Distributed coordinated attitude tracking control for spacecraft formation with communication delays

Accepted Manuscript Distributed coordinated attitude tracking control for spacecraft formation with communication delays Wenjia Wang, Chuanjiang Li, Yanchao Sun, Guangfu Ma

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S0019-0578(18)30407-5 https://doi.org/10.1016/j.isatra.2018.10.028 ISATRA 2929

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ISA Transactions

Received date : 8 March 2018 Revised date : 2 June 2018 Accepted date : 15 October 2018 Please cite this article as: Wang W., et al. Distributed coordinated attitude tracking control for spacecraft formation with communication delays. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.10.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Distributed Coordinated Attitude Tracking Control for Spacecraft Formation with Communication Delays Wenjia Wanga, Chuanjiang Lia, Yanchao Suna, b*, Guangfu Maa a Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China b National Key Laboratory of Military Underwater Intelligent Robot, Harbin Engineering University, Harbin 150001, People’s Republic of China

* Corresponding author: Yanchao Sun E-mail addresses: [email protected] (W. Wang), [email protected] (C. Li), [email protected] (Y. Sun), [email protected] (G. Ma)

*Highlights (for review)

Highlights •

The distributed coordinated attitude tracking control problem is investigated for spacecraft formation with time-varying communication delays.

•

Parametric uncertainties and unknown disturbances are coped with via the adaptive technique.

•

Under the condition that the attitude information of the dynamic leader spacecraft is available to only a subset of follower spacecraft, a distributed estimator and an improved distributed observer are proposed for each follower spacecraft.

•

The Lyapunov-Krasovskii functional approach and LMI approach are employed for the stability analysis.

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Distributed Coordinated Attitude Tracking Control for Spacecraft Formation with Communication Delays Abstract

This paper investigates the distributed coordinated attitude tracking control problem for spacecraft formation with time-varying communication delays under the condition that the dynamic leader spacecraft is a neighbor of only a subset of follower spacecrafts. We consider two cases for the leader spacecraft: i) the attitude derivative is constant, and ii) the attitude derivative is time-varying. In the first case, a distributed estimator is proposed for each follower spacecraft by using its neighbors’ information with communication delays. In the second case, to express the dynamic leader’s attitude, an improved distributed observer is developed to estimate the leader’s information. Based on the estimated values, adaptive coordinated attitude tracking control laws are designed to compensate for parametric uncertainties and unknown disturbances. By employing the Lyapunov-Krasovskii functional approach, the attitude tracking errors and estimation errors are proven to converge to zero asymptotically. Numerical simulations are presented to illustrate the effectiveness of theoretical results. Keywords: Spacecraft formation, Attitude tracking, Coordinated control, Distributed observer, Communication delays

1.

Introduction With more and more special requirements of space missions, spacecraft formation

has attracted much attention of researchers [1-3]. By replacing the single large spacecraft with a group of simpler and smaller spacecrafts, spacecraft formation has advantages, such as increased robustness, improved flexibility, reduced cost, and enhanced reliability [4,5]. One of the most important issues for spacecraft formation is the coordinated attitude control. The coordinated attitude control, according to where the control decisions are made, can be generally categorized into centralized control and distributed control [6,7]. Compared with the centralized control, the distributed control only uses local measurement and information communicated from neighbors, which has less communication requirements and higher system performance. In light

1

of this, the distributed coordinated attitude control for spacecraft formation has been a significant research topic. In fact, the distributed coordinated attitude control is one of consensus problems [8,9]. Generally speaking, the consensus problem refers to making the attitude of spacecraft synchronize to a common value either determined by agreement between spacecrafts in a leaderless scenario or commanded by a leader. The leaderless consensus control problem for multiple spacecraft systems was investigated in [10-12]. For the leaderless case, the controllers in [10-12] only make the attitude of each spacecraft synchronize to a same attitude which may be unknown. The coordinated attitude tracking control can broaden the application of the leaderless consensus control, so that not only the attitude synchronization can be achieved but also the final attitude can synchronize to a specified attitude. By using the behavior-based method, a decentralized coordinated controller was developed and proven that the attitude of each spacecraft could asymptotically converge to a given desired trajectory in [13]. The attitude tracking problem for the spacecraft formation system with a desired time-varying attitude was studied in [14], where the proposed decentralized adaptive sliding-mode control method could achieve attitude synchronization. The neural network and the sliding-mode theory were applied to deal with the attitude tracking problem for spacecraft formation system in [15]. A decentralized finite-time attitude synchronization control strategy by utilizing the quaternion-descripted model of multiple spacecraft was developed in [16]. The aforementioned literatures mainly assume that all follower spacecrafts in a formation system can obtain the information from the leader spacecraft. Therefore, the decentralized attitude control in [13-16] is not fully distributed control scheme. By the influence of the communication bandwidth limitation and space circumstance, the assumption is restrictive and unrealistic. To relax the restriction of this assumption, a feedback control strategy was designed to maintain the constant relative attitude orientation for multiple rigid bodies in [17], where the information of the leader was available to only a subset of follower spacecrafts. Based on neural network and robust control technique, the distributed attitude coordinated control problem with partial 2

