Bipartite Consensus of Heterogeneous Multiagent Systems with Diverse Input Delays

This paper investigates the bipartite consensus problem of heterogeneous multiagent systems with diverse input delays. Based on the systems composed of ﬁrst-order and second-order agents, the novel control protocols are designed. Using frequency-domain analysis and matrix theory, the corresponding upper bounds of the allowable delays are obtained under the undirected topology and directed topology, respectively. Finally, simulation examples are given to verify the theoretical analysis.

It is noteworthy that most of the problems about the consensus of multiagent systems are focused on cooperative network. In fact, cooperative and competitive relationships exist extensively in both natural and engineered network systems, such as opinion dynamics in social networks [12] and biological systems [13]. e bipartite consensus was firstly proposed in [14], which defined a signed graph the edges with positive and negative weights to describe the cooperative and competitive relations between agents. Bipartite consensus can be used for formation control [15], obstacle avoidance of wheeled robots [16], and nanoquadcopters formation [17].
ere are many factors that affect the stability of agents. e time delay problem is one of the important problems that affect the consensus of a multiagent system. e bipartite consensus problem of second-order multiagent systems with fixed time delays was studied in [18]. Based on the second-order multiagent systems, Tian et al. [19] discussed the bipartite consensus problem of the system under different disturbances. e bipartite consensus with arbitrary finite communication delay was discussed in [20]. Most of the above work use undirected graphs as communication networks. Compared with undirected graphs [21], directed graphs [22][23][24][25] are more versatile and cost effective when the edge weights can be arbitrary between two agents. is paper pays attention to bipartite consensus of heterogeneous MAS [26][27][28] with diverse input delays. As far as we know, there are few studies on this aspect, which is the motivation of this work. e theoretical analysis and simulation are presented under undirected topology and directed topology. e upper bounds of the allowable delays are given. e rest of this article is structured as follows. Section 2 introduces some concepts and basic lemmas of graph theory. In Section 3, the bipartite consensus analysis of heterogeneous multiagent systems with multiple input delays under undirected and directed topologies is presented. In Section 4, two numerical examples are given to illustrate the validity of theoretical analysis. Finally, Section 5 draws some conclusions.

Preliminaries.
In this section, the concepts and lemmas of some preliminary diagrams are introduced. We consider a heterogeneous multiagent system with n agents. e connection between agents is represented by an undirected graph or directed graph G � (V, ε, A), where V � ζ 1 , ζ 2 , . . . , ζ n represents the set of nodes, ε ⊆ V × V represents the set of edges, and A � [a ij ] ∈ R n×n represents the adjacency matrix of G. As the multiagent system of cooperation and competition is studied in this paper, the value can be either positive or negative. Here, we choose a ii ≠ 0 for all i ∈ 1, 2, . . . , n { }. If a ij ≠ 0, nodes ζ j and ζ i have information exchange; then, node ζ j is said to be the neighbor of node ζ i . e set of neighbors of node ζ i is denoted by Graph G is strongly connected if there is a path between any two nodes in graph Assumption 1 (see [29]). j∈N l |a ij | � 0, ∀i ∈ N i , and Lemma 1 (see [30]). For Assumption 1, Laplace's matrix L s has at least two roots of zero. Give the time-delay system where y(t) ∈ R n , A i ∈ R n×n , τ i ∈ R, and N is a positive integer.
Taking the Laplace transform, we can get the characteristic equation: Lemma 2 (see [30]). According to the characteristic equation of Lemma 1, if it has only two zero roots and the rest of its roots are on the left half-plane of the complex plane, then it has lim t⟶∞ y(t) � α + βt, where α ∈ R n and β ∈ R n are constant vectors.

Problem Formulation.
In this section, we will consider a heterogeneous multiagent system composed of agents, where m agents are second-order agents and n-m agents are first-order agents. e information transmission of each agent in the heterogeneous system is represented by G, and each agent represents a node.
Suppose the input delay of each agent in the system is not consistent. e dynamics of each second-order agent is given as follows: where x i , v i , u i ∈ R represent the position, velocity, and control input of the second-order agent i, respectively. T i > 0 represents the input delay. e dynamics of each first-order agent is given as follows: where x l , u l ∈ R represent the position and control input of the first-order agent l, respectively. T l > 0 represents the input delay. e bipartite consensus means all agents converge to a value which is the same for all in modulus but not in sign through distributed protocols. Similar to [33], the bipartite consensus protocols for the second-order agents are given by where k 1 , k 2 > 0.
Based on the dynamic neighbor estimation rule in [31,[34][35][36], an estimated speed is added to the first-order agent. e bipartite consensus protocols are given by where k 1 , k 2 > 0. e main purpose of this paper is to study the bipartite consensus protocol for heterogeneous multiagents with diverse input delays under the undirected topology and directed topology, respectively.

