Modeling and Control of Wide-Area Networks

Abstract

This paper provides a survey of recent research progress in mathematical modeling and distributed control of wide-area networks. Firstly, the modeling is introduced for two types of wide-area networks, i.e., coopetitive networks and cooperative networks, with the help of algebraic graph theory. Particularly, bipartite network topologies and cluster network topologies are introduced for coopetitive networks. With respect to cooperative networks, an intermittent clustered network modeling is presented. Then, some classical distributed control strategies are reviewed for wide-area networks to ensure some desired collective behaviors, such as consensus (or synchronization), bipartite consensus (or polarization), and cluster consensus (or fragmentation). Finally, some conclusions and future directions are summarized.

1. Introduction

With the development of science and technology, large-scale networks have gradually replaced local networks with simple structures and single functions, and thus, they have been increasingly used in industry and academia. Such large-scale networks are often called wide-area networks [1], which are generally composed of multiple groups or clusters to exhibit the sparsity of most complex networks [2]. The major characteristic of a wide-area network is the tight communication within clusters and sparse communication between clusters. This is particularly evident in post-disaster emergency communication systems [3,4,5], power grid optimization [6,7,8], internet of things applications [9,10,11,12,13], cloud computing [14,15,16], etc. In addition to the complex network topologies of wide-area networks, behavior emergence on such networks is also very fascinating. Multi-agent systems (MASs) have become a powerful approach to deal with the complexity and diversity of such large-scale networks. Distributed control in MASs is commonly used to study the underlying interaction mechanisms for the emergence of collective intelligent behaviors on wide-area networks. Desired behaviors and local interaction mechanisms are two distinctive features of distributed controls [17].

In general, wide-area networks can be divided into two categories, one of which is a class of networks with both cooperative and competitive interactions, known as coopetitive networks [18]. Cooperation and competition coexist in complex and subtle ways in natural evolution, human activities, and engineering applications [19,20,21]. For example, the group foraging of mixed species, which need to work together to explore the environment during foraging while also competing for limited resources [19]. The personal opinion is updated by taking the average of the beighbors’ opinions [20], however, attempting to change someone else’s opinion may also be seen as hostile or competitive. In engineering and military fields, friendly robots cooperate to capture and intercept enemy robots [21]. In terms of multi-objective convergence forms, bipartite consensus and cluster consensus are two typical types of emergent collective behavior in coopetitive networks [22,23,24]. Specifically, the former requires that the agents achieve a form of “modular consensus”, while the latter is associated with a network topology partition that de-synchronizes the behavior among agents of different groups. Regarding bipartite consensus in networked MASs, structural balance is an indispensable concomitant of the network topology. Under this assumption, many studies have worked to understand the bipartite consensus of cooperative competitive networks across various contexts [25,26,27,28]. On the other hand, when achieving cluster consensus, acyclic partition and balanced couple partition are two typical network structures that ensure cluster consensus. These two communication topologies require the satisfying of the in-degree balance condition between different groups [29,30,31]. It is worth noting that in this type of clustered network, each subgraph only contains positive edges, i.e., cooperative relationships [32]. Recently, the concept of group-bipartite consensus has been introduced by combining acyclic partition and sign function [33]. The group-bipartite network topology removes the limitation that negative links are only allowed to exist between different groups, which is more in line with practical application scenarios. For example, there exist cooperative and competitive relationships between different provinces in terms of economics, politics, culture, etc., while different cities, regions, and enterprises within each province also cooperate and compete.

Life is not always a zero-sum game, where having a winner means having a loser. In fact, most successes do not come from competition, but from cooperation. Then, another type of wide-area network is called a cooperative network. For example, in post-disaster emergency communication networks where multiple subnetworks may become isolated [34], it is common for each group to be responsible for specific communication tasks, and to work quickly to collect, transmit, and process information. Intermittent communication may be established between groups. For simplicity, we refer to a MAS composed of multiple subnetworks as a cluster MAS (CMAS). For cooperative CMASs, consensus (or synchronization) emerges as a prominent collective behavior, whereby all agents ultimately converge to the same state. Bragagnolo et al. first studied a type of intermittent clustered network with a continuous–discrete communication mechanism in [35]. For instance, in a spatially clustered robot formation, robots that belong to the same cluster interact continuously. However, due to constraints such as limited energy and communication range, interactions between robots that are far apart are discrete. Until now, substantial research advancements have been achieved in the context of intermittent clustered networks, particularly focusing on synchronization or output synchronization [36,37,38,39,40,41,42,43].

