Journal of Robotics, Networking and Artificial Life
Vol. 6(2); September (2019), pp. 89–96
DOI: https://doi.org/10.2991/jrnal.k.190828.004; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Mean-square Quasi-composite Rotating Formation Control
of Second-order Multi-agent Systems under Stochastic
Communication Noises
Lipo Mo1, Xiaolin Yuan1, Yingmin Jia2,*, Shaoyan Guo3
School of Science, Beijing Technology and Business University, Beijing 100048, China
The Seventh Research Division and the Center for Information and Control, School of Automation Science andElectrical Engineering, Beihang University (BUAA), 37 Xueyuan Road, Haidian District, Beijing 100191, China
3
Shien-Ming Wu School of Intelligent Engineering, South China University of Technology, Guangzhou 510641, China
1
2

ARTICLE INFO
Article History
Received 20 October 2018
Accepted 15 November 2018
Keywords
Quasi-composite
rotating formation
multi-agent systems
stochastic communication noises

ABSTRACTThis paper mainly concerns the mean-square quasi-composite rotating formation problem of second-order multi-agent systems(MASs) with stochastic communication noises. Firstly, the definition of mean-square quasi-composite rotating formation
is proposed. Afterwards, a distributed control protocol with time-varying control gains is designed. And then, the origin
closed-loop system is changed into an equivalent closed-loop system by taking coordinate transformation. Under some mild
assumptions, sufficient conditions are deduced. Eventually, a numerical simulation is provided to confirm the effectiveness ofthe proposed theoretical results.
© 2019 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
Over the past few decades, distributed coordination control ofMulti-Agent Systems (MASs) is a research focus and attracts lotsof attentions from many fields. Among them, consensus problem
is a fundamental problem, aiming at this problem, numerous ofgood results have been reported. A theoretical framework for analysis of consensus for MASs was built in Olfati-Saber et al. [1], andhad great influence. Besides, others of excellent results have beenreported, such as asynchronous consensus [2], leader following
consensus [3], finite-time consensus [4] and so on [5,6].
It is common to find that there always exist lots of collective motionsin nature and reality engineering, such as the distributed formationflight of satellites around the moon while the moon moving aroundthe earth, flocks of birds flying around a closed circle and so on.
Recently, to imitate or explain such kinds of motions, the collective
rotating motions of second-order MASs was investigated in Lin and
Jia [7], based on which, Lin et al. [8], made a further study on the
distributed composite-rotating consensus of second-order MASs.
Furthermore, rotating consensus [9], common finite-time rotating
encirclement control [10], and the influence of non-­uniform delays
on distributed rotating consensus [11] was investigated. Moreover,notice the fact that, groups of agents may require to maintain
desired formation during this process, therefore, the formation
control of MASs was investigated, such as Xiao et al. [12], Xue et al.
[13], and Meng et al. [14].
Corresponding author. Email: ymjia@buaa.edu.cn*

Notice that the above-mentioned works all assumed that the information exchange between different agents is ideal, that is, eachagent can obtain its neighbor’s information accurately. However,in reality, the communications among different agents may beinterfered by stochastic communication noise, which may lead togreat negative influence on the stability of the system. It is meaningful to design proper control protocol such that the stochastic
communication noises can be countered. The consensus control
problem of MASs with communication noises were solved in Li
and Zhang [15], Cheng et al. [16], Guo et al. [17], and Sun et al.
[18]. The mean-square composite-rotating consensus problem of
second-order MASs with communication noises was addressedby introducing a time-varying consensus gain in the control protocol [19]. Since the formation is considered and a time-varying
consensus gain is introduced into the control protocol to attenuate
the stochastic communication noises, though the origin closedloop system is changed into an equivalent closed-loop system bytaking a coordinate transformation, it is still difficult to obtain the
eigenvector of the matrix. Therefore, the methods proposed in Li
and Zhang [15], Cheng et al. [16], Guo et al. [17], Sun et al. [18],and Mo et al. [19] cannot be directly extended to achieve the quasi-­composite rotating formation control of MASs under stochastic
communication noises. Up-to-date, it is a pity to find that there is
no result on composite-rotating formation control of MASs withstochastic communication noises.
Motivated by the above discussions, the mean-square quasi-­composite rotating formation control of second-order MASs
with stochastic communication noises is considered in this paperL. Mo et al. / Journal of Robotics, Networking and Artificial Life 6(2) 89–96

