Finite-Time Synchronization Control of Spacecraft ... - IEEE Xplore

SPECIAL SECTION ON ADVANCED CONTROL AND HEALTH MANAGEMENT FOR AIRCRAFT AND ITS PROPULSION SYSTEM Received October 7, 2017, accepted November 6, 2017, date of publication November 14, 2017, date of current version December 22, 2017. Digital Object Identifier 10.1109/ACCESS.2017.2772319

Finite-Time Synchronization Control of Spacecraft Formation With Network-Induced Communication Delay RUIXIA LIU , XIBIN CAO , AND MING LIU Research Center of Satellite Technology, Harbin Institute of Technology, Harbin 150001, China

Corresponding author: Xibin Cao ([email protected]) This work was supported in part by the National Natural Science Foundation of China under Grant 91438202, Grant 61333003, Grant 51375109, and Grant 61473096, in part by the New Century Excellent Talents Program of the Ministry of Education of China under Grant NCET-13-0170, and in part by the Open Fund of National Defense Key Discipline Laboratory of Micro-Spacecraft Technology under Grant HIT.KLOF.MST.201505.

ABSTRACT This paper addresses the distributed orbit synchronization control of spacecraft formation flying under an undirected connected graph and in the presence of unknown external disturbances and communication time-delay. A nonsingular fast terminal sliding mode (NFTSM) control strategy, which can solve the singularity and slow convergence to the equilibrium problems of terminal sliding mode (TSM) control, is developed for spacecraft formation. Considering only desired signals are needed for the basis functions of Chebyshev neural networks (CNN) implemented, a CNN is employed to approximate the nonlinear function and bounded external disturbances. Based on the NFTSM and CNN approximation, a distributed finite-time synchronization control law is designed and its finite-time convergence property is proven in theory. Moreover, in order to guarantee good performance for the spacecraft formation control with communication delay, a distributed finite-time synchronization control scheme with communication delay is also given and the uniform ultimate boundedness of all signals in the closed-loop control system is proven. Finally, a numerical example is illustrated to demonstrate the effectiveness of the proposed control strategies. INDEX TERMS Spacecraft formation flying, adaptive control, model uncertainty, communication delay, finite-time control.

I. INTRODUCTION

Spacecraft formation flying (SFF) problem is a primary factor that determines the success of the many space mission such as spacecraft on-orbit servicing, deep space imaging and exploration. The concept SFF is to replace the functionality of large spacecraft with a group of less-expensive, smaller, and cooperative spacecraft. Recently, there have been a great number of results reported on the investigation of the spacecraft formation control [1]–[4]. Among these existing results, the relative dynamic model are generally classified as linear and nonlinear type for SFF. Considering the limitations of linear control, there have been various nonlinear control schemes designed for SFF based on the nonlinear model [5]–[8]. In practical aerospace engineering, various uncertainties and disturbances are inherently present in the spacecraft formation flying, so it is desirable to develop new 27242

control approaches to deal with these issues. Since Neural networks (NN) can approximate uncertain information and any continuous functions, the neural networks (NN) has been developed for spacecraft formation control [9]–[11]. In particular, a Chebyshev neural network (CNN) is a single-layer neural network [12], it is broadly adopted in tracking control due to the intensity of complexity and amount of calculation are less than the multilayered NN. It is worth pointing out that, sliding mode (SM) control is an effective robust control method for uncertain systems, whose has various attractive features such as distinguished robustness against external disturbances and parameter uncertain, fast response [13]–[16]. In the past decades, sliding mode control strategy has been applied for SFF. It should be pointed out that, although they realize effective control for spacecraft, they cannot maintain the closed-loop system convergence with finite time. In order to cope this

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VOLUME 5, 2017

R. Liu et al.: Finite-Time Synchronization Control of Spacecraft Formation

disadvantage, terminal SM (TSM) control was presented by Venkataramman and Gulati [17] and Man et al. [18] for robots. However, there are two disadvantages of TSM control approach which mainly lie in the following two-folds: (1) singularity problem; (2) when the system state vector is far away from the equilibrium, it has lower convergence rate than the linear sliding mode control (LSMC) [2]. In order to solve these problems, a nonsingular fast TSMC (NFTSMC) was presented in [19]. NFTSMC has been widely studied for many practical systems in engineering applications in the past few years, such as robot manipulators systems [20], fourth-order nonlinear systems [21], and nonlinear magnetic bearing system [22]. However, there exists little work focused on the finite-time control based on NFTSMC and CNN in spacecraft formation. A high convergence rate is an important requirement of spacecraft formation synchronization control, and the control law with finite-time convergence is more desirable [23]–[25]. In addition, in order to realize complex spacecraft formation flying mission, information exchange between individual spacecraft are needed for a group of spacecraft coordination control. Based on the position of control strategy, the spacecraft coordination control strategy are generally divided into centralized and decentralized control strategy for SFF. Decentralized control strategy is widely adopted in SFF control due to high precision and fault-tolerant compared to the centralized control strategy. The neighbour spacecraft’s information is needed for every spacecraft itself control for decentralized control strategy. Algebraic graph theory has been widely employed for decentralized control of SFF using limited information change with neighbour spacecraft. Due to connections over wireless communication channels, the wireless network inevitably induces communication delay between spacecraft [10], [26]–[28]. Many efforts have been devoted to studying this phenomenon [29]–[31]. Unfortunately, up to now, the decentralized control problem with communication delay for SFF has not been investigated well. Motivated by the above observations, in this paper, we study an orbital finite-time synchronization control problem for SFF under an undirected connected graph with unknown external disturbances and communication delay. The main contributions in this paper are stated as follows: (1) Considering the two disadvantages of the initial terminal sliding mode control, a NFTSM control strategy is developed for SSF; (2) Considering only desired signals are needed for the basis functions of CNN implemented. Therefore, the proposed CNN-based NFTSM control law has the characteristics of easy implementation and computational simplicity. Moreover, the law can ensure the orbital tracking error converge to the regions in finite time; (3) Considering the communication delay among neighbour spacecraft, a distributed finite-time control law with communication delay is designed, which can ensure the all signals in the resulting closed-loop control system is uniformly ultimately bounded (UUB).

