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Dynamical group consensus of heterogenous multi-agent systems with input time delays
Author’s Accepted Manuscript Dynamical Group Consensus of Heterogenous Multi-Agent Systems with Input Time Delays Guoguang Wen, Yongguang Yu, Zhaoxia Peng, Hu Wang www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)01519-2 http://dx.doi.org/10.1016/j.neucom.2015.10.060 NEUCOM16238
To appear in: Neurocomputing Received date: 7 July 2015 Revised date: 19 August 2015 Accepted date: 20 October 2015 Cite this article as: Guoguang Wen, Yongguang Yu, Zhaoxia Peng and Hu Wang, Dynamical Group Consensus of Heterogenous Multi-Agent Systems with Input Time Delays, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.10.060 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 galley proof before it is published in its final citable 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.
Dynamical Group Consensus of Heterogenous Multi-Agent Systems with Input Time Delays Guoguang Wen a , Yongguang Yu a , Zhaoxia Peng b,∗, Hu Wangc a Department
of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China of Transportation Science and Engineering, Beihang University, Beijing 100191, P.R.China c School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, 100081, China b School
Abstract This paper investigates the dynamics group consensus problem of heterogenous multi-agent systems with time delays, in which agents’ dynamics are modeled by single integrators and double integrators. To achieve group consensus, a class of dynamics group consensus protocols is proposed for heterogenous multi-agent systems with input time delays, which can also be used to solve group consensus for heterogenous multi-agent systems without input time delays. By using frequently-domain analysis method and matrix theory, some sufficient group consensus conditions, which are dependent on the input delays and the control parameters, are obtained for heterogenous multi-agent systems under directed and undirected communication topologies with and without input time delays, respectively. Simulation results are also provided to illustrate the effectiveness of the obtained results. Keywords: Group Consensus, Heterogenous Multi-Agent Systems, Cooperative control, Time delay, Graph theory.
1. Introduction Cooperative control of multi-agent systems has been extensively studied in last decade. Consensus problems, as a branch of cooperative control, has received considerable attention recently in many fields, such as biology, physics, robotics and control engineering[1–6]. Consensus of multi-agent systems means to design some appropriate protocols and controllable strategies via local interaction such that all the agents reach an agreement on their common interest quantity. The interest quantity might represent attitude, position, velocity, temperature, voltage, and so on. In the literature, there are many interesting research results on consensus problems for multi-agent systems, whose agents with first-order dynamics [7–11], second-order dynamics [12–14], high-order dynamics [15–17] and general linear dynamics [18]. Note that almost all papers concern the complete consensus that the states of all agents converge to a common consistent state. However, in practice a phenomena frequently occurs that agreements are different with the change of environments, situations, cooperative tasks or even time. Consequently, a critical problem is to design appropriate protocols such that agents in systems can reach more than on consistent states. This is called group (or cluster) consensus problem in multi-agent systems, in which the agents in a system are divided into multiple subgroups and different subgroups can reach different consistent states. Recently, great deals of excellent research results on group consensus have emerged constantly. In Ref. [19] Yu et al. solved the group consensus problem for multi-agent systems with fixed undirected topology. Sequently, in Ref. [20] Yu et al. extended their results in Ref. [19] to multi-agent systems with switching topologies and communication delays by using the method of double-tree-form. In [21], the authors proposed a hybrid protocol to solve the couple-group average-consensus problem of multi-agent systems with a fixed topology. In [22], Feng et al. studied the static and time-varying group consensus problems of second-order multi-agent systems. In [23], Liao et al. investigated group consensus of dynamical multi-agent networks via pinning scheme. ∗ Corresponding
author: Zhaoxia Peng Email address: [email protected] (Zhaoxia Peng)
Preprint submitted to Neurocomputing
October 27, 2015
Most of aforementioned works on consensus problems are on the homogenous multi-agent systems, in which all the agents have the same dynamics. In practical applications, the dynamics of agents may be different because of common goals of mixed agents or various restrictions of communication costs[24]. Recently, more and more attention has been paid to the consensus problems for heterogenous multi-agent systems, where agents have different dynamics. In [25], Zheng and Wang studied the consensus problem of the heterogeneous multi-agent systems consisted of first-order integrator agents and second-order integrator agents, in which a linear consensus protocol and a saturated consensus protocol were proposed. In [26], Zhu et al. studied the finite time consensus problems for heterogeneous multi-agent systems. In [27], the consensus problem of heterogeneous multi-agent systems consisted of first-order agents, second-order agents and Euler-Lagrange agents. In [28], the consensus seeking problem of heterogeneous multi-agent systems consisted of first-order agents and second-order agents with input delays was investigated. In [29], Liu et. al investigated second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays by introducing novel decentralized adaptive strategies to both the coupling strengths and the feedback gains. Here, it is worth noting that the almost all of the aforementioned works concern consensus/group consensus problems for homogeneous multi-agent systems or consensus problems for heterogeneous multi-agent systems. However, there are few results on the group consensus of heterogeneous multi-agent systems consisting of first-order and secondorder agents. Moreover, to the best of our knowledge, there is no paper concerning group consensus of heterogeneous multi-agent systems with time delays. Motivated by the above discussion, this paper investigates the dynamics group consensus problem of heterogenous multi-agent systems with time delays. The main contributions of this paper can be stated as follows: A class of distributed group consensus protocols is proposed for heterogenous multi-agent systems with input time delays, which also can be used to solve group consensus for heterogenous multi-agent systems without input time delays. By using frequently-domain analysis method and matrix theory, some sufficient group consensus conditions, which are dependent on the input delays and the control parameters, are obtained for heterogenous multi-agent systems under directed and undirected communication topologies with and without time delays, respectively. Simulation results are also provided to illustrate the effectiveness of the obtained results. Compared with the existing references, this paper has the following advantages: Firstly, in contrast of consensus/group consensus for homogeneous multi-agent systems [10–15, 17–23] , we investigate the group consensus problems for heterogenous multi-agent systems. Secondly, in contrast to the existing results in [25, 26, 28, 29], where consensus problems were considered for heterogenous multi-agent systems only under undirected communication topologies, in this paper we consider the group consensus under directed and undirected communication topologies. Thirdly, in contrast to the existing results in [23–26], where consensus/ group consensus problems were investigated without considering time delays, in this paper, group consensus problems are investigated with and without considering time delays, respectively. The rest of this paper is organized as follows. In Section 2, some preliminaries on graph theory are presented. In section 3, the problem description is given. In section 4, the main results on dynamics group consensus problem of heterogenous multi-agent systems with time delays are presented. In section 5, numerical examples are simulated. Conclusions are finally drawn in section 6. 2. Preliminaries 2.1. Algebraic graph theory Algebraic graph theory is a natural framework to study cooperative control problems of multi-agent systems. Using algebraic graph theory, the information exchanged among agents can be modeled by an interaction graph. Let G = (V, E, A) be a weighted directed or undirected graph of order n with the finite nonempty set of nodes V = {v 1 , ..., vn}, the set of edges E ⊆ V × V, and a weighted adjacency matrix A = (a i j )n×n . Here, each node v i in V corresponds to an agent i, and each edge (v i , v j ) ∈ E in a weighted directed graph corresponds to an information link from agent j to agent i, which means that agent i can receive information from agent j. In contrast, the pairs of nodes in weighted undirected graph are unordered, where an edge (v j , vi ) ∈ E denotes that agent i and j can receive information from each other. The weighted adjacency matrix A of a weighted directed graph is defined such that a ii = 0 for all vi ∈ V, ai j 0 if (v j , vi ) ∈ E, and ai j = 0 otherwise. The weighted adjacency matrix A of a weighted undirected graph is defined analogously except that a i j = a ji , ∀i j, since (vi , v j ) ∈ E implies (v j , vi ) ∈ E. We can say vi is a neighbor vertex of v j , 2
if (vi , v j ) ∈ E. A directed path is a sequence of ordered edges of the form (v i1 , vi2 ), vi2 , vi3 ,...,where (v i j , vi j+1 ) ∈ E. If there is a path in G from one node v i to another node v j , then v j is said to be reachable from v i . If not, then v j is said to be not reachable from v i . If there is a node that is reachable from every other node in the digraph, then we say the node is global reachable. L = (li j )n×n is the Laplacian matrix of graph G, and is defined by l i j = −ai j for i j, and lii = nj=1, ji ai j , i, j ∈ {1, ..., n}. For an undirected graph, L is symmetric positive semi-definite. However, L is not necessarily symmetric for a directed graph. Lemma 2.1. [30] 0 is a simple eigenvalue of the Laplacian matrix L, and 1 n = [1, ..., 1]T is the corresponding right eigenvector, that is, L1 n = 0, if and only if the digraph G = (V, E, A) has a globally reachable node. 2.2. Notations We use standard notations throughout this paper. I n represents the n × n identity matrix, 0 m×n represents the m × n zero matrix, and 1 n = [1, 1, . . . , 1]T ∈ Rn (1 for short, when there is no confusion). λ min (A) and λmax (A) are the smallest and the largest eigenvalues of the matrix A respectively. ρ(·) , det(·) represent the spectral radius,determinant of a matrix, respectively. R, C and Z + denote the sets of real numbers, complex numbers and positive integers. For s ∈ C, |s|, Re(s) and Im(s) denote its modulus, real and imaginary part, respectively. arg(·) denotes the phase of a complex number. 3. Problem description In this paper, we consider the group consensus problems under input time delays for heterogeneous multi-agent systems composed of double integrator agents and single integrator agents. Without loss of generality, we suppose there are m double integrator agents described by x˙i (t) = vi (t) v˙ i (t) = ui (t − τ),
i = 1, . . . , m,
(1)
i = m + 1, . . . , n,
(2)
and n − m single integrator agents described by x˙i (t) = ui (t − τ),
where xi ∈ R p , vi ∈ R p and ui ∈ R p are the position, velocity and control input of agent i, respectively. For the simplicity of analysis, in the following, we only consider the case where p = 1. The analysis and main results still hold for any dimension p by using Kronecker product. Group consensus requires that the first m agents reach one consistent state while the last n − m agents reach another consistent state in the presence of information exchange between the two groups. Denote 1 = {1, 2, ..., m}, 2 = {m + 1, m + 2, ..., n}. Definition 1. The heterogeneous multi-agent system (1)-(2) is said to be reach group consensus asymptotically if for any initial state values x i (0) and v i (0) , we have lim xi (t) − x j (t) = 0, lim vi (t) − v j (t) = 0, ∀i, j ∈ k , k = 1, 2. (3) t→∞ t→∞ Now the following necessary assumptions are made. Assumption 3.1.
n j=m+1
ai j = 0, ∀i ∈ 1 , and
m j=1
ai j = 0 ,∀i ∈ 2 .
