Group consensus of multi-agent systems with communication delays

Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Group consensus of multi-agent systems with communication delays Hong Xia, Ting-Zhu Huang, Jin-Liang Shao n, Jun-Yan Yu School of Mathematical Sciences/Research Center for Complex Systems and Control, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 27 February 2015 Received in revised form 28 June 2015 Accepted 27 July 2015 Communicated by Jiahu Qin

In this paper, we investigate group consensus problems for continuous-time multi-agent systems with time delays, where the agents' dynamics are controlled based on information from their neighbors at some discrete time instants. By utilizing nonnegative matrix theory and graph theory, the case with fixed topology and time-invariant delays is first considered, then the general case with switching topologies and time-varying delays is discussed. Some group consensus criteria are obtained in both cases, it is also shown that the agents in nonzero in-degree groups finally converge to the convex hull of those in zero in-degree groups. Simulation examples are provided to demonstrate the effectiveness of the theoretical results. & 2015 Elsevier B.V. All rights reserved.

Keywords: Group consensus Time delays Multi-agent systems Switching topologies

1. Introduction The past few years have witnessed dramatic advances concerned with distributed coordination of multi-agent systems. As one type of critical problems for distributed coordination, consensus problem, with the objective of reaching an agreement regarding a certain quantity of interest, has been studied from various perspectives [1–10]. And it has been found to possess broad applications in sensor networks, robotics teams, unmanned air vehicle formations, biological systems, and group decision making problems. Recently, a more general kind of consensus, group/cluster consensus, has attracted increasing attention from researchers [11–19]. In a complex network consisting of multiple subgroups, group consensus means that the agents in each subgroup reach consensus asymptotically, but the consistent states of different subgroups may not be the same. In fact, group consensus is more appealing for nature and human society, since the agreements in some real-life scenarios, such as the pattern formation of bacteria colonies and the cluster formation of personal opinions, are often different with the changes of environments, situations, cooperative tasks or even time [20,21]. Under the in-degree balanced assumption, Yu and Wang [12] presented some sufficient conditions in terms of linear matrix inequalities (LMIs) for guaranteeing the group consensus with fixed topology, the result was later extended to the multi-agent network with switching topologies

n

Corresponding author. E-mail address: [email protected] (J.-L. Shao).

[13]. Qin and Yu [14] further showed that under directed acyclic topology, the group consensus can be achieved regardless of the magnitudes of the coupling strengths among the agents. Moreover, by utilizing a distributed protocol with Cartesian coordinate coupling matrix, Xie et al. [15] established some necessary and sufficient group consensus criteria for continuous-time multiagent systems. Note that most of the aforementioned works did not take into consideration communication delays, and the agents were assumed to be able to interact with their neighbors continuously. In practical applications, however, it is more realistic especially when time delays are accounted for, and the continuity of the state information of each agent's neighbors cannot be ensured, since the information channels are often unreliable and bandwidth is usually limited. At present, most of research results with time delays and intermittent interaction are concerned about consensus problem such as [22–28] and so on. For example, based on the frequency-domain analysis, Tian and Liu [22] studied the consensus problem for multi-agent systems with input and communication delays. Xiao and Wang [24] investigated asynchronous consensus problems of continuous-time multi-agent systems with discontinuous information transmission. Ulrich et al. [23] discussed the robustness of consensus for multi-agent systems with coupling delays and switching topologies. It would be interesting to explore the group consensus problem with time delays and intermittent interaction. Communication topologies, on the other hand, can influence the dynamic behavior of agents and then play an important role in the study of the group consensus problems. Indeed, Xiao and Wang [29] introduced a “pre-leader-follower” decomposition to

http://dx.doi.org/10.1016/j.neucom.2015.07.108 0925-2312/& 2015 Elsevier B.V. All rights reserved.

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characterize the structures of communication topologies, and discussed the dynamic behavior of networks with general communication topologies in [30]. In [17], Han et al. extended the concept of spanning tree to that of group spanning trees, and discussed the group consensus problem of continuous-time multiagent systems. The group consensus problems under group spanning trees were also considered in [18,19]. Since the concept of group spanning tree is a generalization of that of spanning tree, a natural question is what relationship exists among the final consistent states of different subgroups, which is not figured out in [17–19]. Inspired mainly by the above analysis, this paper focuses on the group consensus problem for continuous-time multi-agent systems with time delays and discontinuous information transmission, where the group consensus control strategy is based on the state information of each agent's neighbors at some discrete time instants. By using nonnegative matrix theory and graph theory, we provide some sufficient or necessary conditions for the solvability of group consensus problem. First, according to the topology structure, the groups under group spanning trees are classified into two categories: zero in-degree groups and nonzero in-degree groups. Then it is proved that the multi-agent systems with fixed topology and time-invariant delays can achieve group consensus, if the graph has group spanning trees and the adjacency matrix has inter-group common influence. For the case with switching topologies and time-varying delays, it is proved that under the assumption of inter-group common influence and bounded time delays, the multi-agent systems can also realize group consensus, if there exists T 4 0 such that the union graph across any T-length time interval has group spanning trees. Furthermore, the nonzero in-degree groups are shown to finally converge to convex combination of the consensus states of the zero in-degree groups. The rest of the paper is organized as follows. Some preliminaries and the problem formulation are provided in Section 2. In Sections 3, we establish our main results. Several simulation examples are provided in Section 4 to demonstrate the effectiveness of the theoretical findings. Finally, some conclusion remarks are made in Section 5.