VOLUME 5, 2017

II. PROBLEM FORMULATION A. SPACECRAFT ORBIT DYNAMICS

The nonlinear relative motion dynamic model for SFF can be written as [12]  µxi  xï − 2n˙yi − n˙ yi − n2 xi +    kR + qk3   ulx uix   + = wix +   ml mif    µ(yi + r) µr 2 y¨ i + 2n˙xi + n˙ xi − n yi + − (1) 3 kR + qk kRk3    u u iy ly   = wiy + +   m m  l if   µzi ulz uiz   z ¨ + + = wiz +  i ml mif kR + qk3 where xi , yi , and zi are the components of the position of the ith (i = 1, 2, . . . , N ) follower spacecraft relative to the leader spacecraft in corresponding axes in the local coordinate frame; µ is the Earthąŕs gravitational constant; ulx , uly , and ulz are the components of the control input acting on leader spacecraft in corresponding axes; uix , uiy , and uiz are the components of the control input acting on ith follower spacecraft in corresponding axes; ml and mif are the masses of the leader and ith follower spacecraft; wix , wiy and wiz are the components of the bounded external disturbance in corresponding axes; n is the orbital angular velocity of the leader spacecraft; R = [0, r, 0]T is the position vector from the center of the Earth to the leader spacecraft. Given the desired trajectory of the ith follower spacecraft qid = [xid , yid , zid ]T , the position tracking error ei1 and velocity tracking error ei2 are defined as ei1 = qi − qid ;

ei2 = q˙ i − q˙ id

where qi = [xi , yi , zi Then ( e˙ i1 = ei2 e˙ i2 = fi (qi , q˙ i , q¨ id ) + wid +

(2)

]T

1 mif

ui

(3)

where fi (qi , q˙ i , q¨ id ) ∈ R3 is a nonlinear function defined as   2n˙yi + n˙ yi + n2 xi + fi1  ˙ xi + n2 yi + fi2  i−n fi (qi , q˙ i , q¨ id ) =  −2n˙xµz (4)  ulz i − − − z ¨ id ml kR + qk3 where

µxi ulx − xïd − 3 ml kR + qk uly µ (yi + r) µr fi2 = − + − − y¨ id ml kR + qk3 kRk3 fi1 = −

The nonlinear function fi (qi , q˙ i , q¨ id ) ∈ R3 converges to the desired nonlinear function fi (qid , q˙ id , q¨ id ) when the initial states converges to desired states.   2n˙ydi + n˙ ydi + fid1 fi (qid , q˙ id , q¨ id ) =  −2n˙xid − n˙ xid + fid2  (5) ulz µzid − kR+qk ïd 3 − ml − z 27243


where fi (qid , q˙ id , q¨ id ) ≡ fid and ulx µxdi − fid1 = n2 xdi − − xïd 3 ml kR + qk uly µr µ (yid + r) + − − y¨ id fid2 = n2 ydi − ml kR + qk3 kRk3

we can obtain the output of the CNN as Fˆ i = fî (qi , q˙ i , q¨ id ) = Wi ς (qid , q˙ id , q¨ id )

In order to facilitate the subsequent control design, the following assumption is made about desired trajectory qid : Assumption 1: The desired trajectory qid and their first two-order time derivatives are assumed to be bounded.

where Wi is the estimation of Wi∗ .Since the external disturbances and the desired states can be treated as bounded, the nonlinear function F i is bounded such that kF i k ≤ FiM , in which FiM is a positive constant. Assumption 2: The optimal weight matrix Wi∗ of the CNN is bounded satisfying tr{Wi∗T Wi∗ } ≤ WiM