Remark 3.2. As in [19, 31], Assumeption 3.1 means a balance of effect between two subsystems. Lemma 3.3. Under Assumption 3.1, L has a zero eigenvalue with the geometric multiplicity of at least two. 3
Consider the following time-delay system: y˙ (t) =
N
Ai y(t − τi )
(4)
i=1
where y(t) ∈ Rn , Ai ∈ Rn×n , τi ∈ R, and N is a positive integer. Applying the Laplace transform to system (4) , the characteristic equation can be obtained as det(sI −
N
Ai e−τi s ) = 0
(5)
i=1
Lemma 3.4. [28] If the roots of equation (5) have negative real parts except for two roots at s = 0, then limt→∞ y(t) = α + βt where α ∈ Rn and β ∈ Rn are two constant vectors. Lemma 3.5. If m > 0, then y/(1 + y 2 ) < arctan(y) + m holds for y > 0; If − π2 < m < 0, then exists y0 > 0, such that y/(1 + y2 ) < arctan(y) + m holds for y > y 0 . Proof: Let q(y) = y/(1 + y 2 ) − arctan(y) − m. Then we can get the derivative of q(y) on the y as q(y) ˙ = −2y 2 /[(1 + y2 )]2 . Obviously, q(y) is monotonously decreasing for y > 0. If m > 0, then q(0) = −m < 0. It implies that y/(1 + y 2 ) < arctan(y) + m holds for y > 0. If 0 > m > − π2 , then q(0) = −m > 0. Also note that y/(1 + y 2 ) − arctan(y) ∈ [− π2 , 0], so there exists y0 > 0 such that q(y 0 ) = y0 /(1 + y20 ) − arctan(y0 ) − m = 0, It implies that y/(1 + y 2 ) < arctan(y) + m holds for y > y0 . The proof is completed. Lemma 3.6. Let h(λ) = (arctan(δμ(λ)) + m)/μ(λ), where δ > 0, and μ(λ) > 0 is monotonously increasing for λ > 0 If m > 0, then h(λ) is monotonously decreasing for λ > 0 If m < 0 and m > − π2 , then exists λ0 > 0, then h(λ) is monotonously decreasing for λ > λ 0 . Proof: Calculating the derivative of h(λ) on λ yields μ(λ) ˙ δμ(λ) ˙ − arctan(δμ(λ)) − m). h(λ) = 2 ( μ (λ) 1 + (δμ(λ))2 Because μ(λ) > 0 and μ(λ) ˙ > 0, if m > 0, we obtain that h(λ) < 0 holds with λ > 0 from Lemma 3.5. If m < 0 and m > − π2 , based on Lemma 3.5, we obtain that h(λ) is monotonously decreasing for λ > λ 0 . Lemma 3.6 is proved. 4. Main results To solve the group consensus of heterogeneous multi-agent system (1)-(2), the following group consensus protocols are proposed, ai j (x j (t − τ) − xi (t − τ)) + k ai j x j (t − τ) + γ ai j ( x˙ j (t − τ) − x˙i (t − τ)) + γ ai j x˙ j (t − τ), ui (t − τ) = k j∈1
j∈2
∀i ∈ 1
j∈1
j∈2
(6) and ui (t − τ) = vi (t − τ) + γ ai j (x j (t − τ) − xi (t − τ)) + γ ai j x j (t − τ), j∈2 j∈1 v˙ i (t) = k ai j (x j (t) − xi (t)) + k ai j x j (t), ∀i ∈ 2 , j∈2
j∈1
where k > 0 and γ > 0 are the control parameters. 4
∀i ∈ 2 , (7)
With the consensus protocols (6) and (7), multi-agent systems (1) and (2) can be written as x˙i (t) = vi (t) ai j (x j (t − τ) − xi (t − τ)) + k ai j x j (t − τ) + γ ai j ( x˙ j (t − τ) − x˙i (t − τ)) + γ ai j x˙ j (t − τ), v˙ i (t) = k j∈1
and
j∈2
j∈1
j∈2
∀i ∈ 1
ai j (x j (t − τ) − xi (t − τ)) + γ ai j x j (t − τ), x˙i (t) = vi (t − τ) + γ j∈2 j∈1 v˙ i (t) = k ai j (x j (t) − xi (t)) + k ai j x j (t), ∀i ∈ 2 . j∈2
(8)
∀i ∈ 2 , (9)
j∈1
Theorem 4.1. The heterogeneous multi-agent systems (8) and (9) with a fixed directed communication topology can reach the group consensus asymptotically if and only if L only has exactly two simple zero eigenvalues, the rest of the eigenvalues of L have positive real parts, and |λi | k2 + ω2ci γ2 <1 ω2ci holds for every nonzero eigenvalue λ i of the Laplacian matrix L and the corresponding ω ci satisfies ωci τ = arctan(
Im(λi ) ωci γ ) + arctan( ). k Re(λi )
Proof: Applying the Laplace transforms to systems (8) and (9), we have sXi (s) = Vi (s), sVi (s) = (k + γs)
n j=1
ai j (X j (s) − Xi (s))e−sτ ,
∀i ∈ 1 ,
(10)
and sXi (s) = Vi (s)e−sτ + γ sVi (s) = k
n j=1
n j=1
ai j (X j (s) − Xi (s))e−sτ ,
ai j (X j (s) − Xi (s)),
∀i ∈ 2 ,
(11)
where Xi (s) and Vi (s) are the Laplace transforms of x i (t) and vi (t) respectively. Let X(s) = [X 1 (s), . . . , Xn (s)]T . From equations (10) and (11), we have s2 X(s) = −(k + γs)e−sτ LX(s).
(12)
Then we can get the characteristic equation of systems (8) and (9) as det(s2 I + (k + γs)e−sτ L) = 0.
(13)
Let λi (i = 1, ..., n) be the eigenvalues of L. Without loss of generality, we assume that λ 1 = λ2 = 0. Hence, equation (13) can be written as s4 Π (s2 + λi (k + γs)e−sτ ) = 0. i=3,...,n
(14)
Note that the equation (14) has four roots at s = 0. Next, let us analysis all the other roots of equation (14) by the following equations s2 + λi (k + γs)e−sτ = 0, 5
i = 3, ..., n.
(15)
Rewrite the Equation (15) as 1 + g i (s) = 0, where g i (s) = [λi (k + γs)/s2 ]e−sτ . According to the Nyquist stability criterion, the roots of (15) have negative real parts if and only if the curve g i ( jw) does not enclose the point (−1, j0) for ω ∈ R. After some manipulation, we can get √ Im(λi ) |λi | k2 +ω2ci γ2 − j(ωτ+π−arctan( ωγ k )−arctan( Re(λi ) )) (16) e . gi ( jω) = 2 ωci
It then follows that, |gi ( jω)| = |λi | k2 + ω2ci γ2 /ω2 ,
(17)
and arg(gi ( jω)) = −ωτ + arctan(
ωγ Im(λi ) ) + arctan( ) = 0, k Re(λi )
(18)
where arg(·) denotes the phase. It can be found that |g i ( jw)| is monotonously decreasing for ω ∈ (0, +∞) and g i ( jw) Im(λi ) crosses the real axis for the first time at ω ci , where ωci is the solution of equation −ωT + arctan( ωγ k ) + arctan( Re(λi ) = 0, that is to say, ωci γ Im(λi ) ) + arctan( ). k Re(λi ) Thus, the roots of (15) have negative real parts if and only if |g i ( jω)| = |λi | k2 + ω2ci γ2 /ω2ci < 1, that is to say, ωci τ = arctan(
|λi | k2 + ω2ci γ2 ω2ci
(19)
(20)
<1
holds. Hence, all the roots of (13) have negative real parts except for four roots at s = 0. Based on Lemma 3.4, we have limt→∞ x(t) = α + βt, where α = [α1 , α2 , . . . , αn ]T ∈ Rn and β = [β1 , β2 , . . . , βn ]T ∈ Rn are two constant vectors. Indeed, from (8) and (9), we can get that lim t→∞ v(t) = β , where v(t) = [v 1 (t), v2 (t), . . . , vn (t)]T . Taking into account the system (8) we obtain k ai j (α j − αi ) + (k(t − τ) + γ) ai j (β j − βi ) = 0, ∀i ∈ 1 . (21) j∈1
j∈1
Since for arbitrary t the above equation (21) holds , it then follows that ai j (α j − αi ) = 0, ∀i ∈ 1 ,
(22)
j∈1
and
ai j (β j − βi ) = 0,
∀i ∈ 1 .
(23)
j∈1
Taking into account the system (9) we obtain ai j (α j − αi ) + t al j (β j − βi ) = 0, j∈2
∀i ∈ 2 .
(24)
j∈2
Since for arbitrary t the above equation (24) holds , it then follows that ai j (α j − αi ) = 0, ∀i ∈ 2 , j∈2
6
(25)
and
ai j (β j − βi ) = 0,
∀i ∈ 2 .