2. Preliminaries and formulation Directed graphs will be used to model the interaction topology among agents. Let G ¼ ðV; EÞ be a directed graph composed of a vertex set V ¼ fv1 ; v2 ; …; vn g and an edge set E D V  V, where vertex vi represents agent i. An edge in E is denoted by an ordered pair ðvj ; vi Þ, where ðvj ; vi Þ A E if and only if that agent i can access the state information of agent j. If ðvi ; vi Þ A E, it is called a self-loop on vertex vi. Let ½n ¼ f1; …; ng be the index set of the vertices, and the
index set of neighbors of vertex vi is denoted by  N i ¼ vj A V : ðvj ; vi Þ A E; j a i . A directed path is a sequence of edges in a directed graph of the form ðvi1 ; vi2 Þ, ðvi2 ; vi3 Þ; …, where vij A V. If there exists a path from vertex vj to vertex vi, we say that vi is reachable from vj. A directed tree is a directed graph, where there exists a vertex, called the root, such that any other vertex of the digraph can be reached by and only one path starting at the root. A spanning tree of a digraph is a directed tree formed by graph edges that connect all the vertexes of the graph. A nonnegative matrix A ¼ ðaij Þ A Rnn can be associated with a directed graph GðAÞ in such a way that A is specified as the weighted adjacency matrix. In detail, for vi ; vj A V, vj A N i 3 aij 4 0, and aii 40 if there is a self-loop on vertex vi. Let I denote an identity matrix with a compatible dimension, 1n denotes an all 1 vector with dimension n, and 0n denotes an allzero vector with dimension n. A vector norm of a vector x is denoted by J x J . Given a matrix A A Rnn , ρðAÞ and J A J represent

the spectral radius of A and the matrix norm of A induced by the vector norm J  J , respectively. A square nonnegative matrix is called a (row) stochastic matrix if all its rows sums are 1. Let ∏sk ¼ v AðkÞ denotes the successive matrix product from s to v, that is, ∏sk ¼ v AðkÞ ¼ AðsÞAðs  1Þ⋯AðvÞ. To study the group consensus problem, we first give a grouping of multi-agent systems. Without loss of generality, we assume that the multi-agent systems consisting of n agents are divided into K disjoint groups. Specifically, a partition of the multi-agent systems, denoted by C ¼ fC1 ; …; CK g, is a sequence of subsets of ½n such that ⋃Kp ¼ 1 Cp ¼ ½n and Cp ⋂Cq ¼ ∅ for p a q. Definition 1 (Han et al. [18]). For a given partition C, a graph G is said to have group spanning trees with respect to C if for each group Cp , p ¼1,…,K, there is a vertex vp A V such that there exist paths in G from vp to all vertexes in Cp . The vertex vp is called the root of the group Cp . It is worth pointing out that the root vertex of Cp does not necessarily belong to Cp . In fact, the group spanning tree is a weaker definition than the spanning tree since the roots of all the groups are not necessarily same. Definition 2 (Rockafellar [31]). A set H  Rμ is called convex if, for any x, y A H, the point ð1  γÞx þ γy A H for any γ A ½0; 1. The convex hull of a finite set X ¼ fx1 ; x2 ; …; xn g, xi A Rμ , i¼ 1,…,n, is the minimal convex set containing all points in X, denoted by CofXg. P P Particularly, CofXg¼f ni¼ 1 αi xi ∣αi Z 0; ni¼ 1 αi ¼ 1g. Let xi ðtÞ A R represent the state of agent i at time t, then the system is said to solve a group consensus problem [12] if for any initial states, the states of agents satisfy: lim J xi ðtÞ  xj ðtÞ J ¼ 0;

t-1

for all i; j A Cp and p ¼1,…,K. In this paper, suppose that each agent receives its neighbors' information at discrete time instants tk, which satisfy 0 ¼ t 0 ot 1 o …o t k o …, and the received information is with time delays, we study the following continuous-time model of n agents: for t A ½t k ; t k þ 1 , X 1 wij ðt k Þðxj ðt k  τij ðt k ÞÞ  xi ðtÞÞ; ð1Þ x_ i ðtÞ ¼ P w ðt Þ ij k j A N ðt Þ j A N i ðt k Þ i

k

where wij ðt k Þ is the weighting factor, τij ðt k Þ is the transmitted information delays from agent j to i at time tk, i a j, and GðtÞ corresponds to the communication topology at time t. For simplicity, we assume that t k þ 1  t k ¼ h, where h is a positive real number. To solve the group consensus problem, the definition of Hajnal diameter [32] is extended to the group case in [18]. In detail, given a matrix A with row vectors A1 ; A2 ; …; An and a partition C, the group Hajnal diameter of A is ΔG ðAÞ ¼ max max J Ai  Aj J ; p ¼ 1;…;K i;j A Cp

for some vector norm J  J . It is clear that ΔG ðAÞ measures the differences between the rows of A. Similarly, given a vector x ¼ ðx1 ; x2 ; …; xn ÞT and a partition G, the group Hajnal diameter of x is ΔG ðxÞ ¼ max max J xi  xj J : p ¼ 1;…;K i;j A Cp

Also we need to recall the definition of inter-group common influence. Definition 3 (Han et al. [18]). Given a partition C ¼ fC1 ; …; CK g, a stochastic matrix A is said to have inter-group common influence if P for each pair of p and q, p; q ¼ 1; 2; …; K, j A Cq aij only depends on the group indices p and q but is independent of i with vi A Cp .

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ðB3Þ there exist an integer τ^ Z 0 such that 0 r τij ðt k Þ r τd ¼ τ^ h holds for any i; j; k. For the normal consensus problem in which all the agents converge to a common consistent state, Xiao et al. [24] transformed the continuous-time systems into their discrete-time counterparts and derived the following result:

Fig. 1. Group spanning trees topologies G1 and G2 .

Definition 3 implies that for any pair of groups Cp and Cq , either there are no edges from Cp to Cq ; or for each vertex in Cq , there is at P least one edge from Cp to it. We can then define j A Cq aij ¼ ςp;q for any i A Cp .

