# Bipartite Consensus of High-Order Edge Dynamics on Coopetition Multiagent Systems

Authors: Yilong Yang; Zhijian Ji; Lei Tian; Huizi Ma; Qingyuan Qi
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The bipartite consensus of high-order edge dynamics is investigated for coopetition multiagent systems, in which the cooperative and competitive relationships among agents are characterized by positive weight and negative weight, respectively. By mapping the initial graph to a line graph, the distributed control protocol is proposed for the strongly connected, digon sign-symmetric structurally balanced line graph; and then we give sufficient conditions for the third-order multi-gent system to achieve both the bipartite consensus of edge dynamics and the final value of bipartite consensus. By transforming the coefficients of characteristic polynomial from complex domain to real number domain, the sufficient conditions for the bipartite consensus of high-order edge dynamics are also proposed, and the final values of the high-order edge dynamics on multiagent systems are obtained.


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## Body

## 1. Introduction

In recent years, researchers from different fields have done many works for the networked system thanks to the wide applications both to formation control, such as mobile robots, unmanned air vehicles (UAVs), and autonomous underwater vehicles (AUVs), and to distributed control, such as distributed forecasting, monitoring, and diagnostics [1–6]. For multiagent network systems, controllability and consensus problems are the focus of researchers [7–15]. In this paper, we investigated the bipartite consensus problems of multiagent systems.The consensus problem is of particular concern because most actual systems can be described by the node dynamic multiagent system. Most of the results on consensus were gained by cooperative systems for the case of nonnegative edge weight, see, e.g., [16–22]. In particular, the consensus of second-order multiagent systems with absolute velocity damping under directed topologies was investigated by transforming it to the first-order multiagent system [18]. Some sufficient and necessary conditions were provided for the consensus of multiagent systems with sampler and zero-order holder in [19], as well as for the consensus of hybrid multiagent systems [22, 23]. Recently, the consensus result of first-order multiagent systems with time delay was improved in [17].Since the relationship between agents can be cooperative and competitive in reality, the real world can be described more precisely by coopetition systems. The sign graphs are employed to represent the cooperative and competitive relationship between agents. The application of this model is generally inspired by, e.g., the social science [24], electrical power network [25, 26], and biological network [27]. For the coopetition system, the bipartite consensus was proposed, which means that all the agents of a subsystem can reach a common state. Many results on bipartite consensus have also been obtained. For example, the bipartite consensus was considered under connected and structurally balanced topology [28–31]. Sufficient conditions making the second-order multiagent system with time delay asymptotically achieve the bipartite consensus were proposed in [28]. For high-order multiagent systems, Valcher proposed necessary and sufficient conditions for consensus and bipartite consensus under structural balance graphs [29]. In [30], Ma designed a control protocol for high-order power integrator network multiagent systems by backstepping technique. After introducing potential function it was proved that the multiagent system under the protocol could reach bipartite consensus. For the process of reaching consensus, Qu introduced a new superposition system to the original system and discussed how to get the fastest convergence rate without destroying the connectivity [32]. In [33], Liu found the optimal topology, by which multiagent systems achieved bipartite consensus with lower control cost than other topologies.In some real systems, e.g., internet, biological network, and electrical power network, the links between agents might be dynamic as the agents. So the consensus problems of edge dynamic are necessary to be studied. By introducing adjacency matrix and defining Laplacian matrix, the consensus problem of the node system was considered by the information of nodes and edges [34]. In [35], the edge dynamics were denoted by the node dynamics on cyclic pursuit graph. Then the conditions of reaching consensus and consensus value were proposed. For directed multiagent systems, Wang transformed initial digraph to its line graph and established both continuous-time and discrete-time protocols. The systems under both of the protocols could asymptotically reach consensus and the consensus values were also given [36]. Meanwhile, for undirected graph, the sufficient conditions referring to the number of edges were gained and the feedback gain matrix was simplified [37]. In [38], Wang proposed distributed edge quasi-consensus protocol to achieve edge quasi-consensus and the sign of states of all edges can be kept for system with nonnegative initial edge states. In [39], consensus of edge dynamics on the strongly connected digraph was investigated after mapping the initial digraph to its line graph. Tian obtained the conditions of reaching bipartite consensus of edge dynamics on coopetition multiagent systems and calculated the final values of edge dynamics on the structurally balanced digraph by using the Jordan canonical form method given in [40].The results above show that the relationships between agents still need to be further considered for higher-order multiagent systems, which can be cooperative and competitive. Our aim in this paper is to find conditions to make the edge dynamics on coopetition multiagent systems asymptotically reach high-order bipartite consensus. First of all, we use digraph to represent the multiagent system with positive and negative edges indicating cooperative and competitive relationship between the agents, respectively. Then the initial digraph is mapped to its line graph, and distributed consensus protocols are designed for the edge dynamics multiagent system. In this paper, we assume that the digraph is structurally balanced. After introducing gauge transformation, we transform the multiagent system to a new one whose adjacency matrix has only nonnegative elements. This means that the problems of bipartite consensus are transformed to the ordinary consensus problems. For the third-order edge dynamics multiagent system, sufficient conditions are obtained for the new system to asymptotically reach edge consensus, which means that the original multiagent system can asymptotically reach bipartite consensus of edge dynamics. Finally, for general high-order multiagent systems, we give the result of eigenvalues of Laplacian matrix and accordingly obtain the condition for the new system to asymptotically reach consensus of edge dynamics. Thus, the multiagent system can asymptotically reach high-order bipartite consensus of edges. For the third-order and high-order bipartite consensus of edge dynamics, the final values are discussed regarding distinct feedback gains.The rest of structure of the paper is as follows. Section2 explains the preliminary knowledge of graph theory and line graphs. Section 3 gives the main research results. Section 4 gives some simulation results. Section 5 concludes this article.


## 2. Preliminaries

### 2.1. Theory of Graph

LetV=1,2,…,n and E=i,j∣ificanreceiveinformationfromj be the node set and edge set, respectively. We denote the signed digraph by G=V,E,A. A=aij∈Rn×n is the adjacency matrix of G, where aij≠0 for i,j∈E. In this paper, we only discuss the digraph without self-loop, which means aii=0, ∀i∈1,2,…,n. Furthermore, if the connection from j to i is cooperative, aij=1; if the connection from j to i is competitive, aij=-1. In digraph, we use dotted lines and solid lines to represent competition connection and cooperative connection, respectively. We assume that the signed digraph is digon sign-symmetric, that is, aijaji≥0. We denote the set of neighbors and the in-degree of node i by Ni=j∣aij≠0 and dini=∑j=1n‍aij, respectively. C is the diagonal degree matrix with cii=dini and L=C-A is the Laplacian matrix of G. Meanwhile, the adjacency matrix of undirected graph is symmetric, so GAu is considered to be an undirected graph derived from the digraph G with Au=(A+AT)/2.A series of interrelated edges inE, (1)P=i2,i1,i3,i2,…,ip,ip-1⊂E,is called a directed path of G, where nodes i1,i2,…,ip are different, and the length of the directed path P is p-1. If there are at least one path from i to j and another one from j to i, ∀i,j∈V, i≠j, the digraph is strongly connected. A path becomes a cycle if the end point of P coincides with the starting point (i.e., i1=ip). This means that s node in strongly connected digraph is in at least one cycle. If a cycle contains even negative edges, i.e., ai2i1⋯ai1ip>0, it is positive. If ai2i1⋯ai1ip<0, the cycle is negative.


