Journal of Robotics, Networking and Artificial Life

Volume 6, Issue 2, September 2019, Pages 89 - 96

Mean-square Quasi-composite Rotating Formation Control of Second-order Multi-agent Systems under Stochastic Communication Noises

Authors
Lipo Mo1, Xiaolin Yuan1, Yingmin Jia2, *, Shaoyan Guo3
1School of Science, Beijing Technology and Business University, Beijing 100048, China
2The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang University (BUAA), 37 Xueyuan Road, Haidian District, Beijing 100191, China
3Shien-Ming Wu School of Intelligent Engineering, South China University of Technology, Guangzhou 510641, China
*Corresponding author. Email: ymjia@buaa.edu.cn
Corresponding Author
Yingmin Jia
Received 20 October 2018, Accepted 15 November 2018, Available Online 18 September 2019.
DOI
10.2991/jrnal.k.190828.004How to use a DOI?
Keywords
Quasi-composite; rotating formation; multi-agent systems; stochastic communication noises
Abstract

This 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 of the proposed theoretical results.

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
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 of Multi-Agent Systems (MASs) is a research focus and attracts lots of attentions from many fields. Among them, consensus problem is a fundamental problem, aiming at this problem, numerous of good results have been reported. A theoretical framework for analysis of consensus for MASs was built in Olfati-Saber et al. [1], and had great influence. Besides, others of excellent results have been reported, 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 motions in nature and reality engineering, such as the distributed formation flight of satellites around the moon while the moon moving around the 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].

Notice that the above-mentioned works all assumed that the information exchange between different agents is ideal, that is, each agent can obtain its neighbor’s information accurately. However, in reality, the communications among different agents may be interfered by stochastic communication noise, which may lead to great 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 addressed by 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 closed-loop system is changed into an equivalent closed-loop system by taking 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 with stochastic 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 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 communication noises, the results of this paper are more realistic. (ii) The definition of the mean square quasi-composite rotating formation of MASs is first proposed. (iii) A novel distributed control protocol with a time-varying control gain is designed, by using which the stochastic communication noises can be attenuated. (iv) By using the theory of stochastic differential equation and proper coordination transformation, it is proved that under some mild conditions, the desired 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 system models 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 real matrix; (·)−1 denotes the inverse of a invertible matrix; λmax(·) denotes the maximum eigenvalue of a real symmetric matrix.

2. PRELIMINARIES

Theories of algebraic graph [21]. Let (𝒱, ε, 𝒜) be an undirected graph with n nodes, 𝒱 = {1, 2, ..., n} represents the set of nodes, and ε𝒱 × 𝒱 represents the set of edges, 𝒜 = [aij] represent the weighted adjacency matrix. Let Ni = {j𝒱: (j, i) ∈ ε} denotes the neighbors set of node i, = [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 of 𝒢 has a simple zero eigenvalue, and 1n is the corresponding eigenvector. Additionally,

0=λ1()λ2()λn().

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, λ2,..., λn}, and the last column of U is 1n/n.

3. MODEL AND PROBLEM STATEMENT

Consider a multi-agent system with n agents, the dynamics of each agent are given as follows [Equations (1) and (2)]:

r˙i(t)=v˜i(t),
v˜˙i(t)=ui(t),
where i = 1, 2, ..., n, ri(t) ∈ ℂ, 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 surround a moving point with a desired structure and a constant angular velocity ω1 > 0 while the moving point surrounds the origin with a constant angular velocity ω2 > 0.

4. MAIN RESULTS

In this section, the mean-square quasi-composite rotating formation control of MASs (1) and (2) is investigated.

Let vi(t)=ω1ω1+ω2(v˜i(t)+jω2ri(t)), ci(t)=ri(t)+jω 11vi(t), i = 1, 2, ..., n denotes the relative velocity and the center of the rotation of the ith agent respectively.

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:

(Yrik(t)Yv˜ik(t))=(rk(t)+σrikηrikv˜i(t)+σv˜ikηv˜ik),
where kNi, {ηrik, ηṽik}, i, k = 1, 2, ..., n are independent standard white noises, {σrik, σṽik} are the finite noise intensities.

Remark 1.