B. GRAPH THEORY

The topology of the information flow between individual follower spacecraft is described by a weight undirected graph G = ($ , E, A ), where $ = {$1 , $2 , . . . $n } is the set of nodes, E ⊆ $ × $ is the set of edges, and A = (aij ) ∈ Rn×n the weighted adjacency matrix of the graph G with nonnegative elements. ($i , $j ) ∈ E means that if and only there is an edge between node $i and $j satisfying ($i , $j ) ∈ E ⇔ ($j , $i ) ∈ E. The weighted adjacency element aij representing the communication quality between the ith and jth follower spacecraft satisfying ($i , $j ) ∈ E ⇔ aij > 0. Throughout this paper, it is assumed that aij = aji ,thus A is a symmetric matrix. Lemma 1 [2]: Suppose that there exists a continuous differential positive definite function V (t), real numbers α > 0,β > 0 and 0 < r < 1, the function V (t) satisfies the following differential inequality V˙ + αV + βV r ≤ 0

∀ > t0

(6)

Then, V (t) converges to the equilibrium point in finite time tf given by αV 1−r (x0 ) + β 1 ln (7) α(1 − r) β Lemma 2 [2]: If the matrix R is symmetric and positive definite, and there exist some positive constant r1 and r2 such that the matrix R can be bounded by tf ≤ t0 +

r2 kxk2 ≤ x T Rx ≤ r1 kxk2

x ∀ ∈ R3

(8)

III. MAIN RESULTS A. NFTSM BASED CONTROL LAW FOR SPACECRAFT FORMATION

In this subsection, we shall discuss the NFTSM tracking control law design problem for SFF based on CNN. Remark 1: For sake of good capabilities of approximation the nonlinear function and external disturbances, a single-layer CNN technique is employed to approximate the unknown function f i (qi , q˙ i , q¨ id ). By using CNN, the unknown function f i (qi , q˙ i , q¨ id ) can be approximated by F i = f i (qi , q˙ i , q¨ id ) = Wi∗ ς (qid , q˙ id , q¨ id ) + εi

(9)

where f i (qi , q˙ i , q¨ id ) = fi (qi , q˙ i , q¨ id )+wid , εi is approximation error of CNN and Wi∗ optimal weight matrix. Consequently, 27244

(10)

(11)

where WiM is a positive constant. Assumption 3: The approximation error εi of the CNN is bounded satisfying kεi k ≤ εiM

(12)

where εiM is a positive constant. In order to achieve spacecraft formation control with high precision and fast convergent rate, a continuous NFTSM based control law for spacecraft formation will be derived, wherein the terminal sliding manifold si ∈ R3 is defined as g/h

p/q

si = ei1 + aei1 + bei2

(13)

where a, b > 0, 1 < p/q < 2, g/h > p/q. Now, consider the spacecraft formation flying error system (3) and the NFTSM surface (13), the control law for ui is designed as mif q ag g/h−1 2−p/q (I3×3 + ei1 )ei2 − mif ui = − bp h · [Ki1 si + Ki2 sνi + µi (t)(Wi ς + ψi ) n X aij (si − sj ) (14) + (1 − µi (t))ψ i ] − mif j=1

where aij denotes the ith row jth column element of A , Ki1 = diag{ki11 , ki12 , ki13 } and Ki2 = diag{ki21 , ki22 , ki23 } are positive definite constant matrices, ν = p/q is a positive constant and sνi = (sνi1 , sνi2 , sνi3 )T ; ψi ∈ R and ψ i ∈ R are the robust control terms; The switch function µi (t) is given by 0, if k Wi ς k> FiM µi (t) = (15) 1, if k Wi ς k≤ FiM Remark 2: A switching between the robust control ψ i and the adaptive NN can be performed by using function µi (t). The adaptive law for Wi is given by ˙ i = µi (t)(σi1 si ς T − σi1 σi2 Wi ) W

(16)

where σi1 and σi2 are positive constants. Then, the robust controller ψi in the control law (14) is used to compensate the approximation errors of the CNN, and it is defined as ψik ≡ κ1 tanh(

3ku κ1 sik ) k = 1, 2, 3 ιi

(17)

where sik is the k − th component of sliding mode si , ιi is a positive scalar, and ku = 0.2785, κ1 is a positive constant VOLUME 5, 2017


satisfying κ1 ≥ εiM . The robust control strategy ψ ik is defined as 3ku κ2 sik ψ ik ≡ κ2 tanh( ) k = 1, 2, 3 (18) ιi

−

i=1

≤

where ku = 0.2785, κ2 is a positive constant satisfying κ2 ≥ FiM . It is easy to verity that 0 ≤ sTi εi − sTi ψi ≤ ≤ 0≤

3 X (|sik |kεi k − sik ψik )

− (19) ≤

Theorem 1: Considering the spacecraft formation control system is described by (3), with the designed control law (14) and adaptive law (16). If Assumption 1-3 are satisfied and the initial conditions satisfy eiT (0)W ei (0)} ≤ Vi max sTi (0)si (0) + tr{W