(26)
j∈2
From (22) and (25), we have Lα = 0. From (23) and (26), we have Lβ = 0. Because rank(L) = n − 1 and L[1, . . . 1]T = 0 based on Lemma 2.1, we have α = a[1, . . . 1] T and β = b[1, . . . 1] T . This completes the proof. Corollary 4.2. The heterogeneous multi-agent systems (8) and (9) with a fixed directed communication topology can ⎞ ⎛ √ 4 ⎟⎟ ⎜⎜ 2 2 reach the group consensus asymptotically if τ < τ ∗ = min(τ1 , τ2 ), where τ1 =
⎜⎜ arctan⎜⎜⎜⎜ γk ⎝
max {ω ˜ ci } and ω ˜ ci being the root of the equation
3≤i≤n
− arctan(ηωci ) −
√
⎟⎟⎟ ⎟⎟⎟+θmax ⎠
with
2 2 4 |λ|4 max γ +4|λ|max k 2 γ arctan( ω˜ max )−θmax k
2 |λ|2 max γ +
i) θmax = arctan( Im(λ Re(λi ) ) and |λ| max being the maximum value of modulus of λ i , and τ2 =
1 ( ηωci ω2ci 1+(ηωci )2
|λ|max γ 4 +4|λ|max k2 2
|λ|max γ 2 +
i) arctan( Im(λ Re(λi ) ))
ω ˜ max
with ω ˜ max =
= 0.
Proof: Taking account into (19) we have τ=
i) arctan( ωkci γ ) + arctan( Im(λ Re(λi ) )
ωci
.
(27)
The derivative of τ in (27) on ω ci is given by dτ 1 ηωci Im(λi ) )) = ( − arctan(ηωci ) − arctan( dωci ω2ci 1 + (ηωci )2 Re(λi ) i) where η = γ/k. Based on Lemma 3.5, if m = arctan( Im(λ ) > 0, then dτ/dω ci < 0 with ωci > 0, i.e., τ decreases when i) √ 2 2 Re(λ |λi | k +ωci γ2 < 1 is equivalent to ωci increases. Also note that the condition ω2 ci
ωci > Therefore, the conditions
|λi |
√
k2 +ω2ci γ2 ω2ci
|λi |2 γ2 +
|λi |4 γ4 + 4|λi |2 k2 . 2
i) < 1 and ωci τ = arctan( ωkci γ ) + arctan( Im(λ Re(λi ) ) in Theorem 4.1 equal
⎞ ⎛ √ 4 4 ⎜⎜⎜ 2 γ2 + 2 k2 ⎟ ⎟⎟⎟ |λ | |λ | γ +4|λ | γ i i i i) ⎟⎟⎠ + arctan( Im(λ arctan ⎜⎜⎜⎝ k 2 Re(λi ) ) . τ< √ |λi |2 γ2 +
(28)
|λi |4 γ4 +4|λi |2 k2 2
i) If − π2 < m = arctan( Im(λ ˜ ci > 0, such as dτ/dωci < 0 holds with ωci > ω ˜ ci , Re(λi ) ) < 0, based on Lemma 3.6, there exists ω ηωci Im(λi ) 1 ˜ ci is the root of the equation ω2 ( 1+(ηωci )2 −arctan(ηωci )−arctan( Re(λi ) )) = 0. i.e., τ decreases when ω ci increases, where ω ci √2 2 2 |λi | k +ωci γ ωci γ Im(λi ) < 1 and ω τ = arctan( ) + arctan( In this case, the conditions ci k Re(λi ) ) in Theorem 4.1 equal ω2 ci
τ<
Therefore, let τ 1 =
i) arctan( ω˜ kci γ ) + arctan( Im(λ Re(λi ) )
⎞ ⎛ √ 4 ⎜⎜⎜ ⎟⎟ 2⎟ |λ|2max γ 2 + |λ|max γ 4 +4|λ|2 ⎜ max k ⎟ ⎟⎟⎟+θmax arctan⎜⎜⎜⎜ γk 2 ⎟⎠ ⎝ √ |λ|2max γ 2 +
min(τ1 , τ2 ). This completes the proof.
ω ˜ ci
2 |λ|4max γ 4 +4|λ|2 max k 2
7
and τ2 =
.
γ arctan( ω˜ max )−θmax k . ω ˜ max
(29)
Hence, we can choose τ ∗ =
Remark 4.3. In Theorem 4.1 and Corollary 4.2, we have obtained the conditions to achieve the group consensus of heterogenous multi-agent systems with directed communication topology. Indeed, if the communication topology is undirected, then all the eigenvalues of L are real. In this case, by the similar proof in 4.1 and Corollary 4.2, we can obtain the conditions to achieve the group consensus of heterogenous multi-agent systems with undirected communication topology. Corollary 4.4. The heterogeneous multi-agent systems (8) and (9) with a fixed undirected communication topology can reach the group consensus asymptotically if and only if the fixed undirected communication topology is connected, and λi k2 + ω2ci γ2 <1 ω2ci holds for every nonzero eigenvalue λ i of the Laplacian matrix L and the corresponding ω ci satisfies ωci τ = arctan(
ωci γ ). k
Proof: The proof is similar to the analysis of Theorem 4.1 and is omitted here. Corollary 4.5. The heterogeneous multi-agent systems (8) and (9) with a fixed undirected communication topology can reach⎛ the group consensus asymptotically if the fixed undirected communication topology is connected, and ⎞ √4 ⎟⎟ ⎜⎜ 2 2 τ<
⎜⎜ arctan⎜⎜⎜⎜ γk ⎝
λmax γ 4 +4λmax k2
λmax γ 2 +
2 λ2 max γ +
2
√4
λmax γ 4 +4λ2max k2
⎟⎟⎟ ⎟⎟⎟ ⎠
with λmax being the maximum eigenvalue of λ i of the Laplacian matrix L.
2
Proof: The proof is similar to the analysis of Corollary 4.2 and is omitted here. Remark 4.6. From the above analysis, we have obtained the conditions to achieve the group consensus of heterogenous multi-agent systems with time delay by using the proposed protocols (6) and (7). Indeed, the proposed protocols (6) and (7) are still valid for achieving the group consensus of heterogenous multi-agent systems without time delay. Corollary 4.7. Under Assumption 3.1, when τ = 0, the heterogeneous multi-agent system (8) and (9) with a fixed communication topology can achieve the group consensus asymptotically if Im2 (λi ) γ2 > max 2≤i≤n Re(λi )[Re2 (λi ) + Im2 (λi )] k where λi , i = 1, 2, ..., n are the eigenvalues of L. Proof: Similar to the proof of Theorem 4.1, here we just need to prove that all the roots of (13) have negative real parts except for four roots at s = 0 when τ = 0. From equation (15), when τ = 0 we have s2 + λi (k + γs) = 0,
i = 3, ..., n,
Hence, si1 =
−γλi +
(−γλi )2 − 4kλi , 2
(30)
and si2 =
−γλi −
8
(γλi )2 − 4kλi , 2
(31)
√ i ∈ {1, 2, · · · , n}. Let (−γλi )2 − 4kλi = c + id, where c and d are real, and i = −1. From (30)and (31), Re(s i j ) < 0(i = 2, 3, ..., n; j = 1, 2) if and only if −γRe(λ i ) < c < γRe(λi ), which is equivalent to Re(λ i ) > 0 and c2 < γ2 Re2 (λi )(i= 2, 3, ..., n). Since (−γλi )2 − 4kλi = c + id, it then follows γ2 λ2i − 4kλi = (c + id)2 Separating the real part and imaginary part gives c2 − d2 = γ2 Re2 (λi ) − Im2 (λi ) − 4kRe(λi ), cd = γ2 Re(λi )Im(λi ) − 2kIm(λi ) By some computation, we have c4 − {γ2 [Re2 (λi ) − Im2 (λi )] − 4kRe(λi )}c2 − Im2 (λi )[γ2 Re(λi ) − 2k] = 0 It is easy to check that c 2 < γ2 Re2 (λi ) if and only if γ2 Im2 (λi ) > max 2≤i≤n Re(λi )[Re2 (λi ) + Im2 (λi )] k
(32)
Hence, we have prove that if the condition (32) holds, then all the roots of (13) have negative real parts except for four roots at s = 0 when τ = 0. The proof is completed. Corollary 4.8. Under Assumption 3.1, when τ = 0, the heterogeneous multi-agent system (8) and (9) with a fixed undirected communication topology can achieve the group consensus asymptotically if the fixed undirected communication topology is connected, and γ > 0 and k > 0. Proof: Under Assumption 3.1, when τ = 0,the fixed undirected communication topology is connected, and γ > 0 and k > 0, then it follows that L has two simple zero eigenvalues and all other eigenvalues of L are positive. Hence, the inequality (32) holds. It is a special case of Corollary 4.7. The proof is completed. 5. Simulations In this section, two examples are given to illustrate the effectiveness of the theoretical results. Example 1: Consider a heterogeneous multi-agent system consisting of 6 agents indexed by 1 to 6, respectively, in which Agents 1 to 4 are first-order integrator agents and Agents 5 and 6 are second-order integrator agents. The topology of the system is given in Fig. 1(a). We partition all agents into two groups G 1 and G 2 . Agents 1 to 4 belong to group G 1 , and Agents 5 and 6 belong to group G 2 . Hence, the Laplacian matrix L of the system is given by ⎡ ⎤ 0 ⎥⎥ ⎢⎢⎢ 2 −2 0 0 0 ⎥ ⎢⎢⎢⎢ 0 2 −2 0 −1 1 ⎥⎥⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢ −2 0 2 0 1 −1 ⎥⎥⎥⎥ L = ⎢⎢⎢⎢⎢ ⎥. 0 1 0 0 ⎥⎥⎥⎥ ⎢⎢⎢ −1 0 ⎢⎢⎢ 0 −1 1 1 3 −3 ⎥⎥⎥⎥ ⎢⎣ ⎥⎦ 0 0 0 0 0 0 It is easy to obtain that L has eigenvalues λ 1 = λ2 = 0, λ3 = 1,λ4 = 3,λ5 = 3 + i and λ6 = 3 − i. L has zero eigenvalues whose algebraic multiplicity is exactly two, and the rest of the eigenvalues have positive real parts. Let’s choose k = 0.9 and γ = 1. By some computation, we can obtain τ 1 = 0.3973 and τ 2 = 0.4757. Then τ ∗ = min(τ1 , τ2 ) = 0.3973. From Fig.2(a) , it can be observed that when τ = 0.2 < 0.3973, the states x 1 to x4 converge to one time-varying consensus value, whereas the states x 5 to x6 converge to another time-varying consensus value. Similarly, it can be found from 9
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(a) A directed topology with six agents.
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Fig. 1: The directed and undirected communication topologies of six agents
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Fig. 2: The states trajectories the agents of the under directed topologies in Fig. 1(a) when τ = 0.2.