When there exist no communication noises, i.e., ηrik = 0 and ηṽik = 0, the problem considered herein degenerate into the quasi-composite rotating formation problem which was considered in Mo et al. [20]. Furthermore, when ω2 = 0, it degenerates into the quasi-rotating formation problem which was investigated in Lin and Jia [7]. When ω2 = 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, the problem 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 = (ρ1e1, ρ2e2, ..., ρnen)T ∈ ℂn denotes the desired formation, where ρi > 0, θi ∈ [0, 2π), 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.

limt+E ci(t)ck(t) 2=0,
limt+E vi(t)ρiejθivk(t)ρkejθk 2=0,
limt+E v˙i(t)jω1vi(t) 2=0,
limt+E c˙i(t)+jω2ci(t) 2=0,
and limt+Var[ ri(t) ]<, limt+Var[ vi(t) ]<, for given ω1 > 0 and ω2 > 0.

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 agents achieve quasi formation h eventually in mean square. Equations (6) and (7) respectively means that all agents and all rotating centers were guaranteed to move in a circle with angular velocity ω1 and ω2 eventually in mean square. Therefore, when conditions (4)(7) are all satisfied, as time tends to infinity, all agents are guaranteed to surround a moving point with a desired formation h and a constant angular velocity ω1 > 0 while all moving rotating centers are guaranteed to surround the origin with a constant angular velocity ω2 > 0 in mean square.

In this paper, the following distributed control protocol is proposed:

ui(t)=j(ω1ω2)v˜i(t)ω1ω2ri(t)a(t)kNiaik[ ri(t)Yrik(t)+jω 11(v˜i(t)Yv˜ik(t)) ]a(t)kNiaik[ v˜i(t)ρiρkej(θiθk)Yv˜ik(t)+jω2(ri(t)ρiρkej(θiθk)Yrik(t)) ],
where a(t) > 0 is a time-varying consensus control gain.

To continue, we need the following assumptions.

Assumption 1.

The communication topology graph 𝒢 is connected.

Assumption 2.

i=1nρiejθi0 when |θiθk| = 0 or |θiθk| = π, i, k = 1, 2, ..., n.

Assumption 3.

Li and Zhang [15]. Convergence condition: 0a(s)ds=.

Assumption 4.

Li and Zhang [15]. Robust condition: 0a2(s)ds<.

Remark 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 against by the control protocol by taking the Assumption 4.

From vi(t)=ω1ω1+ω2(v˜i(t)+jω2ri(t)) and ci(t)=ri(t)+jω 11vi(t), we have [Equation (9)]

vi(t)=v˜i(t)+jω2ci(t).

Furthermore, together (1) we have [Equation (10)]

r˙i(t)=v˜i(t)=vi(t)jω2ci(t).

Then, we have

v˙i(t)=v˜˙i(t)+jω2c˙i(t)=ui(t)+jω2[ r˙i(t)+jω 11v˙i(t) ]=ui(t)+jω2[ vi(t)jω2ci(t) ]ω 11ω2v˙i(t)=j(ω1ω2)[ vi(t)jω2ci(t) ]ω1ω2[ ci(t)jω 11vi(t) ]a(t)kNiaik[ci(t)jω 11vi(t)ck(t)+jω 11vk(t)σrikηrik+jω 11(vi(t)jω2ci(t)vk(t)+jω2ck(t)σv˜ikηv˜ik)]a(t)kNiaik[vi(t)jω2ci(t)ρiρkej(θiθk)×(vk(t)jω2ck(t)+σv˜ikηv˜ik)+jω2(ci(t)jω 11vi(t)ρiρkej(θiθk)×(ck(t)jω 11vk(t)+σrikηrik)]+jω2vi(t)+ω 22ci(t)ω 11ω2v˙i(t),
which implicates that
v˙i(t)=jω1vi(t)a(t)kNiaik[ ci(t)ck(t) ]a(t)kNiaik[ vi(t)ρiρkej(θiθk)vk(t) ]+a(t)1+ω 11ω2kNiaik[ (1+jω2ρiρkej(θiθk))σrikηrik ]+a(t)1+ω 11ω2kNiaik[ (jω 11+ρiρkej(θiθk))σv˜ikηv˜ik ].

