Group Consensus of Second-Order Dynamic Multi-Agent Systems with Time-Varying Communication Delays and Directed Networks

Shuanghe Meng; Lü Xu; Liang Chen

doi:10.21078/JSSI-2016-258-11

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Group Consensus of Second-Order Dynamic Multi-Agent Systems with Time-Varying Communication Delays and Directed Networks

Shuanghe Meng , Lü Xu and Liang Chen

Published/Copyright: June 25, 2016

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From the journal Journal of Systems Science and Information Volume 4 Issue 3

Abstract

This paper studies the group consensus problem for second-order multi-agent dynamic systems with time-varying delays, where the agents in a network may reach one more consistent values asymptotically. The fixed network topology is in case of being directed and weakly connected. Based on algebraic graph theory and Lyapunov function approach, we propose some sufficient conditions for reaching group consensus. All the results are presented in the form of linear matrix inequalities(LMIs). A simulation example is provided to demonstrate the effectiveness of the theoretical analysis.

Keywords: group consensus; second-order multi-agent systems; directed network; time-varying communication delays; LMIs

1 Introduction

In recent years, we have witnessed the rapid development and growing interest in cooperative control of multi-agent systems. Due to its extensive applications in unmanned aerial vehicle formation, robot formation control rendezvous in space, and other areas, many research have been focused on this project. In these applications, the states of all the agents are demand of reaching an agreement on some consistent value, which is called the consensus issue.

Consensus problem has been one of the vital issues in coordination control of multi-agent systems. By the pilot research work of Reynolds[1] and Viseck[2], periodical progress of the multi-agent system distributed coordination control has been made. Most of the researchers have studied consensus problems based on the point that how to design a suitable consensus algorithm or protocol. Olfati-Saber and Murray firstly introduced the theoretical framework for networked dynamic systems about the problems of solving consensus problems[3]. Many problems are related to consensus problem for multi-agent systems, for instance, collective behavior of flocks and swarms, formation control of multi-robots, and synchronization of coupled oscillators[4]. In [5–7], the consensus problems of multi-agent systems with first-order dynamics have been investigated from manifold perspectives. Due to the practical applications where multi-agent systems are with second-order dynamics, the more complicated and challenging consensus problem has received a great deal of attention, where agents are governed by both position and velocity states. The dynamical consensus algorithm for the second-order multi-agent systems was proposed[8], where all agents achieved the same consensus value. Based on discrete-time multi-agent system, where each agent stopped in the same positions with zero velocity, the static consensus algorithm under directed switching topologies with nonuniform time delays was given[9]. Ren showed that the existence of a directed spanning tree is a necessary rather than a sufficient condition to reach second-order consensus by giving a set of algorithms[10]. In [11], Yu found that both the real and imaginary parts of the eigenvalues of the Laplacian matrix play key roles in reaching second-order consensus. Furthermore, a disagreement vector was introduced to analyze double-integrator dynamical multi-agent systems[12]. [13] studied the leader-following consensus problem of second-order multi-agent systems with switching and fixed topoplogies with non-uniform time-varying delays. High-order consensus of multi-agent systems attracted many researchers attention as well; see, for example, [14–19] and references there in.

Recently, a new concept of multi-agent group consensus problem, which differs from the classical consensus problem, has been pointed out and attracted lots and lots of attention within the control academical community[20–26]. In group consensus category, all agents in a multi-agent system are divided into multiple subgroups while each group will reach their own consensus values, which could be various for the changes of cooperative tasks and environment. At the side of traditional consensus problem, group consensus appears more complicated, which need some restriction and assumption. Yu and Wang firstly introduced group consensus to describe this kind of consensus[20]. The primary target of this issue is to desigh suitable consensus protocols or algorithms to make sure the states of the agents in each subgroup can reach the same value. In [20], Yu and Wang considered the situation with undirected topology under some assumptions, where algebratic condictions were derived and influence from any other group to the node is equal to zero. Later, the authors extended their research achievement to some other situations[21, 22]. [21] discussed the average group consensus problem for multi-agent systems with directed topology. [22] considered the switching topologies and time delays for group consensus multi-agent systems, where the agents are described by single-intergrator dynamics, by promoting double-tree-form transformation. Leader-following approach and pinning control methods were investigated in [23] as well as group consensus problem was solved within nonlinear dynamics. Xie and Liang[24] studied second-order group consensus with time delay by Lyapunov first method and Hopf bifurcation theory.