### 2.2. Structural Balance

We defineV1,V2 as two sets of nodes. A digraph is structurally balanced if all of its nodes can be divided into V1,V2, where V1∪V2=V, V1∩V2=ϕ and aij≥0, ∀i,j∈Vp, p∈1,2, aij≤0, ∀i∈Vp, j∈Vq, p,q∈1,2,p≠q. Otherwise, the digraph is structurally unbalanced. According to the property of structurally balanced digraph, we introduce a set of gauge transformations D={D∣D=diag{d1,…,dn},di=±1,i=1,2,…,n} and then have the following lemma.Lemma 1 (see [41]).
A strongly connected, digon sign-symmetric digraphGA is structurally balanced if and only if any of the following equivalent conditions hold: (1)
GAu is structurally balanced;(2)
all directed cycles ofGA are positive;(3)
∃D∈D such that DAD has all nonnegative entries;(4)
0 is an eigenvalue of L.From Lemma1, a strongly connected, digon sign-symmetric digraph G is structurally balanced if and only if Laplacian matrix L has a simple eigenvalue 0 (i.e., rankL=n-1).Lemma 2 (see [42]).
LetG=V,E,A be a weighted digraph with Laplacian L. Denote the maximum node in-degree of the digraph G by dmaxG=maxidini. Then, all the eigenvalues of L are located in the following disk: (2)DG=z∈C:z-dmaxG≤dmaxG,which is centered at z=dmaxG+0j in the complex plane. In other words, except for the single eigenvalue 0, all of the other eigenvalues of L have positive real parts.Lemma 3 (see [42]).
AssumeG is a strongly connected digraph with Laplacian matrix L satisfying Lwr=0, wlTL=0, and wlTwr=1. Then, limt→∞e-Lt=wrwlT.


### 2.3. Line Graph

Before mapping the initial digraph to its line graph, the out-neighbors of edgei,j are defined as k,i∈E, k∈V. In the process of transforming the initial digraph G=V,E,A with n nodes and m edges to its line graph LG, we follow the following rules [43]:(1)
convert a directed edgei,j of G to a node i,j of LG;(2)
generate edges from nodei,j to its out-neighbors k,i∈E, k∈V;(3)
if edgei,j in initial digraph is positive, the generated edges are positive;(4)
if edgei,j in initial digraph is negative, the generated edges are negative.Similar to initial digraph, we denote the Laplacian matrix of line graph byL′=C′-A′, where A′=a′ijm×m and C′=diagdinii=1m are the adjacency matrix and in-degree matrix of line graph, respectively. The elements of L′ are (3)l′st=-ast,s≠t,∑t∈Nsa′st,s=t.Lemma 4 (see [43]).
If a digraphG contains more than one node and is strongly connected, then its line graph LG is also strongly connected.Lemma 5 (see [30]).
For a strongly connected, digon sign-symmetric digraphG, its line graph LG is structurally balanced if and only if G is structurally balanced.In Figure1, (a) is a strongly connected and digon sign-symmetric digraph. From the rules, we map the digraph to its line graph. For example, the edges in (a) are converted into nodes in (b). There is an edge from nodes (1,2) to (6,1) and the edge is positive because the edge (1,2) in (a) is positive. Following this procedure for all nodes, Figure 1(b) is gained. From Lemma 4, the line graph (b) is also strongly connected and digon sign-symmetric. Nodes in (a) can be divided into V1=1,2,3, V2=4,5,6 and all the negative edges are between different sets. The nodes in (b) (i.e., the edges in (a)) can be divided into E1=1,2,2,3,4,2,5,2,6,1 and E2=2,6,3,4,4,5,5,6. Thus (a) is structurally balanced. The adjacency matrix A of (a) and the Laplacian matrix L′ of (b) are (4)A=01000000100-1000-1000-100100-11000-100000,L′=2-1100000001010000000100000100021-10000-100100000000021-100-11000200000000011-100000001.Changing the interaction between nodes in (a), we have the structurally unbalanced digraph (c). Following the rules, we gain the line graph (d) in Figure 1. There is at least one negative cycle, so the line graph (d) is structurally unbalanced. From Lemma 5, we get the same conclusion.Figure 1
Structurally balanced/unbalanced digraph (a)/(c) and their line graph (b)/(d).
(a) (b) (c) (d)By Lemma1, there are eigenvalue 0 of L′ and matrix D=diag1,1,-1,-1,1,1,-1 such that L-′=DL′D has positive diagonal elements and nonpositive nondiagonal elements. From Lemma 2, the other eigenvalues of L′ except the simple eigenvalue 0 are all located in the right half complex plane. Moreover, due to the structure of D, D=D-1.


## 2.1. Theory of Graph

LetV=1,2,…,n and E=i,j∣ificanreceiveinformationfromj be the node set and edge set, respectively. We denote the signed digraph by G=V,E,A. A=aij∈Rn×n is the adjacency matrix of G, where aij≠0 for i,j∈E. In this paper, we only discuss the digraph without self-loop, which means aii=0, ∀i∈1,2,…,n. Furthermore, if the connection from j to i is cooperative, aij=1; if the connection from j to i is competitive, aij=-1. In digraph, we use dotted lines and solid lines to represent competition connection and cooperative connection, respectively. We assume that the signed digraph is digon sign-symmetric, that is, aijaji≥0. We denote the set of neighbors and the in-degree of node i by Ni=j∣aij≠0 and dini=∑j=1n‍aij, respectively. C is the diagonal degree matrix with cii=dini and L=C-A is the Laplacian matrix of G. Meanwhile, the adjacency matrix of undirected graph is symmetric, so GAu is considered to be an undirected graph derived from the digraph G with Au=(A+AT)/2.A series of interrelated edges inE, (1)P=i2,i1,i3,i2,…,ip,ip-1⊂E,is called a directed path of G, where nodes i1,i2,…,ip are different, and the length of the directed path P is p-1. If there are at least one path from i to j and another one from j to i, ∀i,j∈V, i≠j, the digraph is strongly connected. A path becomes a cycle if the end point of P coincides with the starting point (i.e., i1=ip). This means that s node in strongly connected digraph is in at least one cycle. If a cycle contains even negative edges, i.e., ai2i1⋯ai1ip>0, it is positive. If ai2i1⋯ai1ip<0, the cycle is negative.