Based on the above observations, this paper aims to provide a preliminary exploration of group behavior over wide-area networks and introduce some of the important research advances and application scenarios. Figure 1 shows a statistical analysis of several papers in Scopus between the years of 2010–2023 about four types of consensus, i.e., bipartite consensus, cluster consensus, group-bipartite consensus, and synchronization. The structure of this paper is as follows. Section 2 introduces two types of modeling methods for wide-area networks, i.e., coopetitive networks and cooperative networks. Section 3 reviews three collective behaviors on coopetitive networks, namely, bipartite consensus, cluster consensus, and group-bipartite consensus, and introduces some classical distributed control algorithms. Section 4 reviews recent distributed control in coopetitive networks. Section 5 summarizes the challenging issues in future relevant areas. Finally, the conclusion of this article is provided in Section 6.

2. Notations and Modeling of Wide-Area Networks

Graph theory is a powerful tool for the modeling of wide-area networks. In this section, the modeling of coopetitive networks and cooperative networks over wide-area networks are introduced separately using graph theory. The symbols used throughout this article are listed in

Table 1. Nomenclature.

Symbol	Definition
${\mathbb{R}}_{+}$	The sets of non-negative real numbers
$\mathbb{N}$	The sets of non-negative integers
${\mathbb{R}}^n$	The sets of n-dimensional real column vector space
${\mathbb{R}}^{m\times n}$	The sets of $m\times n$-dimensional real matrix space
${\mathbb{C}}^{m\times n}$	The sets of $m\times n$-dimensional complex matrix space
$diag\left\{·\right\}$	The block diagonal matrix
$col\left(·\right)$	The column vectors
‖·‖	The Euclidean norm
⊗	The Kronecker product for matrices
$A^T$	The transpose matrix of A
$C^{+}$	The generating inverse of matrix C
$\mathbf{0}$	The zero matrix with compatible dimensions
$\lambda \left(A\right)$	The eigenvalues of matrix A
$x\left(t_k^{-}\right)$	The left limit of $x\left(t_k\right)$
Hurwitz matrix	All eigenvalues of a matrix have negative real parts

. The relationships between coopetitive networks and cooperative networks are shown in

Table 2. Relationships among coopetitive networks and cooperative networks.

Network Types	Coopetitive Networks			Cooperative Networks
	Bipartite Networks	Cluster Networks	Group-Bipartite Networks
Group number	2	more than 2		more than 2
Typical structure	structural balance	in-degree balance	acyclic partition, sign-balanced couple	a directed spanning tree
Intra-cluster communication	continuous, cooperation	continuous, cooperation	continuous, coopetition	continuous, cooperation
Inter-cluster communication	continuous, competition	continuous, cooperation	continuous, coopetition	discrete, cooperation
Global behavior	Bipartite consensus	cluster consensus	group-bipartite consensus	synchronization

.

2.1. Coopetitive Networks

This subsection employs graph theory to construct models for three distinct types of competitive networks: bipartite networks, cluster networks, and group-bipartite networks. The mathematical framework presented in this subsection provides a valuable tool for analyzing and modeling cooperative–competitive phenomena in both natural and human systems.

2.1.1. Bipartite Networks with Structural Balance Assumption

In bipartite networks, positive edges (represented by blue solid lines) and negative edges (represented by red dashed lines) are used to denote cooperative and competitive relationships between individuals, respectively. Cooperative relationships occur within clusters, while competition exists only between clusters. Herein, Figure 2 provides an intuitive illustration of the concept of bipartite networks, applied to describe healthcare coopetition networks [44]. In this network, there are two core hospitals, each modeled as a subnetwork. Within each subnetwork, hospitals cooperate by sharing healthcare resources, while simultaneously engaging in competition within the healthcare market across different subnetworks.

Directed signed graphs are employed to describe this particular type of network.

Definition 1.

Directed signed graph.

Given a digraph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$, where the set of nodes represents a collection of individuals, denoted as $\mathcal{V}=\{1,\cdots ,N\}$; the set of directed edges is denoted as $\mathcal{E}\in \mathcal{V}\times \mathcal{V}$; $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{N\times N}$ stands for the adjacency matrix of the graph, with element $a_{ij}$ representing the strength of the interaction between node i and node j. The sign function $sgn(•)$ is used to represent the coopetitive relationship between node i and node j, that is,

$$sgn\left(a_{ij}\right)=\left\{\begin{array}{cc}1,a_{ij}>0friendlyandcooperative\hfill & \\ 0,a_{ij}=0noconnection\hfill & \\ -1,a_{ij}<0hostileandcompetitive\hfill \end{array}\right.$$

To give the definition of the structural balance of signed graphs, we first introduce the concept of positive and negative cycles. Generally, the cycles in a signed graph contain both positive and negative edges. If the product of the weights $a_{ij}$ of the edges in a cycle is positive, the cycle is called a positive cycle; otherwise, it is called a negative cycle.

