# Consensus of Second-Order Multiagent Systems with Directed Signed Networks and Communication Delays

Authors: Huiqin Pei; Qiang Lai
Journal: Complexity
 (2020)

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## Abstract

This paper is concerned with the consensus problem for second-order multiagent systems (MASs) with directed signed networks and local communication delays. First, under a strongly connected directed graph, some properties for signed digraph and the model of second-order MASs are provided. A distributed nonlinear consensus control protocol is proposed for implementing consensus of second-order MASs with signed digraph and communication delay. Then, for structurally balanced and unbalanced networks, the consensus of second-order MASs with directed signed networks and local delays is proved via the Lyapunov method and Barbalat lemma. Simulation examples are given to demonstrate the effectiveness of our results.


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## Body

## 1. Introduction

The cooperative control idea of MASs generally originates from gregarious behaviors of organisms in nature and society, such as flocking of migrating birds [1], swarming of foragers, and the synchronized behavior of a shoal of fish. These behaviors can help organisms to avoid predators and look for food, which contribute to the implementation of complex global behaviors. Consensus is known as an important fundamental branch of cooperative control of MASs, which means all agents can be assured to reach the same state value for any initial states of agents. Moreover, it has a wide range of applications, such as mobile robots [2], unmanned aerial vehicle [3], and sensor networks [4].For consensus of MASs [5–10], many studies involve system graphs with all positive edge weights. When the MASs with signed graphs having positive and negative edge weights are taken into account, it is not easy to make all agents achieve the same state value. There are coordination and antagonistic contradiction between agents. In reality, there normally exist networks with signed graphs, e.g., the UAV network with bidirectional flying and the human social network. So the system studied can be seen as a signed network, and it is necessary to consider bipartite consensus of the system, which requests the agents to reach an agreement for agents in modulus but not in sign [11]. In [12], the bipartite consensus of MASs with first-order dynamics was investigated under undirected signed graphs and signed digraphs; e.g., all agents converge to an absolute value with two opposite fronts. Furthermore, the interval bipartite consensus of networked agents was studied, where the signed digraphs only need to include spanning trees in [13]. Subsequently, the finite time bipartite consensus issue of first-order MASs was researched under undirected signed graph in [14–17].In practice, it is known that there are communication delays between agents, which affect the stability and performance of MASs. As we know, there are correlatively few research studies on the bipartite consensus of MASs with time delays. Hence, in this paper, the consensus problem of second-order MASs is investigated under directed signed networks and local communication delays. Compared with research contents in [13, 17], the second-order dynamical models of agents are considered. Meanwhile, a new distributed nonlinear consensus control protocol with local communication delays is put forward. The main contributions of the paper are as follows: (1) a distributed nonlinear consensus control protocol is put forward for implementing consensus of second-order MASs with signed digraph and communication delays; (2) for structurally balanced and unbalanced networks, the corresponding theorem is provided for the consensus of second-order MASs with directed signed networks and local delays about any initial conditions; (3) it is proved through the Lyapunov method and Barbalat lemma.The reminder of the paper is organized as follows. Section2 includes preliminaries of signed graph and problem statement. Section 3 includes main results of this paper, which contain the proposed distributed nonlinear control protocol to achieve the consensus of second-order MASs with directed signed networks and communication delays and the corresponding theorem and so on. Section 4 gives simulation examples to illustrate the validity of our results. Finally, Section 5 is the conclusion of this paper.Notation 1.
Some standard notations are listed as follows that are used in this paper.R denotes real numbers. RN and RN×N are the N-dimensional real vector space and N×N real matrix space, respectively. The signσ shows the sign function with σ∈R, as follows:(1)signσ=1,σ>0,0,σ=0,−1,σ<0.