Motivated by the above discussions, we consider group consensus in second-order dynamic multi-agent system with communication time-varying delays. Firstly we extend and complement the group consensus results in [20] from single-order dynamic system to second-order dynamic system and obtain a consensus protocol based on group structure. Furthermore, the proposed protocol is extended from protocol with fixed time delay to protocol with communication time-varying delay. The Lyapunov function approach is used to analyze the group consensus problem for second-order multi-agent dynamical systems with communication time-varying delay, then a delay-dependent sufficient condition in form of LMI is given for reaching consensus such that we can judge the processed multi-agent system whether can reach a consensus by a feasibility testing of LMIs. The move has a significant significance for applying the theory to the practical applications due to the use of LMI technology.

The rest of this paper is organized as follows. In Section 2, we give some preliminaries and crucial formulation of group consensus system models. The main results of this paper are proposed in Section 3. Numerical simulations and conclusion are given in Sections 4 and 5, respectively.

2 Problem Formulation and Preliminaries

In this section, some basic concepts, results about graph theory are introduced and some problem formulations, definitions, assumptions and supporting lemmas are given to obtain the main results of the paper.

2.1 Graph Theory

The topology of a group multi-agent system is usually described by a graph. Let a weighted directed G(V, E, A) model the interaction communication between n + m agents, where V = {v₁, v₂, ⋯, v_n+m} is the set of nodes, ε ⊆ V × V is the set of edges, and A = [a_ij] ∈ R^{(n + m) × (n + m)} is a weighted adjacency matrix with real adjacency elements a_ij. The node indexes belong to a finite index set Γ = {1, 2, ⋯, n + m}. An edge of G is denoted by e_ij = (v_i, v_j). The adjacency elements associated with the edges of the graph are nonzero, i.e., e_ij ∈ ε (G) ⇔ a_ij ≠ 0. Moreover, we assume a_ii = 0 for all i ∈ Γ. The set of neighbors of the node v_i is denoted by N_i = {v_j ∈ V : (v_i, v_j) ∈ ε (G)}. The Laplacian matrix of topology G is defined by

lij=−aij,j≠i,∑k=1,k≠in+maik,j=i.

2.2 Problem Formulation

Suppose that the multi-agent system consists of n + m agents and each agent has the dynamics as follows:

x˙i(t)=vi(t),v˙i(t)=ui(t),(1)

where x_i ∈ R, v_i ∈ R denote the position and velocity of agent i, respectively, u_i ∈ R is control input.

Without loss of generality, we consider the case of two groups, which can be expand to multi-agent systems with more groups easily. Define Γ₁ = {1, 2, ⋯, n}, Γ₂ = {n + 1, n + 2, ⋯, n + m}, V₁ = {v₁, v₂, ⋯, v_n} and V₂ = {v_n+1, v_n+2, ⋯, v_n+m} as the index sets and node sets to describe the two groups. Then, Γ = Γ₁ ∪ Γ₂ and V = V₁ ∪ V₂, respectively. Furthermore, let N_1i = {v_j ∈ V₁ : (v_i, v_j) ∈ ε (G)}, N_2i = {v_j ∈ V₂ : (v_i, v_j) ∈ ε (G)} and N_i = N_1i ∪ N_2i be the neighbor sets of node v_i in V₁, V₂, and V, respectively. Considering the communication time delay of agents would never be fixed, we improve the protocol [24] and adopt the following control input:

ui(t)=−2kvi(t)+k∑j∈N1iaij[xj(t−τij(t))−xi(t−τij(t))]+k∑j∈N2iaijxj(t−τij(t)),i∈Γ1,−2kvi(t)+k∑j∈N2iaij[xj(t−τij(t))−xi(t−τij(t))]+k∑j∈N1iaijxj(t−τij(t)),i∈Γ2,(2)

where k > 0, τ_ij(t) > 0 means time-varying communication delay between agents. a_ij ≥ 0, ∀i, j ∈ Γ₁, a_ij ≥ 0, ∀ i, j ∈ {Γ₂, a_ij ∈ R, ∀ (i, j) ∈ I = {(i, j) : i ∈ Γ₁, j ∈ Γ₂} ∪ {(i, j) : j ∈ Γ₁, i ∈ Γ₂}. If a_ij < 0, we consider that agent j has a negative effect on agent i.