access to the reference attitude was investigated in [18]. The backstepping technique was applied to design a distributed cooperative control law in [19], which realized the attitude tracking under the undirected communication topology. Due to the heavy communication burden of undirected communication networks, the directed communication topology is more practical. In contrast to the undirected graph, the Laplacian matrix of a directed graph has the nonsymmetric structure. Due to the loss of symmetry, the Lyapunov function design and stability analysis become more difficult under the directed graph. Distributed finite-time attitude cooperative control algorithms under directed communication networks were proposed by employing the fast terminal sliding mode in [20,21], which needed the neighbors’ attitude accelerations for each follower spacecraft. Under the directed communication topology, a distributed attitude tracking controller was designed by integrating a distributed observer to solve the attitude consensus problem in [22]. Besides dealing with the partial access to the leader’s information under the directed communication topology in [20-22], time delay is another problem for spacecraft formation due to the information interaction via the communication networks, which may degrade the control performance or even influence the stability of the entire system. Two approaches are available for investigating the stability of systems with communication delays, namely, the frequency domain approach [23-25] and the time domain approach [26-28]. However, the frequency domain approach only applies to the linear systems with constant delays. In comparison, the time domain approach can be used for nonlinear spacecraft formation systems with time-varying delays. Lyapunov-Krasovskii functional approach is one of the widely used time domain approach due to the less conservatism [29,30]. The decentralized coordinated attitude tracking problem with time-varying communication delays was studied via the Lyapunov-Krasovskii functional approach in [31]. The coordinated attitude control problem with time-varying communication delays was extended to flexible spacecraft formations in [32]. The coordinated attitude tracking control algorithms based on the terminal sliding mode were proposed for spacecraft formation systems in the presence of communication delays in [33]. Actually, most existing 3

results for the coordinate attitude control with communication delays require that all follower spacecrafts have access to the leader spacecraft. Therefore, the distributed coordinated attitude control with communication delays remains an open issue. Motivated by the above facts, we focus on the coordinated attitude control problem for spacecraft formation system with communication delays under a directed communication topology. Two distributed coordinated tracking control strategies are proposed, where only a subset of follower spacecrafts can obtain the dynamic leader’s attitude information. The main contributions of this study are given as follows. (1) The coordinated attitude tracking control laws combined with the adaptive technique are proposed to deal with parametric uncertainties and unknown disturbances in this study. While in [17] and [19], the systems are modeled without uncertainties and external disturbances. In contrast, the spacecraft formation system in the presence of uncertainties and external disturbances is more realistic. (2) Inspired by [34], under the constraint that only a subset of follower spacecrafts can obtain the dynamic leader’s information, the distributed estimator and observer are designed for each follower spacecraft, which only rely on the neighbors’ information. In comparison to the coordinated attitude control with the assumption that the leader’s information is available for all follower spacecrafts [27,31-33], this study only needs the leader spacecraft access to a subset of follower spacecrafts under the directed communication topology. Thus, the communication burden via the designed distributed estimator and observer can be obviously reduced. (3) To solve the time-delay problem, the Lyapunov-Krasovskii functional approach and the linear matrix inequality (LMI) approach are used for the stability analysis in this study. In comparison to the systems with time delays by using the Lyapunov-Krasovskii functional approach [29,30], we consider the distributed coordinated tracking control with a dynamic leader spacecraft in this study. We extend the results in [26-28] to the coordinated attitude tracking control for spacecraft formation systems with time-varying communication delays. 4

The remainder of this research is organized as follows. In Section 2, preliminaries and the problem formulation are presented. The main work is stated in Section 3, where two distributed coordinated attitude tracking control laws and stability analysis are given. In Section 4, numerical simulations are presented to illustrate the effectiveness of the proposed algorithms. Finally, the conclusion is given in Section 5. 2.

Preliminaries and problem formulation

2.1. Spacecraft Attitude Dynamics Consider a group of n rigid spacecraft indexed by 1, , n . The attitude dynamics of the ith spacecraft are described by

 i  G ( i )i

(1)

J i i   i J i i  ui  d i

(2)

where  i   represents the Modified Rodrigues Parameters (MRPs) of the ith 3

spacecraft describing the attitude orientation with respect to the inertial frame,

 i   3 is the angular velocity of the ith spacecraft in the body frame, J i   33 is the inertia matrix of the ith spacecraft, and ui  3 and d i  3 are the control torque and the external disturbance torque, respectively. The matrix G ( i ) is expressed as

The notation  

1 ||  i ||2  G ( i )  1  I 3   i   i iT  2 2  is used to denote the skew-symmetric matrix:

3 2   0     3 0 1    2 1 0  Remark 1. MRPs is defined by  i  i tan(i 4) . It should be noted that MRPs goes 

singularly when i approaches 2 . As shown in [35], it can introduce a shadow counterpart  is   i  iT i of the original MRPs and switch the MRPs when

 iT i  1 to guarantee the global rotation representation without singularity. Therefore, MRPs  i and its shadow counterpart  is can provide a bounded attitude representation.

5

By combining (1) and (2), the attitude dynamics of the ith spacecraft can be transformed into the Lagrangian expression M i ( i )i  C i ( i ,  i ) i   i   di

(3)

where M i ( i )  G  T ( i ) J iG 1 ( i ),  i  G  T ( i ) ui ,  di  G  T ( i )d i , C i ( i ,  i )  G  T ( i ) J iG 1 ( i )G ( i )G 1 ( i )  G  T ( i )( J i i ) G 1 ( i )

Some fundamental properties of system (3) are given as follows [29]: Property 1. There exist positive constants

mi ,

mi

and

ci

such that

mi I 3  M i ( i )  mi I 3 , || Ci ( i ,  i ) || ci ||  i || , where ||  || denotes the Euclidean norm, and I 3 denotes the 3  3 identity matrix.  ( )  2C ( ,  ) is skew-symmetric. It means that Property 2. The matrix M i i i i i  ( )  2C ( ,  )] x  0 , for any vector x  3 . xT[M i i i i i

Property 3. For all x , y   3 , M i ( i ) x  C i ( i ,  i ) y  Yi ( i ,  i , x, y )i , where i   p

is

a

constant

vector

containing

the

system

parameters

and

Yi ( i ,  i , x, y )   3 p is a known regression matrix.