## 2.2. Structural Balance

We defineV1,V2 as two sets of nodes. A digraph is structurally balanced if all of its nodes can be divided into V1,V2, where V1∪V2=V, V1∩V2=ϕ and aij≥0, ∀i,j∈Vp, p∈1,2, aij≤0, ∀i∈Vp, j∈Vq, p,q∈1,2,p≠q. Otherwise, the digraph is structurally unbalanced. According to the property of structurally balanced digraph, we introduce a set of gauge transformations D={D∣D=diag{d1,…,dn},di=±1,i=1,2,…,n} and then have the following lemma.Lemma 1 (see [41]).
A strongly connected, digon sign-symmetric digraphGA is structurally balanced if and only if any of the following equivalent conditions hold: (1)
GAu is structurally balanced;(2)
all directed cycles ofGA are positive;(3)
∃D∈D such that DAD has all nonnegative entries;(4)
0 is an eigenvalue of L.From Lemma1, a strongly connected, digon sign-symmetric digraph G is structurally balanced if and only if Laplacian matrix L has a simple eigenvalue 0 (i.e., rankL=n-1).Lemma 2 (see [42]).
LetG=V,E,A be a weighted digraph with Laplacian L. Denote the maximum node in-degree of the digraph G by dmaxG=maxidini. Then, all the eigenvalues of L are located in the following disk: (2)DG=z∈C:z-dmaxG≤dmaxG,which is centered at z=dmaxG+0j in the complex plane. In other words, except for the single eigenvalue 0, all of the other eigenvalues of L have positive real parts.Lemma 3 (see [42]).
AssumeG is a strongly connected digraph with Laplacian matrix L satisfying Lwr=0, wlTL=0, and wlTwr=1. Then, limt→∞e-Lt=wrwlT.


## 2.3. Line Graph

Before mapping the initial digraph to its line graph, the out-neighbors of edgei,j are defined as k,i∈E, k∈V. In the process of transforming the initial digraph G=V,E,A with n nodes and m edges to its line graph LG, we follow the following rules [43]:(1)
convert a directed edgei,j of G to a node i,j of LG;(2)
generate edges from nodei,j to its out-neighbors k,i∈E, k∈V;(3)
if edgei,j in initial digraph is positive, the generated edges are positive;(4)
if edgei,j in initial digraph is negative, the generated edges are negative.Similar to initial digraph, we denote the Laplacian matrix of line graph byL′=C′-A′, where A′=a′ijm×m and C′=diagdinii=1m are the adjacency matrix and in-degree matrix of line graph, respectively. The elements of L′ are (3)l′st=-ast,s≠t,∑t∈Nsa′st,s=t.Lemma 4 (see [43]).
If a digraphG contains more than one node and is strongly connected, then its line graph LG is also strongly connected.Lemma 5 (see [30]).
For a strongly connected, digon sign-symmetric digraphG, its line graph LG is structurally balanced if and only if G is structurally balanced.In Figure1, (a) is a strongly connected and digon sign-symmetric digraph. From the rules, we map the digraph to its line graph. For example, the edges in (a) are converted into nodes in (b). There is an edge from nodes (1,2) to (6,1) and the edge is positive because the edge (1,2) in (a) is positive. Following this procedure for all nodes, Figure 1(b) is gained. From Lemma 4, the line graph (b) is also strongly connected and digon sign-symmetric. Nodes in (a) can be divided into V1=1,2,3, V2=4,5,6 and all the negative edges are between different sets. The nodes in (b) (i.e., the edges in (a)) can be divided into E1=1,2,2,3,4,2,5,2,6,1 and E2=2,6,3,4,4,5,5,6. Thus (a) is structurally balanced. The adjacency matrix A of (a) and the Laplacian matrix L′ of (b) are (4)A=01000000100-1000-1000-100100-11000-100000,L′=2-1100000001010000000100000100021-10000-100100000000021-100-11000200000000011-100000001.Changing the interaction between nodes in (a), we have the structurally unbalanced digraph (c). Following the rules, we gain the line graph (d) in Figure 1. There is at least one negative cycle, so the line graph (d) is structurally unbalanced. From Lemma 5, we get the same conclusion.Figure 1
Structurally balanced/unbalanced digraph (a)/(c) and their line graph (b)/(d).
(a) (b) (c) (d)By Lemma1, there are eigenvalue 0 of L′ and matrix D=diag1,1,-1,-1,1,1,-1 such that L-′=DL′D has positive diagonal elements and nonpositive nondiagonal elements. From Lemma 2, the other eigenvalues of L′ except the simple eigenvalue 0 are all located in the right half complex plane. Moreover, due to the structure of D, D=D-1.