Fig.2(b) that the states v 1 to v4 converge to one consensus value, whereas the states v 5 to v6 converge to another consensus value. That is to say, the group consensus can be achieved. From Fig. 3 , it can be observed that when τ = 0.4 > 0.3973 the group consensus cannot be achieved. 2 2 i) When τ = 0, γk = 1.1111 > max Re(λ )[ReIm2 (λ(λ)+Im 2 (λ )] = 0.0333 that satisfies the condition of Corollary 4.7. Fig. 2≤i≤n
i
i
i
6(a) shows the position trajectories of agents, and Fig. 6(b) shows the velocity trajectories of agents. From Fig. 6, the states x1 to x4 converge to one time-varying consensus value, whereas the states x 5 to x6 converge to another time-varying consensus value. Similarly, the states v 1 to v4 converge to one consensus value, whereas the states v 5 to v6 converge to another consensus value. That is to say, the group consensus can be achieved. Example 2: In this example, we still consider a heterogenous multi-agent system consisting of 6 agents indexed by 1 to 6, respectively, in which Agents 1 to 4 are first-order integrator agents and Agents 5 and 6 are second-order integrator agents. The topology of the system is given in Fig. 1(b), which is undirected and connected. Similarly, we still partition all agents into two groups G 1 and G 2 . Agents 1 to 4 belong to group G 1 , and Agents 5 and 6 belong to
10
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Fig. 3: The states trajectories the agents of the under directed topologies in Fig. 1(a) when τ = 0.4. .
group G 2 . The Laplacian matrix L of the system is given by ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎣
⎤ 5 −2 −2 −1 0 0 ⎥⎥ ⎥ −2 4 −2 0 −1 1 ⎥⎥⎥⎥⎥ ⎥ −2 −2 4 0 1 −1 ⎥⎥⎥⎥ ⎥ −1 0 0 1 0 0 ⎥⎥⎥⎥ ⎥ 0 −1 1 0 3 −3 ⎥⎥⎥⎥ ⎦ 0 0 −1 0 −3 3
Hence, we can obtain the eigenvalues of L as λ 1 = λ2 = 0, λ3 = 1.1716,λ 4 = 4,λ5 = 6.8284 and λ 6 = 8. L has zero eigenvalues whose algebraic multiplicity is exactly two, and the rest of the eigenvalues are positive. Let’s choose k = 0.9 and γ = 1. ⎞ ⎛ √4 ⎟⎟ ⎜⎜ 2 2 By some computation, we can obtain
⎜⎜ arctan⎜⎜⎜⎜ γk ⎝
λmax γ 2 +
2 λ2 max γ +
λmax γ 4 +4λmax k2 2
√4
2 λmax γ 4 +4λ2 max k
⎟⎟⎟ ⎟⎟⎟ ⎠
= 0.2969. From Fig.4(a) , it can be observed
2
that when τ = 0.2 < 0.2969, the states x 1 to x4 converge to one time-varying consensus value, whereas the states x 5 to x6 converge to another time-varying consensus value. Similarly, from Fig. 4(b), the states v 1 to v4 converge to one consensus value, whereas the states v 5 to v6 converge to another consensus value. That is to say, the group consensus can be achieved. From Fig. 5 , it can be observed that when τ = 0.3 > 0.2969 the group consensus cannot be achieved. When τ = 0, k = 0.9 > 0 and γ = 1 > 0 that satisfies the condition of Corollary 4.8. Fig. 7(a) shows the position trajectories of agents, and Fig. 7(b) shows the velocity trajectories of agents. From Fig. 7, the states x 1 to x4 converge to one time-varying consensus value, whereas the states x 5 to x6 converge to another time-varying consensus value. Similarly, the states v 1 to v4 converge to one consensus value, whereas the states v 5 to v6 converge to another consensus value. That is to say, the group consensus can be achieved. 6. Conclution In this paper, we have investigated the dynamics group consensus problem of heterogenous multi-agent systems with input time delays. We have proposed a class of distributed protocols for achieving group consensus of heterogenous multi-agent systems with input time delays. Some sufficient group consensus conditions, which are dependent on the input delays and the control parameters, have been obtained for heterogenous multi-agent systems under directed and undirected communication topologies with and without time delays, respectively. Rigorous proofs are given by using frequently-domain analysis method, graph theory and matrix theory. Finally numerical simulations are given to 11
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Fig. 7: The states trajectories the agents of the under undirected topologies in Fig. 1(b) without time delay.
verify the theoretical analysis. Note that the results herein are just for the multi-agent systems with fixed communication topology and time-invariant input delays. For future research, it is interesting to further study group consensus problem of heterogenous multi-agent systems under switching topologies and time-varying input time delays. 7. Acknowledgments Authors would like to thank the associate editor and the anonymous reviewers for their valuable comments and suggestions that have improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China under Grants 61403019,61503016 and 11371049. and the Fundamental Research Funds for the Central Universities under Grants YWF-14-RSC-03, 2015JBM106 and 2015JBM1062. References [1] W. Ren, R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, Automatic Control, IEEE Transactions on 50 (2005) 655–661. [2] J. Wang, H. Wu, Leader-following formation control of multi-agent systems under fixed and switching topologies, International Journal of Control 85 (2012) 695–705. [3] Z. Peng, G. Wen, A. Rahmani, Y. Yu, Leadercfollower formation control of nonholonomic mobile robots based on a bioinspired neurodynamic based approach, Robotics and Autonomous Systems 61 (2013) 988 – 996. [4] Z. Peng, G. Wen, A. Rahmani, Y. Yu, Distributed consensus-based formation control for multiple nonholonomic mobile robots with a specified reference trajectory, International Journal of Systems Science 46 (2015) 1447–1457. [5] J. Wang, H. Wu, T. Huang, S. Ren, Pinning control strategies for synchronization of linearly coupled neural networks with reaction-diffusion terms, Neural Networks and Learning Systems, IEEE Transactions on PP (2015) 1–1. [6] H. Wang, Y. Tian, C. Vasseur, Piecewise continuous hybrid systems based observer design for linear systems with variable sampling periods and delay output, Signal Processing 114 (2015) 75 – 84. [7] W. Ren, R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, Automatic Control, IEEE Transactions on 50 (2005) 655–661. [8] G. Wen, W. Yu, J. Wang, D. Xu, J. Cao, Distributed node-to-node consensus of multi-agent systems with time-varying pinning links, Neurocomputing 149, Part C (2015) 1387 – 1395. [9] J. Wang, H. Wu, T. Huang, Passivity-based synchronization of a class of complex dynamical networks with time-varying delay, Automatica 56 (2015) 105 – 112. [10] G. Wen, Z. Duan, G. Chen, W. Yu, Consensus tracking of multi-agent systems with lipschitz-type node dynamics and switching topologies, Circuits and Systems I: Regular Papers, IEEE Transactions on 61 (2014) 499–511. [11] Y. Cao, W. Ren, Distributed formation control for fractional-order systems: Dynamic interaction and absolute/relative damping, Systems & Control Letters 59 (2010) 233 – 240. [12] W. Yu, W. Zheng, G. Chen, W. Ren, J. Cao, Second-order consensus in multi-agent dynamical systems with sampled position data, Automatica 47 (2011) 1496 – 1503. [13] L. Cheng, Y. Wang, Z. Hou, M. Tan, Z. Cao, Sampled-data based average consensus of second-order integral multi-agent systems: Switching topologies and communication noises, Automatica 49 (2013) 1458 – 1464.
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[14] J. Mei, W. Ren, G. Ma, Distributed coordination for second-order multi-agent systems with nonlinear dynamics using only relative position measurements, Automatica 49 (2013) 1419 – 1427. [15] G. Wen, G. Hu, W. Yu, J. Cao, G. Chen, Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs, Systems & Control Letters 62 (2013) 1151 – 1158. [16] G. Wen, Y. Zhao, D. Zhisheng, W. Yu, G. Chen, Containment of higher-order multi-leader multi-agent, Automatic Control, IEEE Transactions on (2015) 1–6. [17] J. Xi, N. Cai, Y. Zhong, Consensus problems for high-order linear time-invariant swarm systems, Physica A: Statistical Mechanics and its Applications 389 (2010) 5619 – 5627. [18] H. Su, M. Chen, J. Lam, Z. Lin, Semi-global leader-following consensus of linear multi-agent systems with input saturation via low gain feedback, Circuits and Systems I: Regular Papers, IEEE Transactions on 60 (2013) 1881–1889. [19] J. Yu, L. Wang, Group consensus of multi-agent systems with undirected communication graphs, Proceedings of the Asian Control Conference, Hong Kong, China (2009) 105–110. [20] J. Yu, L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Systems & Control Letters 59 (2010) 340 – 348. [21] H. Hu, L. Yu, W. Zhang, H. Song, Group consensus in multi-agent systems with hybrid protocol, Journal of the Franklin Institute 350 (2013) 575 – 597. [22] Y. Feng, S. Xu, B. Zhang, Group consensus control for double-integrator dynamic multiagent?systems with fixed communication topology, International Journal of Robust and Nonlinear Control 24 (2014) 532–547. [23] X. Liao, L. Ji, On pinning group consensus for dynamical multi-agent networks with general connected topology, Neurocomputing 135 (2014) 262 – 267. [24] Y. Feng, S. Xu, F. L. Lewis, B. Zhang, Consensus of heterogeneous first- and second-order multi-agent systems with directed communication topologies, International Journal of Robust and Nonlinear Control 25 (2015) 362–375. [25] Y. Zheng, Y. Zhu, L. Wang, Consensus of heterogeneous multi-agent systems, Control Theory Applications, IET 5 (2011) 1881–1888. [26] Y. Zhu, X. Guan, X. Luo, Finite-time consensus of heterogeneous multi-agent systems, Chinese Physics B 22 (2013) 38901. [27] Y. Liu, H. Min, S. Wang, Z. Liu, S. Liao, Distributed consensus of a class of networked heterogeneous multi-agent systems, Journal of the Franklin Institute 351 (2014) 1700 – 1716. [28] C. Liu, F. Liu, Dynamical consensus seeking of heterogeneous multi-agent systems under input delays, International Journal of Communication Systems 26 (2013) 1243–1258. [29] B. Liu, X. Wang, H. Su, Y. Gao, L. Wang, Adaptive second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays, Neurocomputing 118 (2013) 289 – 300. [30] Z. Lin, B. Francis, M. Maggiore, Necessary and sufficient graphical conditions for formation control of unicycles, Automatic Control, IEEE Transactions on 50 (2005) 121–127. [31] J. Yu, L. Wang, Group consensus of multi-agent systems with directed information exchange, International Journal of Systems Science 43 (2012) 334–348.