## 2. Preliminaries and Problem Statement

### 2.1. Preliminaries of the Signed Graph

A signed graphG=V,E,A includes a vertex set V=v1,v2,…,vN, an edge set E⊆V×V, and a signed weighted adjacency matrix A=Aij∈RN×N. Thereinto, an edge Eij=vj,vi∈E denotes the flow of information from vj to vi. The corresponding signed weight Aij is positive or negative; that is, Aij≠0, otherwise Aij=0. Suppose that the graph G has no self-loop, i.e., Aii=0,i∈1,2,…,N. When vj,vi∈E, vi,vj∈E, and AT=A, the graph G is undirected. Conversely, it is called a directed graph or digraph. When V¯⊆V and E¯⊆E, a directed graph G¯=V¯,E¯,A¯ is a subgraph of G. A path is a series of edges vk,vk1,vk1,vk2,…,vkn,vl from vertex vl to vk in G with distinct vertices vki,i=1,2,3,…,n. A cycle is a special path with the starting and ending vertex at the same vertex. For a cycle, when the product of all signed edge weights is positive, the cycle is positive, otherwise negative.Lemma 1 (see [18]).
For a strongly connected directed graphG with positive A=Aij∈RN×N, the Laplacian matrix is represented as L=lij∈RN×N, where(2)lij=−∑l=1,l≠iNAil,i=j,Aij,i≠j.Then, there is s>0 to make sTL=0.Definition 1 (see [12]).
When a signed graphG allows a bipartition of the vertexes, that is, V1 and V2 with V1∪V2=V and V1∩V2=0, the signed graph G is known as structurally balanced, i.e., Aij>0,vi,vj∈V1, and Aij<0,vi∈V1,vj∈V2.Definition 2.
LetD¯ define a diagonal matrix: D¯=diagδδ=δ1,δ2,…,δi,…,δN,δi∈±1.Lemma 2 (see [12, 15, 18]).
For a strongly connected signed digraphG, the conditions of structural balance are shown as follows:(1)
All directed cycles are positive inG(2)
Arbitrary matrixD∈D¯, and there exists DAD being nonnegative(3)
L has an eigenvalue 0, and rankL=N−1Lemma 3 (see [12, 15, 18]).
For a strongly connected signed digraphG, the conditions of structural unbalance are shown as follows:(1)
At least one directed cycle is negative inG(2)
There exists no matrixD∈D¯, and there is DAD being nonnegative(3)
L has an eigenvalue 0, and rankL=N−1


### 2.2. Problem Statement

Consider a multiagent system with second-order agents and local communication delays under the directed signed network. Each agent has the following dynamics:(3)p⌢˙it=v⌢it,v⌢˙it=u⌢it,i=1,2,…,N,where p⌢it∈Rn and v⌢it∈Rn represent the position state and velocity state of agent i, separately, and u⌢it∈Rn is the control input of agent i. In order to facilitate the analysis, assume that all agents are in 1-D space, namely, p⌢i,v⌢i,u⌢i∈R. However, the results of the paper can be extended to multidimensional space.Definition 3.
The consensus of second-order multiagent system (3) is considered to be achieved for any initial state values p⌢i0 and v⌢i0 if there exist(4)limt⟶∞p⌢i−p⌢j=0, andp⌢i=p⌢j,limt⟶∞v⌢i−v⌢j=0, andv⌢i=v⌢j,i,j=1,2,…,N.
The objective of this paper is to design a distributed nonlinear consensus control protocol which is used to find the consensus of second-order MASs with signed digraph and communication delays and to prove the corresponding consensus of second-order MASs for structurally balanced and unbalanced networks through the Lyapunov method and Barbalat lemma, that is, the stability analysis of second-order MASs.