For the sake of facilitating the subsequent analysis, the following assumption is made for system (3): Assumption 1. [36] The disturbance  di is bounded, i.e. there exists an unknown positive constant ki such that ||  di || ki . 2.2. Graph theory We introduce a directed graph   { ,  ,  } to describe the communication topology among the spacecraft. The graph



consists of a node set

  {v1 , v 2 ,  , v n } , an edge set      and an adjacency matrix   [aij ]   nn .

The edge (vi , v j )   denotes that the jth spacecraft can obtain the information from the ith spacecraft, but not vice versa. A directed path is a sequence of ordered edges of the form (i1 , i2 ), (i2 , i3 ),  , where i j   . Let the adjacency matrix  of the graph

 with aij =1 if (v j , vi )   , and aij  0 otherwise. The Laplacian matrix   [lij ]   nn associated with  is defined as lij   i  j aij and lij   aij , i  j . 6

A directed graph is said to have a directed spanning tree if there is at least one node which has directed paths to all the other nodes. Suppose that there exists a leader in the system, labeled as spacecraft 0. Consider an augmented graph  used to denote the communication topology for the spacecraft formation system in this study, which consists of a leader spacecraft, n follower spacecrafts (1)-(2), and the edges among all spacecraft. The access of follower spacecraft to the leader spacecraft is represented by a diagonal matrix B  diag(a10 , a20 ,, an 0 ) , where ai 0  1 if the ith spacecraft can obtain the information from the leader, and ai 0  0 otherwise. Define H    B . Assumption 2. For spacecraft formation system, the communication topology graph

 has a directed spanning tree. Lemma 1. [37] If the graph  has a directed spanning tree, all eigenvalues of the matrix H have positive real parts. Lemma 2. [38] Define a positive definite matrix P . If the graph  has a directed spanning tree, the matrix R  PH  H T P is positive definite. 2.3. Problem formulation The leader’s attitude and its derivative are denoted by  0 and  0 , respectively. The attitude tracking error and its derivative for each follower spacecraft are defined as

 i =  i   0 (4)  i =  i   0 (5) Consider the spacecraft formation system under the directed communication graph  . The control objective of this study is to design distributed coordinated attitude tracking control schemes that ensure each follower spacecraft to track the dynamic leader spacecraft, that is, for all i  1, 2, , n ,

lim  i  0 , lim  i  0 t 

t 

(6)

In particular, we address this problem considering the leader’s information is available to only a subset of follower spacecrafts in the presence of time-varying communication delays. 3.

Coordinated attitude control with communication delays

3.1. Distributed attitude control when the leader’s attitude derivative is constant 7

In this subsection, we consider the case that the leader’s attitude derivative  0 is constant. To proceed further, a useful lemma is given as follow. Lemma 3. [39] For any symmetric positive definite matrix M   mm , scalar and vector function  :[1 ,  2 ]   m

 2  1  0

such that the following

integration holds



2

1

T

 ( ) d 

 M 

2

1



 ( ) d   ( 2  1 )



2

1

 T ( ) M ( ) d 



(7)

Since not all follower spacecrafts can obtain the leader’s information, each follower spacecraft can only use the neighbors’ information to track the leader’s attitude by the local communication. Moreover, the time delay is inevitable during the information interaction. Define the following auxiliary variables: n

 ri  vˆi    [aij ( i (t  T )   j (t  T )]

(8)

j0

si   i   ri (9) where  is a positive constant, T is the time-varying communication delay satisfying 0  T  T0 with T0 being the upper bound of T , and vˆi is the ith follower spacecraft’s estimate of the leader’s attitude derivative  0 to be designed. By Property 3, we can obtain M i ( i )ri  C i ( i ,  i ) ri  Yi ( i ,  i , ri ,  ri ) i

(10)

For the ith follower spacecraft, the following coordinated attitude tracking control law and the distributed estimator are proposed ˆ  kˆ sgn( s )  i   K i si  Yi  i i i

(11)

 n  (12) vˆi     aij [vˆi (t  T )  vˆ j (t  T )]  ai 0 [vˆi (t  T )   0 (t   )] j  1   where K i is a positive definite diagonal matrix,  is a positive constant, ˆ is the estimate of  .  ˆ and kˆ are updated by Yi  Yi ( i ,  i , ri ,  ri ) , and  i i i i

the following adaptive tuning laws:

ˆ T  (13) i   iYi si  kˆi   i || si ||1 (14) where  i is a positive definite diagonal matrix,  i is a positive constant, and ||  ||1 represent the sum norm of a vector.