## 3. Main Results

### 3.1. Bipartite Consensus of Third-Order Edge Dynamics

Letξij,θij,ψij represent, respectively, the position, velocity, and acceleration of edge i,j at time t. The third-order continuous-time edge dynamics model is (5)ξ˙ij=θij,θ˙ij=ψij,ψ˙ij=uij.For system (5), we design the following control protocol (6)uij=-k1θij-k2ψij+∑r∈NjAjrsgnAjrξjrt-ξijt+∑r∈NjAjrβ1sgnAjrθjrt-θijt+∑r∈NjAjrβ2sgnAjrψjrt-ψijt,∀i,j∈E,where k1,k2,β1,β2≥0 are feedback gains.Lemma 6 (see [44]).
Given a third-order complex coefficient polynomial, in the form of(7)p3s=s3+ζ2+jη2s2+ζ1+jη1s+ζ0+jη0,then p3s is stable if and only if ζ2>0, ζ2η2η1+ζ22ζ1-η12-ζ2ζ0>0 and (8)ζ2·detη2-ζ1-η00ζ2η1-ζ001η2-ζ1-η00ζ2η1-ζ0-detη1-ζ000ζ2η1-ζ001η2-ζ1-η00ζ2η1-ζ0>0.Letξ=ξij, θ=θij, ψ=ψij∈Rm×1, i=1,2,…,n; j∈Ni; m=∑i=1n‍dini. Then system (5) can be simplified as ξ˙Tθ˙Tψ˙TT=MξTθTψTT, where (9)M=0Im000Im-L′-k1Im-β1L′-k2Im-β2L′,and matrix L′ is defined as above. Let ξ-=Dξ, θ-=Dθ, ψ-=Dψ, D∈D, we have (10)ξ-˙Tθ-˙Tψ-˙TT=M-ξ-Tθ-Tψ-TTwhere (11)M-=0Im000Im-L-′-k1Im-β1L-′-k2Im-β2L-′,with L-′=DL′D. Notice that (12)detλI3m-M-=detλIm-Im00λIm-ImL-′k1Im+β1L-′λ+k2Im+β2L-′=detλ3+k2λ2+k1λIm+β2λ2+β1λ+1L-′=∏i=1mλ3+k2-β2μiλ2+k1-β1μiλ-μiwhere μi, i=1,2,…,m, are, respectively, the eigenvalues of -L′. There is an eigenvalue μi=0 of -L′. Thus matrix M- has at least one 0 eigenvalue.Assumption. For l∈1,h-1, where h is the order of system, if kl≠0, then ki≠0, i≥l.Theorem 7.
For a continuous-time edge dynamic multiagent system (5), let the corresponding digraph G be strongly connected, digon sign-symmetric, and structurally balanced. When the assumption holds, system (5) can asymptotically reach the bipartite consensus of edge dynamics under protocol (6) for μi≠0, if the following inequalities are satisfied (13)k2-β2Reμi>0,(14)β1β2Imμi2+Reμi+k2-β2Reμik1-β1Reμi·k2-β2Reμi-β12Imμi2>0,and(15)k2-β2Reμidet-β2Imμi-k1-β1ReμiImμi0k2-β2Reμi-β1ImμiReμi01-β2Imμi-k1-β1ReμiImμi0k2-β2Reμi-β1ImμiReμi-det-β1ImμiReμi00k2-β2Reμi-β1ImμiReμi01-β2Imμi-k1-β1ReμiImμi0k2-β2Reμi-β1ImμiReμi>0.Moreover, if k1=k2=0, the final values of bipartite consensus are (16)limt→∞ξt=wlTDξ0Dwr+twlTDθ0Dwr+12t2wlTDψ0Dwr,limt→∞θt=wlTDθ0Dwr+twlTDψ0Dwr,limt→∞ψt=wlTDψ0Dwr.where wl and wr are the left and right eigenvectors associated with the single eigenvalue 0 of the Laplacian matrix L-′, respectively, and wlTwr=1. If k1=0,k2≠0, the final values of bipartite consensus are (17)limt→∞ξt=wlTDξ0Dwr+1k2+twlTDθ0Dwr+tk2wlTDψ0Dwr,limt→∞θt=wlTDθ0Dwr+1k2wlTDψ0Dwrlimt→∞ψt=0.If k1≠0, the consensus values are as follows, respectively, (18)limt→∞ξt=wlTDξ0Dwr+k2k1wlTDθ0Dwr+1k1wlTDψ0Dwr,limt→∞θt=0,limt→∞ψt=0.Proof.
Forμi=0, when the assumption holds, the roots of polynomial λ2+k2λ+k1 are 0 or complex numbers with negative real parts. For μi≠0, when conditions (13), (14), and (15) are satisfied, the roots of polynomial all have negative real parts, that is, all eigenvalues except 0 of M- have negative real parts. Therefore, we have the following cases. (i)
Ifk1=k2=0, we see from Lemma 6 that matrix M- has exactly three zero eigenvalues and all the other eigenvalues have negative real parts. From the stability theorem and the results in [29], system (10) can asymptotically reach the consensus of edge dynamics, and limt→∞ξ-t=wrwlTξ-0+twrwlTθ-0+1/2t2wrwlTψ-0, limt→∞θ-t=wrwlTθ-0+twrwlTψ-0, and limt→∞ψ-t=wrwlTψ-0, where wr and wl are the right and left eigenvectors associated with μi=0 of Laplacian matrix L-′=DL′D, respectively, and wlTwr=1. Thus, from D=D-1, system (5) can asymptotically reach the bipartite consensus of edge dynamics under protocol (6), and (19)limt→∞ξt=DwrwlTDξ0+twrwlTDθ0+12t2wrwlTDψ0=wlTDξ0Dwr+twlTDθ0Dwr+12t2wlTDψ0Dwr,limt→∞θt=DwrwlTDθ0+twrwlTDψ0=wlTDθ0Dwr+twlTDψ0Dwr,limt→∞ψt=DwrwlTDψ0=wlTDψ0Dwr.(ii)
Ifk1=0,k2≠0, similar to the analysis above, matrix M- has exactly two zero eigenvalues and all the other eigenvalues have negative real parts. Then (10) can asymptotically reach the consensus of edge dynamics, and the final values of consensus are limt→∞ξ-t=wrwlTξ-0+1/k2+twrwlTθ-0+t/k2wrwlTψ-0, limt→∞θ-t=wrwlTθ-0+1/k2wrwlTψ-0, limt→∞ψ-t=0, where wr and wlT are defined as above. Thus, (5) can asymptotically reach the bipartite consensus of edge dynamics under protocol (6), and (20)limt→∞ξt=DwrwlTDξ0+1k2+twrwlTDθ0+tk2wrwlTDψ0=wlTDξ0Dwr+1k2+twlTDθ0Dwr+tk2wlTDψ0Dwr,limt→∞θt=DwrwlTDθ0+1k2wrwlTDψ0=wlTDθ0Dwr+1k2wlTDψ0Dwr,limt→∞ψt=0(iii)
Ifk1≠0, similar to the analysis in part one, matrix M- has exactly one zero eigenvalue and all the other eigenvalues have negative real parts. Then (10) can asymptotically reach the consensus of edge dynamics, and the final values of consensus are limt→∞ξ-t=wrwlTξ-0+k2/k1wrwlTθ-0+1/k1wrwlTψ-0, limt→∞θ-t=0, limt→∞ψ-t=0, where wr and wlT are defined as above. Therefore (5) can asymptotically reach the bipartite consensus of edge dynamics under protocol (6), and (21)limt→∞ξt=DwrwlTDξ0+k2k1wrwlTDθ0+1k1wrwlTDψ0=wlTDξ0Dwr+k2k1wlTDθ0Dwr+1k1wlTDψ0Dwr,limt→∞θt=0,limt→∞ψt=0.(iv)
If signed digraphG is structurally unbalanced, its line graph LG is also structurally unbalanced, which means that all the eigenvalues of Laplacian matrix -L′ have negative real parts. Therefore, all the eigenvalues of matrix M have negative real parts under conditions (13), (14), and (15). Thus, system (5) is asymptotically stable under protocol (6). Accordingly, we have limt→∞ξt=0, limt→∞θt=0, limt→∞ψt=0.