## 2.1. Preliminaries of the Signed Graph

A signed graphG=V,E,A includes a vertex set V=v1,v2,…,vN, an edge set E⊆V×V, and a signed weighted adjacency matrix A=Aij∈RN×N. Thereinto, an edge Eij=vj,vi∈E denotes the flow of information from vj to vi. The corresponding signed weight Aij is positive or negative; that is, Aij≠0, otherwise Aij=0. Suppose that the graph G has no self-loop, i.e., Aii=0,i∈1,2,…,N. When vj,vi∈E, vi,vj∈E, and AT=A, the graph G is undirected. Conversely, it is called a directed graph or digraph. When V¯⊆V and E¯⊆E, a directed graph G¯=V¯,E¯,A¯ is a subgraph of G. A path is a series of edges vk,vk1,vk1,vk2,…,vkn,vl from vertex vl to vk in G with distinct vertices vki,i=1,2,3,…,n. A cycle is a special path with the starting and ending vertex at the same vertex. For a cycle, when the product of all signed edge weights is positive, the cycle is positive, otherwise negative.Lemma 1 (see [18]).
For a strongly connected directed graphG with positive A=Aij∈RN×N, the Laplacian matrix is represented as L=lij∈RN×N, where(2)lij=−∑l=1,l≠iNAil,i=j,Aij,i≠j.Then, there is s>0 to make sTL=0.Definition 1 (see [12]).
When a signed graphG allows a bipartition of the vertexes, that is, V1 and V2 with V1∪V2=V and V1∩V2=0, the signed graph G is known as structurally balanced, i.e., Aij>0,vi,vj∈V1, and Aij<0,vi∈V1,vj∈V2.Definition 2.
LetD¯ define a diagonal matrix: D¯=diagδδ=δ1,δ2,…,δi,…,δN,δi∈±1.Lemma 2 (see [12, 15, 18]).
For a strongly connected signed digraphG, the conditions of structural balance are shown as follows:(1)
All directed cycles are positive inG(2)
Arbitrary matrixD∈D¯, and there exists DAD being nonnegative(3)
L has an eigenvalue 0, and rankL=N−1Lemma 3 (see [12, 15, 18]).
For a strongly connected signed digraphG, the conditions of structural unbalance are shown as follows:(1)
At least one directed cycle is negative inG(2)
There exists no matrixD∈D¯, and there is DAD being nonnegative(3)
L has an eigenvalue 0, and rankL=N−1


## 2.2. Problem Statement

Consider a multiagent system with second-order agents and local communication delays under the directed signed network. Each agent has the following dynamics:(3)p⌢˙it=v⌢it,v⌢˙it=u⌢it,i=1,2,…,N,where p⌢it∈Rn and v⌢it∈Rn represent the position state and velocity state of agent i, separately, and u⌢it∈Rn is the control input of agent i. In order to facilitate the analysis, assume that all agents are in 1-D space, namely, p⌢i,v⌢i,u⌢i∈R. However, the results of the paper can be extended to multidimensional space.Definition 3.
The consensus of second-order multiagent system (3) is considered to be achieved for any initial state values p⌢i0 and v⌢i0 if there exist(4)limt⟶∞p⌢i−p⌢j=0, andp⌢i=p⌢j,limt⟶∞v⌢i−v⌢j=0, andv⌢i=v⌢j,i,j=1,2,…,N.
The objective of this paper is to design a distributed nonlinear consensus control protocol which is used to find the consensus of second-order MASs with signed digraph and communication delays and to prove the corresponding consensus of second-order MASs for structurally balanced and unbalanced networks through the Lyapunov method and Barbalat lemma, that is, the stability analysis of second-order MASs.