Definition 1

(Group consensus[20]) The multi-agent system (1) under protocol (2) is said to achieve second-order group consensus, if

(i)limt→+∞xi(t)−xj(t)=0,∀i,j∈Γ1,(ii)limt→+∞xi(t)−xj(t)=0,∀i,j∈Γ2,(iii)limt→+∞vi(t)=0,∀i∈Γ.(3)

Definition 2

(Group average-consensus[20]) The multi-agent system (1) under protocol (2) is said to achieve second-order group average-consensus, if

(i)limt→+∞xi(t)−xj(t)=1n∑j=1nxj(0),∀i,j∈Γ1,(ii)limt→+∞xi(t)−xj(t)=1m∑j=n+1n+mxj(0),∀i,j∈Γ2,(iii)limt→+∞vi(t)=0,∀i∈Γ.(4)

Assumption 1

We make the following propositions which means a balance of effect between two sub-networks:

∑j=n+1n+maij=0,∀i∈Γ1,∑j=1naij=0,∀i∈Γ2.(5)

Assumption 2

L has exactly two simple zero eigenvalues and all the other eigenvalues have positive real parts.

Assumption 3

The time-varying delays satisfy 0 ≤ τ_ij(t) ≤ τ and τ˙ij(t) ≤ d < 1 for some known constants τ and d.

Remark 1

Assumption 1 means the sum of adjacent weights from every node in one group to all nodes in another group is identical at each time while Assumption 2 make the protocol of the system realisable for group consensus of multi-agent dynamic systems. Furthermore, while the communication time delays among the sub-groups are often long and should not be ignored due to the long transporting line, Assumption 3 is reasonable from the practical application attitude.

3 Main Results

In this section we use state transformation method and establish theorem to solve the group consensus problem of second-order systems with time-varying delays in directed and fixed topology.

In order to achieve the objective, we defined some error vector. Denote a_i(t) = vi(t)k + x_i(t), b₁(t) = [x₂(t) − x₁(t), a₂(t) − a₁(t), ⋯, x_n(t) − x₁(t), a_n(t) − a₁(t)]^T, b₂(t) = [x_{n + 2}(t) − x_{n + 1}(t),a_{n + 2}(t) − a_{n + 1}(t), ⋯, x_{n + m}(t) − x_{n + 1}(t), a_n+m(t) − a_n+1 (t)]^T and δ(t)=[b1T(t)b2T(t)]T. By using reduced-order Laplacian method proposed in [24], we can rewrite the system (1) under protocol control (2) as

δ˙(t)=(In+m−2⊗A)δ(t)−(Lk⊗B)δ(t−τij(t)),(6)

where A=−kkk−k,B=0010,

Lk=l22−l12⋯l2n−l1nl2,n+2−l1,n+2⋯l2,n+m−l1,n+m⋮⋱⋮⋮⋱⋮ln2−l12⋯lnn−l1nln+2,2−l1,n+2⋯ln,n+m−l1,n+mln+2,2−ln+1,2⋯ln+2,n−ln+1,nln+2,n+2−ln+1,n+2⋯ln+2,n+m−ln+1,n+m⋮⋱⋮⋮⋱⋮ln+m,2−ln+1,2⋯ln+m,n−ln+1,nln+m,n+2−ln+1,n+2⋯ln+m,n+m−ln+1,n+m.

Lemma 1

(Liang, et al.[24]) L_K has the same eigenvalues as L except two zero eigenvalues.

Lemma 2

(Boyd, et al.[25]) For a given matrixS=S11S12S12TS22,where S₁₁and S₂₂are square matrices, the following conditions are equivalent:

S < 0;
S11<0,S22−S12TS11−1S12<0;
S22<0,S11−S12S22−1S12T<0.

Lemma 3

(Kwon, et al.[26]) For any given positive-definite matrix M ∈ R^n×nand a vector functionδ : [β, α] → Rⁿ, the following inequalities hold:

(α−β)∫βαδT(s)Mδ(s)ds≥(∫βαδ(s)ds)TM(∫βαδ(s)ds),(α−β)22∫βα∫sαδT(u)Mδ(u)duds≥(∫βα∫saδ(u)duds)TM(∫βα∫saδ(u)duds).