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PII: DOI: Reference:
S0925-2312(15)01519-2 http://dx.doi.org/10.1016/j.neucom.2015.10.060 NEUCOM16238
To appear in: Neurocomputing Received date: 7 July 2015 Revised date: 19 August 2015 Accepted date: 20 October 2015 Cite this article as: Guoguang Wen, Yongguang Yu, Zhaoxia Peng and Hu Wang, Dynamical Group Consensus of Heterogenous Multi-Agent Systems with Input Time Delays, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.10.060 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 galley proof before it is published in its final citable 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.
Dynamical Group Consensus of Heterogenous Multi-Agent Systems with Input Time Delays Guoguang Wen a , Yongguang Yu a , Zhaoxia Peng b,∗, Hu Wangc a Department
of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China of Transportation Science and Engineering, Beihang University, Beijing 100191, P.R.China c School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, 100081, China b School
Abstract This paper investigates the dynamics group consensus problem of heterogenous multi-agent systems with time delays, in which agents’ dynamics are modeled by single integrators and double integrators. To achieve group consensus, a class of dynamics group consensus protocols is proposed for heterogenous multi-agent systems with input time delays, which can also be used to solve group consensus for heterogenous multi-agent systems without input time delays. By using frequently-domain analysis method and matrix theory, some sufficient group consensus conditions, which are dependent on the input delays and the control parameters, are obtained for heterogenous multi-agent systems under directed and undirected communication topologies with and without input time delays, respectively. Simulation results are also provided to illustrate the effectiveness of the obtained results. Keywords: Group Consensus, Heterogenous Multi-Agent Systems, Cooperative control, Time delay, Graph theory.
1. Introduction Cooperative control of multi-agent systems has been extensively studied in last decade. Consensus problems, as a branch of cooperative control, has received considerable attention recently in many fields, such as biology, physics, robotics and control engineering[1–6]. Consensus of multi-agent systems means to design some appropriate protocols and controllable strategies via local interaction such that all the agents reach an agreement on their common interest quantity. The interest quantity might represent attitude, position, velocity, temperature, voltage, and so on. In the literature, there are many interesting research results on consensus problems for multi-agent systems, whose agents with first-order dynamics [7–11], second-order dynamics [12–14], high-order dynamics [15–17] and general linear dynamics [18]. Note that almost all papers concern the complete consensus that the states of all agents converge to a common consistent state. However, in practice a phenomena frequently occurs that agreements are different with the change of environments, situations, cooperative tasks or even time. Consequently, a critical problem is to design appropriate protocols such that agents in systems can reach more than on consistent states. This is called group (or cluster) consensus problem in multi-agent systems, in which the agents in a system are divided into multiple subgroups and different subgroups can reach different consistent states. Recently, great deals of excellent research results on group consensus have emerged constantly. In Ref. [19] Yu et al. solved the group consensus problem for multi-agent systems with fixed undirected topology. Sequently, in Ref. [20] Yu et al. extended their results in Ref. [19] to multi-agent systems with switching topologies and communication delays by using the method of double-tree-form. In [21], the authors proposed a hybrid protocol to solve the couple-group average-consensus problem of multi-agent systems with a fixed topology. In [22], Feng et al. studied the static and time-varying group consensus problems of second-order multi-agent systems. In [23], Liao et al. investigated group consensus of dynamical multi-agent networks via pinning scheme. ∗ Corresponding
author: Zhaoxia Peng Email address: [email protected] (Zhaoxia Peng)
Preprint submitted to Neurocomputing
October 27, 2015
Most of aforementioned works on consensus problems are on the homogenous multi-agent systems, in which all the agents have the same dynamics. In practical applications, the dynamics of agents may be different because of common goals of mixed agents or various restrictions of communication costs[24]. Recently, more and more attention has been paid to the consensus problems for heterogenous multi-agent systems, where agents have different dynamics. In [25], Zheng and Wang studied the consensus problem of the heterogeneous multi-agent systems consisted of first-order integrator agents and second-order integrator agents, in which a linear consensus protocol and a saturated consensus protocol were proposed. In [26], Zhu et al. studied the finite time consensus problems for heterogeneous multi-agent systems. In [27], the consensus problem of heterogeneous multi-agent systems consisted of first-order agents, second-order agents and Euler-Lagrange agents. In [28], the consensus seeking problem of heterogeneous multi-agent systems consisted of first-order agents and second-order agents with input delays was investigated. In [29], Liu et. al investigated second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays by introducing novel decentralized adaptive strategies to both the coupling strengths and the feedback gains. Here, it is worth noting that the almost all of the aforementioned works concern consensus/group consensus problems for homogeneous multi-agent systems or consensus problems for heterogeneous multi-agent systems. However, there are few results on the group consensus of heterogeneous multi-agent systems consisting of first-order and secondorder agents. Moreover, to the best of our knowledge, there is no paper concerning group consensus of heterogeneous multi-agent systems with time delays. Motivated by the above discussion, this paper investigates the dynamics group consensus problem of heterogenous multi-agent systems with time delays. The main contributions of this paper can be stated as follows: A class of distributed group consensus protocols is proposed for heterogenous multi-agent systems with input time delays, which also can be used to solve group consensus for heterogenous multi-agent systems without input time delays. By using frequently-domain analysis method and matrix theory, some sufficient group consensus conditions, which are dependent on the input delays and the control parameters, are obtained for heterogenous multi-agent systems under directed and undirected communication topologies with and without time delays, respectively. Simulation results are also provided to illustrate the effectiveness of the obtained results. Compared with the existing references, this paper has the following advantages: Firstly, in contrast of consensus/group consensus for homogeneous multi-agent systems [10–15, 17–23] , we investigate the group consensus problems for heterogenous multi-agent systems. Secondly, in contrast to the existing results in [25, 26, 28, 29], where consensus problems were considered for heterogenous multi-agent systems only under undirected communication topologies, in this paper we consider the group consensus under directed and undirected communication topologies. Thirdly, in contrast to the existing results in [23–26], where consensus/ group consensus problems were investigated without considering time delays, in this paper, group consensus problems are investigated with and without considering time delays, respectively. The rest of this paper is organized as follows. In Section 2, some preliminaries on graph theory are presented. In section 3, the problem description is given. In section 4, the main results on dynamics group consensus problem of heterogenous multi-agent systems with time delays are presented. In section 5, numerical examples are simulated. Conclusions are finally drawn in section 6. 2. Preliminaries 2.1. Algebraic graph theory Algebraic graph theory is a natural framework to study cooperative control problems of multi-agent systems. Using algebraic graph theory, the information exchanged among agents can be modeled by an interaction graph. Let G = (V, E, A) be a weighted directed or undirected graph of order n with the finite nonempty set of nodes V = {v 1 , ..., vn}, the set of edges E ⊆ V × V, and a weighted adjacency matrix A = (a i j )n×n . Here, each node v i in V corresponds to an agent i, and each edge (v i , v j ) ∈ E in a weighted directed graph corresponds to an information link from agent j to agent i, which means that agent i can receive information from agent j. In contrast, the pairs of nodes in weighted undirected graph are unordered, where an edge (v j , vi ) ∈ E denotes that agent i and j can receive information from each other. The weighted adjacency matrix A of a weighted directed graph is defined such that a ii = 0 for all vi ∈ V, ai j 0 if (v j , vi ) ∈ E, and ai j = 0 otherwise. The weighted adjacency matrix A of a weighted undirected graph is defined analogously except that a i j = a ji , ∀i j, since (vi , v j ) ∈ E implies (v j , vi ) ∈ E. We can say vi is a neighbor vertex of v j , 2
if (vi , v j ) ∈ E. A directed path is a sequence of ordered edges of the form (v i1 , vi2 ), vi2 , vi3 ,...,where (v i j , vi j+1 ) ∈ E. If there is a path in G from one node v i to another node v j , then v j is said to be reachable from v i . If not, then v j is said to be not reachable from v i . If there is a node that is reachable from every other node in the digraph, then we say the node is global reachable. L = (li j )n×n is the Laplacian matrix of graph G, and is defined by l i j = −ai j for i j, and lii = nj=1, ji ai j , i, j ∈ {1, ..., n}. For an undirected graph, L is symmetric positive semi-definite. However, L is not necessarily symmetric for a directed graph. Lemma 2.1. [30] 0 is a simple eigenvalue of the Laplacian matrix L, and 1 n = [1, ..., 1]T is the corresponding right eigenvector, that is, L1 n = 0, if and only if the digraph G = (V, E, A) has a globally reachable node. 2.2. Notations We use standard notations throughout this paper. I n represents the n × n identity matrix, 0 m×n represents the m × n zero matrix, and 1 n = [1, 1, . . . , 1]T ∈ Rn (1 for short, when there is no confusion). λ min (A) and λmax (A) are the smallest and the largest eigenvalues of the matrix A respectively. ρ(·) , det(·) represent the spectral radius,determinant of a matrix, respectively. R, C and Z + denote the sets of real numbers, complex numbers and positive integers. For s ∈ C, |s|, Re(s) and Im(s) denote its modulus, real and imaginary part, respectively. arg(·) denotes the phase of a complex number. 3. Problem description In this paper, we consider the group consensus problems under input time delays for heterogeneous multi-agent systems composed of double integrator agents and single integrator agents. Without loss of generality, we suppose there are m double integrator agents described by x˙i (t) = vi (t) v˙ i (t) = ui (t − τ),
i = 1, . . . , m,
(1)
i = m + 1, . . . , n,
(2)
and n − m single integrator agents described by x˙i (t) = ui (t − τ),
where xi ∈ R p , vi ∈ R p and ui ∈ R p are the position, velocity and control input of agent i, respectively. For the simplicity of analysis, in the following, we only consider the case where p = 1. The analysis and main results still hold for any dimension p by using Kronecker product. Group consensus requires that the first m agents reach one consistent state while the last n − m agents reach another consistent state in the presence of information exchange between the two groups. Denote 1 = {1, 2, ..., m}, 2 = {m + 1, m + 2, ..., n}. Definition 1. The heterogeneous multi-agent system (1)-(2) is said to be reach group consensus asymptotically if for any initial state values x i (0) and v i (0) , we have lim xi (t) − x j (t) = 0, lim vi (t) − v j (t) = 0, ∀i, j ∈ k , k = 1, 2. (3) t→∞ t→∞ Now the following necessary assumptions are made. Assumption 3.1.
n j=m+1
ai j = 0, ∀i ∈ 1 , and
m j=1
ai j = 0 ,∀i ∈ 2 .