Furthermore, we have

c˙i(t)=jω2ci(t)jω 11a(t)kNiaik[ ci(t)ck(t) ]jω 11a(t)kNiaik[ vi(t)ρiρkej(θiθk)vk(t) ]+jω 11a(t)1+ω 11ω2kNiaik[ (1+jω2ρiρkej(θiθk))σrikηrik ]+jω 11a(t)1+ω 11ω2kNiaik[ (jω 11+ρiρkej(θiθk))σv˜ikηv˜ik ]
Let δ(t) = (v1(t), c1(t), ..., vn(t), cn(t))T, then the closed-loop system (9) and (10) can be rewritten as
δ˙(t)=(InA+a(t)˜B1+a(t)B2)δ(t)+a(t)Ωη,
where A=(jω100jω2), B1=(10jω 110), B2=(010jω 11), ˜=F1HH*F1, F = diag{ρ1, ..., ρn}, H = diag{e1, ..., en}, η=(η 1T,,η nT)T2n2, where ηi = (σri1, σṽi1, ..., σrin, σṽin)T ∈ ℂ2n, Ω = diag{Ω1, ..., Ωn} ∈ ℂ2n×2n2, Ωi=ω1ω1+ω2(ai1(ρ1+jω2ρiej(θiθ1)ρ1jω 11ρ1+ρiej(θiθ1)ρ1jρ1ω2ρiej(θiθ1)ω1ρ1ω 11ρ1+jρiej(θiθ1)ω1ρ1),,ain(ρn+jω2ρiej(θiθn)ρnjω 11ρn+ρiej(θiθn)ρnjρnω2ρiej(θiθn)ω1ρnω 11ρn+jρiej(θiθn)ω1ρn)), i = 1, 2, ..., n.

Lemma 3.

Suppose that Assumptions 1 and 2 are fulfilled. For a matrix Φ=(˜jω 11˜jω 11), zero is the eigenvalue of Φ and its algebraic multiplicity is two, all other eigenvalues of Φ have negative real parts.

Proof.

Denote β¯1=[ FH1n0 ], β¯2=[ 01n ]2n, it is clear that Φβ¯1=0 and Φβ¯2=0.

Suppose Φ has a grade 2 generalized eigenvector associated with β¯1, denoted by [ z1z2 ]2n, such that Φ[ z1z2 ]=[ FH1n0 ], i.e.,

˜z1z2=FH1n,
jω 11˜z1jω 11z2=0.
Equation (16) lead to
˜z1+z2=0,
then (15) plus (17), we arrive at FH1n = 0, which is a contradiction.

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

Next, let us prove that other eigenvalues of Φ have negative real part. Let λ0 ≠ 0 be an arbitrary eigenvalue of Φ and [ x 1Tx 2T ]T2n is its eigenvector associated to λ0. Thus

˜x1x2=λ0x1,
jω 11˜x1jω 11x2=λ0x2,
which leads to λ0x1 = 1 λ0x2 and x2=jω 11x1. Together with (18), we have ˜x1+jω 11x1=λ0x1, i.e.,
(˜x1jω 11)x1=λ0x1.

Consider the following auxiliary system:

x˙=(˜jω 11)x,xn.

Take 𝒱(t) = x*(t) x(t), along system (21), we have

V˙(t)=2x*(t)˜x(t)0.

Therefore, Re(λ0) ≤ 0. If λ0 = jb, b ≠ 0 and V˙(t)=0. By Lasalle’s invariant theorem, we have x1 ∈ span{FH1n}, thus x1=jω 11λ0x1, i.e., FH1n=jω 11λ0FH1n. Noted that zero is an eigenvalue of , we have 1 nTx1=0, i.e., 1 nTFH1n=k=1nρkejθk=0, which is contradicted with Assumption 2. Therefore, we have Re(λ0) < 0.

Lemma 4.

Under Assumptions 14, ˜B1+a(t)B2 has zero eigenvalue with algebraic multiplicity 2, and all other eigenvalues have negative real part.

Proof.

Since ˜B1+a(t)B2 is similar to Φ, from Lemma 3, the conclusion is obvious.

Theorem 1.

Suppose that Assumptions 14 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).

Proof.

Let β1=(FH1n[ 10 ]), β2=(1n[ 01 ]), it is clear that

(InA+˜B1+B2)β1=FH1n[ jω10 ]=jω1β1,
(˜B1+B2)β1=0,
(InA+˜B1+B2)β2=1n[ 0jω2 ]=jω2β2,
(˜B1+B2)β2=0.