### 3.2. Bipartite Consensus of High-Order Edge Dynamics

Theh order continuous-time edge dynamics model is (22)ξ˙ij0=ξij1,ξ˙ij1=ξij2,⋮,ξ˙ijh-1=uij,where ξij1, l∈0,h-1 are the states of edges at time t. For system (22), we design the high-order control protocol (23)uij=-∑l=1h-1klξijl+∑r∈NjAjrsgnAjrξjr0t-ξij0t+∑l=1h-1∑r∈NjβlsgnAjrξjrlt-ξijltwhere k1,k2,…,kh-1≥0 and β1,β2,…,βh-1>0 are feedback gains. Denote ξl=ξijlt∈Rm×1, j∈Ni, m=∑i=1ndini, and ω=ξ0Tξ1T⋯ξh-1TT. Then we rewrite (22) with (23) as (24)ω˙=Mω,where (25)M=0Im0⋯000Im⋯0⋮⋮⋮⋱⋮000⋯Im-L′-k1Im-β1L′-k2Im-β2L′⋯-kh-1Im-βh-1L′,and matrix L′ is the Laplacian matrix of LG. Let ξ-l=Dξl, l∈0,h-1, D∈D, and the state vector ω-=ξ-0Tξ-1T⋯ξ-h-1TT. We have (26)ω-˙=M-ω-,where (27)M-=0Im0⋯000Im⋯0⋮⋮⋮⋱⋮000⋯Im-L-′-k1Im-β1L-′-k2Im-β2L-′⋯-kh-1Im-βh-1L-′with L-′=DL′D.Notice that(28)detλIhm-M-=detλIm-Im0⋯00λIm-Im⋯0⋮⋮⋮⋱⋮000⋯-ImL-′k1Im+β1L-′k2Im+β2L-′⋯λ+kh-1Im+βh-1L-′=detλh+kh-1λh-1+⋯+k1λIm+βh-1λh-1+⋯+β1λ+1L-′=∏i=1mλh+kh-1-βh-1μiλh-1+⋯+k2-β2μiλ2+k1-β1μiλ-μiwhere μi, i=1,2,…,m are eigenvalues of -L′ and there is a simple eigenvalue μi=0. Then matrix M- has at least one 0 eigenvalue.By Lemma2, all the eigenvalues of -L′ except the simple eigenvalue 0 have negative real parts, which shows that the eigenvalues of -L′ are 0, negative real numbers, and conjugate complex numbers with negative real parts. Therefore, we reorder the eigenvalues of -L′ to (29)0,μ2,…,μl,μl+1,μ-l+1,…,μl+k,μ-l+k,l,k≥0,l+2k=m.Theorem 8.
Given a high-order complex coefficient polynomial(30)fx=an+bnixn+an-1+bn-1ixn-1+⋯+a1+b1ix+a0+b0i,and another complex coefficient polynomial gx with the same order and conjugate coefficients as the above polynomial fx. (31)gx=an-bnixn+an-1-bn-1ixn-1+⋯+a1-b1ix+a0-b0i.Their product is (32)hx=fx×gx=∑l=1n∑i+j=2l-12aiaj+bibjx2l-1+∑l=0n∑i+j=2li≠j2aiaj+bibj+al2+bl2x2lProof.
The result can be proved by the multiplication rule of complex coefficient polynomial and the fact that every coefficient ofhx is greater than 0.Denoteλh+kh-1-βh-1μiλh-1+⋯+k2-β2μiλ2+k1-β1μiλ-μi by Phμi, and denote Phμi·P′hμi by P2hμi. Then from Theorem 8, (26) can be written as (33)detλIhm-Γ=∏i=1lPhμi·∏i=l+1l+kP2hμi.Lemma 9 (see [45]).
Given a system with characteristic polynomial(34)Ds=ansn+an-1sn-1+⋯+a1s+a0,ai>0,i∈0,n,then the system is stable if and only if the determinant consists of coefficients of characteristic polynomials (35)Δn=an-1an-3⋯000anan-2⋯0000an-1⋯0000an⋯000⋮⋮⋱⋮⋮⋮00⋯a3a1000⋯a4a2a0and its order principal minor determinant Δii=1,2,…,n-1 is greater than 0, namely, (36)Δ1=an-1>0,Δ2=an-1an-3anan-2>0,⋮,Δn>0.Theorem 10.
For a continuous-time edge dynamics multiagent system (22), let the corresponding signed digraph G be strongly connected, digon sign-symmetric and structurally balanced. When the Assumption holds, system (22) can asymptotically reach the bipartite consensus of edge dynamics under protocol (23), if the determinants and their order principal minor determinants of Phμi and P2hμi in (33) are all greater than 0 for μi≠0, and Δj>0, j∈1,h-l of Ph0 with kl=0, kl+1≠0 for μi=0.Proof.
From Lemma9 and the results in [40], if the determinants and their order principal minor determinants of Phμi and P2hμi in (33) are all greater than 0 for μi≠0, all roots of the corresponding subpolynomial (28) have negative real parts. If Δj>0, j∈1,h-l with kl=0, kl+1≠0 for μi=0, the roots except 0 of the corresponding subpolynomial (28) have negative real parts. Thus system (26) can reach the consensus of edges. We have the following cases. (i)
Ifki=0, i∈1,h-1, matrix M- has exactly h zero eigenvalues and all the other eigenvalues have negative real parts. By the results of [29], system (26) can asymptotically reach the consensus of edge dynamics, and limt→∞ξ-i=∑k=ih-1‍tk-i/k-i!wrwlTξ-k0, i∈0,h-1, where wr and wlT are the same as above. By the property of D, system (22) can asymptotically reach the bipartite consensus of edge dynamics under protocol (23), and the final values of bipartite consensus are (37)limt→∞ξi=∑k=ih-1tk-ik-i!DwrwlTDξk0=∑k=ih-1tk-ik-i!wlTDξk0Dwr,i∈0,h-1.(ii)
Ifki=0,ki+1≠0, i∈1,h-2, matrix M- has exactly i+1 zero eigenvalues and all the other eigenvalues have negative real parts. Therefore, system (26) can asymptotically reach the consensus, and limt→∞ξ-i=wrwlTξ-i0+∑j=i+1h-2‍kl+1/ki+1wrwlTξ-l0+1/ki+1wrwlTξ-h-10, limt→∞ξ-l=0, l>i. Similar to the analysis in the first part, system (22) can asymptotically reach the bipartite consensus of edges dynamics under protocol (23), and (38)limt→∞ξi=wlTDξi0Dwr+∑l=i+1h-2kl+1ki+1wlTDξl0Dwr+1ki+1wlTDξh-10Dwr,limt→∞ξl=0,l>iFurthermore, the limit ofξl at t→∞ is i-l order function of t when l<i.(iii)
Ifk1≠0, matrix M- only has exactly one zero eigenvalue and all the other eigenvalues have negative real parts. Thus, system (26) can asymptotically reach the consensus of edge dynamics, and limt→∞ξ-0=wrwlTξ-00+∑i=1h-2‍ki+1/k1wrwlTξ-i0+1/k1wrwlTξ-h-10, limt→∞ξ-i=0, i∈1,h-1. System (22) can asymptotically reach the bipartite consensus of edge dynamics under protocol (23), and the final values of bipartite consensus are (39)limt→∞ξ0=wlTDξ00Dwr+∑i=1h-2ki+1k1wlTDξi0Dwr+1k1wlTDξh-10Dwr,limt→∞ξi=0,i∈1,h-1.(iv)
IfLG is structurally unbalanced, all eigenvalues of Laplacian matrix -L′ have negative real parts. All eigenvalues of matrix M have negative real parts, which means (22) can be asymptotically stable under protocol (23) and limt→∞ξi=0, i∈0,h-1.Remark 11.
Although the general conditions on higher-order systems are valid for third-order systems, the conditions derived for third-order systems have their own characteristics and-values. To be more specific, the judgment method of the third-order system is more direct than that of the general higher-order system. More importantly, the results of third-order systems are much more detailed. Thus, the results of third-order systems are useful for detailing the effect of feedback gains on the consensus end-value and for understanding the implications of the results of general higher-order systems. For example, one can see from the third-order system thatξij0 is the function of t in the presence of kl=0 with the order equal to the number of kl equal to zero.