3. Main results For a multi-agent system consisting of K groups, we divide the agents under group spanning trees into zero in-degree groups and nonzero in-degree groups. Without loss of generality, suppose the number of zero in-degree groups is m, denoted by C1 ; …; Cm : C1 ¼ f1; 2; …; l1 g, C2 ¼ fl1 þ 1; l1 þ 2; …; l2 g…; Cm ¼ flm  1 þ 1; lm  1 þ 2; …; lm g, respectively. And let the agents of nonzero in-degree group be Cm þ 1 ¼½n⧹ð [ m i ¼ 1 C i Þ ¼ flm þ 1, lm þ 2, …, ng. For convenience, let n1 ¼ l1 , n2 ¼ l2  l1 , …, nm þ 1 ¼ n lm . Example 1. We consider a multi-agent system consisting of 8 agents, with the direct interaction topologies described in Fig. 1. Given a partition: C ¼ fC1 ; C2 ; C3 g : C1 ¼ f1; 2g, C2 ¼ f3; 4; 5g; C3 ¼ f6; 7; 8g. It is clear that both G1 and G2 have group spanning trees, where the roots of C1 ; C2 and C3 in G1 are 1, 3 and 3 respectively, and the roots of C1 ; C2 and C3 in G2 are 16 and 6 respectively. Moreover, the zero indegree groups of G1 are C1 and C2 , while the zero in-degree groups of G2 are C1 and C2 ⋃C3 . If all the groups in the topology GðAÞ have zero in-degree, then there is no information exchange between each other, so they can be considered separately. Therefore, we assume that there are vertexes which belong to nonzero in-degree groups, that is, lm o n. Remark 1. For each agent in nonzero in-degree group under group spanning trees topology, there exists at least one agent in zero in-degree groups that has a directed path to it, therefore the definition of group spanning trees can be regarded as a generalization of that of united spanning tree [33] or spanning forest [34]. Based on the classification presented above, we study the group consensus problem for system (1). For convenience, define a new matrix Aðt k Þ ¼ ðaij ðt k ÞÞ as: ð1Þ if N i ðt k Þ a ∅, 8 w ðt Þ < P ij k j A N i ðt k Þ w ðt Þ s A N i ðt k Þ is k ; aij ðt k Þ ¼ :0 j2 = N i ðt k Þ ð2Þ if N i ðt k Þ ¼ ∅, ( 0 jai aij ðt k Þ ¼ 1 j¼i Then make the following assumptions about Aðt k Þ and τij ðt k Þ: ðB1Þ the matrix Aðt k Þ has inter-group common influence for any k; ðB2Þ there exists a constant e 4 0 such that either aij ðt k Þ ¼ 0 or aij ðt k Þ Z e holds for any i; j; k;

Lemma 1 (Xiao and Wang [24]). If there exists T Z 0 such that for all t Z 0, the union of graph GðtÞ across time interval ½t; t þ T contains a spanning tree, then the time-delayed system (1) satisfying assumptions ðB2Þ  ðB3Þ solves the consensus problem. 3.1. Systems with fixed topology and time-invariant delays As a first step toward system (1), we first consider the case that topology GðtÞ is fixed and time delays are invariant, in which both of assumptions ðB2Þ and ðB3Þ hold due to the limited number of agents and invariant time delays in the system. Under the circumstance, system (1) can be written as x_ i ðtÞ ¼ P

1

X

j A N i wij j A N

wij ðxj ðt k τij Þ xi ðtÞÞ; i ¼ 1; 2; …; n:

ð2Þ

i

From 2, we have xi ðt k þ 1 Þ ¼ e  h xi ðt k Þ þð1 e  h Þ

n X

aij xj ðt k  τij Þ:

ð3Þ

j¼1

Remark 2. When system (2) is free of time delays (that is τij  0), it follows from Eq. (3) that xi ðt k þ 1 Þ ¼ e  h xi ðt k Þ þð1 e  h Þ

n X

aij xj ðt k Þ:

ð4Þ

j¼1

We let the state vector of ith zero in-degree group denoted by x Ti ; i ¼ 1; 2; …; m, and the state vector of nonzero in-degree group denoted by x Tm þ 1 . In detail, x 1 ¼ ðx1 ; x2 ; …; xl1 ÞT ; x 2 ¼ ðxl1 þ 1 ; xl1 þ 2; …; xl2 ÞT ; …; x m þ 1 ¼ ðxlm þ 1 ; xlm þ 2 ; …; xn ÞT , then by rearranging the indices of the n agents, Eq. (4) can be rewritten as 1 10 0 1 0 x 1 ðt k þ 1 Þ x 1 ðt k Þ ⋯ 0 0 A~ 11 C CB B C B ⋮ ⋮ C CB B C B ⋮ ⋱ ⋮ ⋮ C CB B C¼B CB x m ðt k Þ C; B x m ðt k þ 1 Þ C B 0 0 ⋯ A~ mm A A@ @ A @ x m þ 1 ðt k þ 1 Þ x m þ 1 ðt k Þ A~ m þ 1;1 ⋯ A~ m þ 1;m A~ m þ 1;m þ 1 ð5Þ where A~ ii ¼ e  h I þ ð1  e  h ÞAii ; Aii A R is the adjacency matrix corresponding to the ith group, i ¼ 1; 2; …; m þ 1; A~ m þ 1;j ¼ ð1  e  h Þ Am þ 1;j ; j ¼ 1; 2; …; m. Using the notation xðt k Þ ¼ ðx T1 ðt k Þ; …; x Tm ðt k Þ; x Tm þ 1 ðt k ÞÞT , system (5) can be written in matrix form as ni ni

~ xðt k þ 1 Þ ¼ Axðt k Þ:

ð6Þ

For system (2) with time delays, we denote a finite set ΓðAÞ by ΓðAÞ ¼ fB ¼ ðbij Þ A Rnn : bij ¼ aij

or

bij ¼ 0g:

Since the dynamics of agents in each zero in-degree group are influenced only by the agents in the same group, it follows from Eq. (3) that the state vector of ith zero in-degree group x i ; i ¼ 1; 2; …; m, satisfies x i ðt k þ 1 Þ ¼ ½e  h I þ ð1  e  h ÞAii;0 x i ðt k Þ þ ð1  e  h ÞAii;1 x i ðt k  1 Þ þ ⋯ þð1  e  h ÞAii;^τ x i ðt k  τ^ Þ;