Remark 3.2. As in [19, 31], Assumeption 3.1 means a balance of effect between two subsystems. Lemma 3.3. Under Assumption 3.1, L has a zero eigenvalue with the geometric multiplicity of at least two. 3
Consider the following time-delay system: y˙ (t) =
N
Ai y(t − τi )
(4)
i=1
where y(t) ∈ Rn , Ai ∈ Rn×n , τi ∈ R, and N is a positive integer. Applying the Laplace transform to system (4) , the characteristic equation can be obtained as det(sI −
N
Ai e−τi s ) = 0
(5)
i=1
Lemma 3.4. [28] If the roots of equation (5) have negative real parts except for two roots at s = 0, then limt→∞ y(t) = α + βt where α ∈ Rn and β ∈ Rn are two constant vectors. Lemma 3.5. If m > 0, then y/(1 + y 2 ) < arctan(y) + m holds for y > 0; If − π2 < m < 0, then exists y0 > 0, such that y/(1 + y2 ) < arctan(y) + m holds for y > y 0 . Proof: Let q(y) = y/(1 + y 2 ) − arctan(y) − m. Then we can get the derivative of q(y) on the y as q(y) ˙ = −2y 2 /[(1 + y2 )]2 . Obviously, q(y) is monotonously decreasing for y > 0. If m > 0, then q(0) = −m < 0. It implies that y/(1 + y 2 ) < arctan(y) + m holds for y > 0. If 0 > m > − π2 , then q(0) = −m > 0. Also note that y/(1 + y 2 ) − arctan(y) ∈ [− π2 , 0], so there exists y0 > 0 such that q(y 0 ) = y0 /(1 + y20 ) − arctan(y0 ) − m = 0, It implies that y/(1 + y 2 ) < arctan(y) + m holds for y > y0 . The proof is completed. Lemma 3.6. Let h(λ) = (arctan(δμ(λ)) + m)/μ(λ), where δ > 0, and μ(λ) > 0 is monotonously increasing for λ > 0 If m > 0, then h(λ) is monotonously decreasing for λ > 0 If m < 0 and m > − π2 , then exists λ0 > 0, then h(λ) is monotonously decreasing for λ > λ 0 . Proof: Calculating the derivative of h(λ) on λ yields μ(λ) ˙ δμ(λ) ˙ − arctan(δμ(λ)) − m). h(λ) = 2 ( μ (λ) 1 + (δμ(λ))2 Because μ(λ) > 0 and μ(λ) ˙ > 0, if m > 0, we obtain that h(λ) < 0 holds with λ > 0 from Lemma 3.5. If m < 0 and m > − π2 , based on Lemma 3.5, we obtain that h(λ) is monotonously decreasing for λ > λ 0 . Lemma 3.6 is proved. 4. Main results To solve the group consensus of heterogeneous multi-agent system (1)-(2), the following group consensus protocols are proposed, ai j (x j (t − τ) − xi (t − τ)) + k ai j x j (t − τ) + γ ai j ( x˙ j (t − τ) − x˙i (t − τ)) + γ ai j x˙ j (t − τ), ui (t − τ) = k j∈1
j∈2
∀i ∈ 1
j∈1
j∈2
(6) and ui (t − τ) = vi (t − τ) + γ ai j (x j (t − τ) − xi (t − τ)) + γ ai j x j (t − τ), j∈2 j∈1 v˙ i (t) = k ai j (x j (t) − xi (t)) + k ai j x j (t), ∀i ∈ 2 , j∈2
j∈1
where k > 0 and γ > 0 are the control parameters. 4
∀i ∈ 2 , (7)
With the consensus protocols (6) and (7), multi-agent systems (1) and (2) can be written as x˙i (t) = vi (t) ai j (x j (t − τ) − xi (t − τ)) + k ai j x j (t − τ) + γ ai j ( x˙ j (t − τ) − x˙i (t − τ)) + γ ai j x˙ j (t − τ), v˙ i (t) = k j∈1
and
j∈2
j∈1
j∈2
∀i ∈ 1
ai j (x j (t − τ) − xi (t − τ)) + γ ai j x j (t − τ), x˙i (t) = vi (t − τ) + γ j∈2 j∈1 v˙ i (t) = k ai j (x j (t) − xi (t)) + k ai j x j (t), ∀i ∈ 2 . j∈2
(8)
∀i ∈ 2 , (9)
j∈1
Theorem 4.1. The heterogeneous multi-agent systems (8) and (9) with a fixed directed communication topology can reach the group consensus asymptotically if and only if L only has exactly two simple zero eigenvalues, the rest of the eigenvalues of L have positive real parts, and |λi | k2 + ω2ci γ2 <1 ω2ci holds for every nonzero eigenvalue λ i of the Laplacian matrix L and the corresponding ω ci satisfies ωci τ = arctan(
Im(λi ) ωci γ ) + arctan( ). k Re(λi )
Proof: Applying the Laplace transforms to systems (8) and (9), we have sXi (s) = Vi (s), sVi (s) = (k + γs)
n j=1
ai j (X j (s) − Xi (s))e−sτ ,
∀i ∈ 1 ,
(10)
and sXi (s) = Vi (s)e−sτ + γ sVi (s) = k
n j=1
n j=1
ai j (X j (s) − Xi (s))e−sτ ,
ai j (X j (s) − Xi (s)),
∀i ∈ 2 ,
(11)
where Xi (s) and Vi (s) are the Laplace transforms of x i (t) and vi (t) respectively. Let X(s) = [X 1 (s), . . . , Xn (s)]T . From equations (10) and (11), we have s2 X(s) = −(k + γs)e−sτ LX(s).
(12)
Then we can get the characteristic equation of systems (8) and (9) as det(s2 I + (k + γs)e−sτ L) = 0.
(13)
Let λi (i = 1, ..., n) be the eigenvalues of L. Without loss of generality, we assume that λ 1 = λ2 = 0. Hence, equation (13) can be written as s4 Π (s2 + λi (k + γs)e−sτ ) = 0. i=3,...,n
(14)
Note that the equation (14) has four roots at s = 0. Next, let us analysis all the other roots of equation (14) by the following equations s2 + λi (k + γs)e−sτ = 0, 5
i = 3, ..., n.
(15)
Rewrite the Equation (15) as 1 + g i (s) = 0, where g i (s) = [λi (k + γs)/s2 ]e−sτ . According to the Nyquist stability criterion, the roots of (15) have negative real parts if and only if the curve g i ( jw) does not enclose the point (−1, j0) for ω ∈ R. After some manipulation, we can get √ Im(λi ) |λi | k2 +ω2ci γ2 − j(ωτ+π−arctan( ωγ k )−arctan( Re(λi ) )) (16) e . gi ( jω) = 2 ωci
It then follows that, |gi ( jω)| = |λi | k2 + ω2ci γ2 /ω2 ,
(17)
and arg(gi ( jω)) = −ωτ + arctan(
ωγ Im(λi ) ) + arctan( ) = 0, k Re(λi )
(18)
where arg(·) denotes the phase. It can be found that |g i ( jw)| is monotonously decreasing for ω ∈ (0, +∞) and g i ( jw) Im(λi ) crosses the real axis for the first time at ω ci , where ωci is the solution of equation −ωT + arctan( ωγ k ) + arctan( Re(λi ) = 0, that is to say, ωci γ Im(λi ) ) + arctan( ). k Re(λi ) Thus, the roots of (15) have negative real parts if and only if |g i ( jω)| = |λi | k2 + ω2ci γ2 /ω2ci < 1, that is to say, ωci τ = arctan(
|λi | k2 + ω2ci γ2 ω2ci
(19)
(20)
<1
holds. Hence, all the roots of (13) have negative real parts except for four roots at s = 0. Based on Lemma 3.4, we have limt→∞ x(t) = α + βt, where α = [α1 , α2 , . . . , αn ]T ∈ Rn and β = [β1 , β2 , . . . , βn ]T ∈ Rn are two constant vectors. Indeed, from (8) and (9), we can get that lim t→∞ v(t) = β , where v(t) = [v 1 (t), v2 (t), . . . , vn (t)]T . Taking into account the system (8) we obtain k ai j (α j − αi ) + (k(t − τ) + γ) ai j (β j − βi ) = 0, ∀i ∈ 1 . (21) j∈1
j∈1
Since for arbitrary t the above equation (21) holds , it then follows that ai j (α j − αi ) = 0, ∀i ∈ 1 ,
(22)
j∈1
and
ai j (β j − βi ) = 0,
∀i ∈ 1 .
(23)
j∈1
Taking into account the system (9) we obtain ai j (α j − αi ) + t al j (β j − βi ) = 0, j∈2
∀i ∈ 2 .
(24)
j∈2
Since for arbitrary t the above equation (24) holds , it then follows that ai j (α j − αi ) = 0, ∀i ∈ 2 , j∈2
6
(25)
and
ai j (β j − βi ) = 0,
∀i ∈ 2 .