90

as a response to the lack of corresponding researches, by using
which the actual physical system can be described more precisely.
The main contributions of this paper can be summarized as follows:(i) In contrast with Mo et al. [20], where the quasi-composite rotating formation control of MASs was solved without communicationnoises, the results of this paper are more realistic. (ii) The definitionof the mean square quasi-composite rotating formation of MASsis first proposed. (iii) A novel distributed control protocol witha time-varying control gain is designed, by using which the stochastic communication noises can be attenuated. (iv) By using thetheory of stochastic differential equation and proper coordinationtransformation, it is proved that under some mild conditions, thedesired formation can be achieved.
The rest of this paper is organized as follows. In Section 2, the theories of algebraic graph are introduced. Section 3 describes the systemmodels and the problem under investigation. In Section 4, the definition of mean-square quasi-composite rotating formation is given,besides a novel distributed control protocol and sufficient conditions
are deduced for achieving mean-square quasi-composite rotating
formation of MASs. A numerical simulation is given in Section 5
to verify the effectiveness of theoretical results we proposed.
Finally, a conclusion is made for this paper in Section 6. Notations:Throughout this paper, let ℝ and ℂ denote the sets of real numbers
and complex numbers; Let In ∈ ℝn×n (On ∈ ℝn×n) be the n × n identity(zero) matrix, 1n ∈ ℝn denote the n × 1 column vector of all ones; Let
diag{s1, s2, ..., sn} ∈ ℝn×n denotes the diagonal matrix with diagonal
entries s1, s2, ..., sn; j represents the imaginary unit; A* represents the
conjugate transpose of matrix A; For complex vector x ∈ ℂn, ||x||2 =x*x; tr(P) represents the trace of P; (·)T denotes the transpose of a realmatrix; (·)−1 denotes the inverse of a invertible matrix; lmax(·) denotes
the maximum eigenvalue of a real symmetric matrix.

2. PRELIMINARIES
Theories of algebraic graph [21]. Let (V, e, A) be an undirected
graph with n nodes, V = {1, 2, ..., n} represents the set of nodes,
and e ⊆ V × V represents the set of edges, A = [aij] represent theweighted adjacency matrix. Let Ni = {j ∈ V: (j, i) ∈ e} denotes theneighbors set of node i, ℒ = [lij] denotes the Laplacian of graph G.
Graph G is said to be connected if there exists a path between every
two distinct nodes.
Lemma 1. Ren and Beard [22]. If G is a connected undirected graph,then the Laplacian ℒ of G has a simple zero eigenvalue, and 1n is thecorresponding eigenvector. Additionally,
0 = l1 () £ l2 () £  £ ln ().
Lemma 2. Ren and Beard [22]. If G is a connected undirected graph,
there exists an orthogonal matrix U such that UT ℒU = diag{0,l2, ..., ln}, and the last column of U is 1n / n .

×			 vi (t ) = ui (t ), (2)where i = 1, 2, ..., n, ri(t) ∈ ℂ, vi(t) ∈ ℂ, ui(t) ∈ ℂ respectively denotes
the position, velocity and control input of the ith agent.

In this paper, our objective is to design a distributed control protocol by local information to assure that the effect of stochastic communication noises can be counteracted, and all agents surrounda moving point with a desired structure and a constant angularvelocity w1 > 0 while the moving point surrounds the origin with aconstant angular velocity w2 > 0.