ð7Þ

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where Aii;τ A ΓðAii Þ; i ¼ 1; 2; …; m; τ ¼ 0; 1; …; τ^ , corresponding to the part of dynamics of ith zero in-degree group with time delay τ, P satisfies ττ^ ¼ 0 Aii;τ ¼ Aii , and Aii is the adjacency matrix of ith zero in-degree group. Define new state vectors as follows: y 1 ðkÞ ¼ ðx T1 ðt k Þ; x T1 ðt k  1 Þ; …; x T1 ðt k  τ^ ÞÞT ; y 2 ðkÞ ¼ ðx T2 ðt k Þ; x T2 ðt k  1 Þ; …; x T2 ðt k  τ^ ÞÞT ; ⋯

Then for y i ; i ¼ 1; 2; …; m, it follows from (7) that y i ðk þ 1Þ ¼ Ωii y i ðkÞ;

ð8Þ

where B B B Ωii ¼ B B B @

e  h I þð1  e  h ÞAii;0 I

ð1 e  h ÞAii;1 0

⋯ ⋯

ð1 e  h ÞAii;^τ  1 0

0 ⋮ 0

I ⋮ 0

⋯ ⋱ ⋯

0 ⋮ I

Proposition 1 can be viewed as an extension of proposition 3 in [24] to the group consensus, the proof can be derived in a similar way accordingly. Also, taking a similar line to the proof of Theorem 4.1 in [29], we can get that ~ has group spanLemma 2. GðAÞ has group spanning trees ⟺GðAÞ ning trees ⟺GðΩÞ has group spanning trees, where A~ and Ω are, respectively, defined in (6) and (12).

y m þ 1 ðkÞ ¼ ðx Tm þ 1 ðt k Þ; x Tm þ 1 ðt k  1 Þ; …; x Tm þ 1 ðt k  τ^ ÞÞT :

0

Proposition 1. Continuous-time system (2) solves the group consensus problem if and only if discrete-time system (12) solves the group consensus problem.

1 ð1  e  h ÞAii;^τ C 0 C C C: 0 C C A ⋮ 0

Since, from Proposition 1, the research of discrete-time system (12) is adequate for the investigation of continuous-time system (2), we present some results concerning discrete-time system for later use. For the discrete-time system xðt þ 1Þ ¼ AxðtÞ;

ð14Þ

where the diagonal entries of A are allowed to be zeros, the following results on normal consensus have been derived.

Obviously, Ωii is a stochastic matrix with ð^τ þ 1Þ  ð^τ þ1Þ blocks, i ¼ 1; 2; …; m. The dynamics of agents in nonzero in-degree group are influenced not only by the agents in the same group, but also by the agents in the m zero in-degree groups, without or with time P delays. For y m þ 1 , let F~ i;τ A ΓðA~ m þ 1;i Þ and τ ¼ 0 τ F~ i;τ ¼ A~ m þ 1;i ; i ¼ 1; 2; …; m; τ ¼ 0; 1; …; τ^ , where A~ m þ 1;i ¼ ð1 e  h ÞAm þ 1;i has the same meaning with that in (5). Besides, let F~ m þ 1;τ A Γðð1  e  h Þ Am þ 1;m þ 1 Þ where Am þ 1;m þ 1 is the adjacency matrix of nonzero P in-degree group, and e  h I þ ττ ¼ 0 F~ m þ 1;τ ¼ e  h I þ ð1  e  h Þ ~ Am þ 1;m þ 1 ¼ A m þ 1;m þ 1 means the same as that in (5). Further, denote Ωm þ 1;i as 0 1 F~ i;0 F~ i;1 ⋯ F~ i;^τ  1 F~ i;^τ B C B 0 0 ⋯ 0 0 C C; ð9Þ Ωm þ 1;i ¼ B B ⋮ ⋮ ⋱ ⋮ ⋮ C @ A 0 0 ⋯ 0 0

Lemma 3 (Xiao and Wang [35] and Xiao et al. [36]). Let A be a stochastic matrix. If GðAÞ has a spanning tree with the property that the root vertex of the spanning tree has self-loop in GðAÞ, then system (14) solves a consensus problem. Besides, λ ¼ 1 is an algebraically (and hence geometrically) simple eigenvalue of A, and is the unique eigenvalue of maximum modulus. Furthermore, if system (14) solves a consensus problem, then there exists a nonnegative left eigenvector f A Rn of A associated with eigenvalue 1 such that limt-1 At ¼ T T 1n f and f 1n ¼ 1.

and denote Ωm þ 1;m þ 1 as 0 h e I þ F~ m þ 1;0 B I B B Ωm þ 1;m þ 1 ¼ B 0 B B @ ⋮

Theorem 1. Suppose that stochastic matrix A has inter-group common influence, and GðAÞ has group spanning trees, then system (2) solves the group consensus problem.

0

F~ m þ 1;1 0

⋯ ⋯

F~ m þ 1;τ  1 0

I

⋯

0

⋮

⋱

⋮

0

⋯

I

1

F~ m þ 1;τ C 0 C C 0 C C; C ⋮ A 0 ð10Þ

then we have y m þ 1 ðk þ 1Þ ¼

m þ1 X

Ωm þ 1;i y i ðkÞ:

ð11Þ

i¼1

To sum up, let yðkÞ ¼ ðy T1 ðkÞ; y T2 ðkÞ; …; y Tm þ 1 ðkÞÞT , then combining (8) with (11), we can obtain that yðk þ 1Þ ¼ ΩyðkÞ; where 0 B B Ω¼B B @

ð12Þ

Ω1;1

⋯

0

0

⋮

⋱

⋮

⋮

0

⋯

Ωm;m

0

Ωm þ 1;1

⋯

Ωm þ 1;m

Ωm þ 1;m þ 1

1 C C C: C A

ð13Þ

Because Aii is stochastic, so is Ω, hence system (12) can be regarded as a discrete-time multi-agent system with ð^τ þ 1Þn agents with the associated communication topology denoted by GðΩÞ, we further present the following proposition for it.

Lemma 4. Suppose that GðAÞ has group spanning trees, then ρðΩm þ 1;m þ 1 Þ o 1 holds in (13). The proof follows from the definition of group spanning trees and Lemma 3, and is omitted here. We now state the result in the case of multi-agents systems with fixed topology and time-invariant delays.