(20)

ei = W ∗ −Wi is the CNN weight approximation error where W i ei are UUB. matrix. Then si and W Proof: we construct the following Lyapunov function candidate: n n 1 X q X T 1−p/q fi T W fi } (21) si ei2 si + tr{W V = 2pb 2σi1 i=1

i=1

(22)

i=1

n

ag g/h−1 q X T 1−p/q si ei2 (ei2 + ei1 V˙ = ei2 pb h i=1

n 1 1 X bp p/q−1 ˙ } eiT W e (F i + ui )) + tr{W + e2i i q mif σi2 i=1

n

q X T 1−p/q ag g/h−1 bp p/q−1 = si ei2 (ei2 + ei1 ei2 + ei2 (F i pb h q i=1

1 + (−mif [Ki1 si + Ki2 sνi + µi (t)(Wi ς + ψi ) mif mif q ag g/h−1 2−p/q − (1 − µi (t))ψ] − (I3×3 + ei1 )ei2 bp h n n X 1 X ˙ } eiT W e − mif aij (si − sj )) + tr{W i σi2 j=1

i=1

sTi (F i − [Ki1 si + K2 sνi + µi (t)(Wi ς + ψi )

− (1 − µi (t))ψ i ]) −

i=1 j=1 VOLUME 5, 2017

i=1 j=1 n X

1 σi1

n X

aij (si − sj )T (si − sj )

˙ i} eiT W tr{W

i=1

ei ς + εi − ψi ) + (1 − µi (t))(F i − ψ i ) sTi [µi (t)(W

i=1

− sTi (Ki1 si + K2 sνi )] −

n 1 X ˙ i} eiT W tr{W σ1 i=1

≤−

n X

sTi (Ki1 si + Ki2 sνi ) +

n X

i

i=1

i=1

+ σi2 µi (t)

n X

eiT Wi } tr{W

i=1 n X

≤ −λmin (Ki1 )

sTi si +

i=1 n X

n X i=1

i +

n σi2 X WiM 2 i=1

sTi si + c1

aij sTi (si

− sj )

(23)

Note that 1 T 1 s si + (si − sj )T (si − sj ) 2 i 2 σ µ (t) eiT W ei } eiT Wi } ≤ − i2 i tr{W σi2 µ(t)tr{W 2 σi2 µi (t) + WiM 2 σ2 ≤ WiM 2 are applied, λmin (Ki1P ) is the minimum eigenvalue of the P matrix Ki1 , and c1 = ni=1 i + σ2i2 ni=1 WMi . Thus, we can further obtain V˙ < 0 outside the set r c1 si = {si |k si k≤ (24) λmin (Ki1 ) sTi (si − sj ) ≤

Based on a standard Lyapunov theory discussion in [2], it can ei are UUB. be conclude that s and W Theorem 2: Considering the spacecraft formation control system is described by (3), with the designed control law (14) and adaptive law (16). If Assumption 1-3 are satisfied and the initial conditions satisfy eiT (0)W ei (0)} ≤ Vi max sTi (0)si (0) + tr{W

i=1 n X n X

n

i=1

Substituting (14), (16) and (13) into (22) yields

=

2

= −λmin (Ki1 )

The derivative of the Lyapunov function Vi is defined as

n X

n n 1 XX

i=1

n n q X T 1−p/q 1 X ˙ i} fi T W V˙ = si ei2 tr{W s˙i + pb σi1

n

1 XX aij sTi si 2 i=1 j=1

−

− sTi ψi ≤ ιi

ei ς + εi − ψi ) + (1 − µi (t))(F i − ψ i ) sTi [µi (t)(W

− sTi (Ki1 si + K2 sνi )] −

ιi = ιi 3

k=1 sTi F i

n X i=1

k=1 3 X

n 1 X ˙ i} eiT W tr{W σ1

(25)

and then the NFTSM manifold si converges to the region 1si in finite time, then the tracking errors ei1k and 27245


ei2k (k = 1, 2, 3) converge to the regions 1ei1 and 1ei2 in finite time, respectively. Where 1si = min{21 , 22 } (26) 1si p/q 1si g/h ) ; 1ei2 = ( ) (27) 1ei1 = ( a b where −λmin (Ki2 ) is the minimum eigenvalue of the matrix Ki2 , and i is a positive constant, and r r i r1 r1 i 21 = , 22 = ( )2/(v+1) λmin (Ki1 )r2 λmin (Ki2 )r2 r2 Proof: Construct the following Lyapunov function candidate: n n X q X T 1−p/q si ei2 si (28) V = Vi = 2pb i=1

i=1

FIGURE 1. Communication graph between the follower spacecraft.

TABLE 1. The initial states of the three follower spacecraft.