8

Remark 2. Due to the uncertainties for each follower spacecraft, we use the estimate ˆ by adaptive tuning law (13) instead of  . In Eq. (11), kˆ sgn( s ) is used to  i i i i

compensate for the external disturbances with the time-varying gain kˆi updating by Eq. (14).    ˆ . Substituting (11) into (3), the closed-loop system can be Define  i i i

expressed as the following form   kˆ sgn( s )   M i ( i ) si  Ci ( i ,  i ) si   K i si  Yi  (15) i i i di Theorem 1. Consider the spacecraft formation attitude system governed by Eq. (3)

under Assumptions 1-2. If the time derivative of T satisfies 0  T (t )  d  1 and there exist positive definite matrices P , Q and R such that the following LMI holds T PF  (1  d )Q  2  (G  F )   T  R G  F F   0 (16) 0  T  R  (1  d )Q   F  0  ( H   3 )  U  ( F  G )T P + P ( F  G ) , F =  and  ， 0  ( H   )  3 

 U + dQ M  T  F P  (1  d )Q

where

 0 I 3n  G=  ,  represents the Kronecker product. Then the distributed estimator 0 0  and the coordinated control law in Eqs. (11)-(14) can ensure the follower spacecrafts converge to the leader spacecraft asymptotically.

 be the column stack vectors of  , s ,  and   , Proof: Let  , s ,  d , and  i i di i respectively. Meanwhile, let K , Y , kˆ , M ( ) and C ( , ) be the block diagonal matrices of K i , Yi , kˆi , M i ( i ) and C i ( i ,  i ) , respectively. The closed-loop system (15) can be written in a vector form as   (kˆ  I )sgn( s )   M ( ) s  C ( ,  ) s   Ks  Y  3 d

(17)

Consider the Lyapunov function candidate n

 T  1    1 k 2 V1  1 s T M ( ) s  1  (18) 2 2 2 i i i 1 where  is the block diagonal matrix of  i , and ki  ki  kˆi . Take the derivative of V1 and we can obtain n  ( ) s    T  1    1 k k V1  s T M ( ) s + 1 s T M 2  i i i 1 i

9

(19)

By Property 2 and Assumption 1, substituting (13)-(14) and (17) into (19) yields ˆ n 1  ˆ   (kˆ  I ) sgn( s )   ]    T  1 V1  s T [ Ks  Y    ki ki 3 d  i 1 i n

  s T    TY T s  k || s ||   s T Ks  s TY  i i 1 d i 1

(20)

n

  s T Ks  s T d   ki || si ||1 i 1

T

  s Ks Since K is positive definite, we can obtain V1  0 , which means V1 (t )  V1 (0)  , k  L . From (9),  ,   L . By using when t  0 . According to Eq. (18), s,   i ri 

Property 1 and Eq. (17), we have s, M ( ), C ( ,  )  L . By noting that V1  0 , it follows that lim V1 (t )  V1 () for V1 ()  [0,V1 (0)] . Integrating both sides of (20), t 

we can get 

min ( K )  s T (t ) s(t )dt  V1 (0)  V1 () 0

(21)

where min () represents the minimum singular value of a matrix. From (21), it implies that s  L2 . Thus, we have s  L2  L and s  L . By Barbalat’s Lemma in Ref. [40], it can be concluded that s  0 as t   . Define vi = vˆi   0 . Let  and v be the column stack vectors of  i and vi . Note that  0 is constant. Then, we can write (9) and (12) in the vector form as v    ( H   3 )v (t  T )

(22)

   ( H  Ι 3 ) (t  T )  v  s

(23)

Consider the following system    ( H   3 ) (t  T )  v  v    ( H   3 )v (t  T )

(24)

Let x  [ T , v T ]T . Then Eq. (24) can be transformed into the following form

x  Gx  Fx (t  T ) (25) In the light of Assumption 2, all eigenvalues of H have positive real parts according to Lemma 1. From the expressions of G and F , we can note that all eigenvalues of F  G have positive real parts. By Lemma 2, there exists a positive definite matrix P

such that U  ( F  G )T P + P ( F  G ) is positive definite.

10

Considering system (24) with communication delays, we choose the following Lyapunov-Krasovskii functional:

V2  x T Px  

t

t T

x T ( )Qx ( )d  T0 

t

t T0

(  t  T0 ) x T ( ) Rx ( )d

(26)

The derivative of V2 along the trajectories of (25) is V2  2 x T Px  x T Qx  (1  T )x T (t  T )Qx (t  T ) T02 x T Rx  T0 

t

t T0

(27)

x T ( ) Rx ( )d

By using the Leibniz-Newton formula, we have



t

t T0

x ( )d  x (t )  x (t  T0 ) .

Define x  x  x (t  T0 ) . By Lemma 3, the following inequality holds

x T (t ) Rx (t )  T0 

t

t T0

x T ( ) Rx ( )d

(28)

Note that T has the upper bound and 0  T (t )  d  1 . By using (25) and (28), it follows from (27) that V2  2 x T P[Gx  Fx (t  T )]  x T Qx  (1  T )x T (t  T )Qx (t  T )  T02 x T Rx  x T Rx  x T [ P (G  F ) + (G  F )T P + dQ ] x  T02 [(G  F ) x  Fx ]T R[(G  F )x  Fx ]  2 x T [ PF  (1  d )Q ] x  x T [ R + (1  d )Q ] x

(29)

 [ x T x T ]M [ x T x T ]T According to Eq. (16), it follows that system (25) is asymptotically stable, which implies that v  0 as t   . From the above analysis, we note that si  0 as t   . By Lemma 2.4 in Ref. [41], system (23) is also asymptotically stable, which means that  i  0 as t   . From (23), we can conclude that  i  0 as t   . This completes the proof of Theorem 1. 3.2. Distributed attitude control when the leader’s attitude derivative is time-varying In this subsection, the leader’s attitude derivative  0 is allowed to be time-varying. Since  0 changes over time, the follower spacecrafts are more difficult to track the leader. In this case, we assume that the leader’s attitude  0 is generated by the following system [34]

v  Sv  0  Cv

(30)

where S   mm and C   3m are constant matrices, v   m . In the remainder of this subsection, we need the following assumption for the leader 11