Definition 2.

Structurally balanced.

As presented in [45], if all the cycles in a signed graph $\mathcal{G}$ are positive cycles, then $\mathcal{G}$ is called structurally balanced; if there is at least one negative cycle, then $\mathcal{G}$ is called structurally unbalanced; if there are no cycles in $\mathcal{G}$, then it is called a vacuum-balanced graph. Different balanced structures are given in Figure 3, where the blue solid edges have positive weights, indicating a cooperative relationship; the red dashed edges have negative weights, indicating competitive relationships.

From a linear algebra perspective, if the signed graph exhibits structural balance, the set of nodes $\mathcal{V}$ can be partitioned into two groups: ${\mathcal{V}}_1=\{1,\cdots ,N_0\}$ and ${\mathcal{V}}_2=\{N_0+1,\cdots ,N\}$. The relationship is cooperative within subgroups, and competition exists only between two subgroups. Bipartite consensus is a typical feature of this type of network, i.e., the individuals in the two groups have the same absolute value of the final state, but opposite signs.

By assigning appropriate numbers to each individual, the adjacency matrix associated with the signed graph $\mathcal{G}$ can be transformed into the following block matrix form.

$$A=\left(\begin{array}{cc}{A}_{11}\hfill & A_{12}\hfill \\ A_{21}\hfill & A_{22}\hfill \end{array}\right)$$

where $A_{11}\in {\mathbb{R}}^{N_0\times N_0}$ and $A_{22}\in {\mathbb{R}}^{\left(N-N_0\right)\times \left(N-N_0\right)}$ are non-negative matrices, while $A_{12}\in {\mathbb{R}}^{N_0\times (N-N_0)}$ and $A_{21}\in {\mathbb{R}}^{(N-N_0)\times N_0}$ are non-positive matrices.

The Laplacian matrix $\mathcal{L}$ of a signed graph plays an important role in analyzing the collective behavior evolution of coopetitive systems, and its definition is as follows.

$$\mathcal{L}=\mathcal{D}-\mathcal{A}$$

(1)

where $\mathcal{D}=diag\{d_1,\cdots ,d_N\}$ denotes the degree matrix, $d_i={\sum }_{j\in N_i}\left|a_{ij}\right|$ with $N_i=\{j\mid (i,j)\in \mathcal{E}\}$.

2.1.2. Cluster Networks with In-Degree Balance Condition

In practice, systems need to be divided into multiple clusters to accomplish different tasks due to differences in performance and task requirements. Therein, individuals within a group are in a cooperative relationship, while individuals belonging to different subgroups can choose to compete or cooperate with each other [46]. For instance, a strategic alliance model in which companies cooperate and compete with each other in order to share research and development costs, mitigate risks, learn, and acquire complementary resources [22].

In cluster networks, the graph has a partition of the node set $\mathcal{V}$ that takes the form $\{{\mathcal{V}}_1,\cdots ,{\mathcal{V}}_k\}$ such that ${\mathcal{V}}_i\ne ⌀,{\cup }_{\ell =1}^k{\mathcal{V}}_{\ell }=\mathcal{V}$ and ${\mathcal{V}}_i\cap {\mathcal{V}}_j=⌀,i\ne j,i,j\in \{1,2,\cdots ,k\}$. Let ${\mathcal{G}}_{\ell }$ denote the underlying topology of cluster ${\mathcal{V}}_{\ell },\ell =1,\cdots ,k$, i.e., ${\mathcal{V}}_{\ell }=\mathcal{V}\left(G_{\ell }\right)$. Without loss of generality, the node set of each cluster can be represented by $\mathcal{V}\left(G_{\ell }\right)=\left\{{\sum }_{j=0}^{\ell -1}{n}_j+1,\cdots ,{\sum }_{j=0}^{\ell }{n}_j\right\},1\le \ell \le k$, where $n_0=0,{\sum }_{\ell =1}^k{n}_{\ell }=N$. For convenience, let $\overline{i}$ denote the index of the subset where node i belongs, that is, $i\in {\mathcal{V}}_{\overline{i}}$. Obviously, $1\le \overline{i}\le k$. Nodes i and j are said to be in the same subgroup if $\overline{i}=\overline{j}$.

Assumption 1.

