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

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.


Introduction
e 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. ese 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][6][7][8][9][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. ere 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][15][16][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. e 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. e reminder of the paper is organized as follows. Section 2 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. R N and R N×N are the N-dimensional real vector space and N × N real matrix space, respectively. e sign(σ) shows the sign function with σ ∈ R, as follows:

Preliminaries of the Signed Graph.
A signed graph G � (V, E, A) includes a vertex set V � v 1 , v 2 , . . . , v N , an edge set E⊆V × V, and a signed weighted adjacency matrix A � (A ij ) ∈ R N×N . ereinto, an edge E ij � (v j , v i ) ∈ E denotes the flow of information from v j to v i . e corresponding signed weight A ij is positive or negative; that is, A ij ≠ 0, otherwise A ij � 0. Suppose that the graph G has no self-loop, i.e., A ii � 0, i ∈ 1, 2, . . . , N. When (v j , v i ) ∈ E, (v i , v j ) ∈ E, and A T � 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, tEn, qA) is a subgraph of G. A path is a series of edges 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 graph G with positive A � (A ij ) ∈ R N×N , the Laplacian matrix is represented as L � (l ij ) ∈ R N×N , where en, there is s > 0 to make s T L � 0.
Definition 1 (see [12]). When a signed graph G allows a bipartition of the vertexes, that is, Definition 2. Let D define a diagonal matrix: Lemma 2 (see [12,15,18]). For a strongly connected signed digraph G, the conditions of structural balance are shown as follows: (1) All directed cycles are positive in G (2) Arbitrary matrix D ∈ D, and there exists DAD being nonnegative (3) L has an eigenvalue 0, and rank(L) � N − 1 Lemma 3 (see [12,15,18]). For a strongly connected signed digraph G, the conditions of structural unbalance are shown as follows: (1) At least one directed cycle is negative in G (2) ere exists no matrix D ∈ D, and there is DAD being nonnegative (3) L has an eigenvalue 0, and rank(L) � 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: where p ⌢ i (t) ∈ R n and v ⌢ i (t) ∈ R n represent the position state and velocity state of agent i, separately, and u ⌢ i (t) ∈ R n is the control input of agent i. In order to facilitate the analysis, assume that all agents are in 1-D space, namely, However, the results of the paper can be extended to multidimensional space.

Definition 3.
e consensus of second-order multiagent system (3) is considered to be achieved for any initial state values p e 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 secondorder MASs.

Main Results
In this section, the consensus is investigated for the secondorder multiagent system with strongly connected directed signed network. e 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: where a and b are nonnegative coupling coefficients, τ ij denotes the communication delay from vertex v j to v i with i ≠ j, and τ ii � 0. Proof. For a signed digraph G with strong connectivity, the diagonal matrix rough (6), expression (7) is obtained from system (3) and nonlinear control protocol (5): When the directed signed network of the system is structurally balanced, according to Lemma 2, there exists DA D with all nonnegative elements, and one has (7) is rewritten as e corresponding signed digraph is marked as G + with strongly connected and nonnegative weights. en, by (8), there is Complexity 3 According to Lemma 1, one has s T L � 0, which implies that In addition, on account of Take into account the following Lyapunov function: where Take the derivative of V 1 (t) and V 2 (t), and there exist Hence, by (15) and (16), we have erefore, V(t) is descending and V(t) ≥ 0 shows that lim t⟶∞ V(t) is finite. Furthermore, through (12), it is known that ξ i (t) and η i (t) are bounded for i ∈ 1, 2, . . . , N { }. By system (9), for arbitrary i ∈ 1, 2, . . . , N { }, we see that _ ξ i (t) and _ η i (t) are bounded. So, it is concludedd that € V(t) is also bounded. According to Barbalat lemma [19], it is obtained that lim t⟶∞ as and η i (t) tend to be steady values c i , d i ∈ R, respectively. Since the matrix A + is irreducible, the 4 Complexity largest invariant sets of system (9) are Φ 1 � ξ 1 (t), ξ 2 (t), . . . , ξ N (t) | ξ 1 (t) � ξ 2 (t) � · · · � ξ N (t)} and Φ 2 � η 1 (t), η 2 (t), . . . , η N (t) | η 1 (t) � η 2 (t) � · · · � η N (t) . ere exist constants c ∈ R and d ∈ R, such that ξ i (t) ⟶ c and η i (t) ⟶ d as t ⟶ ∞ for i ∈ 1, 2, . . . , N { }. erefore, for system (9) with any finite communication delay τ ij and initial values, we have lim t⟶∞ ξ i (t) ⟶ c and lim t⟶∞ η i (t) ⟶ d � 0. As a result, the position p ⌢ i (t) and velocity v ⌢ i (t) states of system (3) are satisfied with expression (4) for i � 1, 2, . . . N { }. en, 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 graph G 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 (v j0 , v i0 ), A i0,j0 < 0. us, the subgraph G of G is structurally balanced to exclude the edge (v j 0 , v i 0 ). From Definition 1, the signed digraph G has a bipartition of the vertexes V 1 and V 2 . Meanwhile, v j 0 and v i 0 belong to V 1 or V 2 . Note that V 1 � V 1 and V 2 � V 2 . At present, adding the edge (v j 0 , v i 0 ) to the subgraph G, the bipartition of the vertexes V 1 and V 2 for G is obtained. So, it is known that D � diag(δ) with δ i � 1 as v i ∈ V 1 and δ i � −1 as v i ∈ V 2 . en, one has DAD � en, from (7), there exists Similarly, consider the Lyapunov function Here, define a nonnegative matrix Take the derivative of V 1 (t) and V 2 (t), and it is obtained that us, via (22) and (23), we have Hence, V(t) is nonincreasing, and V(t) ≥ 0 indicates that lim t⟶∞ V(t) is finite. Ulteriorly, via V(t), it is known that ξ i (t) and η i (t) are bounded for i ∈ 1, 2, . . . , N { }. rough system (19), for ∀i ∈ 1, 2, . . . , N { }, _ ξ i (t) and _ η i (t) are bounded. It is concluded that € V(t) is likewise bounded. Via Barbalat lemma [19], it is obvious that lim t⟶∞ 6 Complexity Because of a, b > 0, s i , s i 0 > 0, B + ji > 0, and (26) Moreover, for system (19), one has en, there are _ ξ i (t) ⟶ 0 and _ η i (t) ⟶ 0 for i ∈ 1, 2, . . . , N { }. Here, G(A + ) is a strongly connected digraph, according to Lemma 4 in [16], and it is obtained that G(B + ) is a weakly connected digraph. Hence, one has ξ 1 (t) � ξ 2 (t) � · · · � ξ N (t) and η 1 (t) � η 2 (t) � · · · � η N (t) as t ⟶ ∞.
Meanwhile, there are 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, τ ij or τ ji , 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.

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. e corresponding adjacency matrix is represented as Here, proposed nonlinear control protocol (5) with communication delays is used to find the consensus of system (3). Choose that all communication delays are τ 15  When all communication delays are τ 15 � 0.6, τ 21 � 0.6, τ 32 � 0.65, τ 43 � 0.65, τ 52 � 0.6, and τ 54 � 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.

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. e corresponding adjacency matrix is denoted as Choose that all communication delays are τ 15 � 0.1, τ 21 � 0.3, τ 32 � 0.25, τ 43 � 0.15, τ 52 � 0.1, and τ 54 � 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.

Conclusion
In this paper, the consensus of second-order MASs with strongly connected signed digraph and communication delays has been studied. e 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.