4. MAIN RESULTS
In this section, the mean-square quasi-composite rotating formation control of MASs (1) and (2) is investigated.
w1(v (t ) + jw 2ri (t )), ci (t ) = ri (t ) + jw 1-1vi (t ), i = 1, 2, ..., nw1 + w 2 idenotes the relative velocity and the center of the rotation of the ith
agent respectively.
Let vi (t ) =Suppose that the information exchange among the agents is
affected by the stochastic communication noises. The information of the ith agent received from its neighbor agents is defined
as follows:æ Y (t )ö æ r (t ) + s rikhrik ö		 ç rik ÷ = ç k÷ , (3)è Yvikè vi (t ) + s vik
 (t )ø
 h vik ø
where k ∈ Ni, {hrik, h vik}, i, k = 1, 2, ..., n are independent standard
white noises, {srik, s vik} are the finite noise intensities.
Remark 1. When there exist no communication noises, i.e., hrik = 0
and h vik = 0, the problem considered herein degenerate into the quasi-­composite rotating formation problem which was considered in Mo
et al. [20]. Furthermore, when w2 = 0, it degenerates into the quasi-­rotating formation problem which was investigated in Lin and Jia [7].
When w2 = 0 and have no formation requirements for the agents,
the problem considered herein degenerate into the mean-square consensus problem which was investigated [8]. Generally speaking, theproblem considered in Lin and Jia [7], Guo et al. [17], and Mo et al.
[20] all can be seen as a special case of this paper.
The definition of the mean-square quasi-composite rotating formation is given as follows:
Definition 1. Let h = (h1, h2, ..., hn)T = (r1ejq , r2ejq , ..., rnejq )T ∈ ℂndenotes the desired formation, where ri > 0, qi ∈[0, 2p), i = 1, 2,..., n. The MASs (1) and (2) are said to achieve mean-square quasi-­composite rotating formation if for each i, k = 1, 2, ..., n.
1

2

n

2

lim E ci (t ) − ck (t ) = 0, (4)
		 
t →+∞

3. MODEL AND PROBLEM STATEMENT
Consider a multi-agent system with n agents, the dynamics of eachagent are given as follows [Equations (1) and (2)]:			 ri (t ) = vi (t ), (1)2v (t ) v (t )lim E i jqi - k jq k
		 
t ®+¥
rie
rk e

= 0, (5)2		 lim E vi (t ) - jw 1vi (t ) = 0, (6)
t ®+¥

2

		 lim E ci (t ) + jw 2ci (t ) = 0, (7)t ®+¥

L. Mo et al. / Journal of Robotics, Networking and Artificial Life 6(2) 89–96

and lim Var[ri (t )] < ∞, lim Var[vi (t )] < ∞ , for given w1 > 0 and
t →+∞

t →+∞

w2 > 0.

91

Then, we have×vi (t ) = vi (t ) + jw 2ci (t )= ui (t ) + jw 2[ri (t ) + jw 1-1vi (t )]= ui (t ) + jw 2[vi (t ) - jw 2ci (t )] - w 1-1w 2vi (t )= j(w 1 - w 2 )[vi (t ) - jw 2ci (t )]- w 1w 2[ci (t ) - jw 1-1vi (t )]- a(t ) å aik [ci (t ) - jw 1-1vi (t ) - ck (t )Remark 2. Here we make more explanations for the Definition 1.
Equation (4) means that the rotating centers of all agents reach consensus eventually in mean square. Equation (5) means that all agentsachieve quasi formation h eventually in mean square. Equations (6)and (7) respectively means that all agents and all rotating centerswere guaranteed to move in a circle with angular velocity w1 and w2
eventually in mean square. Therefore, when conditions (4)–(7) are
all satisfied, as time tends to infinity, all agents are guaranteed tosurround a moving point with a desired formation h and a constantangular velocity w1 > 0 while all moving rotating centers are guaranteed to surround the origin with a constant angular velocity w2 > 0
in mean square.

k ÎN i+ jw 1-1vk (t ) - s rikhrik + jw 1-1 (vi (t )- jw 2ci (t ) - vk (t ) + jw 2ck (t ) - s vik
 h vik )] (11)r j q -q
- a(t ) å aik [vi (t ) - jw 2ci (t ) - i e ( i k )rkk ÎN i´ (vk (t ) - jw 2ck (t ) + s vik
 h vik )rj q -q+ jw 2 (ci (t ) - jw 1-1vi (t ) - i e ( i k )rk´ (ck (t ) - jw 1-1v k (t ) + s rikhrik )]+ jw 2vi (t ) + w 22ci (t ) - w 1-1w 2vi (t ),In this paper, the following distributed control protocol is
proposed:ui (t ) = j(w 1 - w 2 )vi (t ) - w 1w 2ri (t )ù-a(t ) å aik éëri (t ) - Yrik (t ) + jw 1-1 (vi (t ) - Yvik
 (t ))û
k ÎN iri j (qi -q k )ùéYvik (t )
êvi (t ) - r eúkú,-a(t ) å aik êæöú
ri j (qi -q k )êk ÎN i
j
r
(
t
)
e
Y
(
t
)
+
w
rik
2ç i
ê
÷ø ú
rk
è
ë
û