Proof. We first show that the agents in each zero in-degree group can reach consensus. Noticing that the corresponding topology with each zero in-degree group has a spanning tree, and assumptions ðB1Þ  ðB3Þ hold in the case of fixed topology and timeinvariant delays, it then follows from Lemma 1 that system (2) solves the consensus problem. Correspondingly, the augmented system (8) which holds the same consensus property with system (2) solves the consensus problem. More specifically, given initial values y 1 ð0Þ; y 2 ð0Þ; …; y m ð0Þ, there exists a nonnegative left eigenvector f i A Rð^τ þ 1Þni of Ωii associated with eigenvalue λ ¼ 1 such that T T limk-1 y i ðkÞ ¼ 1ð^τ þ 1Þni f i y i ð0Þ and f i 1ð^τ þ 1Þni ¼ 1. Next, we prove that the agents in nonzero in-degree group can reach group consensus. From the above statement and Lemma 4, we know that Ω has an eigenvalue λ¼ 1 with both algebraic multiplicity and geometrical multiplicity equal to m, and all the other eigenvalues of Ω satisfy j λj o 1. Let wr1 ¼ ð1ð^τ þ 1Þn1 ; 0ð^τ þ 1Þn2 ; …; 0ð^τ þ 1Þnm ; ðI  Ωm þ 1;m þ 1 Þ  1 Ωm þ 1;1 1ð^τ þ 1Þn1 ÞT ;

⋯ wrm ¼ ð0ð^τ þ 1Þn1 ; 0ð^τ þ 1Þn2 ; ⋯; 1ð^τ þ 1Þnm ; ðI  Ωm þ 1;m þ 1 Þ  1 Ωm þ 1;m 1ð^τ þ 1Þnm ÞT ;

it can be verified that wri ; i ¼ 1; …; m; are m right eigenvectors of Ω associated with eigenvalue λ ¼ 1. Similarly, it can also be proved T T that wl1 ¼ ðf 1 ; 0; …; 0; 0ÞT ; …; wlm ¼ ð0; 0; …; f m ; 0ÞT are m left eigenvectors of Ω associated with eigenvalue λ ¼1. As a result, Ω can be

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written in Jordan canonical form as 0 B B Ω ¼ WJW  1 ¼ wr1 ; …; wrm ; wr;m þ 1 ; …; wrn B B @

1 ⋱



1

0 1 wTl1 1B C B ⋮ C C CB T B CB wlm C C CB C; CB wT AB l;m þ 1 C C C J1 B @ ⋮ A T wln

where wri and wli ; i ¼ m þ 1; …; n, are the right and left eigenvectors or generalized eigenvectors of Ω corresponding to the eigenvalues j λj o1, respectively, and J 1 is the Jordan upper diagonal block matrix corresponding to the eigenvalues j λj o 1. Then for system (12), we have 0 B B B B lim yðkÞ ¼ lim Ωk yð0Þ ¼ B B k-1 k-1 B @

T

⋯

0

1ð^τ þ 1Þn2 f 2

⋯

0

⋮ 0

⋮ 0

⋱ ⋯

⋮ 1ð^τ þ 1Þnm f m

Θm þ 1;1 ð1Þ

Θm þ 1;2 ð1Þ

⋯

Θm þ 1;m ð1Þ

1ð^τ þ 1Þn1 f 1

0

0

T

0

T

1

C 0C C C ⋮ Cyð0Þ C 0C A 0

where Θm þ 1;i ð1Þ ¼ ðI Ωm þ 1;m þ 1 Þ  1 Ωm þ 1;i 1ð^τ þ 1Þni f i ; T

i ¼ 1; …; m:

Accordingly, the final state of agents in nonzero in-degree group is lim y m þ 1 ðkÞ ¼

k-1

m X

ðI  Ωm þ 1;m þ 1 Þ  1 Ωm þ 1;i 1ð τ^ þ 1Þni f i y i ð0Þ: T

ð15Þ

i¼1

As shown in Lemma 4, ρðΩm þ 1;m þ 1 Þ o1 holds, which in turn implies that the matrix ðI  Ωm þ 1;m þ 1 Þ  1 is well defined. Note that 1 0 1 0 10 1ni F~ i;0 F~ i;1 ⋯ F~ i;^τ  1 F~ i;^τ A~ m þ 1;i 1ni C B C B CB B 1n C B C B 0 0 ⋯ 0 0 C 0 C: CB i C ¼ B Ωm þ 1;i 1ð^τ þ 1Þni ¼ B B C B C B ⋮ C ⋮ ⋮ ⋱ ⋮ ⋮ A@ ⋮ A @ A @ 1ni 0 0 ⋯ 0 0 0 ð16Þ After some computation, we can get that lim y m þ 1 ðkÞ ¼ 1τ^ þ 1 k-1 " m X  ðI  A~ m þ 1;m þ 1 Þ  1 A~

#

5

Q is a nonnegative matrix with the property that all its row sums are 1, then the conclusion follows directly from Eq. (17) and Definition (2). Remark 4. In [18], Han et al. have proved that the discrete-time system xðt þ 1Þ ¼ AxðtÞ solves the group consensus problem if GðAÞ has group spanning trees and every vertex in GðAÞ has a self-loop. From Lemma 3, the second condition can be relaxed to that only the root vertexes of the zero in-degree groups have self-loops in GðAÞ. Furthermore, we can also show that, for the above discretetime system, the agents in nonzero in-degree group finally converge to the convex hull of those in zero in-degree groups. 3.2. Systems with switching topologies and time-varying delays We now consider the case with switching topologies and timevarying delays, as described in system (1). It can be noted that GðtÞ and Gðt k Þ denote the same communication graph when t A ½t k ; t k þ 1 Þ, so we use Gðt k Þ to represent the unchanging communication topology during the time interval ½t k ; t k þ 1 Þ. Because of the limited number of vertexes in Gðt k Þ, all the possible communication topologies constitute a finite set, denoted by Γ ¼ fG1 ; G2 ; …; Gl g. Given each pair of groups: Cp and Cq , p; q ¼ 1; …; K, each graph in Γ should satisfy the following assumptions: (C1) either there is no edge connecting Cp and Cq ; (C2) or for each vertex in Cp and each graph Gi ; i ¼ 1; …; l, there is at least an edge from Cq to it. Following the same analysis with the case of fixed topology and time-invariant delays, system (1) can be written as yðk þ1Þ ¼ Ωðt k ÞyðkÞ; where