## 3.1. Bipartite Consensus of Third-Order Edge Dynamics

Letξij,θij,ψij represent, respectively, the position, velocity, and acceleration of edge i,j at time t. The third-order continuous-time edge dynamics model is (5)ξ˙ij=θij,θ˙ij=ψij,ψ˙ij=uij.For system (5), we design the following control protocol (6)uij=-k1θij-k2ψij+∑r∈NjAjrsgnAjrξjrt-ξijt+∑r∈NjAjrβ1sgnAjrθjrt-θijt+∑r∈NjAjrβ2sgnAjrψjrt-ψijt,∀i,j∈E,where k1,k2,β1,β2≥0 are feedback gains.Lemma 6 (see [44]).
Given a third-order complex coefficient polynomial, in the form of(7)p3s=s3+ζ2+jη2s2+ζ1+jη1s+ζ0+jη0,then p3s is stable if and only if ζ2>0, ζ2η2η1+ζ22ζ1-η12-ζ2ζ0>0 and (8)ζ2·detη2-ζ1-η00ζ2η1-ζ001η2-ζ1-η00ζ2η1-ζ0-detη1-ζ000ζ2η1-ζ001η2-ζ1-η00ζ2η1-ζ0>0.Letξ=ξij, θ=θij, ψ=ψij∈Rm×1, i=1,2,…,n; j∈Ni; m=∑i=1n‍dini. Then system (5) can be simplified as ξ˙Tθ˙Tψ˙TT=MξTθTψTT, where (9)M=0Im000Im-L′-k1Im-β1L′-k2Im-β2L′,and matrix L′ is defined as above. Let ξ-=Dξ, θ-=Dθ, ψ-=Dψ, D∈D, we have (10)ξ-˙Tθ-˙Tψ-˙TT=M-ξ-Tθ-Tψ-TTwhere (11)M-=0Im000Im-L-′-k1Im-β1L-′-k2Im-β2L-′,with L-′=DL′D. Notice that (12)detλI3m-M-=detλIm-Im00λIm-ImL-′k1Im+β1L-′λ+k2Im+β2L-′=detλ3+k2λ2+k1λIm+β2λ2+β1λ+1L-′=∏i=1mλ3+k2-β2μiλ2+k1-β1μiλ-μiwhere μi, i=1,2,…,m, are, respectively, the eigenvalues of -L′. There is an eigenvalue μi=0 of -L′. Thus matrix M- has at least one 0 eigenvalue.Assumption. For l∈1,h-1, where h is the order of system, if kl≠0, then ki≠0, i≥l.Theorem 7.
For a continuous-time edge dynamic multiagent system (5), let the corresponding digraph G be strongly connected, digon sign-symmetric, and structurally balanced. When the assumption holds, system (5) can asymptotically reach the bipartite consensus of edge dynamics under protocol (6) for μi≠0, if the following inequalities are satisfied (13)k2-β2Reμi>0,(14)β1β2Imμi2+Reμi+k2-β2Reμik1-β1Reμi·k2-β2Reμi-β12Imμi2>0,and(15)k2-β2Reμidet-β2Imμi-k1-β1ReμiImμi0k2-β2Reμi-β1ImμiReμi01-β2Imμi-k1-β1ReμiImμi0k2-β2Reμi-β1ImμiReμi-det-β1ImμiReμi00k2-β2Reμi-β1ImμiReμi01-β2Imμi-k1-β1ReμiImμi0k2-β2Reμi-β1ImμiReμi>0.Moreover, if k1=k2=0, the final values of bipartite consensus are (16)limt→∞ξt=wlTDξ0Dwr+twlTDθ0Dwr+12t2wlTDψ0Dwr,limt→∞θt=wlTDθ0Dwr+twlTDψ0Dwr,limt→∞ψt=wlTDψ0Dwr.where wl and wr are the left and right eigenvectors associated with the single eigenvalue 0 of the Laplacian matrix L-′, respectively, and wlTwr=1. If k1=0,k2≠0, the final values of bipartite consensus are (17)limt→∞ξt=wlTDξ0Dwr+1k2+twlTDθ0Dwr+tk2wlTDψ0Dwr,limt→∞θt=wlTDθ0Dwr+1k2wlTDψ0Dwrlimt→∞ψt=0.If k1≠0, the consensus values are as follows, respectively, (18)limt→∞ξt=wlTDξ0Dwr+k2k1wlTDθ0Dwr+1k1wlTDψ0Dwr,limt→∞θt=0,limt→∞ψt=0.Proof.
Forμi=0, when the assumption holds, the roots of polynomial λ2+k2λ+k1 are 0 or complex numbers with negative real parts. For μi≠0, when conditions (13), (14), and (15) are satisfied, the roots of polynomial all have negative real parts, that is, all eigenvalues except 0 of M- have negative real parts. Therefore, we have the following cases. (i)
Ifk1=k2=0, we see from Lemma 6 that matrix M- has exactly three zero eigenvalues and all the other eigenvalues have negative real parts. From the stability theorem and the results in [29], system (10) can asymptotically reach the consensus of edge dynamics, and limt→∞ξ-t=wrwlTξ-0+twrwlTθ-0+1/2t2wrwlTψ-0, limt→∞θ-t=wrwlTθ-0+twrwlTψ-0, and limt→∞ψ-t=wrwlTψ-0, where wr and wl are the right and left eigenvectors associated with μi=0 of Laplacian matrix L-′=DL′D, respectively, and wlTwr=1. Thus, from D=D-1, system (5) can asymptotically reach the bipartite consensus of edge dynamics under protocol (6), and (19)limt→∞ξt=DwrwlTDξ0+twrwlTDθ0+12t2wrwlTDψ0=wlTDξ0Dwr+twlTDθ0Dwr+12t2wlTDψ0Dwr,limt→∞θt=DwrwlTDθ0+twrwlTDψ0=wlTDθ0Dwr+twlTDψ0Dwr,limt→∞ψt=DwrwlTDψ0=wlTDψ0Dwr.(ii)
Ifk1=0,k2≠0, similar to the analysis above, matrix M- has exactly two zero eigenvalues and all the other eigenvalues have negative real parts. Then (10) can asymptotically reach the consensus of edge dynamics, and the final values of consensus are limt→∞ξ-t=wrwlTξ-0+1/k2+twrwlTθ-0+t/k2wrwlTψ-0, limt→∞θ-t=wrwlTθ-0+1/k2wrwlTψ-0, limt→∞ψ-t=0, where wr and wlT are defined as above. Thus, (5) can asymptotically reach the bipartite consensus of edge dynamics under protocol (6), and (20)limt→∞ξt=DwrwlTDξ0+1k2+twrwlTDθ0+tk2wrwlTDψ0=wlTDξ0Dwr+1k2+twlTDθ0Dwr+tk2wlTDψ0Dwr,limt→∞θt=DwrwlTDθ0+1k2wrwlTDψ0=wlTDθ0Dwr+1k2wlTDψ0Dwr,limt→∞ψt=0(iii)
Ifk1≠0, similar to the analysis in part one, matrix M- has exactly one zero eigenvalue and all the other eigenvalues have negative real parts. Then (10) can asymptotically reach the consensus of edge dynamics, and the final values of consensus are limt→∞ξ-t=wrwlTξ-0+k2/k1wrwlTθ-0+1/k1wrwlTψ-0, limt→∞θ-t=0, limt→∞ψ-t=0, where wr and wlT are defined as above. Therefore (5) can asymptotically reach the bipartite consensus of edge dynamics under protocol (6), and (21)limt→∞ξt=DwrwlTDξ0+k2k1wrwlTDθ0+1k1wrwlTDψ0=wlTDξ0Dwr+k2k1wlTDθ0Dwr+1k1wlTDψ0Dwr,limt→∞θt=0,limt→∞ψt=0.(iv)
If signed digraphG is structurally unbalanced, its line graph LG is also structurally unbalanced, which means that all the eigenvalues of Laplacian matrix -L′ have negative real parts. Therefore, all the eigenvalues of matrix M have negative real parts under conditions (13), (14), and (15). Thus, system (5) is asymptotically stable under protocol (6). Accordingly, we have limt→∞ξt=0, limt→∞θt=0, limt→∞ψt=0.