	 

(8)

where a(t) > 0 is a time-varying consensus control gain.
To continue, we need the following assumptions.

which implicates thatvi (t ) = jw 1vi (t ) - a(t ) å aik éëci (t ) - ck (t )ùû
k ÎN i  

Assumption 1. The communication topology graph G is connected.
njq
Assumption 2. ∑rie i ≠ 0 when |qi – qk| = 0 or |qi – qk| = p, i, k =1, 2, ..., n.

i =1Furthermore, we have

Assumption 3. Li and Zhang [15]. Convergence condition:ci (t ) = − jw 2ci (t ) − jw 1−1 a(t ) ∑ aik [ci (t ) − ck (t )] − jw 1−1a(t )¥k ∈N iòa(s) ds = ¥ .
0éùr jq -q- a(t ) å aik êvi (t ) - i e ( i k )v k (t )ú
rkk ÎN iëûéù (12)æ
ri j (qi - q k ) öa(t )s rikhrik ú
aik êç 1 + jw 2 e+
å-1÷
1 + w 1 w 2 k ÎNi êëèrkøúûéæ -1 ri j (qi - q k ) ö
ù
a(t )
+
 h vik ú.
å aik ê jw1 + r e
÷ø s vik
1 + w 1-1w 2 k ÎNi êëçèúûkr j (q −q )
jw 1−1a(t )aik vi (t ) − i e i k v k (t ) +∑−1
rkk ∈N i 1 + w1 w 2raik  1 + jw 2 i e j (qi −q k )  s rikhrik 
∑
rk



k ∈N i −1 ri j (q −q ) 
jw 1−1a(t )aik  jw 1 + e i k  s vik+
 h vik ∑−1
1 + w 1 w 2 k ∈Ni 
rk



¥

2
Assumption 4. Li and Zhang [15]. Robust condition: òa (s)ds < ¥.
0Remark 3. For the purpose of suppressing the stochastic communication noises, the time-varying control gain a(t) is introduced into
the control protocol, i.e., making it more robust to the stochastic
communication noises. All agents are assured to achieve quasi-composite rotating formation motion with a not too fast rate by taking
Assumption 3, and the stochastic communication noises are againstby the control protocol by taking the Assumption 4.

w1(v (t ) + jw 2ri (t )) and ci (t ) = ri (t ) + jw 1-1vi (t ) ,w1 + w 2 i
we have [Equation (9)]From vi (t ) =vi (t ) = vi (t ) + jw 2ci (t ). (9)		 Furthermore, together (1) we have [Equation (10)]		 ri (t ) = vi (t ) = vi (t ) - jw 2ci (t ). (10)

  

(13)

Let d(t) = (v1(t), c1(t), ...,vn(t), cn(t))T, then the closed-loop system(9) and (10) can be rewritten as×  d (t ) = (I n Ä A + a(t )  Ä B1 + a(t )  Ä B2 )d (t ) + a(t )Wh ,(14)
−1 
0
 −1
0
0 
 jw
where A =  1, B1 = , B2 = 
−1
−1 ,


 0 − jw 2  − jw 1 0
 0 − jw 1  = F −1H H * F −1, F = diag{r , ..., r }, H = diag{ejq , ..., ejq },
1

(

)

T

2

n

1

n

h = h1T , ,hnT Î 2n , where hi = (sri1, s vi1, ..., srin, s vin)T ∈ ℂ2n,
Ω = diag{Ω , ..., Ω } ∈ ℂ2n×2n ,21nL. Mo et al. / Journal of Robotics, Networking and Artificial Life 6(2) 89–96