Take invertible matrix U = [β1, β2, Ū] ∈ ℂ2n×2n with U1=[ γ 1T,γ 2T,V¯T ]T2n×2n such that

U1(˜B1+B2)U=diag{ 0,0,Λ },
where Λ ∈ ℂ(2n–2)×(2n–2) is a Jordan matrix. From Lemma 4, all eigenvalues of Λ have negative real parts.

Let ζ(t) = U−1δ(t), ζ(t)=(ζ1(t),ζ2(t),ζ¯T(t))T, ζ¯(t)=(ζ3(t),,ζ2n(t))T. Then, we have

ζ˙(t)=U1δ˙(t)=U1(InA+a(t)˜B1+a(t)B2)Uζ(t)+U1a(t)Ωη=[(jω1000jω2000jV¯(InA¯)U¯+a(t)Λ)ζ(t)+(γ1γ2V¯)a(t)Ωη,
where Ā = jA.

Then, the origin system can be decoupled into the following system:

ζ˙1(t)=jω1ζ1(t)+γ1a(t)Ωη,
ζ˙2(t)=jω2ζ2(t)+γ2a(t)Ωη,
ζ¯˙(t)=[ jV¯(InA¯)U¯+a(t)Λ ]ζ¯(t)+V¯a(t)Ωη.

In view of the theory of stochastic differential equation, the solution of (29)(31) is

ζ1(t)=ejω1tζ1(0)+0tejω1tγ1a(τ)ΩdW(τ),
ζ2(t)=ejω2tζ2(0)+0tejω2tγ2a(τ)ΩdW(τ),
ζ¯(t)=e0t[ jV¯(InA¯)U¯+a(τ)Λ ]dτζ¯(0)+0teτt[ jV¯(InA¯)U¯+a(s)Λ ]dsV¯a(τ)ΩdW(τ),
where W(t) = (Wr11(t), W11(t),...,Wr1n(t), W1n(t),...,Wrn1(t), Wṽn1(t), ..., Wrnn(t), Wṽnn(t))T and Wrik(t), Wṽik(t), i, k = 1, 2, ..., n are standard Brownian motions. Therefore, we have
limt+Eζ¯(t)2limt+E||ejV¯(InA¯)U¯t+Λ0t[ a(τ) ]dτζ¯(0)||2+limt+E{tr [ (0teτt[ jV¯(InA¯)U¯+a(s)Λ ]dsV¯a(τ)ΩdW(τ)) ×(0teτt[ jV¯(InA¯)U¯+a(s)Λ ]dsV¯a(τ)ΩdW(τ))* ] }limt+E[ ζ¯(0) 2e2Real(λmax(Λ))0ta(τ)dτ ]+limt+E[ 0t||eτt[ jV¯(InA¯)U¯+a(s)Λ ]dsV¯a(τ)Ω||2dτ ]limt+E[ ζ¯(0) 2e2Real(λmax(Λ))0ta(τ)dτ ]+limt+E0ta2(τ)e2Real(λmax(Λ))τta(s)dstr(V¯ΩΩ*V¯*)dτlimt+E[ ζ¯(0) 2e2Real(λmax(Λ))0ta(τ)dτ ]+limt+E[(2n2)0ta2(τ)e2Real(λmax(Λ))τta(s)dsdτ]×λmax(V¯ΩΩ*V¯*).

Since all eigenvalues of Λ have negative real parts, this together Lemma 3 with Assumptions 3, we have limt+E[ ζ¯(0) 2e2Real(λmax(Λ))0ta(τ)dτ ]=0. Furthermore, together with Assumption 4, we have limt+E[(2n2)0ta2(τ)e2Real(λmax(Λ))τta(s)dsdτ=0. Therefore, we have

limt+E ζ¯(t) 2=0.

Together δ(t) = (t) with (32) and (33), we have

limt+E δ(t)β1ζ1(t)β2ζ2(t) 2=limt+E||δ(t)FH1n[ 10 ](ejω1tζ1(0)+0tejω1tγ1a(τ)ΩdW(τ))1n[ 10 ](ejω2tζ2(0)+0tejω2tγ2a(τ)ΩdW(τ))||2=0.

Let v(t) = (v1(t), v2(t),...,vn(t))T, c(t) = (c1(t), c2(t),...,cn(t))T, we can obtain that limt+E||v(t)FH1nejω1tζ1(0)FH1n0tejω1tγ1a(τ)ΩdW(τ)||2=0, and limt+E||c(t)1nejω2tζ2(0)1n0tejω2tγ2a(τ)ΩdW(τ)||2=0.

