the approximation error, i=1,2,…,6.The estimation offnn can be expressed as follows:(56)f^nn=W^iTϕixi,where W^i denotes the estimation of Wi.Theorem 2.
Consider the spacecraft formation flying system described by (15) and (16), with the following control law and the adaptive updating law of the RBF neural network:(57)u=C1−1f^nn−u3−C1−1Mu0+u1+u2,where(58)u0=1∈2mnxded2−n/msignxde,(59)u1=c1∈1∈2mnxded2−n/mxedc1−1signxde,(60)u2=α1signs+α2sηsigns,(61)u3=MM−1C11signsdm,(62)W^.i=Πwϕixiϱi,where ϱi is ith element of the term sT∈2n/mxdedn/m−1M−1, i=1,2,…,6, and Πw is a positive symmetrical matrix. The tracking errors of the spacecraft formation flying system will converge to zero.Proof.
Define the error of the weighting vector asW˜i=Wi−W^i. Consider the following Lyapunov function:(63)V2t=12sTs+12∑i=1ntrW˜iTΠw−1W˜i.
Taking the derivative of (63) with respect to time and substituting the proposed control law (57)–(62) and the adaptive updating law (62), then (63) can be converted as(64)V˙2t=sTs.−∑i=1ntrW˜iTΠw−1W˜˙i=sT−∈2nmxdedn/m−1M−1C21signsdm−M−1C2d+∈2nmxdedn/m−1M−1f˜nn+∈2nmxdedn/m−1α1signs+α2sηsigns−∑i=1ntrW˜iTΠw−1W˜˙i.
Since(65)sT∈2nmxdedn/m−1−M−1C21signsdm+M−1C2d=−∑i=16si∈2nmxdedn/m−1M−1C2signsidm+∑i=16si∈2nmxdedn/m−1M−1C2di≤−∑i=16si∈2nmxdedn/m−1M−1C2dm+∑i=16si∈2nmxdedn/m−1M−1C2di≤0.
Therefore,(66)V˙2t≤sT∈2nmxdedn/m−1M−1f˜nn+sT∈2nmxdedn/m−1−α1signs−α2sηsigns−∑i=1ntrW˜iTΠw−1W˜˙i≤sT∈2nmxdedn/m−1M−1∑i=1nW˜iTϕixi+ξi+sT∈2nmxdedn/m−1−α1signs−α2sηsigns−∑i=1ntrW˜iTϕixisT∈2nmxdedn/m−1M−1.
Since(67)sT∈2nmxdedn/m−1M−1∑i=1nW˜iTϕixi=−∑i=1ntrW˜iTϕixisT∈2nmxdedn/m−1M−1,(66) can be rewritten as follows:(68)V˙2t≤−sT∈2nmxdedn/m−1α1signs+α2sηsigns+∈2nmxdedn/m−1sTM−1∑i=1nξi≤−α¯1s−α¯2sη+1+∈2nmsTxdedn/m−1xdedn/m−1M−1∑i=1nξi≤−α¯1s−α¯2sη+1+∈2nmsxdedn/mn/m−1−1M−1∑i=1nξi.
Defineξh=∈2n/mxdedn/m−1M−1∑i=1nξi, then we have(69)V˙2t≤−ξ¯s,where ξ¯=α¯1−ξh.
Therefore, it can be concluded that the tracking errors of the spacecraft formation flying system converge to zero and the proof is completed.


## 4. Numerical Simulation

In this section, the simulation result that shows the effectiveness of the supertwisting disturbance observer, the nonsingular fast terminal sliding mode control law, and the adaptive neurosliding mode control law are presented, respectively, in each section.The parameters in the formation flying system are given as follows: the masses of the target and the chaser spacecraft aremt=mc=110kg, the initial orbit parameters of the target spacecraft are set as the semimajor axis a=7000km, the eccentricity e=0.01, the orbit inclination i=30°, and the ascending node is Ω=45°, and apogee and perigee are both 0°.The inertial matrix of the chaser spacecraft isJ=310.7;13.51;0.514kg⋅m2, and the disturbance torque dτ=0.1sint−cos0.3tsin0.5tTNm. The desired attitude and position are given as ρcd=2−2−0.5Tm and σd=0.001sin10t−cos10tsin10tT.


### 4.1. Supertwisting Disturbance Observer

The gains of the supertwisting disturbance observer are chosen asλ1=2 and λ2=4. In order to demonstrate the advantage of the proposed supertwisting disturbance observer, we compare its performance with the traditional linear observer as below:(70)x^˙1=−L1x1−x^1+x2,x^˙2=−L2x2−x^2.TheL1 and L2 in (70) are selected as(71)L1=diag0.023,0.011,0.014,0.012,0.010,0.008×104,L2=diag1.320,0.300,0.400,0.270,0.160,0.070×104.The disturbance estimation error given by the linear disturbance observer (70) and the proposed supertwisting disturbance observer (26) are depicted in Figures 5(a) and 5(b), respectively.Figure 5
Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbance observer.
(a)(b)Remark 5.
It can be seen from Figure5 that the supertwisting disturbance observer estimation error converges to zero in finite time, whereas the estimation error from the linear disturbance observer takes longer time to converge to zero. Therefore, the proposed supertwisting observer can estimate the disturbance in finite time, which demonstrates its advantage over the linear one in the spacecraft formation flying system.