Assumption 3. All the eigenvalues of matrix S are semi-simple with zero real parts. Remark 3. Under Assumption 3, system (30) can generate  0 as a class of sinusoidal signals of arbitrary amplitudes and initial phases [34]. Without loss of generality, we assume that there exist non-negative integers m0 , m1 satisfying m0  2m1  m such that S  diag{0 m0 m0 ,diag{01 ,  ,0m1 }  a}

(31)

 0 1 where  01 , ,0m1 are positive constants, and a   .  1 0 According to Remark 3, we note that

S

is skew-symmetric. Define

  [ 01 , , 0m1 ]T . In order to solve the problem that the leader’s information is available to only a subset of follower spacecrafts, we propose the following distributed observers n

ˆi   1  aij [ˆi (t  T )  ˆ j (t  T )]

(32)

j0

n

vˆi  Sˆ i vˆi   2  aij [vˆi (t  T )  vˆ j (t  T )]

(33)

j 0

where ˆi and vˆi are the estimates of the leader’s information  and v for the ith follower spacecraft, 1 and  2 are positive constants, and

Sˆi  diag{0 m0 m0 ,diag{ˆi1 , ,ˆim1 }  a}

(34)

We assume that vˆ0   0 and Sˆ 0  S . To facilitate the subsequent analysis, we need the following lemma. Lemma 4. Consider system (30) under Assumptions 2-3. If the time derivative of T satisfies 0  T (t )  d  1 and there exist positive definite matrices P1 , P2 , Q and R such that

Q  1 ( P1 H  I m1 )   P2 2 ( H  I m )   N1    0,   0, N 2   T T (1  d )Q   (1  d ) P2    1 ( H P1  I m1 )   2 ( H  I m )  2（P1  S )  R  2 ( P1 H  I m )  N3   0 T (1  d ) R     2 ( H P1  I m ) (35) ˆ By using Eqs. (32)-(33) for 1 , 2  0 , then lim( Si  S )  0 and lim(vˆi  v )  0 . t 

12

t 

Proof: Define Si  Sˆi  S and vi = vˆi  v . Then Eqs. (32)-(33) can be written as the following form

 S   1 ( H  I m ) S (t  T )

(36)

v  S d vˆ  ( I n  S )v  2 ( H  I m )v (t  T ) (37) where v is the column vector of vˆi , S d is the block diagonal matrix of Si , and S  col( S1 , , S n ) .

First, we will prove that system (36) is asymptotically stable. We note that the vector  can be directly observed by (32). Let i  ˆi   . Eq. (32) can be written in the following vector form

  1 ( H  I m1 ) (t  T ) (38) where  is the column vector of i . From (34), it is worth noting that only ˆi1 , ,ˆim1 are nonzero elements in Sˆi . Thus, the stability problem of system (36) can be transformed into the stability problem of system (38). Under Assumption 2 and Lemma 1, all eigenvalues of H have positive real parts. The Lyapunov-Krasovskii functional is selected as t

V1   T ( P1  I m1 )    T ( )Q ( )d t T

(39)

By using (38), the derivative of (39) is given as V1  2 T ( P1  I m1 )   T Q  (1  T ) T (t  T )Q (t  T )  2 1 T ( P1 H  I m1 ) (t  T )   TQ  (1  d ) T (t  T )Q (t  T )

(40)

 [ T  T (t  T )] N1[ T  T (t  T )]T If N1  0 holds, system (38) is asymptotically stable. Thus, system (36) is also

asymptotically stable, that is lim( Sˆ i  S )  0 . t 

Then, we give the stability analysis of system (37). Choose the following Lyapunov-Krasovskii functional V2  vˆ T vˆ  

t

t T

vˆ T ( ) P2 vˆ ( )d

(41)

Since Sˆi is skew-symmetric, the derivative of (41) is

V2  2vˆ T vˆ  vˆ T P2 vˆ  (1  T )vˆ T (t  T ) P2 vˆ (t  T )  2 2 vˆ T ( H  I m )vˆ (t  T )  vˆ T P2vˆ  (1  d )vˆ T (t  T ) P2 vˆ (t  T ) =[vˆ T vˆ T (t  T )] N 2 [vˆ T vˆ T (t  T )]T If N 2  0 holds, V2  0 . From (41), we can conclude that vˆ is bounded. 13

(42)

We choose the following Lyapunov-Krasovskii functional T V3  v（ P1  I m )v  

t

t T

v T ( ) Rv ( )d

(43)

According to Eq. (37), taking the derivative of V3 can obtain T V3  2v（ P1  I m )v  v T Rv  (1  T )v T (t  T ) Rv (t  T ) T T T  2v（ P1  I m ) S d vˆ + 2v（ P1  S )v  2 2 v（ P1 H  I m )v (t  T )  v T Rv  (1  T )v T (t  T ) Rv (t  T )

(44)

 T  ˆ Let q  2v（P 1  I m ) S d v , while W  V3   q ( )d is a continuous scaler function. 0

From (40) and (42), we have proven that S  0 as t   and vˆ is bounded. The vector v of the leader is bounded in the practical case, which can imply that v is bounded. Thus,





0

q ( )d exists and is bounded. As a result, W is lower bounded.