The derivative of the Lyapunov function Vi is defined as V˙ =

n X i=1

n

q X T 1−p/q V˙ i = si ei2 s˙i pb

(29)

i=1

Substituting (14), (16) and (13) into (22) yields n

q X T 1−p/q ag g/h−1 V˙ = si ei2 (ei2 + ei1 ei2 pb h

TABLE 2. The desired states of the three follower spacecraft.

i=1

1 bp p/q−1 + ei2 (F i + ui )) q mif n

bp p/q−1 q X T 1−p/q ag g/h−1 ei2 + ei2 = si ei2 (ei2 + ei1 pb h q i=1

By using Lemma 2, the term sTi Ki2 sνi of the equation (30) is calculated as

1 (−mfi [Ki1 si + Ki2 sνi + µi (t) mif mif q (I3×3 · (Wi ς + ψi ) − (1 − µi (t))ψ i ] − bp

· (F i +

ag g/h−1 2−p/q )ei2 + ei1 − mif h =

n X

n X

−sTi K2 sνi ≤ −λmin (Ki2 )

n 2pb (ν+1)/2 X qr1 2 (ν+1)/2 ≤ −λmin (Ki2 )( ) ( s ) r1 q 2pb ik

aij (si − sj ))

k=1

j=1

2pb (ν+1)/2 ) ≤ −λmin (Ki2 )( r1 q q T 1−p/q (ν+1)/2 ·( s e si ) 2pb i i2

i=1 n X n X

(ν+1)/2

aij sTi (si − sj )

≤

where b2 = λmin (Ki2 )(2pb/r1 q)(ν+1)/2 , then we can obtain

ei ςi + εi − ψi ) + (1 − µi (t)) sTi [µi (t)(W

V˙ ≤ −b1

i=1

≤−


i=1

≤ −λmin (Ki1 )

n X

i=1

sTi si −

n X

i

i=1

sTi Ki2 sνi +

V˙ ≤ − n X i=1

i

n X

n X

(ν+1)/2

Vi

i=1

+

n X

i

(32)

i=1

n

(b1 −

i=1

(30)

V˙ ≤ −b1

n X i=1

27246

Vi − b2

where b1 = λmin (Ki1 )(2pb/r1 q). Then, (32) can be rewritten as the following two forms:

i=1 n X

n X i=1

(F i − ψ i ) − sTi (Ki1 si + Ki2 sνi )] n X

(31)

≤ −b2 Vi

i=1 j=1 n X

sν+1 ik

k=1

sTi (F i − [Ki1 si + Ki2 sνi + µi (t)(Wi ς + ψi )

− (1 − µi (t))ψ i ]) −

3 X

X (ν+1)/2 i )Vi − b2 Vi Vi

(33)

i=1

Vi −

n X (b2 − i=1

i (ν+1)/2 )V (ν+1)/2 i Vi

(34)

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FIGURE 2. Trajectories of position tracking error of controller (14) and (43). (a) The follower spacecraft 1. (b) The follower spacecraft 1. (c) The follower spacecraft 2. (d) The follower spacecraft 2. (e) The follower spacecraft 3. (f) The follower spacecraft 3.

Case 1: from (33), if b1 − i /Vi > 0, it is not difficult to obtain the closed-loop control system is finite-time stability by using Lemma 1, and therefore si will reach the region r i r1 ksi k ≤ 1si = (35) λmin (Ki1 )r2 in finite time. VOLUME 5, 2017

(ν+1)/2

Case 2: from (34), if b2 − i /Vi > 0, it is not difficult to obtain the closed-loop control system is finite-time stability using Lemma 1, and therefore si will reach the region

ksi k ≤ 1si =

r (

i r1 )2/(v+1) λmin (Ki2 )r2 r2

(36) 27247


FIGURE 3. Trajectories of velocity tracking error of controller (14) and (43). (a) The follower spacecraft 1. (b) The follower spacecraft 1. (c) The follower spacecraft 2. (d) The follower spacecraft 2. (e) The follower spacecraft 3. (f) The follower spacecraft 3.

in finite time. Thus, it can be concluded that the sliding mode si will reach the region ksi k ≤ 1si = min{21 , 22 } (37) in finite time, which means that ksi k ≤ 1si . Then, from (13) we can obtain g/h

p/q

ei1k + aei1k + bei2k = δik , |δik | ≤ 1si 27248

(38)

Equation (38) can be rewritten in two forms as follows δik g/h p/q ei1k + (a − g/h )ei1k + bei2k = 0 (39) ei1k δik p/q g/h e1ik + aei1k + (b − p/q )ei2k = 0 (40) ei2k VOLUME 5, 2017


FIGURE 4. Trajectories of thrust force of controller (14) and (43). (a) The follower spacecraft 1. (b) The follower spacecraft 1. (c) The follower spacecraft 2. (d) The follower spacecraft 2. (e) The follower spacecraft 3. (f) The follower spacecraft 3.

g/h

From (13), when a − δik /ei1k > 0, (39) is still satisfies the presented NFTSM form. Therefore, it is not difficult to see that the position tracking error ei1k will converge to the region

|ei1k | ≤ 1ei1 = ( VOLUME 5, 2017

|δik | g/h 1si g/h ) ≤( ) a a

p/q

in finite time.From (13), if b − δik /ei2k > 0 the inequality (40) is also satisfies the presented NFTSM form. Obviously, the tracking error ei1k will converge to the region |ei2k | ≤ 1ei2 = (

(41)

|δik | p/q 1si p/q ) ≤( ) b b

(42)

in finite time. 27249


FIGURE 5. Formation configuration of the 1st∼3rd follower spacecraft in the leaderąŕs frame of controller (14).