0

B B Ωðt k Þ ¼ B B @

ð18Þ

Ω1;1 ðt k Þ

⋯

0

0

⋮

⋱

⋮

⋮

1

0

⋯

Ωm;m ðt k Þ

0

Ωm þ 1;1 ðt k Þ

⋯

Ωm þ 1;m ðt k Þ

Ωm þ 1;m þ 1 ðt k Þ

C C C: C A

ð17Þ

Note that, given time tk, Ωðt k Þ in (18) and Ω in (12) have the same form. We can, therefore, have the following result.

Since A has inter-group common influence, it can be verified that ðI  A~ m þ 1;m þ 1 Þ  1 A~ m þ 1;i ¼ ðI Am þ 1;m þ 1 Þ  1 Am þ 1;i has inter-group T common influence. Also note that ΔG ð1ni f i y i ð0ÞÞ ¼ 0; it then follows from Eq. (17) that

Theorem 2. If there exists T Z 0 such that for all t Z 0, the union of graph GðtÞ across time interval ½t; t þ T contains group spanning trees, then the time-delayed system (1) satisfying assumptions (B1)–(B3) and (C1)–(C2) solves the group consensus problem.

lim ΔG ðy m þ 1 ðkÞÞ ¼ 0:

Proof. For the agents in each zero in-degree group, it follows from Lemma 1 that y i ðk þ1Þ ¼ Ωii ðt k Þy i ðkÞ, i ¼ 1; 2; …; m, solves the consensus problem. We next show that the agents in nonzero indegree group can achieve group consensus. Let

T

m þ 1;i 1ni f i y i ð0Þ :

i¼1

k-1

In other words, the agents in nonzero in-degree group achieve group consensus. To sum up, system (2) solves the group consensus problem.□ Remark 3. In fact, we can not only prove that system (2) solves the group consensus problem, but also prove that the agents in nonzero in-degree group finally converge to the convex hull of those in zero in-degree groups, and the coefficients of the convex combination are given by ðI  A~ m þ 1;m þ 1 Þ  1 ðA~ m þ 1;1 ; …; A~ m þ 1;m Þ 9 Q : P ~i In detail, since ðI  A~ m þ 1;m þ 1 Þ  1 ¼ 1 i ¼ 0 A m þ 1;m þ 1 is a nonnegative matrix, Q is a nonnegative matrix. Also noticing that Q 1lm ¼ ðI  A~ m þ 1;m þ 1 Þ ðA~ m þ 1;1 1n1 þ A~ m þ 1;2 1n2 þ ⋯ þ A~ m þ 1;m 1nm Þ ¼ ðI  A~ m þ 1;m þ 1 Þ  1 ðI  A~ m þ 1;m þ 1 Þ1n 1

mþ1

¼ 1nm þ 1 ;

0

B s B Φðs; vÞ ¼ ∏ Ωðt k Þ ¼ B B @ k¼v

Φ1;1 ðs; vÞ ⋮ 0

⋯ ⋱ ⋯

Φm þ 1;1 ðs; vÞ

⋯

1

0 ⋮

0 ⋮

Φm;m ðs; vÞ Φm þ 1;m ðs; vÞ

0 Φm þ 1;m þ 1 ðs; vÞ

C C C; C A

then we have y m þ 1 ð2vÞ ¼

m X

½Φm þ 1;i ð2v; vÞΦii ðv; 1Þ þ Φm þ 1;m þ 1 ð2v; vÞΦm þ 1;i ðv; 1Þy i ð0Þ

i¼1

þ Φm þ 1;m þ 1 ð2v; 1Þy m þ 1 ð0Þ;

ð19Þ

it follows that ΔG ðy m þ 1 ð2vÞÞ r

m X

ΔG ðΦm þ 1;i ð2v; vÞΦii ðv; 1Þy i ð0ÞÞ

i¼1

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6

þ

m X

þ Φm þ 1;m þ 1 ð2v; 1Þy m þ 1 ð0Þ 

ΔG ðΦm þ 1;m þ 1 ð2v; vÞΦm þ 1;i ðv; 1Þy i ð0ÞÞ

i¼1

ð20Þ

ΔG ðΦii ðv; 1Þy i ð0ÞÞ þ b

i¼1

m X

ð21Þ

lim Φii ðv; 1Þ ¼ 1ðτ^ þ 1Þni ωTi :

ð22Þ

In addition, since Ωðt k Þ in (18) and Ω in (12) have the same form at any time tk and system (1) satisfies assumptions (C 1)–(C 2), following a similar proof with that of Lemma 4, we can get that lim Φm þ 1;m þ 1 ðv; 1Þ ¼ lim Φm þ 1;m þ 1 ð2v; vÞ ¼ 0:

ð23Þ

v-1

Note that ΔG ð1ð^τ þ 1Þni ωTi Þ ¼ 0, it follows from (21) that limv-1 ΔG ðy m þ 1 ð2vÞÞ ¼ 0, which implies that the agents in nonzero in-degree group finally achieve group consensus.□ Remark 5. For system (1) in Theorem 2, the agents in nonzero indegree group can be proved to finally converge to the convex hull of those in zero in-degree groups. It follows from (23) that for any ε 4 0, there exists t1 such that J Φm þ 1;m þ 1 ð2v; vÞ J o ε holds when v 4 t 1 . Similarly, J Φm þ 1;m þ 1 ð2v; 1Þ J o ε holds when v 4 t 1 . For the above ε, it then follows from (22) that there exists t2 such that J Φii ðv; 1Þ  1ð^τ þ 1Þni ωTi J o ε when v 4 t 2 . Consequently, let β ¼ maxi ¼ 1;…;m þ 1 J y i ð0Þ J , for any t 4 max ft 1 ; t 2 g, we have that Φm þ 1;i ð2v; vÞ1ðτ^ þ 1Þni ωTi y i ð0Þ J

m X

Φm þ 1;i ð2v; vÞΦii ðv; 1Þy i ð0Þ þ

i¼1

m X

o 2mbβε þ βε:

lim J y m þ 1 ð2vÞ 

m X

Φm þ 1;i ð2v; vÞ1ð^τ þ 1Þni ωTi y i ð0Þ J ¼ 0:

ð24Þ

i¼1

On one hand, for i ¼ 1; 2; …; m, it can be noted from (22) that lim y i ð2vÞ ¼ lim Φii ð2v; 1Þy i ð0Þ ¼ 1ð^τ þ 1Þni ωTi y i ð0Þ:

v-1

v-1

On the other hand, Φð2v; vÞ is obviously a stochastic matrix for any v. Thus, by (23), we have that m X

lim

v-1

Φm þ 1;i ð2v; vÞ  1ð^τ þ 1Þlm ¼ 1ð^τ þ 1Þlm :

i¼1

To sum up, it follows from (24) that the conclusion comes into existence. Remark 6. Theorem 1 can be viewed as a special case of Theorem 2, although for the case with switching topologies and timevarying delays, the coefficients of the convex combination might be time-varying (see Example 3 in the Simulations Section), which is different from the case with fixed topology and time-invariant delays. 4. Simulations In this section, numerical examples are presented to illustrate the validity of the proposed theoretical results. We consider a multi-agent system consisting of eight agents, which is divided into three groups: C1 ¼ f1; 2g; C2 ¼ f3; 4; 5g; C3 ¼ f6; 7; 8g.

i¼1

¼J

J Φm þ 1;m þ 1 ð2v; vÞ J J Φm þ 1;i ðv; 1Þ J J y i ð0Þ J þ J Φm þ 1;m þ 1 ð2v; 1Þ J J y m þ 1 ð0Þ J

i¼1

v-1

v-1

J y m þ 1 ð2vÞ 

J Φm þ 1;i ð2v; vÞ J J y i ð0Þ J J Φii ðv; 1Þ  1ð^τ þ 1Þni ωTi J

That is,

i¼1

According to Lemma 1, there exists ωi A Rð^τ þ 1Þni such that

m X

m X

m X

þ

ΔG ðΦm þ 1;m þ 1 ð2v; vÞy i ð0ÞÞ

þΔG ðΦm þ 1;m þ 1 ð2v; 1Þy m þ 1 ð0ÞÞ:

v-1

r

i¼1

One can find a constant b4 0 such that both J Φm þ 1;i ð2v; vÞ J o b and J Φm þ 1;i ðv; 1Þ J o b hold for any v, thus m X

Φm þ 1;i ð2v; vÞ1ð^τ þ 1Þni ωTi y i ð0Þ J

i¼1

þ ΔG ðΦm þ 1;m þ 1 ð2v; 1Þy m þ 1 ð0ÞÞ:

ΔG ðy m þ 1 ð2vÞÞ r b

m X

Φm þ 1;m þ 1 ð2v; vÞΦm þ 1;i ðv; 1Þy i ð0Þ

i¼1

Example 2. For the continuous-time system (1) with time-varying delays, the topologies are changing between the topologies shown in Fig. 2(a) and (b). While neither of Fig. 2(a) and (b) has group spanning trees, their union has group spanning trees and the roots of groups C1 ; C2 and C3 are 1; 3 and 3, respectively. The above figures also satisfy the assumptions ðC1Þ ðC2Þ. Under assumption ðB1Þ, it is easy to see that G ¼ ðςp;q Þ A RKK is a stochastic matrix. We take h ¼ 0:2; τ^ ¼ 2, and matrix G as 0 1 1 0 0 B0 1 0C Gðt k Þ ¼ @ A: 1 3

Fig. 2. Switching topologies G3 and G4 .

1 3

3

3.5 3

2.5

2.5

2

2 1.5

1.5

1

1

0.5 0

0.5

-0.5 -1

1 3

0

20

40

60 Time

80

100

120

0

0

20

40

60

80

100

120

Time

Fig. 3. State trajectories of agents and behavior of ΔG ðxÞ in Example 2.

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3.5

3

3

2.5

2.5 2

2

1.5

1.5

1 0.5

1

0

0.5

-0.5 -1

7

0

10

20

30

40

0

0

10

Time

20

30

40

Time

Fig. 4. State trajectories of agents and behavior of ΔG ðxÞ in Example 3.

Furthermore, we pick the time-varying delays τij ðkÞ as when k is odd;

τi1 ðt k Þ ¼ τi3 ðt k Þ ¼ τi5 ðt k Þ ¼ τi7 ðt k Þ ¼ 0:2; τi2 ðt k Þ ¼ τi4 ðt k Þ ¼ τi6 ðt k Þ ¼ τi8 ðt k Þ ¼ 0:4;

τi1 ðt k Þ ¼ τi3 ðt k Þ ¼ τi5 ðt k Þ ¼ τi7 ðt k Þ ¼ 0:4; τi2 ðt k Þ ¼ τi4 ðt k Þ ¼ τi6 ðt k Þ ¼ τi8 ðt k Þ ¼ 0:2:

when k is even;

Note that all conditions in Theorem 2 are satisfied. According to Theorem 2, system (1) solves the group consensus problem. Fig. 3 (a) and (b) show, respectively, the state trajectories of the agents and the behavior of group Hajnal diameter ΔG ðxÞ. Both of them indicate that the group consensus is reached asymptotically, and Fig. 3(a), further, shows that the agents in nonzero in-degree group finally converge to the convex hull of those in zero indegree groups. Example 3. For system (1) with time delays, take the same switching topologies as those in Example 2. Here, we consider the case that the matrix C depends on time. Specifically, when k is odd, we take the same Gðt k Þ as that in Example 2; when k is even, we pick the matrix Gðt k Þ as 0 1 1 0 0 B 0 1 0C Gðt k Þ ¼ @ A: 4 15