Obviously, we have T T W  2v（ P1  S )v  2 2v（ P1 H  I m )v(t  T )  v T Rv  (1  T )v T (t  T ) Rv (t  T ) T  v T  2(P1  S )  R  v  2 2v（ P1 H  I m )v (t  T )  (1  d )v T (t  T ) Rv (t  T ) (45)

 [v T

v T (t  T )]N 3[v T v T (t  T )]T If N 3  0 holds, W  0 . Because W is lower bounded, W converges as t   . Since





0

q ( )d exists and lim S  0 by (40), V3 also converges as t   . From t 

(43), v  0 as t   , that is

lim(vˆi  v )  0 . t

Remark 4. In the view of the above analysis, the estimation errors Si and vi can n

converge to zero asymptotically. Let ei  2   aij (vˆi (t  T )  vˆ j (t  T ))  . By Lemma j 0

4, we can obtain lim  Cvˆi (t )   0 (t )   lim  Cvˆi (t )  Cv (t )   lim Cvi (t )  0 t 

t 

t 

lim Cvˆi (t )   0 (t )  lim Cvˆi (t )  Cv (t ) t 





t 





   lim C  Sˆ (t )vˆ (t )  e (t )  S (vˆ (t )  v (t ))   lim C Sˆ i (t )vˆi (t )  ei (t )  Sv (t ) t 

i

t 

i

i

i

 lim C S i (t )vˆi (t )  ei (t )  Svi (t ) t 



(46)

i



0 From Remark 4, we can conclude that Cvˆi   0  0 and Cvˆi   0  0 as

t . 14

For the case where  0 is time-varying, the distributed observers (32)-(33) with communication delays can ensure that the estimates converge to the leader’s attitude

 0 and  0 by Lemma 4 and Remark 4. Before giving the control law, we define the following auxiliary variables

 ri  CSˆi vˆi   ( i  Cvˆi ) (47) si   i   ri (48) where  is a positive constant. By Property 3, the parametric uncertainties can also be expressed as Eq. (10). Then, the adaptive control law is proposed as ˆ  kˆ sgn( s )  i   K i si  Yi  (49) i i i ˆ T  (50) i   iYi si  kˆi   i || si ||1 (51) where K i and  i are positive definite diagonal matrices, and  i is a positive

constant. Theorem 2. Consider the spacecraft formation attitude system governed by Eq. (3) under Assumptions 1-3. If the time derivative of T satisfies 0  T (t )  d  1 and there exist positive definite matrices P1 , P2 , Q and R such that the inequalities (35) hold, then the distributed observers (32)-(33) and the coordinated control law (49) -(51) can ensure the follower spacecrafts converge to the leader spacecraft asymptotically. Proof: Consider the following Lyapunov function candidate n

 T  1    1 k 2 V  1 s T M ( ) s  1  2 2 2 i i i 1 By Property 2 and Assumption 1, the derivative of V is given as

(52)

V   s T Ks (53)  Thus, V  0 . Since the control law (49)-(51) and the Lyapunov function (52) are similar to the case of Theorem 1, it can be easily verified that s  0 as t   . Substituting (47) into (48), we can obtain

 i   i  si  CSˆi vˆi   Cvˆi

(54)

By Eq. (33) and Remark 4, Eq. (54) can be written as  i  Cvˆi   ( i  Cvˆi )  si  CSˆ i vˆi  Cvˆi n

 si   2C   aij (vˆi (t  T )  vˆ j (t  T ))  j 0

 si  Cei 15

(55)

From the above expression, system (55) can be regarded as a first order differential equation with the state  i  Cvˆi and the input si  Cei . Because  is a positive constant, system (55) is globally exponentially stable at the origin  i  Cvˆi  0 when si  0 and ei  0 . By Lemma 2.4 in Ref. [41], system (55) is input-to-state stable. Note from Lemma 4 and Remark 4 that ei  0 as t   . Therefore, si  Cei  0 as t   . Because system (55) is input-to-state stable with the input si  Cei and the state  i  Cvˆi , it can be verified that  i  Cvˆi  0 as t   . As a result, it follows that  i  Cvˆi  0 as t   . From (46), we can conclude that ( i   0 ) and ( i   0 ) will converge to zero, that is  i  0 and  i  0 as t   . With the above analysis, the proof is completed. Remark 5. The coordinated attitude control law (49)-(51) with the distributed observers (32)-(33) can ensure that all follower spacecrafts track the dynamic leader when the system has time-varying communication delays. Despite the hard restrictions describe in Theorem 2, the coordinated control law (49)-(51) with the distributed observers (32)-(33) can solve the coordinated attitude tracking problem. Remark 6. For the case that  0 is constant, the upper bound of communication delay T0 can be calculated by the LMI (16). In Theorem 2, there is no constraint on T0 . Therefore, the calculated value T0 by Eq. (16) is still suitable for the case that

 0 is time-varying. 4.

Numerical simulations In this section, numerical simulations are given to illustrate the effectiveness of the

proposed control algorithms in Section 3. A formation system consisting of four follower spacecrafts and one leader spacecraft is considered. Fig. 1 shows the communication topology among the spacecraft, which satisfies Assumption 2. The inertia matrix and initial conditions of follower spacecrafts are presented in Table 1.

16

Figure 1 Communication topology Table 1 System parameters of follower spacecrafts Inertia matrix 2 J i ( kg  m )

Spacecraft No. i

Initial attitude

Initial attitude derivative  i (0)

 i (0) T

1

[12,0.4,0.2;0.4,10,0.6;0.2,0.6,11]

[0.03,0.04,-0.04]

[0,0,0]T

2

[14,0.2,0.4;0.2,12,0.4;0.4,0.4,10]

[0.04,0.02,0.04]T

[0,0,0]T

3

[13,0.4,0.4;0.4,10,0.4;0.4,0.4,9]

[-0.02,0.08,0.02]T

[0,0,0]T

4

[16,0.6,0.2;0.6,14,0.4;0.2,0.4,12]

[-0.05,0.04,-0.06]T

[0,0,0]T

The external disturbances are given as

 di  (i  3.5)[cos(0.1t ),sin(0.1t ),1]T 103 (Nm)

i  1,  , 4

4.1. The leader’s attitude derivative is constant For the control algorithm (11)-(14), the leader’s attitude derivative is set as