B. NFTSM CONTROL LAW DESIGN OF SPACECRAFT FORMATION FLYING WITH COMMUNICATION DELAY

In this subsection, we aim to solve the finite-time tracking control law design problem based on NFTSM and CNN for SFF in the presence of external disturbances and communication time delay. Remark 3: Considering the possible communication time delay between follower spacecraft, we introduce the parameter Tij , which is the non-negative constant representing communication time delay from the jth to ith follower spacecraft. Inspired by [32], the parameter Tij satisfies Tij ≥ 0, T˙ ij ≤ hij , and hij < 1. Then, the control law for ui is designed as ui = −mif [Ki1 si + Ki2 sνi + µi (t)(Wi ς + ψi ) mif q ag g/h−1 2−p/q −(1 − µi (t))ψi ] − (I3×3 + ei1 )ei2 bp h n X − mif %i si − mif aij (si − sj (t − Tij )) (43)

ei are the node i), ρij = ρji otherwise ρij = 0. Then si and W UUB. Proof: Define the following Lyapunov function candidate: n n 1 X q X T 1−p/q ei } eiT W si ei2 si + tr{W V = 2pb 2σi1 i=1 i=1 Z t n n 1 XX + ρij sTj (τ )sj (τ )dτ 2 t−Tij

(46)

i=1 j=1

its time derivative is n n q X T 1−p/q 1 X ˙ } eiT W e si ei2 tr{W V˙ = s˙i + i pb σi2 i=1 n

i=1

n

1 XX ρij [sTj sj − (1 − T˙ ij ) + 2 i=1 j=1

·sTj (t − Tij )sj (t − Tij )]

(47)

j=1

Substituting (16) , (43) and (13) into (47) yields Theorem 3: Considering the spacecraft formation control system is described by (3), with the designed control law (43) and adaptive law (16). For a given constant aij (i, j = 1, 2, . . . , N ), if Assumption 1-3 are satisfied and the initial conditions satisfy ei (0)} ≤ Vi max eiT (0)W sTi (0)si (0) + tr{W

(44)

and there exists hij < 1 and ρij satisfying ρij (1 − hij ) ≥ aij n n X X ρij < 2%i + aij j=1

i=1

·(F i +

n 1 1 X ˙ } eiT W e ui )) + tr{W i mif σi2 i=1

n

n

1 XX + ρij [sTj sj − (1 − T˙ ij ) 2 i=1 j=1

(45)

j=1

where ρij is defined as: for ∀i = 1, 2, · · · , n, if j ∈ Ni (Ni ⊂ {1, 2, · · · , N } is the set of the adjacency nodes of 27250

n

q X T 1−p/q ag g/h−1 bp p/q−1 V˙ = si ei2 (ei2 + ei1 ei2 + e2i pb h q

·sTj (t − Tij )sj (t − Tij )] n q X T 1−p/q ag g/h−1 bp p/q−1 = si ei2 (ei2 + ei1 ei2 + ei2 pb h q i=1 1 ·(F i + (−mif [Ki1 si + Ki2 sνi + µi (t) mif VOLUME 5, 2017


FIGURE 6. Formation configuration of the 1st∼3rd follower spacecraft in the leaderąŕs frame of controller (43).

mif q (I3×3 bp n X ag g/h−1 2−p/q + ei1 )ei2 − mif %i si − mif aij (si h

·(Wi ς + ψi ) − (1 − µi (t))ψ i ] −

j=1

n

n

1 XX ρij [sTj sj − (1 − T˙ ij ) −sj (t − Tij ))) + 2 i=1 j=1

·sTj (t − Tij )sj (t − Tij )] + =

n X

1 σi2

n X

˙ } e eiT W tr{W i

i=1

sTi (F i − [Ki1 si + Ki2 sνi + µi (t)(Wi ς + ψi )

i=1

−(1 − µi (t))ψ i ]) −

n X n X

aij sTi (si − sj (t − Tij ))

i=1 j=1 n n n n X 1 X 1 XX T ˙ T e − tr{Wi Wi } − %i si si + ρij σi1 2 i=1

≤

≤

i=1

i=1 j=1

·[sTj sj − (1 − T˙ ij )sTj (t − Tij )sj (t − Tij )] n X ei ς + εi − ψi ) + (1 − µi (t))(F i − ψ i ) sTi [µi (t)(W i=1 n X n n X X %i sTi si − sTi (Ki1 si + Ki2 sνi )] − aij sTi si − i=1 i=1 j=1 n X n n X n X X 1 1 + aij sTi si + aij sTj (t − Tij ) 2 2 i=1 j=1 i=1 j=1 n n X n X 1 X ˙ i} + 1 eiT W ·sj (t − Tij ) − tr{W ρij σi1 2 i=1 i=1 j=1 ·[sTj sj − (1 − T˙ ij )sTj (t − Tij )sj (t − Tij )] n X ei ς + εi − ψi ) + (1 − µi (t))(F i − ψ i ) sTi [µi (t)(W i=1