Theorem 1

Consider a second-order multi-agent system (1) and group consensus control protocol (2) satisfyingAssumptions 1, 2and3, all the nodes of each group with comminication time-varying delay reach group consensus, if there exist matrices T₁and T₂with proper dimensions, positive definite and symmetric matrices P, W, M, R, UandΘ=Θ11Θ12∗Θ22>0,such that the following LMIs hold:

Πp=ΠpT=Πij6×6<0,(7)

Θ11Θ12T1∗Θ22T2∗∗τK1TK2>0,(8)

where

Π11=K1TP+PK1+K2TPK2+K1TUK1+T1+T1T−(v−u)2R+τΘ11,Π22=−K2TPK2+K2TUK2−T2−T2T+τΘ22,Π33=−(v−u)2W−U,Π44=−(v−u)24W,Π55=−M,Π66=−R−W,Π13=(v−u)W,Π12=PK2+K1TUK2−T1+T2+τΘ22,K1=In+m−2⊗A,K2=Lk⊗B,

and the remaining terms ofΠ_ij are zeros.

Proof

Choose a Lyapunov-Krasovskii functional candidate as follows:

Vi(t)=∑i=16Vi(t),(9)

where

V1(t)=δT(t)Pδ(t),V2(t)=∫t−τij(t)tδT(s)K2TPK2δ(s)ds,V3(t)=∫t−utδ˙T(s)Uδ˙(s)ds,V4(t)=u∫−u0∫t+utδT(s)Mδ(s)dsdu,V5(t)=(v−u)∫−v−u∫t+utδT(s)Rδ(s)dsdu,V6(t)=(v−u)22∫t−vt−u∫ut−u∫vt−uδ˙T(s)Wδ˙(s)dsdvdu,

where v, u are constants.

Calculating the time derivative of V(t), we obtain

V˙1(t)+V˙2(t)=δ˙T(t)Pδ(t)+δT(t)Pδ˙(t)+δT(t)K2TPK2δ(t)+δT(t−τij(t))K2TPK2δ(t−τij(t))=δT(t)K1TP+PK1+K2TPK2δ(t)−δT(t−τij(t))K2TPK2δ(t−τij(t))+δT(t−τij(t))K2TPδ(t)+δT(t)PK2δ(t−τij(t)),(10)

V˙3(t)=−δ˙T(t−u)Uδ˙(t−u)+δ˙T(t)Uδ˙(t)=−δ˙T(t−u)Uδ˙(t−u)+δT(t)K1TUK1δ(t)+δT(t)K1TUK2δ(t−τij(t))+δT(t−τij(t))K2TUK1δ(t)+δT(t−τij(t))K2TUK2δ(t−τij(t)),(11)

V˙4(t)=−u∫t−utδT(s)Mδ(s)ds≤−∫t−utδ(s)dsTM∫t−utδ(s)ds,(12)

V˙5(t)=(v−u)2δT(s)Rδ(s)−(v−u)∫t−vt−uδT(s)Rδ(s)ds≤(v−u)2δT(s)Rδ(s)−∫t−vt−uδ(s)dsTR∫t−vt−uδ(s)ds,(13)

V˙6(t)=(v−u)24δ˙T(t−u)Wδ˙(t−u)−(v−u)22∫t−vt−u∫ut−uδ˙T(s)Wδ˙(s)dsdu≤(v−u)24δ˙T(t−u)Wδ˙(t−u)−(v−u)δ(t−u)−∫t−vt−uδ(u)duTW(v−u)δ(t−u)−∫t−vt−uδ(u)du.(14)

By the Leibniz-Newton formula, we have

∫t−τij(t)tδ˙(β)dβ=δ(t)−δ(t−τij(t)).(15)

Then, for any T₁ and T₂ with appropriate dimensions, we have

2δT(t)T1+δT(t−τij(t))T2δ(t)−∫t−τij(t)tδ˙(β)dβ−δT(t−τij(t))=0.(16)

Obviously, for any matrix θ ≥ 0, it holds that

τδ(t)δ(t−τij(t))TΘδ(t)δ(t−τij(t))≥∫t−τij(t)tδ(t)δ(t−τij(t))TΘδ(t)δ(t−τij(t))dβ.(17)