2 5

1 3

Besides, we choose τij ðkÞ as τi1 ðt k Þ ¼ τi3 ðt k Þ ¼ τi5 ðt k Þ ¼ τi7 ðt k Þ ¼ 0:2; τi2 ðt k Þ ¼ τi4 ðt k Þ ¼ τi6 ðt k Þ ¼ τi8 ðt k Þ ¼ 0:4: The state trajectories of the agents are then shown in Fig. 4(a), it can be seen that the group consensus is achieved, and the agents in nonzero in-degree group, in particular, converge to the convex hull of those in zero in-degree groups. Note that the coefficients of the convex combinations are not fixed. In Fig. 4(b), the behavior of group Hajnal diameter ΔG ðxÞ is plotted, whose convergence to zero further back up the group consensus. 5. Conclusion In this paper, we discuss the group consensus control with time delays for continuous-time multi-agent systems. The group consensus protocol is implemented based on the state information of each agent's neighbors at some discrete times. More specifically, we divide the agents under group spanning trees into zero indegree groups and nonzero in-degree groups, then by employing graph theory and matrix theory, we derive some criteria for group

consensus in both systems with fixed topology and time-invariant delays, and systems with switching topologies and time-varying delays. In both scenarios considered above, it is shown that the nonzero in-degree groups finally converge to convex combination of the consensus states of the zero in-degree groups. In the future, we will consider more general interaction topology and try to extend the work to asynchronous case and high-order multi-agent systems.

Acknowledgment This research is supported by NSFC (61403064,61170309) and the Fundamental Research Funds for the Central Universities (ZYGX2013Z005). References [1] R. Olfati-Saber, R. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49 (2004) 1520–1533. [2] J. Qin, H. Gao, C. Yu, On discrete-time convergence for general linear multiagent systems under dynamic topology., IEEE Trans. Autom. Control 59 (4) (2014) 1054–1059. [3] W. Ren, R. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (5) (2005) 655–661. [4] J. Qin, W. Zheng, H. Gao, Consensus of multiple second-order vehicles with a time-varying reference signal under directed topology, Automatica 47 (9) (2011) 1983–1991. [5] J. Hu, Y. Lin, Consensus control for multi-agent systems with double-integrator dynamics and time delays, IET Control Theory Appl. 4 (1) (2010) 109–118. [6] J. Qin, H. Gao, W. Zheng, Second-order consensus for multi-agent systems with switching topology and communication delay, Syst. Control Lett. 60 (6) (2011) 390–397. [7] X. Feng, T. Huang, J. Shao, Several consensus protocols with memory of multiagent systems, Math. Comput. Model. 58 (2013) 1625–1633. [8] J. Hu, Y. Hong, Leader-following coordination of multi-agent systems with coupling time delays, Phys. A 374 (2) (2007) 853–863. [9] H. Xia, T. Huang, J. Shao, J. Yu, Leader-following formation control for secondorder multi-agent systems with time-varying delays, Trans. Inst. Meas. Control 36 (5) (2014) 627–636. [10] J. Qin, C. Yu, Exponential consensus of general linear multi-agent systems under directed dynamic topology, Automatica 50 (9) (2014) 2327–2333. [11] Y. Chen, J. Lu, On the cluster consensus of discrete-time multi-agent systems, Syst. Control Lett. 60 (3) (2011) 517–523. [12] J. Yu, L. Wang, Group consensus in multi-agent systems with undirected communication graphs, in: Proceedings of the 7th Asian Control Conference, Hong Kong, 2009, pp. 105–110. [13] J. Yu, L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Syst. Control Lett. 59 (6) (2010) 340–348. [14] J. Qin, C. Yu, Cluster consensus control of generic linear multi-agent systems under directed topology with acyclic partition, Automatica 49 (2) (2013) 2898–2905. [15] D. Xie, Q. Liu, Necessary and sufficient condition for the group consensus of multi-agent systems, Appl. Math. Comput. 243 (15) (2014) 870–878. [16] L. Ji, Q. Liu, X. Liao, On reaching group consensus for linearly coupled multiagent networks, Inf. Sci. 287 (10) (2014) 1–12.

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Ting-Zhu Huang received the B.S., M.S., and Ph.D. degrees in Computational Mathematics from the Department of Mathematics, Xi’an Jiaotong University, Xi’an, China, in 1986, 1992 and 2001, respectively. During 2005, he was a visiting scholar in Department of Computer Science, Loughborough University of UK. He is currently a Full Professor in the School of Mathematical Sciences, UESTC. He is currently an editorial of Advances in Numerical Analysis, Journal of Pure and Applied Mathematics: Advances and Applications, Journal of Electronic Science and Technology of China. He is the author or coauthor of more than 100 research papers. His current research interests include numerical linear algebra and scientific computation with applications, iterative methods of linear systems, numerical algorithms for image processing, preconditioning technologies, and matrix analysis with applications.

Jin-Liang Shao was born in Shanxi Province, China, in 1981. He received the B.Sc. and Ph.D. degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2003 and 2009, respectively. He is currently an associate professor with UESTC. His research interests include multi-agent system, robust control, and matrix analysis with applications in control theory.

Jun-Yan Yu received B.S. degree and M.E. degree from Department of Mathematics, Zhengzhou University, Zhengzhou, China, in 2002 and 2005, respectively, and Ph.D. degree from the Center for Systems and Control, College of Engineering, Peking University, Beijing, China. She is presently working in School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China. Her research interests include hybrid and switched systems, networked control systems and multi-agent systems.

Hong Xia received the B.S. degree in Mathematics from Shandong Normal University in 2003, M.S. degree in Mathematics from Wuhan University in 2006, and Ph. D. degree in Mathematics from University of Electronic Science and Technology in 2014. Her research interests include multi-agent system, matrix analysis with applications.

Please cite this article as: H. Xia, et al., Group consensus of multi-agent systems with communication delays, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.07.108i