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− sTi (Ki1 si + Ki2 sνi )] −

n

n

1 XX (aij − ρij )sTi si 2 i=1 j=1

n n X n X 1 X ˙ i} − 1 eiT W − tr{W (ρij (1 − hij ) − aij ) σi1 2 i=1

i=1 j=1

·sTj (t − Tij )sj (t − Tij ) −

n X

%i sTi si

i=1

≤−

n X


n X

i

i=1

i=1

+ σi2 µi (t)

n X

eiT Wi } tr{W

i=1 n X

≤ −λmin (Ki1 )

= −λmin (Ki1 )

i=1 n X

sTi si

+

n X i=1

n σi2 X i + WiM 2 i=1

sTi si + c1

(48)

i=1

where the facts that n X n X

ρij =

n X n X

i=1 j=1 n X n X

i=1 j=1 n X n X

i=1 j=1

i=1 j=1

ρij sTj sj =

ρji ρij sTi si

1 T 1 s si + sTj (t − Tij )sj (t − Tij ) 2 i 2 σ µ (t) eiT Wi } ≤ − i2 i tr{W eiT W ei } σi2 µi (t)tr{W 2 σi2 µi (t) + WiM 2 σi2 ≤ WiM 2 sTi sj (t

− Tij ) ≤

27251


are applied, c1 = outside the set

Pn

i=1 i

+

σ2 2

Pn

si = {si |k si k≤

i=1 WMi .

r

Therefore, V˙ < 0

c1 λmin (Ki1 )

(49)

Based on a standard Lyapunov theory extension discusei sion in [2], the last conclusion can be obtained si and W are UUB. IV. ILLUSTRATIVE EXAMPLE

In this section, an example is provied to illustrate the effectiveness of the proposed control laws using NFTSM and CNN in this paper. The mass of the leader and the follower spacecraft are ml = 1kg, and mif = 1kg, (i = 1, 2, 3) respectively. The input control force is ui ≤ umax = 2N , (i = 1, 2, 3). For simplicity, the leader spacecraft is assumed in a circular reference orbit of radius 6728km and ith (i=1,2,3) follower spacecraft is represented by "Sat i‘‘. The communication graph between the follower spacecraft is shown in Figure 1. The information exchange among follower spacecraft is described by a weighted adjacency matrix A defined by   0 1, 1 1.1 0 1.1  . (50) A = [aij ]3×3 =  1.1 1, 1 1.1 0 The initial states and desired states of the individual follower spacecraft are shown in Table 1 and Table 2. The parameters of NFTSM controller are taken as g = 1.2, h = 1, p = 10, q = 9,a = 0.002, b = 0.2, κ1 = 10, κ2 = 2, ζ = 0.01, Ki1 = diag(0.2, 0.2, 0.2), Ki2 = diag(0.02, 0.02, 0.02), ιi = 0.008, σi1 = 100, σi2 = 0.001, (i = 1, 2, 3). The external disturbance input wid , (i = 1, 2, 3) is assumed to be wid = (0.001sin(nt), 0.001cos(2nt), 0.001sin(3nt))T mN. The communication delay is T12 = T21 = 1 − 0.2sin(0.2t)(s); T13 = T31 = 1.5 − 0.2sin(0.1t)(s); T23 = T32 = 1 − 0.3sin(0.1t)(s);

(51)

The relative position tracking errors ei1 (i=1,2,3) of the controller (14) and (43) are depicted in Figure 2. Figure 2 show that the relative position tracking errors ei1 (i=1,2,3) of the controller (14) convert to the region 1ei1 in approximately 30s. Due to network-induced communication delay between follower spacecraft, as can be seen in Figure 2, the relative position tracking errors ei1 (i=1,2,3) of controller (43) take longer to be uniformly ultimately bounded. The relative velocity tracking errors ei2 (i=1,2,3) of the controller (14) and (43) are depicted in Figure 3. The variations of control input thrusts ui (i=1,2,3) of the controller (14) and (43) are depicted in Figure 4. The formation configuration of the 1st∼3rd spacecraft in the leader’s frame is depicted in Figure 5. Considering the communication delay between the follower spacecraft, the formation configuration of the 1st∼3rd spacecraft in the leader’s frame is shown in Figure 6. It is obvious the proposed finite-time control strategies based 27252

on CNN have good capability of approximating nonlinear function and external disturbances, and provide good performance for the spacecraft formation synchronization control. V. CONCLUSIONS