## 3. Main Results

In this section, the consensus is investigated for the second-order multiagent system with strongly connected directed signed network. The communication delays of information flow between agents are considered. Taking into account multiagent system (3), the following distributed nonlinear consensus control protocol is put forward:(5)u⌢it=−a∑j=1NAijp⌢it−signAijp⌢jt−τij−b∑j=1NAijv⌢it−signAijv⌢jt−τij,i=1,2,…,N,where a and b are nonnegative coupling coefficients, τij denotes the communication delay from vertex vj to vi with i≠j, and τii=0.Theorem 1.
Consider second-order multiagent system (3) with a strongly connected directed signed graph G for any finite communication delay τij. For any initial conditions, when G is structurally balanced, the bipartite consensus of the system can be asymptotically achieved. Similarly, when G is structurally unbalanced, the consensus of the system can gradually come true.Proof.
For a signed digraphG with strong connectivity, the diagonal matrix D∈D¯ is chosen. Let ξt=Dp⌢tandηt=Dv⌢t; we have(6)ξit=δip⌢it,ηit=δiv⌢it,i=1,2,…,N.
Through (6), expression (7) is obtained from system (3) and nonlinear control protocol (5):(7)ξ˙it=ηit,i=1,2,…,N,δiη˙it=−a∑j=1NAijδiξit−signAijδjξjt−τij−b∑j=1NAijδiηit−signAijδjηjt−τij.
When the directed signed network of the system is structurally balanced, according to Lemma2, there exists DAD with all nonnegative elements, and one has δisignAijδj=signAij=1. Then, via δisignAijδj=1 and δi2=1, expression (7) is rewritten as(8)ξ˙it=ηit,η˙it=−a∑j=1NAijξit−ξjt−τij−b∑j=1NAijηit−ηjt−τij,i=1,2,…,N.
LetA+=Aij+N×N, where Aij+=Aij, for all i,j=1,2,…,N. The corresponding signed digraph is marked as G+ with strongly connected and nonnegative weights. Then, by (8), there is(9)ξ˙it=ηit,η˙it=−a∑j=1NAij+ξit−ξjt−τij−b∑j=1NAij+ηit−ηjt−τij,i=1,2,…,N.
According to Lemma1, one has sTL=0, which implies that(10)silii=−∑j=1,j≠iNsjlji.
In addition, on account oflii=−∑j=1NAij+, it can be acquired that(11)∑j=1NsiAij+=∑j=1NsjAji+,∑i=1NsjAji+=∑i=1NsiAij+.
Take into account the following Lyapunov function:(12)Vt=V1t+V2t,where(13)V1t=12∑i=1NsiηiTtηit,(14)V2t=b2∑i=1N∑j=1N∫t−τjitsjAji+ηiTθηiθdθ.
Take the derivative ofV1t and V2t, and there exist(15)V˙1t=∑i=1NsiηiTtη˙it=∑i=1NsiηiTt−a∑j=1NAij+ξit−ξjt−τij−b∑j=1NAij+ηit−ηjt−τij,(16)V˙2t=b2∑i=1N∑j=1NsjAji+ηiTtηit−ηiTt−τjiηit−τji=b2∑i=1N∑j=1NsiAij+ηiTtηit−b2∑i=1N∑j=1NsiAij+ηjTt−τijηjt−τij.
Hence, by (15) and (16), we have(17)V˙t=V˙1t+V˙2t=−a∑i=1N∑j=1NsiAij+ηiTtξit−ξjt−τij−b2∑i=1N∑j=1NsiAij+ηiTtηit−2ηiTtηjt−τij+ηjTt−τijηjt−τij=−a∑i=1N∑j=1NsjAji+ηiTtξit−ξit−τji−b2∑i=1N∑j=1NsiAij+ηit−ηjt−τijTηit−ηjt−τij≤−a∑i=1N∑j=1NsjAji+ξit−ξit−τjiTτjiξit−ξit−τji−b2∑i=1N∑j=1NsjAji+ηit−ηit−τjiTηit−ηit−τji≤0.