(26)
j∈2
From (22) and (25), we have Lα = 0. From (23) and (26), we have Lβ = 0. Because rank(L) = n − 1 and L[1, . . . 1]T = 0 based on Lemma 2.1, we have α = a[1, . . . 1] T and β = b[1, . . . 1] T . This completes the proof. Corollary 4.2. The heterogeneous multi-agent systems (8) and (9) with a fixed directed communication topology can ⎞ ⎛ √ 4 ⎟⎟ ⎜⎜ 2 2 reach the group consensus asymptotically if τ < τ ∗ = min(τ1 , τ2 ), where τ1 =
⎜⎜ arctan⎜⎜⎜⎜ γk ⎝
max {ω ˜ ci } and ω ˜ ci being the root of the equation
3≤i≤n
− arctan(ηωci ) −
√
⎟⎟⎟ ⎟⎟⎟+θmax ⎠
with
2 2 4 |λ|4 max γ +4|λ|max k 2 γ arctan( ω˜ max )−θmax k
2 |λ|2 max γ +
i) θmax = arctan( Im(λ Re(λi ) ) and |λ| max being the maximum value of modulus of λ i , and τ2 =
1 ( ηωci ω2ci 1+(ηωci )2
|λ|max γ 4 +4|λ|max k2 2
|λ|max γ 2 +
i) arctan( Im(λ Re(λi ) ))
ω ˜ max
with ω ˜ max =
= 0.
Proof: Taking account into (19) we have τ=
i) arctan( ωkci γ ) + arctan( Im(λ Re(λi ) )
ωci
.
(27)
The derivative of τ in (27) on ω ci is given by dτ 1 ηωci Im(λi ) )) = ( − arctan(ηωci ) − arctan( dωci ω2ci 1 + (ηωci )2 Re(λi ) i) where η = γ/k. Based on Lemma 3.5, if m = arctan( Im(λ ) > 0, then dτ/dω ci < 0 with ωci > 0, i.e., τ decreases when i) √ 2 2 Re(λ |λi | k +ωci γ2 < 1 is equivalent to ωci increases. Also note that the condition ω2 ci
ωci > Therefore, the conditions
|λi |
√
k2 +ω2ci γ2 ω2ci
|λi |2 γ2 +
|λi |4 γ4 + 4|λi |2 k2 . 2
i) < 1 and ωci τ = arctan( ωkci γ ) + arctan( Im(λ Re(λi ) ) in Theorem 4.1 equal
⎞ ⎛ √ 4 4 ⎜⎜⎜ 2 γ2 + 2 k2 ⎟ ⎟⎟⎟ |λ | |λ | γ +4|λ | γ i i i i) ⎟⎟⎠ + arctan( Im(λ arctan ⎜⎜⎜⎝ k 2 Re(λi ) ) . τ< √ |λi |2 γ2 +
(28)
|λi |4 γ4 +4|λi |2 k2 2
i) If − π2 < m = arctan( Im(λ ˜ ci > 0, such as dτ/dωci < 0 holds with ωci > ω ˜ ci , Re(λi ) ) < 0, based on Lemma 3.6, there exists ω ηωci Im(λi ) 1 ˜ ci is the root of the equation ω2 ( 1+(ηωci )2 −arctan(ηωci )−arctan( Re(λi ) )) = 0. i.e., τ decreases when ω ci increases, where ω ci √2 2 2 |λi | k +ωci γ ωci γ Im(λi ) < 1 and ω τ = arctan( ) + arctan( In this case, the conditions ci k Re(λi ) ) in Theorem 4.1 equal ω2 ci
τ<
Therefore, let τ 1 =
i) arctan( ω˜ kci γ ) + arctan( Im(λ Re(λi ) )
⎞ ⎛ √ 4 ⎜⎜⎜ ⎟⎟ 2⎟ |λ|2max γ 2 + |λ|max γ 4 +4|λ|2 ⎜ max k ⎟ ⎟⎟⎟+θmax arctan⎜⎜⎜⎜ γk 2 ⎟⎠ ⎝ √ |λ|2max γ 2 +
min(τ1 , τ2 ). This completes the proof.
ω ˜ ci
2 |λ|4max γ 4 +4|λ|2 max k 2
7
and τ2 =
.
γ arctan( ω˜ max )−θmax k . ω ˜ max
(29)
Hence, we can choose τ ∗ =
Remark 4.3. In Theorem 4.1 and Corollary 4.2, we have obtained the conditions to achieve the group consensus of heterogenous multi-agent systems with directed communication topology. Indeed, if the communication topology is undirected, then all the eigenvalues of L are real. In this case, by the similar proof in 4.1 and Corollary 4.2, we can obtain the conditions to achieve the group consensus of heterogenous multi-agent systems with undirected communication topology. Corollary 4.4. The heterogeneous multi-agent systems (8) and (9) with a fixed undirected communication topology can reach the group consensus asymptotically if and only if the fixed undirected communication topology is connected, and λi k2 + ω2ci γ2 <1 ω2ci holds for every nonzero eigenvalue λ i of the Laplacian matrix L and the corresponding ω ci satisfies ωci τ = arctan(
ωci γ ). k
Proof: The proof is similar to the analysis of Theorem 4.1 and is omitted here. Corollary 4.5. The heterogeneous multi-agent systems (8) and (9) with a fixed undirected communication topology can reach⎛ the group consensus asymptotically if the fixed undirected communication topology is connected, and ⎞ √4 ⎟⎟ ⎜⎜ 2 2 τ<
⎜⎜ arctan⎜⎜⎜⎜ γk ⎝
λmax γ 4 +4λmax k2
λmax γ 2 +
2 λ2 max γ +
2
√4
λmax γ 4 +4λ2max k2
⎟⎟⎟ ⎟⎟⎟ ⎠
with λmax being the maximum eigenvalue of λ i of the Laplacian matrix L.
2
Proof: The proof is similar to the analysis of Corollary 4.2 and is omitted here. Remark 4.6. From the above analysis, we have obtained the conditions to achieve the group consensus of heterogenous multi-agent systems with time delay by using the proposed protocols (6) and (7). Indeed, the proposed protocols (6) and (7) are still valid for achieving the group consensus of heterogenous multi-agent systems without time delay. Corollary 4.7. Under Assumption 3.1, when τ = 0, the heterogeneous multi-agent system (8) and (9) with a fixed communication topology can achieve the group consensus asymptotically if Im2 (λi ) γ2 > max 2≤i≤n Re(λi )[Re2 (λi ) + Im2 (λi )] k where λi , i = 1, 2, ..., n are the eigenvalues of L. Proof: Similar to the proof of Theorem 4.1, here we just need to prove that all the roots of (13) have negative real parts except for four roots at s = 0 when τ = 0. From equation (15), when τ = 0 we have s2 + λi (k + γs) = 0,
i = 3, ..., n,
Hence, si1 =
−γλi +
(−γλi )2 − 4kλi , 2
(30)
and si2 =
−γλi −
8
(γλi )2 − 4kλi , 2
(31)
√ i ∈ {1, 2, · · · , n}. Let (−γλi )2 − 4kλi = c + id, where c and d are real, and i = −1. From (30)and (31), Re(s i j ) < 0(i = 2, 3, ..., n; j = 1, 2) if and only if −γRe(λ i ) < c < γRe(λi ), which is equivalent to Re(λ i ) > 0 and c2 < γ2 Re2 (λi )(i= 2, 3, ..., n). Since (−γλi )2 − 4kλi = c + id, it then follows γ2 λ2i − 4kλi = (c + id)2 Separating the real part and imaginary part gives c2 − d2 = γ2 Re2 (λi ) − Im2 (λi ) − 4kRe(λi ), cd = γ2 Re(λi )Im(λi ) − 2kIm(λi ) By some computation, we have c4 − {γ2 [Re2 (λi ) − Im2 (λi )] − 4kRe(λi )}c2 − Im2 (λi )[γ2 Re(λi ) − 2k] = 0 It is easy to check that c 2 < γ2 Re2 (λi ) if and only if γ2 Im2 (λi ) > max 2≤i≤n Re(λi )[Re2 (λi ) + Im2 (λi )] k
(32)
Hence, we have prove that if the condition (32) holds, then all the roots of (13) have negative real parts except for four roots at s = 0 when τ = 0. The proof is completed. Corollary 4.8. Under Assumption 3.1, when τ = 0, the heterogeneous multi-agent system (8) and (9) with a fixed undirected communication topology can achieve the group consensus asymptotically if the fixed undirected communication topology is connected, and γ > 0 and k > 0. Proof: Under Assumption 3.1, when τ = 0,the fixed undirected communication topology is connected, and γ > 0 and k > 0, then it follows that L has two simple zero eigenvalues and all other eigenvalues of L are positive. Hence, the inequality (32) holds. It is a special case of Corollary 4.7. The proof is completed. 5. Simulations In this section, two examples are given to illustrate the effectiveness of the theoretical results. Example 1: Consider a heterogeneous multi-agent system consisting of 6 agents indexed by 1 to 6, respectively, in which Agents 1 to 4 are first-order integrator agents and Agents 5 and 6 are second-order integrator agents. The topology of the system is given in Fig. 1(a). We partition all agents into two groups G 1 and G 2 . Agents 1 to 4 belong to group G 1 , and Agents 5 and 6 belong to group G 2 . Hence, the Laplacian matrix L of the system is given by ⎡ ⎤ 0 ⎥⎥ ⎢⎢⎢ 2 −2 0 0 0 ⎥ ⎢⎢⎢⎢ 0 2 −2 0 −1 1 ⎥⎥⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢ −2 0 2 0 1 −1 ⎥⎥⎥⎥ L = ⎢⎢⎢⎢⎢ ⎥. 0 1 0 0 ⎥⎥⎥⎥ ⎢⎢⎢ −1 0 ⎢⎢⎢ 0 −1 1 1 3 −3 ⎥⎥⎥⎥ ⎢⎣ ⎥⎦ 0 0 0 0 0 0 It is easy to obtain that L has eigenvalues λ 1 = λ2 = 0, λ3 = 1,λ4 = 3,λ5 = 3 + i and λ6 = 3 − i. L has zero eigenvalues whose algebraic multiplicity is exactly two, and the rest of the eigenvalues have positive real parts. Let’s choose k = 0.9 and γ = 1. By some computation, we can obtain τ 1 = 0.3973 and τ 2 = 0.4757. Then τ ∗ = min(τ1 , τ2 ) = 0.3973. From Fig.2(a) , it can be observed that when τ = 0.2 < 0.3973, the states x 1 to x4 converge to one time-varying consensus value, whereas the states x 5 to x6 converge to another time-varying consensus value. Similarly, it can be found from 9
2
1
2
1
1
2
2
−1
2
1
5
2
3 1
3
5
−1
2
3
−1
−1
4
1
2
1
3
4
6
(a) A directed topology with six agents.