In this paper, we have studied the problem of distributed orbit synchronization control for SFF under an undirected connected graph with unknown external disturbances and communication delay. A nonsingular fast terminal siding mode (NFTSM) control scheme based on Chebyshev neural network (CNN) was proposed, which can guarantee the orbital tracking error converge to the regions in finite time. Communication delay is inevitably present among the follower spacecraft of information exchange, a distributed finitetime control scheme considering communication delay is designed, which can ensure all the signals in the resulting closed-loop control system are UUB. REFERENCES [1] H.-T. Liu, J. Shan, and D. Sun, ‘‘Adaptive synchronization control of multiple spacecraft formation flying,’’ J. Dyn. Syst., Meas., Control, vol. 129, no. 3, pp. 337–342, 2007. [2] A.-M. Zou, K. D. Kumar, Z.-G. Hou, and X. Liu, ‘‘ Finite-time attitude tracking control for spacecraft using terminal sliding mode and Chebyshev neural network,’’ IEEE Trans. Syst., Man, B, Cybern., vol. 41, no. 4, pp. 950–963, Aug. 2011. [3] W. Wang, C. Huang, J. Cao, and F. E. Alsaadi, ‘‘Event-triggered control for sampled-data cluster formation of multi-agent systems,’’ Neurocomputing, vol. 267, no. 6, pp. 25–35, Dec. 2017. [4] R. Kristiansen and P. J. Nicklasson, ‘‘Spacecraft formation flying: A review and new results on state feedback control,’’ Acta Astron., vol. 65, nos. 11–12, pp. 1537–1552, Dec. 2009. [5] M. S. De Queiroz, V. Kapila, and Q. Yan, ‘‘Adaptive nonlinear control of multiple spacecraft formation flying,’’ J. Guid., Control, Dyn., vol. 23, no. 3, pp. 385–390, May 2000. [6] J. Yao, Z. Jiao, and D. Ma, ‘‘Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping,’’ IEEE Trans. Ind. Electron., vol. 61, no. 11, pp. 6285–6293, Nov. 2014. [7] Z. Chen, B. Yao, and Q. Wang, ‘‘Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect,’’ IEEE/ASME Trans. Mechatronics, vol. 18, no. 3, pp. 1122–1129, Jun. 2013. [8] H. Min, S. Wang, F. Sun, Z. Gao, and J. Zhang, ‘‘Decentralized adaptive attitude synchronization of spacecraft formation,’’ Syst. Control Lett., vol. 61, no. 1, pp. 238–246, Jan. 2012. [9] J. C. Patra and A. C. Kot, ‘‘Nonlinear dynamic system identification using Chebyshev functional link artificial neural networks,’’ IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 32, no. 4, pp. 505–511, Aug. 2002. [10] X. Zhao, H. Yang, W. Xia, and X. Wang, ‘‘Adaptive fuzzy hierarchical sliding-mode control for a class of MIMO nonlinear time-delay systems with input saturation,’’ IEEE Trans. Fuzzy Syst., vol. 25, no. 5, pp. 1062–1077, Oct. 2017. [11] Z. Chen, B. Yao, and Q. Wang, ‘‘µ-synthesis-based adaptive robust control of linear motor driven stages with high-frequency dynamics: A case study,’’ IEEE/ASME Trans. Mechatronics, vol. 20, no. 3, pp. 1482–1490, Jun. 2015. [12] A.-M. Zou and K. D. Kumar, ‘‘Adaptive output feedback control of spacecraft formation flying using Chebyshev neural networks,’’ J. Aerosp. Eng., vol. 24, no. 3, pp. 361–372, Jul. 2010. [13] H. Li, J. Wang, H.-K. Lam, Q. Zhou, and H. Du, ‘‘Adaptive sliding mode control for interval type-2 fuzzy systems,’’ IEEE Trans. Syst., Man, Cybern., Syst., vol. 46, no. 12, pp. 1654–1663, Dec. 2016, doi: 10.1109/TSMC.2016.2531676. [14] W. Sun, Y. Zhao, J. Li, L. Zhang, and H. Gao, ‘‘Active suspension control with frequency band constraints and actuator input delay,’’ IEEE Trans. Ind. Electron., vol. 59, no. 1, pp. 530–537, Jan. 2012. VOLUME 5, 2017


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RUIXIA LIU was born in Inner Mongolia, China, in 1987. She received the M.S. degree in measurement technique and automation equipment from Harbin Engineering University, Harbin, China, in 2014. She is currently pursuing the Ph.D. degree with the Harbin Institute of Technology, Harbin, China. Her research interests include spacecraft formation flying control, networked control systems, spacecraft rendezvous control, fault tolerant control, robust control/filtering, and Kalman filter.

XIBIN CAO received the B.S., M.S., and Ph.D. degrees from the Harbin Institute of Technology, in 1985, 1988, and 1991, respectively. Since 1991, he has been with the School of Astronautics, Harbin Institute of Technology, where he is currently a Full Professor. Since 2009, he has been the Dean of Astronautics School with the Harbin Institute of Technology, where he has been an Assistant President, since 2015. He was a recipient of the Distinguished Professor of Yangtze River Scholar, Ministry of Education of China, in 2005.

MING LIU was born in Jilin Province, China, in 1981. He received the B.S. degree in information and computing science and the M.S. degree in operational research and cybernetics from Northeastern University, Shenyang, China, in 2003 and 2006, respectively, and the Ph.D. degree in mathematics from the City University of Hong Kong in 2009. He joined the Harbin Institute of Technology in 2010, where he is currently a Professor. His research interests include networked control systems, fault detection and fault tolerant control, and sliding mode control. He was selected as the New Century Excellent Talents in University the Ministry of Education of China in 2013.

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