92

Therefore, Re(l0) ≤ 0. If l0 = jb, b ≠ 0 and V (t ) = 0 . By Lasalle’sæ æ r1 + jw 2 rie j (qi -q1 )ö
jw 1-1 r1 + rie j (qi -q1 ) ö
ç ç
÷
÷
r1
r1
ç ai1 ç÷j (q i -q1 )j (q i -q1 ) ÷-1-w 1 r1 + j rie÷ç ç j r1 - w 2 rie÷÷ø÷
w 1 r1w 1 r1w 1 ç çè÷,çWi =æ rn + jw 2 rie j (qi -qn )w1 + w 2 ç
jw 1-1 rn + rie j (qi -qn ) ö ÷
ç
÷÷
ç
rn
rn
÷
ç ,¼, ain çj (q i -q n )j (q i -q n ) ÷-1-w 1 rn + j rie
ç j rn - w 2 rie
÷÷
ç
çè
÷ø ÷
çè
w 1 rnw 1 rnø−1invariant theorem, we have x1 ∈ span{FH1n}, thus x1 = − jw 1 l0 x1,-1i.e., FH 1n = - jw 1 l0 FH 1n . Noted that zero is an eigenvalue of ℒ,
T

Lemma 4. Under Assumptions 1–4,  Ä B1 + a(t ) Ä B2 has zero
eigenvalue with algebraic multiplicity 2, and all other eigenvalues
have negative real part.
Proof. Since  Ä B1 + a(t ) Ä B2 is similar to Φ, from Lemma 3,
the conclusion is obvious.
Theorem 1. Suppose that Assumptions 1–4 are fulfilled. The desired
mean-square quasi-composite rotating formation control of MASs (1)and (2) can be achieved by taking the control protocol (8).

1algebraic multiplicity is two, all other eigenvalues of Φ have negative
real parts.

ææ
é0 ù ö
é1 ù ö
Proof. Let b1 = ç FH 1n Ä ê ú÷ , b 2 = ç 1n Ä ê ú÷ , it is clear that0èè
ë1 û ø
ë ûø

é FH 1 ùé0ù
2nProof. Denote b 1 = ê 0 n ú , b 2 = ê ú Î  , it is clear that
û
ë
ë1n û
Fb1 = 0 and Fb 2 = 0.

 jw 
    (I n ⊗ A +  ⊗ B1 +  ⊗ B2 )b1 = FH 1n ⊗  1  = jw 1 b1 , (23) 0 

Suppose Φ has a grade 2 generalized eigenvector associated with b1,z 
2ndenoted by  1  ∈ , such that F é z1 ù = é FH 1n ù, i.e.,êz ú ê 0 ú
z2 
ë 2û ëû( Ä B1 +  Ä B2 )b1 = 0, (24)0     (I n ⊗ A +  ⊗ B1 +  ⊗ B2 )b 2 = 1n ⊗ 
 = − jw 2 b 2 , (25) − jw 2 ( Ä B1 +  Ä B2 )b 2 = 0. (26)		      -z1 - z 2 = FH 1n , (15)−1−1		 − jw z − jw z = 0. (16)
1

1

jq

with Assumption 2. Therefore, we have Re(l0) < 0.

Lemma 3. Suppose that Assumptions 1 and 2 are fulfilled. For a − 
 matrix Φ = , zero is the eigenvalue of Φ and its−1 −1 − jw  − jw 

1

T

k =1

i = 1, 2, ..., n.

1nwe have 1n x1 = 0 , i.e., 1n FH 1n = årk e k = 0 , which is contradicted

2

Take invertible matrix U = [b1, b2, U ] ∈ ℂ2n×2n with U -1 = [g 1T , g 2T , V T ]T Î 
U = [g , g , V T ]T Î 2n ´ 2n such that		 z1 + z 2 = 0, (17)	 U -1 ( Ä B1 +  Ä B2 ) U = diag{0, 0, L}, (27)then (15) plus (17), we arrive at FH1n = 0, which is a contradiction.
Equation (16) lead to

-1

T
1

Similarly, Φ does not have two generalized eigenvectors associatedwith b 2 . Hence, zero is the eigenvalue of Φ with algebraic multiplicity two.