Combining Equations (10)–(17) yields

V˙i(t)≤δ(t)δ(t−τij(t))δ(t−u)δ˙(t−u)∫t−utδT(s)ds∫t−vt−uδT(s)dsTΠ11Π12Π13Π14Π15Π16∗Π22Π23Π24Π25Π26∗∗Π33Π34Π35Π36∗∗∗Π44Π45Π46∗∗∗∗Π55Π56∗∗∗∗∗Π66δ(t)δ(t−τij(t))δ(t−u)δ˙(t−u)∫t−utδT(s)ds∫t−vt−uδT(s)ds−∫t−τij(t)tδ(t)δ(t−τij(t))δ˙(β)TΘ11Θ12T1∗Θ22T2∗∗τK1TK2δ(t)δ(t−τij(t))δ˙(β).(18)

Then, by Lemma 2, Equations (7) and (8), we can get V˙(t) < 0. According to the definition of Lyapunov-Krasovskii functional candidate (9), system (6) is asymptotically stable, which implies that system (1) would achieve group consensus. This completes the proof.

Remark 2

For group consensus multi-agent systems with communication time-varying delay and directed topology, protocol (2) ensure couple-group consensus, which indicates the extension of the issue from single-integrator dynamics to double-integrator situations is not trival. Furthermore, it is indicated that both time delay bound and Laplacian matrix play key roles in reaching group consensus.

4 Numerical Example

In this section, to illustrate the effectiveness of the results proposed in previous section, we consider a directed graph with 6 agents indexed by 1 to 6, respectively, in which Agents 1 to 4 are in group I and Agents 5 and Agents 6 belong in group II as shown in Figure 1. Hence, the Laplacian matrix L of the system is given by

L=2−2000002−20−11−20201−1−1001000−1113−40000−11.

Figure 1

The directed topology with six agents

It is easy to obtain that L has eigenvalues λ₁ = λ₂ = 0, λ₃ = 4.4557, λ₄ = 2.7 + 1.177i, and λ₅ = 2.7 − 1.177i. L has zero eigenvalues whose algebraic multiplicity is exactly two apparently, and the rest of the eigenvalues have positive real parts. We choose k = 1.2, μ = 0.1, v = 0.35, d = 0.25 and define τ_ij(t) = τ | sin (t) | for convenience. Let the initial state and velocity be x(0) = [1,100, − 2,50,3, − 1]^T, v(0) = [1, 2, 1.5, 3, 2.2, 0.6]^T. When τ = 0.25, Figure 2 shows the states trajectories of all six agents and Figure 3 shows the velocity trajectories of all agents for the multi-agent system.

Figure 2

The trajectories of position x_i when τ = 0.25

Figure 3

The trajectories of velocity v_i when τ = 0.25

From Figure 2, it can be observed that states x₁ to x₄ converge to one consensus value, whereas the states x₅ and x₆ converge to another consensus value. Moreover, it can be seen from Figure 3 that the states v₁ to v₆ converge to one consensus value asymptotically. That is to say, the group consensus can be achieved. Furthermore, when τ ≤ 0.25, LMIs in Theorem 1 are feasible, which means that the system can achieve group consensus asymptotically. By changing parameters and increasing communication time delay bound, situations would change and when it comes to τ = 0.52, LMIs in Theorem 1 are unfeasible while the systems still can reach group consensus. It represents that our result still has the conservativeness to some extent. When τ = 0.73, it can be concluded from Figure 4 and Figure 5 that neither the position nor velocity can get consensus while LMIs are unfeasible which means the system appears divergency and cannot reach group consensus.

Figure 4

The trajectories of position x_i when τ = 0.87

Figure 5

The trajectories of velocity v_i when τ = 0.87

5 Conclusions

In this paper, the group consensus problem for second-order multi-agent dynamic systems with communication time-varying delays has been investigated, where the agents in a network may reach one more consistent values asymptotically. The delay-dependent sufficient conditions of achieving group consensus have been proposed based on Lyapunov function method and LMIs. Finally, simulation results are provided to verify the effectiveness of the proposed condition for agreement in networks with communication time-delays.

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Received: 2016-1-22

Accepted: 2016-3-4

Published Online: 2016-6-25

Articles in the same Issue

https://doi.org/10.21078/JSSI-2016-258-11

Keywords for this article

group consensus; second-order multi-agent systems; directed network; time-varying communication delays; LMIs