Therefore,Vt is descending and Vt≥0 shows that limt⟶∞Vt is finite. Furthermore, through (12), it is known that ξitandηit are bounded for i∈1,2,…,N. By system (9), for arbitrary i∈1,2,…,N, we see that ξ˙itandη˙it are bounded. So, it is concludedd that V¨t is also bounded. According to Barbalat lemma [19], it is obtained that limt⟶∞asjAji+ξit−ξit−τjiT/τjiξit−ξit−τji=0 and limt⟶∞b/2sjAji+ηit−ηit−τjiTηit−ηit−τji=0. On account of a>0,b>0,s>0,andAji+>0,(18)limt⟶∞ξit−ξit−τji=0,limt⟶∞ηit−ηit−τji=0.
Moreover, there areξ˙it⟶0andη˙it⟶0 for i∈1,2,…,N.
Because the signed network is strongly connected, it is observed thatξ˙it⟶0andη˙it⟶0 when t⟶∞ for i∈1,2,…,N. So, ξitandηit tend to be steady values ci,di∈R, respectively. Since the matrix A+ is irreducible, the largest invariant sets of system (9) are Φ1=ξ1t,ξ2t,…,ξNtξ1t=ξ2t=⋯=ξNtandΦ2=η1t,η2t,…,ηNtη1t=η2t=⋯=ηNt. There exist constants c∈Randd∈R, such that ξit⟶candηit⟶d as t⟶∞ for i∈1,2,…,N. Therefore, for system (9) with any finite communication delay τij and initial values, we have limt⟶∞ξit⟶candlimt⟶∞ηit⟶d=0. As a result, the position p⌢it and velocity v⌢it states of system (3) are satisfied with expression (4) for i=1,2,…N. Then, the bipartite consensus of system (3) can be asymptotically realized for the signed network with structural balance.
When the directed signed network of the system is structurally unbalanced, the corresponding graphG includes one or more negative cycles from Lemma 3. In order to facilitate the analysis, assume that G only include one negative cycle with an negative weight edge vj0,vi0, Ai0,j0<0. Thus, the subgraph G¯ of G is structurally balanced to exclude the edge vj0,vi0. From Definition 1, the signed digraph G¯ has a bipartition of the vertexes V¯1 and V¯2. Meanwhile, vj0andvi0 belong to V¯1 or V¯2. Note that V¯1=V1andV¯2=V2. At present, adding the edge vj0,vi0 to the subgraph G¯, the bipartition of the vertexes V1 and V2 for G is obtained. So, it is known that D=diagδ with δi=1 as vi∈V1 and δi=−1 as vi∈V2. Then, one has DAD=Aij0N×N with one negative element (Ai0j00=δi0Ai0j0δj0=Ai0j0<0).
Then, from (7), there exists(19)ξ˙it=ηit,i=1,2,…,N.η˙it=−a∑j=1NAijξit−δisignAijδjξjt−τij−b∑j=1NAijηit−δisignAijδjηjt−τij.
Similarly, consider the Lyapunov functionVt=V1t+V2t, where(20)V1t=12∑i=1NsiηiTtηit,V2t=b2∑i=1N∑j=1N∫t−τjitsjAjiηiTθηiθdθ.
Here, define a nonnegative matrixB+=Bij+N×N, where(21)Bij+=0,i=i0,j=j0,Aij+,else.
Take the derivative ofV1t and V2t, and it is obtained that(22)V˙1t=∑i=1NsiηiTtη˙it=∑i=1NsiηiTt−a∑j=1NAijξit−δisignAij+δjξjt−τij+∑i=1NsiηiTt−b∑j=1NAijηit−δisignAij+δjηjt−τij+si0ηi0TtAi0j0aξj0t−τi0j0+ξi0t+bηj0t−τi0j0+ηi0t=∑i=1N∑j=1NasiηiTtBij+ξjt−τij−ξit+∑i=1N∑j=1NbsiηiTtBij+ηjt−τij−ηit+asi0ηi0TtAi0j0ξj0t−τi0j0+ξi0t+bsi0ηi0TtAi0j0ηj0t−τi0j0+ηi0t,(23)V˙2t=b2∑i=1N∑j=1NsjAjiηiTtηit−ηiTt−τjiηit−τji=b2∑i=1N∑j=1NsiAij+ηiTtηit−b2si0Ai0j0ηi0Ttηi0t−b2∑i=1N∑j=1NsiAij+ηjTt−τijηjt−τij+b2si0Ai0j0ηj0Tt−τi0j0ηj0t−τi0j0=b2∑i=1N∑j=1NsiBij+ηiTtηit−b2si0Ai0j0ηi0Ttηi0t−b2∑i=1N∑j=1NsiBij+ηjTt−τijηjt−τij+b2si0Ai0j0ηj0Tt−τi0j0ηj0t−τi0j0.