 0  [0.003, -0.001, 0.002]T and the leader’s initial attitude  0 (0) is chosen as  0  [-0.03, 0.06, -0.02]T . The upper bound of time derivative of communication delay T is chosen as d  0.3 . Apply Theorem 1 to calculate the maximal allowed upper bound of T and the result is T0  0.4641 . For satisfying the constraint of upper

bound,

the

time-varying

communication

delay

is

selected

as

T  [0.2  0.05cos(0.2t )]s . The distributed estimator and controller parameters are selected as   0.1 ,   1 , K i  50 I 3 , i  0.1I 6 , and  i  1 . The initial values of ˆ , and kˆ are all set to be zero. To avoid the chattering, we use tanh() to replace vˆi ,  i i

the function sgn() . Figs. 2 and 3 show the attitude tracking errors and the angular velocity tracking errors of each follower spacecraft, respectively. The angular velocity tracking errors are defined as  i   i   0 . It can be seen that the four follower spacecrafts can track the leader asymptotically, while the tracking errors can converge to the region |  i ( m ) | 3  103 and |  i ( m ) | 110 3 rad/ s (m  1, 2,3) after a short period, roughly in 50s. The control torques are shown in Fig. 4 and the maximum 17

values keep in the limitation of | ui ( m ) | 0.2Nm . The adaptive law (14) can provide the estimates of external disturbances so that the third term in (11) is always active. To compensate for external disturbances, the control torques will never be to zero. Fig. 5 shows the estimation errors vi . As we can see, the estimation errors can converge to zero in about 12s. Hence, the distributed estimator (12) can exactly estimate the leader’s attitude derivative  0 . 0.1 i =1

0.05

i =2

~i1 <

0

i =4

-3

4

-0.05

x 10

2

-0.1 -0.15 0 0.04

0 50 10

20

30

60

40

50

70 60

i =1

0.02

80 70

i =2

0

~ i2 <

i =3

90

100

80

90

i =3

100 i =4

-3

1

-0.02

x 10

0 -0.04 -0.06

-1 50 0

10

20

30

60

40

50 i =1

0.05

70 60

80 70

i =2

0

90

100

80

90

i =3

100 i =4

~ i3 <

-3

x 10

-0.05

2

-0.1

-2 50

0

0

10

20

30

40

60 50

70 60

80 70

90

100

80

90

100

t (s)

Figure 2 Attitude tracking errors of follower spacecrafts

18

0.02

! ~i1 (rad/s)

i =1

i =2

i=4

0 -3

1 -0.02

x 10

0 -1 50

-0.04

0

10

20

30

60

40

50 i =1

0.01

! ~ i2 (rad/s)

i =3

70 60

80 70

i =2

0

90

100

80

90

i =3

100 i=4

-4

5

-0.01

x 10

0 -0.02 -0.03

-5 50 0

10

20

30

60

40

50

70 60

80 70

90

100

80

90

100

0.04 i =1

! ~i3 (rad/s)

0.02

i =2

0

i =3

i=4

-4

-0.02

5

x 10

-0.04

0

-0.06

-5 50 0

10

20

30

60

40

50

70 60

80 70

90

100

80

90

100

t (s)

ui1 (Nm)

Figure 3 Angular velocity tracking errors of follower spacecrafts 0.1

i =1

i =2

i =3

i =4

0 -0.1 0

10

20

30

40

50

60

70

80

90

100

ui2 (Nm)

0.1 i =1

i =2

i =3

i =4

0 -0.1

0

10

20

30

40

50

60

70

80

90

100

ui3 (Nm)

0.2 i =1

i =2

i =3

i =4

0 -0.2 0

10

20

30

40

50

60

70

80

90

t (s)

Figure 4 Control torques of follower spacecrafts

19

100

-3

x 10

~vi1

0 i =1

i=2

i =3

i=4

-2 -4

0

10

20

30

40

50

60

70

80

90

100

-4

10

x 10

~vi2

i =1

i =2

i =3

i =4

5 0 0

10

20

30

40

50

60

70

80

90

100

-4

x 10

~vi3

0 i =1

i =2

i =3

i =4

-10 -20 0

10

20

30

40

50

60

70

80

90

100

t (s)

Figure 5 Estimation errors vi

4.2. The leader’s attitude derivative is time-varying For the control algorithm (49)-(51) and the observers (32)-(33), we consider

 0 (t )  [0.05 cos(0.1t ), 0.06 sin(0.1t ), -0.03cos(0.1t )]T , which can be generated by (30) with 0.05   0  S  diag{0.1a} , C   0.06 0    0 0.03

and v (0)  [0,1]T . According to Remark 5, the communication delay in subsection 4.1 is still suitable for this case. For simplicity, we choose the same communication delay as that in subsection 4.1. The observer and controller parameters are selected as ˆ , kˆ , vˆ , 1  1 ， 2  1 , K i  50 I 3 , i  0.1I 6 , and  i  1 . The initial values of  i i i

and ˆi are all set to be zero. Figs. 6 and 7 present the attitude tracking errors and the angular velocity tracking errors of each follower spacecraft, respectively. The four follower spacecrafts can track the leader, while the tracking errors can converge to the region |  i ( m ) | 5  103 and |  i ( m ) | 2  103 rad/ s ( m  1, 2,3) after a short period, roughly in 50s. Since the leader’s attitude derivative  0 is time-varying, the attitude accuracy is a little lower than that in subsection 4.1. The control torques can be seen in Fig. 8, which indicates that the control torques can stay in the limitation of 1Nm . To compensate for the external disturbances and track the dynamic leader, the control torques will never be to zero. Figs. 9 and 10 show the estimation errors vi and i . 20