1
6
(b) A undirected topology with six agents.
Fig. 1: The directed and undirected communication topologies of six agents
80
Positions of agents
60 50 40
8 Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
6 Velocities of agents
70
30 20
Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
0 −2 −4
10 0 0
4 2
−6 5
10 t(s)
15
−8 0
20
(a) Position trajectories of the agents
5
10 t(s)
15
20
(b) Velocities trajectories of the agents
Fig. 2: The states trajectories the agents of the under directed topologies in Fig. 1(a) when τ = 0.2.
Fig.2(b) that the states v 1 to v4 converge to one consensus value, whereas the states v 5 to v6 converge to another consensus value. That is to say, the group consensus can be achieved. From Fig. 3 , it can be observed that when τ = 0.4 > 0.3973 the group consensus cannot be achieved. 2 2 i) When τ = 0, γk = 1.1111 > max Re(λ )[ReIm2 (λ(λ)+Im 2 (λ )] = 0.0333 that satisfies the condition of Corollary 4.7. Fig. 2≤i≤n
i
i
i
6(a) shows the position trajectories of agents, and Fig. 6(b) shows the velocity trajectories of agents. From Fig. 6, the states x1 to x4 converge to one time-varying consensus value, whereas the states x 5 to x6 converge to another time-varying consensus value. Similarly, the states v 1 to v4 converge to one consensus value, whereas the states v 5 to v6 converge to another consensus value. That is to say, the group consensus can be achieved. Example 2: In this example, we still consider a heterogenous multi-agent system consisting of 6 agents indexed by 1 to 6, respectively, in which Agents 1 to 4 are first-order integrator agents and Agents 5 and 6 are second-order integrator agents. The topology of the system is given in Fig. 1(b), which is undirected and connected. Similarly, we still partition all agents into two groups G 1 and G 2 . Agents 1 to 4 belong to group G 1 , and Agents 5 and 6 belong to
10
8000
1.5 Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
Positions of agents
4000 2000
4
Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
1 Velocities of agents
6000
x 10
0 −2000
0.5
0
−4000 −0.5 −6000 −8000 0
2
−1 0
10
8
6
4
2
t(s)
6
4
8
10
t(s)
(a) Position trajectories of the agents
(b) Velocities trajectories of the agents
Fig. 3: The states trajectories the agents of the under directed topologies in Fig. 1(a) when τ = 0.4. .
group G 2 . The Laplacian matrix L of the system is given by ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎣
⎤ 5 −2 −2 −1 0 0 ⎥⎥ ⎥ −2 4 −2 0 −1 1 ⎥⎥⎥⎥⎥ ⎥ −2 −2 4 0 1 −1 ⎥⎥⎥⎥ ⎥ −1 0 0 1 0 0 ⎥⎥⎥⎥ ⎥ 0 −1 1 0 3 −3 ⎥⎥⎥⎥ ⎦ 0 0 −1 0 −3 3
Hence, we can obtain the eigenvalues of L as λ 1 = λ2 = 0, λ3 = 1.1716,λ 4 = 4,λ5 = 6.8284 and λ 6 = 8. L has zero eigenvalues whose algebraic multiplicity is exactly two, and the rest of the eigenvalues are positive. Let’s choose k = 0.9 and γ = 1. ⎞ ⎛ √4 ⎟⎟ ⎜⎜ 2 2 By some computation, we can obtain
⎜⎜ arctan⎜⎜⎜⎜ γk ⎝
λmax γ 2 +
2 λ2 max γ +
λmax γ 4 +4λmax k2 2
√4
2 λmax γ 4 +4λ2 max k
⎟⎟⎟ ⎟⎟⎟ ⎠
= 0.2969. From Fig.4(a) , it can be observed
2
that when τ = 0.2 < 0.2969, the states x 1 to x4 converge to one time-varying consensus value, whereas the states x 5 to x6 converge to another time-varying consensus value. Similarly, from Fig. 4(b), the states v 1 to v4 converge to one consensus value, whereas the states v 5 to v6 converge to another consensus value. That is to say, the group consensus can be achieved. From Fig. 5 , it can be observed that when τ = 0.3 > 0.2969 the group consensus cannot be achieved. When τ = 0, k = 0.9 > 0 and γ = 1 > 0 that satisfies the condition of Corollary 4.8. Fig. 7(a) shows the position trajectories of agents, and Fig. 7(b) shows the velocity trajectories of agents. From Fig. 7, the states x 1 to x4 converge to one time-varying consensus value, whereas the states x 5 to x6 converge to another time-varying consensus value. Similarly, the states v 1 to v4 converge to one consensus value, whereas the states v 5 to v6 converge to another consensus value. That is to say, the group consensus can be achieved. 6. Conclution In this paper, we have investigated the dynamics group consensus problem of heterogenous multi-agent systems with input time delays. We have proposed a class of distributed protocols for achieving group consensus of heterogenous multi-agent systems with input time delays. Some sufficient group consensus conditions, which are dependent on the input delays and the control parameters, have been obtained for heterogenous multi-agent systems under directed and undirected communication topologies with and without time delays, respectively. Rigorous proofs are given by using frequently-domain analysis method, graph theory and matrix theory. Finally numerical simulations are given to 11
50
15 Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
30
Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
10 Velocities of agents
Positions of agents
40
20
5
0
10
−5
0 0
8
6
4
2
−10 0
10
8
6
4
2
t(s)
10
t(s)
(a) Position trajectories of the agents
(b) Velocities trajectories of the agents
Fig. 4: The states trajectories the agents of the under undirected topologies in Fig. 1(b) when τ = 0.2.
x 10
4
8 Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
1.5
Positions of agents
1 0.5 0 −0.5
Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
4
0 −2
−1
−4 −6 2
4
2
−1.5 −2 0
x 10
6 Velocities of agents
2
4
6
8
−8 0
10
2
4
t(s)
6
8
10
t(s)
(a) Position trajectories of the agents
(b) Velocities trajectories of the agents
Fig. 5: The states trajectories the agents of the under undirected topologies in Fig. 1(b) when τ = 0.3.
80
Positions of agents
50
6 Velocities of agents
70 60
8 Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
40 30
4 2 Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
0
20 −2
10 0 0
5
10 t(s)
15
−4 0
20
5
10 t(s)
15
20
(b) Velocities trajectories of the agents
(a) Position trajectories of the agents
Fig. 6: The states trajectories the agents of the under directed topologies in Fig. 1(a) without time delay.
12
8
40
6 Velocities of agents
Positions of agents
50
30 Agent1 Agent2 Agent3 Agent4 Agent5 Agent6
20
10
0 0
2
6
4
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(a) Position trajectories of the agents
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Fig. 7: The states trajectories the agents of the under undirected topologies in Fig. 1(b) without time delay.
verify the theoretical analysis. Note that the results herein are just for the multi-agent systems with fixed communication topology and time-invariant input delays. For future research, it is interesting to further study group consensus problem of heterogenous multi-agent systems under switching topologies and time-varying input time delays. 7. Acknowledgments Authors would like to thank the associate editor and the anonymous reviewers for their valuable comments and suggestions that have improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China under Grants 61403019,61503016 and 11371049. and the Fundamental Research Funds for the Central Universities under Grants YWF-14-RSC-03, 2015JBM106 and 2015JBM1062. References [1] W. Ren, R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, Automatic Control, IEEE Transactions on 50 (2005) 655–661. [2] J. Wang, H. Wu, Leader-following formation control of multi-agent systems under fixed and switching topologies, International Journal of Control 85 (2012) 695–705. [3] Z. Peng, G. Wen, A. Rahmani, Y. Yu, Leadercfollower formation control of nonholonomic mobile robots based on a bioinspired neurodynamic based approach, Robotics and Autonomous Systems 61 (2013) 988 – 996. [4] Z. Peng, G. Wen, A. Rahmani, Y. Yu, Distributed consensus-based formation control for multiple nonholonomic mobile robots with a specified reference trajectory, International Journal of Systems Science 46 (2015) 1447–1457. [5] J. Wang, H. Wu, T. Huang, S. Ren, Pinning control strategies for synchronization of linearly coupled neural networks with reaction-diffusion terms, Neural Networks and Learning Systems, IEEE Transactions on PP (2015) 1–1. [6] H. Wang, Y. Tian, C. Vasseur, Piecewise continuous hybrid systems based observer design for linear systems with variable sampling periods and delay output, Signal Processing 114 (2015) 75 – 84. [7] W. Ren, R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, Automatic Control, IEEE Transactions on 50 (2005) 655–661. [8] G. Wen, W. Yu, J. Wang, D. Xu, J. Cao, Distributed node-to-node consensus of multi-agent systems with time-varying pinning links, Neurocomputing 149, Part C (2015) 1387 – 1395. [9] J. Wang, H. Wu, T. Huang, Passivity-based synchronization of a class of complex dynamical networks with time-varying delay, Automatica 56 (2015) 105 – 112. [10] G. Wen, Z. Duan, G. Chen, W. Yu, Consensus tracking of multi-agent systems with lipschitz-type node dynamics and switching topologies, Circuits and Systems I: Regular Papers, IEEE Transactions on 61 (2014) 499–511. [11] Y. Cao, W. Ren, Distributed formation control for fractional-order systems: Dynamic interaction and absolute/relative damping, Systems & Control Letters 59 (2010) 233 – 240. [12] W. Yu, W. Zheng, G. Chen, W. Ren, J. Cao, Second-order consensus in multi-agent dynamical systems with sampled position data, Automatica 47 (2011) 1496 – 1503. [13] L. Cheng, Y. Wang, Z. Hou, M. Tan, Z. Cao, Sampled-data based average consensus of second-order integral multi-agent systems: Switching topologies and communication noises, Automatica 49 (2013) 1458 – 1464.
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