## 3.2. Bipartite Consensus of High-Order Edge Dynamics

Theh order continuous-time edge dynamics model is (22)ξ˙ij0=ξij1,ξ˙ij1=ξij2,⋮,ξ˙ijh-1=uij,where ξij1, l∈0,h-1 are the states of edges at time t. For system (22), we design the high-order control protocol (23)uij=-∑l=1h-1klξijl+∑r∈NjAjrsgnAjrξjr0t-ξij0t+∑l=1h-1∑r∈NjβlsgnAjrξjrlt-ξijltwhere k1,k2,…,kh-1≥0 and β1,β2,…,βh-1>0 are feedback gains. Denote ξl=ξijlt∈Rm×1, j∈Ni, m=∑i=1ndini, and ω=ξ0Tξ1T⋯ξh-1TT. Then we rewrite (22) with (23) as (24)ω˙=Mω,where (25)M=0Im0⋯000Im⋯0⋮⋮⋮⋱⋮000⋯Im-L′-k1Im-β1L′-k2Im-β2L′⋯-kh-1Im-βh-1L′,and matrix L′ is the Laplacian matrix of LG. Let ξ-l=Dξl, l∈0,h-1, D∈D, and the state vector ω-=ξ-0Tξ-1T⋯ξ-h-1TT. We have (26)ω-˙=M-ω-,where (27)M-=0Im0⋯000Im⋯0⋮⋮⋮⋱⋮000⋯Im-L-′-k1Im-β1L-′-k2Im-β2L-′⋯-kh-1Im-βh-1L-′with L-′=DL′D.Notice that(28)detλIhm-M-=detλIm-Im0⋯00λIm-Im⋯0⋮⋮⋮⋱⋮000⋯-ImL-′k1Im+β1L-′k2Im+β2L-′⋯λ+kh-1Im+βh-1L-′=detλh+kh-1λh-1+⋯+k1λIm+βh-1λh-1+⋯+β1λ+1L-′=∏i=1mλh+kh-1-βh-1μiλh-1+⋯+k2-β2μiλ2+k1-β1μiλ-μiwhere μi, i=1,2,…,m are eigenvalues of -L′ and there is a simple eigenvalue μi=0. Then matrix M- has at least one 0 eigenvalue.By Lemma2, all the eigenvalues of -L′ except the simple eigenvalue 0 have negative real parts, which shows that the eigenvalues of -L′ are 0, negative real numbers, and conjugate complex numbers with negative real parts. Therefore, we reorder the eigenvalues of -L′ to (29)0,μ2,…,μl,μl+1,μ-l+1,…,μl+k,μ-l+k,l,k≥0,l+2k=m.Theorem 8.
Given a high-order complex coefficient polynomial(30)fx=an+bnixn+an-1+bn-1ixn-1+⋯+a1+b1ix+a0+b0i,and another complex coefficient polynomial gx with the same order and conjugate coefficients as the above polynomial fx. (31)gx=an-bnixn+an-1-bn-1ixn-1+⋯+a1-b1ix+a0-b0i.Their product is (32)hx=fx×gx=∑l=1n∑i+j=2l-12aiaj+bibjx2l-1+∑l=0n∑i+j=2li≠j2aiaj+bibj+al2+bl2x2lProof.
The result can be proved by the multiplication rule of complex coefficient polynomial and the fact that every coefficient ofhx is greater than 0.Denoteλh+kh-1-βh-1μiλh-1+⋯+k2-β2μiλ2+k1-β1μiλ-μi by Phμi, and denote Phμi·P′hμi by P2hμi. Then from Theorem 8, (26) can be written as (33)detλIhm-Γ=∏i=1lPhμi·∏i=l+1l+kP2hμi.Lemma 9 (see [45]).
Given a system with characteristic polynomial(34)Ds=ansn+an-1sn-1+⋯+a1s+a0,ai>0,i∈0,n,then the system is stable if and only if the determinant consists of coefficients of characteristic polynomials (35)Δn=an-1an-3⋯000anan-2⋯0000an-1⋯0000an⋯000⋮⋮⋱⋮⋮⋮00⋯a3a1000⋯a4a2a0and its order principal minor determinant Δii=1,2,…,n-1 is greater than 0, namely, (36)Δ1=an-1>0,Δ2=an-1an-3anan-2>0,⋮,Δn>0.Theorem 10.
For a continuous-time edge dynamics multiagent system (22), let the corresponding signed digraph G be strongly connected, digon sign-symmetric and structurally balanced. When the Assumption holds, system (22) can asymptotically reach the bipartite consensus of edge dynamics under protocol (23), if the determinants and their order principal minor determinants of Phμi and P2hμi in (33) are all greater than 0 for μi≠0, and Δj>0, j∈1,h-l of Ph0 with kl=0, kl+1≠0 for μi=0.Proof.
From Lemma9 and the results in [40], if the determinants and their order principal minor determinants of Phμi and P2hμi in (33) are all greater than 0 for μi≠0, all roots of the corresponding subpolynomial (28) have negative real parts. If Δj>0, j∈1,h-l with kl=0, kl+1≠0 for μi=0, the roots except 0 of the corresponding subpolynomial (28) have negative real parts. Thus system (26) can reach the consensus of edges. We have the following cases. (i)
Ifki=0, i∈1,h-1, matrix M- has exactly h zero eigenvalues and all the other eigenvalues have negative real parts. By the results of [29], system (26) can asymptotically reach the consensus of edge dynamics, and limt→∞ξ-i=∑k=ih-1‍tk-i/k-i!wrwlTξ-k0, i∈0,h-1, where wr and wlT are the same as above. By the property of D, system (22) can asymptotically reach the bipartite consensus of edge dynamics under protocol (23), and the final values of bipartite consensus are (37)limt→∞ξi=∑k=ih-1tk-ik-i!DwrwlTDξk0=∑k=ih-1tk-ik-i!wlTDξk0Dwr,i∈0,h-1.(ii)
Ifki=0,ki+1≠0, i∈1,h-2, matrix M- has exactly i+1 zero eigenvalues and all the other eigenvalues have negative real parts. Therefore, system (26) can asymptotically reach the consensus, and limt→∞ξ-i=wrwlTξ-i0+∑j=i+1h-2‍kl+1/ki+1wrwlTξ-l0+1/ki+1wrwlTξ-h-10, limt→∞ξ-l=0, l>i. Similar to the analysis in the first part, system (22) can asymptotically reach the bipartite consensus of edges dynamics under protocol (23), and (38)limt→∞ξi=wlTDξi0Dwr+∑l=i+1h-2kl+1ki+1wlTDξl0Dwr+1ki+1wlTDξh-10Dwr,limt→∞ξl=0,l>iFurthermore, the limit ofξl at t→∞ is i-l order function of t when l<i.(iii)
Ifk1≠0, matrix M- only has exactly one zero eigenvalue and all the other eigenvalues have negative real parts. Thus, system (26) can asymptotically reach the consensus of edge dynamics, and limt→∞ξ-0=wrwlTξ-00+∑i=1h-2‍ki+1/k1wrwlTξ-i0+1/k1wrwlTξ-h-10, limt→∞ξ-i=0, i∈1,h-1. System (22) can asymptotically reach the bipartite consensus of edge dynamics under protocol (23), and the final values of bipartite consensus are (39)limt→∞ξ0=wlTDξ00Dwr+∑i=1h-2ki+1k1wlTDξi0Dwr+1k1wlTDξh-10Dwr,limt→∞ξi=0,i∈1,h-1.(iv)
IfLG is structurally unbalanced, all eigenvalues of Laplacian matrix -L′ have negative real parts. All eigenvalues of matrix M have negative real parts, which means (22) can be asymptotically stable under protocol (23) and limt→∞ξi=0, i∈0,h-1.Remark 11.
Although the general conditions on higher-order systems are valid for third-order systems, the conditions derived for third-order systems have their own characteristics and-values. To be more specific, the judgment method of the third-order system is more direct than that of the general higher-order system. More importantly, the results of third-order systems are much more detailed. Thus, the results of third-order systems are useful for detailing the effect of feedback gains on the consensus end-value and for understanding the implications of the results of general higher-order systems. For example, one can see from the third-order system thatξij0 is the function of t in the presence of kl=0 with the order equal to the number of kl equal to zero.