Thus, via (22) and (23), we have(24)V˙t=V˙1t+V˙2t=a∑i=1N∑j=1NsiηiTtBij+ξjt−τij−ξit+asi0Ai0j0ηi0Ttξj0t−τi0j0+ξi0t−b2∑i=1N∑j=1NsiBij+ηiTtηit−2ηiTtηjt−τij+ηjTt−τijηjt−τij+b2si0Ai0j0ηi0Ttηi0t+2ηi0Ttηj0t−τi0j0+ηj0Tt−τi0j0ηj0t−τi0j0=−a∑i=1N∑j=1NsjBji+ηiTtξit−ξit−τji+asi0Ai0j0ηi0Ttξi0t+ξj0t−τi0j0−b2∑i=1N∑j=1NsiBij+ηit−ηjt−τijTηit−ηjt−τij+b2si0Ai0j0ηi0t+ηj0t−τi0j0Tηi0t+ηj0t−τi0j0≤−a∑i=1N∑j=1NsjBji+ξit−ξit−τjiTτjiξit−ξit−τji+asi0Ai0j0ηi0Ttξi0t+ξj0t−τi0j0−b2∑i=1N∑j=1NsjBji+ηit−ηit−τjiTηit−ηit−τji+b2si0Ai0j0ηi0t+ηj0t−τi0j0Tηi0t+ηj0t−τi0j0≤0.
Hence,Vt is nonincreasing, and Vt≥0 indicates that limt⟶∞Vt is finite. Ulteriorly, via Vt, it is known that ξitandηit are bounded for i∈1,2,…,N. Through system (19), for ∀i∈1,2,…,N, ξ˙itandη˙it are bounded. It is concluded that V¨t is likewise bounded. Via Barbalat lemma [19], it is obvious that limt⟶∞si0Ai0j0ηi0t+ηj0t−τi0j0Tηi0t+ηj0t−τi0j0=0, and(25)limt⟶∞asi0Ai0j0ξi0t+ξj0t−τi0j0=0,limt⟶∞asjBji+ξit−ξit−τjiTτjiξit−ξit−τji=0,limt⟶∞b2sjBji+ηit−ηit−τjiTηit−ηit−τji=0.
Because ofa,b>0,si,si0>0,Bji+>0,andAi0j0<0, there are(26)limt⟶∞ξit−ξit−τji=0,limt⟶∞ηit−ηit−τji=0,limt⟶∞ξi0t+ξj0t−τi0j0=0,limt⟶∞ηi0t+ηj0t−τi0j0=0.
Moreover, for system (19), one has(27)ξ˙i0t=ηi0t,η˙i0t=−a∑j=1NAi0jξi0t−δi0signAi0jδjξjt−τi0j−b∑j=1NAi0jηi0t−δi0signAi0jδjηjt−τi0j=aAi0j0ξi0t+ξjt−τi0j+bAi0j0ηi0t+ηjt−τi0j−a∑j≠j0Bi0jξi0t−ξjt−τi0j−b∑j≠j0Bi0jηi0t−ηjt−τi0j⟶0ast⟶∞.ξ˙it=ηit,i∈1,2,…,N,η˙it=−a∑j=1NBij+ξit−ξjt−τij−b∑j=1NBij+ηit−ηjt−τij⟶0ast⟶∞.
Then, there areξ˙it⟶0andη˙it⟶0 for i∈1,2,…,N.
Here,GA+ is a strongly connected digraph, according to Lemma 4 in [16], and it is obtained that GB+ is a weakly connected digraph. Hence, one has ξ1t=ξ2t=⋯=ξNt and η1t=η2t=⋯=ηNt as t⟶∞. Meanwhile, there are ξi0t=ξj0t=0andηi0t=ηj0t=0 as t⟶∞. Finally, p⌢it=δiξit=0andv⌢it=δiηit=0, as t⟶∞ for i=1,2,…N, which are satisfied with expression (4). So, under structural unbalance, the consensus of system (3) can gradually come true. When the signed digraph G includes more cycles with negative weight, the above analysis approach is applicable.Remark 1.
In this paper, the finite communication delay,τij or τji, is minor. In most cases, the excessive delay causes system instability. However, in some cases, the delay may have a stabilizing effect. In [20], Yu et al. researched consensus and quasi-consensus of multiagent systems. In the delayed position information case, the corresponding consensus can be achieved. In [21,22], Liu et al. proposed Bessel–Laguerre integral inequality, which was applied to stability analysis of linear systems with infinite distributed delays, and gave a survey on time-delay approach to networked control.