### 4.2. The Nonsingular Fast Terminal Sliding Mode Control Law with Supertwisting Disturbance Observer

The parameters for the nonsingular fast terminal sliding mode surface are set as∈1=0.1, ∈2=0.02, c1=27/19, and n/m=21/19. The parameters in the control law are set as α1=100, α2=0.1, ∈=0.5, and η=1.5.The state responses ofxe and xde under the proposed control sliding model control law (43) are given in Figures 6 and 7, respectively.Figure 6
The state responses ofxe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbance observer.Figure 7
The state responses ofxde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbance observer.Furthermore, to illustrate the effectiveness of the proposed control law, in this part, we compare it with a widely used nonlinear PD control law given as follows:(72)u=C1−1M−Kpxe−Kdxde,where the coefficient Kp and Kd are calculated as follows:(73)Kp=diag16.5,3.75,50,13.5,20,8.75,Kd=diag5.75,2.75,3.5,3.0,2.5,2.0.The state responses of vectorsxe and xde under the nonlinear PD control law are given in Figures 8 and 9.Figure 8
The state responses ofxe with the nonlinear PD control law (72).Figure 9
The state responses ofxde with the nonlinear PD control law (72).Remark 6.
From Figures6–9, it can be seen that the proposed control law (43) yields better transient responses as compared to the nonlinear PD control law. Also, it is obvious that the proposed control law (43) guarantees the tracking errors converge to zero in a finite time.


### 4.3. Adaptive Neurosliding Mode Control Law

The parameters of the RBF neural network,ci is evenly selected over the space and gi=10. 200 neurons are used to approximate each fnni. Πw=diag100,100,100,100,100,100,100.The state responses of the system under the RBF neural network control law (57) are given in Figures 10 and 11.Figure 10
The state responses ofxe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neural network.Figure 11
The state responses ofxde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neural network.Remark 7.
Figures10 and 11 indicate that the tracking errors converge to a small region of zero with a good adaptive performance. This concludes that the RBF neural network control law (57) is able to compensate for the unknown dynamics in the spacecraft formation flying system.


## 4.1. Supertwisting Disturbance Observer

The gains of the supertwisting disturbance observer are chosen asλ1=2 and λ2=4. In order to demonstrate the advantage of the proposed supertwisting disturbance observer, we compare its performance with the traditional linear observer as below:(70)x^˙1=−L1x1−x^1+x2,x^˙2=−L2x2−x^2.TheL1 and L2 in (70) are selected as(71)L1=diag0.023,0.011,0.014,0.012,0.010,0.008×104,L2=diag1.320,0.300,0.400,0.270,0.160,0.070×104.The disturbance estimation error given by the linear disturbance observer (70) and the proposed supertwisting disturbance observer (26) are depicted in Figures 5(a) and 5(b), respectively.Figure 5
Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbance observer.
(a)(b)Remark 5.
It can be seen from Figure5 that the supertwisting disturbance observer estimation error converges to zero in finite time, whereas the estimation error from the linear disturbance observer takes longer time to converge to zero. Therefore, the proposed supertwisting observer can estimate the disturbance in finite time, which demonstrates its advantage over the linear one in the spacecraft formation flying system.


## 4.2. The Nonsingular Fast Terminal Sliding Mode Control Law with Supertwisting Disturbance Observer

The parameters for the nonsingular fast terminal sliding mode surface are set as∈1=0.1, ∈2=0.02, c1=27/19, and n/m=21/19. The parameters in the control law are set as α1=100, α2=0.1, ∈=0.5, and η=1.5.The state responses ofxe and xde under the proposed control sliding model control law (43) are given in Figures 6 and 7, respectively.Figure 6
The state responses ofxe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbance observer.Figure 7
The state responses ofxde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbance observer.Furthermore, to illustrate the effectiveness of the proposed control law, in this part, we compare it with a widely used nonlinear PD control law given as follows:(72)u=C1−1M−Kpxe−Kdxde,where the coefficient Kp and Kd are calculated as follows:(73)Kp=diag16.5,3.75,50,13.5,20,8.75,Kd=diag5.75,2.75,3.5,3.0,2.5,2.0.The state responses of vectorsxe and xde under the nonlinear PD control law are given in Figures 8 and 9.Figure 8
The state responses ofxe with the nonlinear PD control law (72).Figure 9
The state responses ofxde with the nonlinear PD control law (72).Remark 6.
From Figures6–9, it can be seen that the proposed control law (43) yields better transient responses as compared to the nonlinear PD control law. Also, it is obvious that the proposed control law (43) guarantees the tracking errors converge to zero in a finite time.


## 4.3. Adaptive Neurosliding Mode Control Law

The parameters of the RBF neural network,ci is evenly selected over the space and gi=10. 200 neurons are used to approximate each fnni. Πw=diag100,100,100,100,100,100,100.The state responses of the system under the RBF neural network control law (57) are given in Figures 10 and 11.Figure 10
The state responses ofxe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neural network.Figure 11
The state responses ofxde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neural network.Remark 7.
Figures10 and 11 indicate that the tracking errors converge to a small region of zero with a good adaptive performance. This concludes that the RBF neural network control law (57) is able to compensate for the unknown dynamics in the spacecraft formation flying system.


## 5. Conclusion

This paper has proposed the nonsingular fast terminal sliding mode control laws to solve the tracking control problem for spacecraft formation flying systems. First, a supertwisting disturbance observer has been introduced to estimate the disturbances in finite time. Then, based on a nonsingular fast terminal sliding mode surface, a novel control law has been presented to guarantee the tracking errors of the spacecraft formation converge to zero in finite time. Furthermore, for the unknown nonlinear parts in the spacecraft formation flying system, a RBF neural network has been proposed to approximate them and then compensate them. Finally, via simulations the effectiveness of the proposed control laws has been demonstrated. In our future work, we will investigate (1) the spacecraft formation system subject to the time-delay on terminal adaptive neurosliding mode control [48, 49] and (2) nonlinear observer design for the spacecraft formation system.



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