## 4. Simulation

We choose digraph (a) with 6 nodes and 9 edges in Figure1, in which the solid line between nodes represents cooperative relationship between agents and the dotted line represents competitive relationship between agents. The weights of solid line and dotted line are taken as 1 and -1, respectively. The line graph of (a) is (b) with 9 nodes. Let ξ0=1.6,-3.7,-1.8,2.5,-2,-1,3.6,1,-3T, θ0=-3.7,-0.5,2,-2.5,1,2.5,-1,-2,0.6T, ψ0=1.6,2,-2,1,-2.5,-3,3.6,-1,0.4T. Let D=diag1,1,-1,-1,1,-1,1,-1,1, then we have (40)L-′=DL′D=2-1-1000000010-10000000100000-10002-1-10000-10010000000002-1-100-1-100020000000001-1-100000001,and 0 is a simple eigenvalue of L-′. The corresponding left eigenvector and right eigenvector are (41)wl=-0.2328-0.6209-0.3105-0.3105-0.3105-0.1552-0.0776-0.1552-0.4657T,wr=-0.3789-0.3789-0.3789-0.3789-0.3789-0.3789-0.3789-0.3789-0.3789T,and wlT,wr satisfy wlTwr=1.Figure2 shows that the absolute value of position and velocity of all the edges in the system converge to limt→∞ξij=0.4234t2-0.2205t-1.4705 and limt→∞θij=0.8468t-0.2205, respectively, with the absolute values of acceleration of edges converging to 0.8468.Figure 2
Position, velocity, and acceleration trajectories of 1(a) withk1=k2=0, β1=β2=3.Ifk1=0, k2=1, β1=β2=3, Figure 3 shows that the absolute value of position trajectories of all the edges in the system converge to limt→∞ξijt=0.6263t-1.691, and the absolute values of velocity and acceleration trajectories of all edges are limt→∞θijt=2.3601, limt→∞ψijt=0.Figure 3
Position, velocity, and acceleration trajectories of 1(a) withk1=0, k2=1, β1=β2=3.In Figure4, for the third-order edge dynamic multiagent system (5), if k1=k2=1, β1=β2=2, we can get the following results by calculation (42)limt→∞ξij=0.8442,limt→∞θij=0,limt→∞ψij=0,i,j∈E.Figure 4
Position, velocity, and acceleration trajectories of 1(a) withk1=1, k2=1, β1=β2=3.If the digraph is structurally unbalanced ((c) in Figure1), system (5) is asymptotically stable. Figure 5 shows that if the digraph is structurally unbalanced, all states of edges converge to 0, that is to say, limt→∞ξijt=0, limt→∞θijt=0, limt→∞ψijt=0, i,j∈E.Figure 5
Position, velocity, and acceleration trajectories of 1(c) withk1=0.5, k2=1, β1=β2=1.


## 5. Conclusion

In this paper, we first considered the information of edges and mapped the digraph to a line graph. Then the protocols of third-order and ordinary high-order multiagent systems were designed on the line graph, respectively. We proposed the conditions such that multiagent systems can asymptotically reach the third-order bipartite consensus of edge dynamics. For the high-order multiagent system, we simplified the characteristic polynomial of the system on the line graph and gain the consensus conditions making system asymptotically reach the bipartite consensus of edge dynamics. By taking advantage of the properties of eigenvalues, we discussed the state values of all the edges for the third-order and ordinary high-order multiagent system. The high-order consensus of edges with other constraints has not been discussed, which is one of our research directions in the future.



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*Source: 1628239-2019-06-19.xml*