The estimation errors can converge to zero in about 20s, which implies that the distributed observers can provide accurate estimates. These results verify that the designed control algorithms can achieve coordinated attitude tracking for spacecraft formation with the dynamic leader in the presence of time-varying communication delays. 0.1 i =1

i =2

i =3

i =4

0

~i1 <

-3

5 -0.1

x 10

0 -5 50

-0.2

0

10

20

30

60

40

50

70 60

80 70

90

100

80

90

100

0.1 i =1

0.05

i =2

~ i2 <

0

i =3

i =4

-3

x 10

5

-0.05

0 -0.1

-5 50 0

10

20

30

60

40

50 i =1

0.05

70 60

80 70

i =2

0

90

100

80

90

i =3

100 i =4

~ i3 <

-3

2

-0.05

x 10

0 -2 50

-0.1 0

10

20

30

40

60 50

70 60

80 70

90

100

80

90

100

t (s)

Figure 6 Attitude tracking errors of follower spacecrafts

21

i =1

! ~i1 (rad/s)

0.1

i =2

0

i =3

i=4

-3

2 -0.1

x 10

0

-0.2 0

10

20

30

-2 50 60 40 50

70 60

80 70

90 80

100 90

100

0.1 i =1

! ~ i2 (rad/s)

0.05

i =2

i =3

i=4

0 -3

-0.05

2

-0.1

0

-0.15

-2 50 0

10

20

30

60

40

0.05

! ~i3 (rad/s)

x 10

50 i =1

70 60

80 70

i =2

0

90

100

80

90

i =3

100 i=4

-3

1 -0.05

x 10

0 -1 50

-0.1 0

10

20

30

60

40

50

70 60

80 70

90

100

80

90

100

t (s)

Figure 7 Angular velocity tracking errors of follower spacecrafts

u i1 (Nm)

1 i =1

-1

0

10

20

30

40

u i2 (Nm)

0.5

i =3

i =4

50 i =1

60

70

i =2

80

90

i =3

100 i =4

0 -0.5 -1

ui3 (Nm)

i =2

0

0

10

20

30

40

50 i =1

0.5

60

70

i =2

80

90

i =3

100 i =4

0 -0.5 0

10

20

30

40

50

60

70

80

90

t (s)

Figure 8 Control torques of follower spacecrafts

22

100

0

~vi1

-0.05 -0.1 i =1

-0.15 0

10

20

30

40

i =2

50

60

i=3

70

80

i =4

90

100

0

v~i2

-0.2 -0.4 -0.6 -0.8 -1

i=1

0

10

20

30

40

i =2

50

60

i =3

70

80

i =4

90

100

Figure 9 Estimation errors vi 0 -0.01 -0.02 -0.03

2~i

-0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.1 0

i=1 10

20

30

40

i=2 50

60

i=3 70

80

i=4 90

100

t (s)

Figure 10 Estimation errors i

5.

Conclusion The distributed coordinated attitude tracking control problem for spacecraft

formation in the presence of time-varying communication delays is investigated under the directed communication topology. Two cases for the dynamic leader spacecraft are considered. In both cases the leader’s information is assumed to be available to only a subset of follower spacecrafts. In the first case where the leader’s attitude derivative is constant, a coordinated tracking control law with a distributed estimator is proposed via the adaptive technique to cope with parametric uncertainties and unknown disturbances. In the second case where the leader’s attitude derivative is time-varying, 23

to solve the problem of local communication, an improved distributed observer for each follower spacecraft is designed to estimate the leader’s attitude information, which only utilizes the neighbors’ information with communication delays. Based on the estimated information, an adaptive coordinated attitude tracking control strategy is proposed. By using Lyapunov-Krasovskii functional approach and the LMI approach, the proposed control algorithms are proven that the attitude of each follower spacecraft can synchronize to the attitude of the leader spacecraft asymptotically. The effectiveness of the proposed control algorithms is demonstrated by numerical simulations. The distributed coordinated attitude tracking control problem for spacecraft formation systems when considering input saturation will be studied as the further work. Acknowledgments This present work was supported by the National Natural Science Foundation of China under Grant nos. 61403103, 61304005, and 61273175. References [1] Chung SJ, Ahsun U, Slotine JJE. Application of synchronization to formation flying spacecraft: Lagrangian approach. J Guid Control Dyn 2009; 32(2): 512-26. [2] Du HB, Li SH, Qian CJ. Finite-time attitude tracking control of spacecraft with application to attitude synchronization. IEEE Trans Autom Control 2011; 56(11): 2711-7. [3] Liu X, Kumar KD. Network-based tracking control of spacecraft formation flying with communication delays. IEEE Trans Aerosp Electron Syst 2012; 48(3): 2302-14. [4] Ren W, Beard RW. Decentralized scheme for spacecraft formation flying via the virtual structure approach. J Guid Control Dyn 2004; 27(1): 73-82. [5] Hu QL, Dong HY, Zhang YM, Ma GF. Tracking control of spacecraft formation flying with collision avoidance. Aerosp Sci Technol 2015; 42(3): 353-64. [6] VanDyke MC, Hall CD. Decentralized coordinated attitude control within a formation of spacecraft. J Guid Control Dyn 2006; 29(5): 1101-9. [7] Lee D, Sanyal AK, Butcher EA. Asymptotic tracking control for spacecraft formation flying with decentralized collision avoidance. J Guid Control Dyn 2015; 38(4): 587-600. 24

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