T2

where Λ ∈ ℂ(2n–2)×(2n–2) is a Jordan matrix. From Lemma 4, all eigenvalues of Λ have negative real parts.

Let z(t) = U−1d (t), z (t ) = (z 1 (t ), z 2 (t ), z T (t ))T , z (t ) = (z 3 (t ),¼,z 2n (t ))Tz (realt ) =part.
(z 3 (t ),¼,z 2n (t ))T . Then, we have
Next, let us prove that other eigenvalues of Φ have negative
Let l0 ≠ 0 be an arbitrary eigenvalue of Φ and  x1
its eigenvector associated to l0. Thus

TT

x2T  ∈2n is		 (x1 - jw 1-1)x1 = - l0 x1 . (20)z(t ) = U -1d (t )= U -1 (I n Ä A + a(t ) Ä B1 + a(t ) Ä B2 )Uz (t )
     
(28)
+ U -1a(t )Whæ jw 1ö00÷ z (t )= [ç 0 - jw 20ç÷0
jV (In Ä A)U + a(t )L ø
è 0
æg 1ö
+ ç g 2 ÷ a(t )Wh ,
ç ÷
èV ø

Consider the following auxiliary system:

where Ā = jA.

		 x = -( - jw 1-1)x , x Î n . (21)Then, the origin system can be decoupled into the following system:

Take V(t) = x*(t) x(t), along system (21), we have		 z1 (t ) = jw 1z 1 (t ) + g 1a(t )Wh , (29)V (t ) = -2 x * (t )  x(t ) £ 0. (22)	 z2 (t ) = - jw 2z 2 (t ) + g 2a(t ) Wh , (30)		 −x1 − x2 = l0 x1 , (18)		 −1 − jw x − jw −1x = l x , (19)
	 
1

1

1

2

0 2

−1

which leads to l0x1 = jw1 l0x2 and x2 = − jw 1 x1. Together with (18),−1
we have −x1 + jw 1 x1 = l0 x1 , i.e.,L. Mo et al. / Journal of Robotics, Networking and Artificial Life 6(2) 89–96

Together d (t) = Uz(t) with (32) and (33), we havez (t ) = éë jV (I n Ä A)U + a(t )L ùûz (t ) + Va(t )Wh . (31)lim E d (t ) - b1z 1 (t ) - b 2z 2 (t )

2

t ®+¥

In view of the theory of stochastic differential equation, the solution of (29)–(31) isé1 ù= lim E || d (t ) - FH 1n Ä ê ú (e jw1tz 1 (0)
t ®+¥
ë0 û
t
(37)
é1 ù
+ òe jw1tg 1a(t )WdW (t )) - 1n Ä ê ú (e - jw 2tz 2 (0)
0
ë û
0

t

z 1 (t ) = e jw1tz 1 (0) + òe jw1tg 1a(t )WdW (t ), (32)t0+ òe - jw 2tg 2a(t )WdW (t )) ||2 = 0.

tz 2 (t ) = e - jw 2tz 2 (0) + òe - jw 2tg 2a(t ) WdW (t ), (33)00

Let v(t) = (v1(t), v2(t),...,vn(t))T, c(t) = (c1(t), c2(t),...,cn(t))T, we can obtainttò[ jV ( In Ä A)U + a(t ) L ]dtz (t ) = e 0z (0)tò[ jV ( In Ä A)U + a( s ) L ]ds+ òe tVa(t ) WdW (t ),
t

93

jw t
jw t
2
that lim E ||v(t ) − FH 1n e 1 z 1 (0) − FH 1n ∫e 1 g 1a(t )ΩdW (t )|| = 0 ,
t →+∞

(34)

0

t

− jw t− jw t2
and lim E ||c(t ) − 1n e 2 z 2 (0) − 1n ∫e 2 g 2a(t )ΩdW (t )|| = 0 .
t →+∞