Since limt+E||FH1n0tejω1tγ1a(τ)ΩdW(τ)||2=0, and limt+E||1n0tejω2tγ2a(τ)ΩdW(τ)||2=0, then we have

limt+E||v(t)FH1nejω1tζ1(0)||2=0,
limt+E||c(t)1nejω2tζ2(0)||2=0,
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 mean square consensus control of MASs was investigated, the mean-square quasicomposite rotating formation control of MASs is investigated in 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-loop system is changed into an equivalent closed-loop system by taking 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 solve the problem proposed herein.

5. A NUMERICAL EXAMPLE

To demonstrate the effectiveness of the proposed theoretical results, 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 as r1(0) = −2.8 – 4.8j, r2(0) = 1.7 + 4.2j, r3(0) = −1.1 + 6.2j, r4(0) = 3 + 5j, r5(0) = −1 + 6j, r6(0) = 3 + 5.3j, 1(0) = 2.1, 2(0) = 2.8, 3(0) = −0.9, 4(0) = −2, 5(0) = −1, 6(0) = −2.2. Take a(t)=21+t, ω1 = 0.8, ω2 = 0.02, θi=πi6, i = 1, 2, ..., 6 and choose ρ1 = 2, ρ2 = 0.5, ρ3 = 1.5, ρ4 = 2.5, ρ5 = 1, ρ6 = 2.5.

Figure 1

The communication topology graph.

Figure 2 shows the position trajectories of the MASs, from which 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 a constant angular velocity. We could see the whole (Figure 4) and local (Figure 5) trajectories of the position and moving rotating centers of the MASs for more intuitive, which implicates that Theorem 1 is effective. Besides, it is clear from Figures 6 and 7 that the control input value of each agent tends to a bounded value as time tends to infinity.

Figure 2

Position trajectories of the MASs.

Figure 3

Trajectories of the moving rotating centers.

Figure 4

Whole trajectories of the position and moving rotating centers of the MASs.

Figure 5

Local trajectories of the position and moving rotating centers of the MASs.

Figure 6

Trajectories of the control inputs of the MASs (the real part).

Figure 7

Trajectories of the control inputs of the MASs (the imaginary part).

6. CONCLUSION

The mean-square quasi-composite rotating formation problem of second-order MASs with stochastic communication noises was investigated in this paper. A novel distributed control protocol with a 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.

Authors Introduction

Prof. Lipo Mo

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

Mrs. Xiaolin Yuan

She received her B.S. degree in School of Science, Beijing Technology and Business University, China, in 2017. Currently, she is pursuing the Master degree in control theory in School of Mathematics and Statistics, Beijing Technology and Business University. Her research interests include coordination control of Fractional order multi-agent systems.

Prof. Yingmin Jia

He received his B.S. degree in Control 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 Professor of automatic control. His current research interests include robust control, adaptive control and intelligent control, and their applications in industrial processes and vehicle systems.

Dr. Shaoyan Guo

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 Shien-Ming Wu School of Intelligent Engineering, South China University of Technology. His research interests include coordination control of multi-agent systems

REFERENCES

[10]L Mo and Y Yu, Finite-time rotating encirclement control of multi-agent systems, Acta Automat. Sin, Vol. 43, 2017, pp. 1665-1672.
[21]C Godsil and G Royle, Algebraic graph theory, Springer-Verlag, New York, 2001.
Journal
Journal of Robotics, Networking and Artificial Life
Volume-Issue
6 - 2
Pages
89 - 96
Publication Date
2019/09/18
ISSN (Online)
2352-6386
ISSN (Print)
2405-9021
DOI
10.2991/jrnal.k.190828.004How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Lipo Mo
AU  - Xiaolin Yuan
AU  - Yingmin Jia
AU  - Shaoyan Guo
PY  - 2019
DA  - 2019/09/18
TI  - Mean-square Quasi-composite Rotating Formation Control of Second-order Multi-agent Systems under Stochastic Communication Noises
JO  - Journal of Robotics, Networking and Artificial Life
SP  - 89
EP  - 96
VL  - 6
IS  - 2
SN  - 2352-6386
UR  - https://doi.org/10.2991/jrnal.k.190828.004
DO  - 10.2991/jrnal.k.190828.004
ID  - Mo2019
ER  -