Bipartite Consensus under Undirected
Topology. We will consider the bipartite consensus of heterogeneous multiagent systems with diverse input delays under undirected topology in this section. is paper uses neighborhood estimation rules to estimate the speed of first-order agents. Using the dynamic change of the position of the agent around the first-order agents, the estimated velocity is substituted for its actual velocity. Based on the heterogeneous system with both cooperative and competitive relations, the second-order agent control protocol is designed as follows: and the first-order agent control protocol is designed as follows: where sgn(·) is the sign function, k 1 > 0 and k 2 > 0 are the control gains, N i and N l represent the neighborhood of the agent i and l, and τ i and τ l represent the input delay of the agent i and l.

Lemma 7.
For the heterogeneous multiagent systems (8) and (9), if and only if the Laplacian matrix L s has at least one zero eigenvalue, the real parts of the rest of eigenvalues are positive, and satisfy the condition of inequality (10), and the systems can achieve bipartite consensus.
where λ i is the eigenvalue of L s and ω ci satisfies ω ci � arctan(ω ci k 2 /k 1 ).
Proof. Take the Laplace transform on (8) and (9); then, we can obtain (12) From (11) and (12), we can obtain where us, the characteristic equation of (8) and (9) is given by Because the topology is undirected connected, 0 is a simple root of matrix L s , and the rest of the roots are positive.

Bipartite Consensus under Directed Topology.
We will study the bipartite consensus of heterogeneous multiagent systems with diverse input delays under directed topology in this section. e control protocols are designed as follows: where k 1 > 0, k 2 > 0 are the control gains, N i and N l represent the neighborhood of the agent i and l, and τ i and τ l represent the input delay of the agent i and l.

Lemma 8.
For the heterogeneous multiagent systems (30) and (31), if and only if the Laplacian matrix L s has at least two zero eigenvalues, the real parts of the rest of eigenvalues 4 Complexity are positive, and satisfy the condition of inequality (32), and the systems can achieve bipartite consensus: where λ i is the eigenvalue of L s , and ω ci satisfies Proof. e Laplace transform of (30) and (31): From Assumption 1, (34), and (35), en, the characteristic root of systems (31) and (32)  (38) Obviously, we know that the equation has four zero roots, and then we can analyze the rest of the characteristic roots of the equation. For let h(s) � 1 + g i (jω) � 0. erefore, equation (39) can be written as 1 + λ i (k 1 + k 2 s)e − τs /s 2 � 0, where g i (jω) � λ i (k 1 + k 2 s)e − τs /s 2 . Based on Nyquist's criterion, if and only if curve g i (jω) does not include point (− 1, j0), the characteristic root of equation (39) is located on the left half-plane of the complex plane. en, ) .

Numerical Examples and Simulations
Example 1. We will test and verify the results obtained by a multiagent system with five agents in this section. e initial conditions are randomly set and the topology is shown in Figure 1. e maximum eigenvalue of L s is λ max � 4.0445. k 1 � 0.45 and k 2 � 0.3. According to eorem 1, we obtain τ * < 0.5057. Let τ 1 � 0.5, τ 2 � 0.4, τ 3 � 0.3, τ 4 � 0.2, and τ 5 � 0.1. It is clear that bipartite consensus can be achieved when the input delays are below the upper bound of the allowable delay (see Figure 2). Let τ 1 � 0.4, τ 2 � 0.3, τ 3 � 0.4, τ 4 � 0.3, and τ 5 � 0.6. e bipartite consensus cannot be achieved when one of input delays exceeds the upper bound of the allowable delay (see Figure 3).

Conclusions
Different from the previous work, we consider bipartite consensus of heterogeneous multiagent systems with diverse input delays. Based on the matrix theory and the frequency domain theory, the maximum input delay for the systems to achieve bipartite consensus is obtained. e future work will extend the existing work to time-varying input delays.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. 8 Complexity