## 4. Simulation

In this section, simulation examples are given to demonstrate the effectiveness of our results.Example 1.
Under structural balance, a second-order strongly connected multiagent system (3) with signed digraph is taken into account as shown in Figure 1(a), which includes 5 agents. The corresponding adjacency matrix is represented as(28)Aa=00001.21.500000−100000100030−1.30.
Here, proposed nonlinear control protocol (5) with communication delays is used to find the consensus of system (3). Choose that all communication delays are τ15=0.1,τ21=0.3,τ32=0.25,τ43=0.15,τ52=0.1,andτ54=0.1 and the nonnegative coupling coefficients are a=1andb=2. The initial position and velocity states are p⌢0=15,−12,−5,3,−7T and v⌢0=7,3,−5,−4,10. From Figure 2, it is seen that the bipartite consensus of system (3) can be asymptotically reached after a certain amount of time. In other words, position states of agents 1,2,3 and agents 4,5 converge to the same value, respectively. Meanwhile, velocity states of all agents converge to zero.
When all communication delays areτ15=0.6,τ21=0.6,τ32=0.65,τ43=0.65,τ52=0.6,andτ54=0.6, it takes more time to achieve the same value for position states of agents in Figure 3(a), respectively. From Figure 3(b), it is seen that the time required is increased for the velocity states of all agents reaching zero.Figure 1
Strongly connected directed signed networks.
(a)(b)Figure 2
Position and velocity states of agents on the structurally balanced network in Figure1(a).
(a)(b)Figure 3
Position and velocity states of agents on the structurally balanced network in Figure1(a).
(a)(b)Example 2.
Under structural unbalance, a second-order strongly connected multiagent system (3) with signed digraph is considered as shown in Figure 1(b), which contains 5 agents. The corresponding adjacency matrix is denoted as(29)Ab=00001.21.500000−1000001000−30−1.30.
Choose that all communication delays areτ15=0.1,τ21=0.3,τ32=0.25,τ43=0.15,τ52=0.1,andτ54=0.1. Similarly, using proposed nonlinear control protocol (5), the consensus of the system can be achieved for the same initial conditions and the parameters. From Figure 4, it is obvious that the position and velocity states of all agents realize the same value gradually through a certain time.Figure 4
Position and velocity states of agents on structurally unbalanced network in Figure1(b).
(a)(b)


## 5. Conclusion

In this paper, the consensus of second-order MASs with strongly connected signed digraph and communication delays has been studied. The corresponding distributed nonlinear consensus control protocol has been designed for the achievement of second-order MASs with directed signed networks. In addition, using the Lyapunov method, the consensus of second-order MASs with signed digraph and local delays has been certified under structurally balanced and unbalanced networks. Finally, the effectiveness of main results has been further verified via simulation examples.



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*Source: 1712643-2020-09-29.xml*