0

0

t

t

jw t
− jw t2
Since lim E || FH 1n ∫e 1 g 1a(t )ΩdW (t )|| = 0 , and lim E || 1n ∫e 2 g 2a(t )Ωdwhere W(t) = (Wr11(t), W v11(t),...,Wr1n(t), W v1n(t),...,Wrn1(t),t →+∞0
W vn1(t), ..., Wrnn(t), W vnn(t))T and Wrik(t), W vik(t), i, k = 1, 2, ..., n aret
− jw 2 t
2lim E || 1n ∫e g 2a(t )ΩdW (t )|| = 0, then we havestandard Brownian motions. Therefore, we have
t →+∞

t →+∞

0

lim E z (t )

2

t ®+¥

t

jV ( In Ä A)Ut + L [ a (t )]dt

ò
0

t ®+¥

2

lim E c(t ) - 1n e - jw 2tz 2 (0) = 0, (39)		 z (0) ||2t ®+¥éæ t t é jV ( In Ä A)U + a( s ) L ù dsöûòëê+ lim E{tr ç e t
Va(t )WdW (t )÷÷
t ®+¥
êçè ò
ø
ë 0
*
t
ü
ù
æ t ò éë jV ( In Ä A)U + a( s ) L ùû dsöúï´ ç òe t
Va(t )WdW (t )÷ ú ý
ç0
÷ úï
è
ø ûþ
t
ù
é
2 Real ( lmax ( L )) òa (t )dt£ lim E ê
2
ú
0
t ®+¥
û
ë|| z (0) || etùé t ò éë jV ( In Ä A)U + a( s ) L ùû ds
2
ú+ lim E ê || e t
	 
(35)
(
t
)
W
||
d
t
Va
t ®+¥ ê ò
ú
û
ë0
t
é
2 Real ( lmax ( L )) òa (t )dt ù£ lim E ê
2
ú
0
t ®+¥
û
ë|| z (0) || et+ lim E òa (t )et ®+¥which indicates that conditions (4)–(7) are satisfied, i.e., the desired
formation control of MASs (1) and (2) is achieved.
Remark 4. Different from Li and Zhang [15], Cheng et al. [16],Guo et al. [17], Sun et al. [18], and Mo et al. [19], where the meansquare consensus control of MASs was investigated, the mean-squarequasicomposite rotating formation control of MASs is investigatedin this paper. Since the formation is considered and a time-varying
consensus gain is introduced into the control protocol to attenuate
the stochastic communication noises, though the origin closed-loopsystem is changed into an equivalent closed-loop system by taking acoordinate transformation, it is still difficult to obtain the eigenvectorof the matrix. Therefore, the methods proposed in Li and Zhang [15],Cheng et al. [16], Guo et al. [17], Sun et al. [18], and Mo et al. [19]cannot be directly extended to solve the problem proposed herein.

t

2

ò

2 Real ( lmax ( L )) a ( s )ds
t

0

(

)

tr V WW*V * dt

5. A NUMERICAL EXAMPLEé
2 Real ( lmax ( L )) òa (t )dt ù£ lim E ê
2
ú
0
t ®+¥
||
z
(
0
)
|
|
e
û
ë
t

To demonstrate the effectiveness of the proposed theoreticalresults, consider MASs with six nodes. The communication topology graph of the MASs is shown in Figure 1, from which we could
see that the communication topology graph is connected such that
the Assumption 1 is satisfied. Without loss of generality, we assume
that the weight of each edge is 1. Then consider the MASs (1) and(2), where the control input in (2) is chosen as the distributed control protocol (8). The initial conditions of the system are taken astt+ lim E[(2n - 2)òa2 (t )e
t ®+¥

2

		 lim E v(t ) - FH 1n e jw1tz 1 (0) = 0, (38)t ®+¥£ lim E || e0

ò

2 Real ( lmax ( L )) a ( s )ds
t

dt ]

0

´ lmax (V WW*V * ).

Since all eigenvalues of Λ have negative real parts, this together Lemma 3t
2 Real ( lmax ( L )) òa (t )dt ùwith Assumptions 3, we have lim E éêt2
ú = 0. t0t ®+¥2 Real ( lmax ( L )) òa ( s )dsû
ë|| z (0) || er1(0)= −2.8t − 4.8j, r2(0) = 1.7 + 4.2j, r3(0) = −1.1 + 6.2j, r4(0) = 3
2
Furthermore, together with Assumption 4, we have lim E[(2n - 2)òa (t )e+ 5j, r (0) = −1 +dt6j,= r0 (0) = 3 + 5.3j, v (0) = 2.1, v (0) = 2.8, v (0)tE[(2n - 2)òa2 (t )e

t ®+¥

t

ò

2 Real ( lmax ( L )) a ( s )dst

dt = 0. Therefore, we have02			 lim E || z (t ) || = 0. (36)
t ®+¥

0

5

6

1

2

3

2
= −0.9, v4(0) = −2, v5(0) = −1, v6(0) = −2.2. Take a(t ) =,w =1+ t 1pi
0.8, w2 = 0.02, q i = , i = 1, 2, ..., 6 and choose r1 = 2, r2 = 0.5,
6
r3 = 1.5, r4 = 2.5, r5 = 1, r6 = 2.5.

94

L. Mo et al. / Journal of Robotics, Networking and Artificial Life 6(2) 89–96

Figure 1 | The communication topology graph.

Figure 4 | Whole trajectories of the position and moving rotating centersof the MASs.

Figure 2 | Position trajectories of the MASs.
Figure 5 | Local trajectories of the position and moving rotating centers of
the MASs.

Figure 3 | Trajectories of the moving rotating centers.

Figure 2 shows the position trajectories of the MASs, fromwhich we could see that all agents finally surround a common
moving point with a desired formation. Figure 3 shows the trajectories of the moving rotating centers, from which we could
see that the moving points finally surround the origin with aconstant angular velocity. We could see the whole (Figure 4) andlocal (Figure 5) trajectories of the position and moving rotating
centers of the MASs for more intuitive, which implicates that

Figure 6 | Trajectories of the control inputs of the MASs (the real part).

Theorem 1 is effective. Besides, it is clear from Figures 6 and 7 thatthe control input value of each agent tends to a bounded value
as time tends to infinity.

L. Mo et al. / Journal of Robotics, Networking and Artificial Life 6(2) 89–96

Figure 7 | Trajectories of the control inputs of the MASs (the imaginarypart).

6. CONCLUSION
The mean-square quasi-composite rotating formation problem of
second-order MASs with stochastic communication noises wasinvestigated in this paper. A novel distributed control protocol witha time-varying control gain was designed. And then, corresponding
sufficient conditions were deduced. Besides, the effectiveness of the
proposed theoretical results was confirmed via a numerical example.

ACKNOWLEDGMENTS
This work was supported by the Beijing Educational Com­­mittee Foundation [No. KM201910011007 and No. PXM2019_014213_000007], the Beijing Natural Science Foundation [No.
Z180005] and the Graduate Research Capacity Improvement
Program of 2019.

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96

Authors Introduction
Prof. Lipo Mo

Prof. Yingmin Jia

He received his B.S. degree in
Department
ofMathematics,
Shihezi University, China, in 2003,
and the PhD degree in School ofMathematics and Systems Sciencefrom the Beihang University, Beijing,in 2010. Currently, he is an Associate
Professor at Beijing Technology andBusiness University. His researchinterests include stochastic systems,
coordination control of multi-agent
systems, and distributed optimization.

He received his B.S. degree inControl Theory from Shandong
University in 1982, and his M.S. and
PhD degrees both in Control Theory
and Applications from Beihang
University (BUAA), Beijing, China,
in 1990 and 1993, respectively. In
1993, he joined the Seventh Research
Division at Beihang University,where he is currently a Professorof automatic control. His current
research interests include robust
control, adaptive control and intelligent control, and their applications in industrial processes and vehicle systems.

Mrs. Xiaolin Yuan

Dr. Shaoyan Guo
She received her B.S. degree in Schoolof Science, Beijing Technology andBusiness University, China, in 2017.
Currently, she is pursuing the Master
degree in control theory in School ofMathematics and Statistics, Beijing
Technology and Business University.
Her research interests include coordination control of Fractional ordermulti-agent systems.

He received his B.S. and M.S.
degree in school of Science, Beijing
Technology and Business University,
China, in 2015 and 2018. Currently,
he is a PhD candidate in control
science and engineering in ShienMing Wu School of Intelligent
Engineering, South China Universityof Technology. His research interests include coordination control ofmulti-agent systems.