In-degree balance.

$$\sum _{j\in \mathcal{V}\left(G_{\ell }\right)}{a}_{ij}=0,\forall i=1,\dots ,N,i\in \mathcal{V}\backslash \mathcal{V}\left(G_{\ell }\right),\ell =1,\dots ,k.$$

Two types of clustered networks satisfying Assumption 1, which have been studied by many scholars [46], are shown in Figure 4. One is Figure 4a with an acyclic partition structure, with the requirement that information from the former subcluster can be passed on to the latter subcluster, but information from the latter subcluster cannot be passed on to the former subcluster. In contrast, the one shown in Figure 4b is not an acyclic partition but a balanced couple partition, because the two subclusters can pass information to each other. Cluster consensus is typical for these examples, i.e., individuals within a group reach the same state while the states of individuals within different groups can end up being different.

The Laplacian matrices $\mathcal{L}=\left[l_{ij}\right]\in {\mathbb{R}}^{N\times N}$ corresponding to $\mathcal{G}$ in Figure 4a,b have the following forms, respectively:

$$\mathcal{L}=\left[l_{ij}\right]=\left[\begin{array}{ccc}{\mathcal{L}}_{11}& \cdots & {\mathbf{0}}_{n_1\times n_k}\\ ⋮& \ddots & ⋮\\ {\mathcal{L}}_{k1}& \cdots & {\mathcal{L}}_{kk}\end{array}\right]$$

(2)

$$\mathcal{L}=\left[l_{ij}\right]=\left[\begin{array}{ccc}{\mathcal{L}}_{11}& \cdots & {\mathcal{L}}_{1k}\\ ⋮& \ddots & ⋮\\ {\mathcal{L}}_{k1}& \cdots & {\mathcal{L}}_{kk}\end{array}\right]$$

(3)

where ${\mathcal{L}}_{ii}$ represents the information exchange within subgroup ${\mathcal{G}}_i$, and ${\mathcal{L}}_{ij}$ represents the information exchange from subgroup ${\mathcal{G}}_i$ to subgroup ${\mathcal{G}}_j$, with $i,j=1,\cdots ,k$.

2.1.3. Group-Bipartite Networks

Apart from occurring between different clusters, competition is possible between agents belonging to the same group in cluster consensus problems (see Figure 5). Consequently, a definition of a group-bipartite network has been proposed in the past year or two. The collective behavior of such networks is termed group-bipartite consensus, and can be used to describe some tasks of multi-objective symmetry. For example, in search and rescue missions, UAVs search the area and perform tasks in formations of symmetric patterns [47].

As described in [33], the group-bipartite network topology combines the structural balance properties of bipartite networks and the acyclic partition approach of cluster networks. Sign-balanced couple and the following assumptions are needed for the group-bipartite networks:

Assumption 2.

Each of ${\mathcal{G}}_i,i\in \{1,\cdots ,k\}$ is structurally balanced.

Assumption 3.

Each of ${\mathcal{G}}_i,i\in \{1,\cdots ,k\}$ has a spanning tree.

Under Assumption 2, the node set ${\mathcal{V}}_i$ of ${\mathcal{G}}_i(i=1,\cdots ,k)$ can be divided into two groups ${\mathcal{V}}_i^{\left(1\right)}$ and ${\mathcal{V}}_i^{\left(2\right)}$ satisfying ${\mathcal{V}}_i^{\left(1\right)}\cup {\mathcal{V}}_i^{\left(2\right)}={\mathcal{V}}_i$, ${\mathcal{V}}_i^{\left(1\right)}\bigcap {\mathcal{V}}_i^{\left(2\right)}=\varnothing$. Moreover, a diagonal matrix ${\Phi }_i=diag\{{\varphi }_{{\sum }_{j=0}^{l-1}{n}_j+1},\cdots ,{\varphi }_{{\sum }_{j=0}^l{n}_j}\}$ is defined, where ${\varphi }_j=1$ for $j\in {\mathcal{V}}_i^{\left(1\right)}$, and ${\varphi }_j=-1$ for $j\in {\mathcal{V}}_i^{\left(2\right)}$.

On the other hand, since the partition $\{{\mathcal{V}}_1,{\mathcal{V}}_2,\cdots ,{\mathcal{V}}_k\}$ is acyclic, a new form of Laplacian matrix $\mathcal{L}=\left[l_{ij}\right]\in {\mathbb{R}}^{N\times N}$ is constructed with a lower block triangular form, as (2), where

$$l_{ij}=\left\{\begin{array}{cc}-a_{ij},\hfill & i\ne j,\hfill \\ \sum _{j\notin {\mathcal{V}}_{\overline{i}}}{\varphi }_j{a}_{ij}+\sum _{j\in {\mathcal{V}}_{\overline{i}}}\left|a_{ij}\right|,\hfill & i=j.\hfill \end{array}\right.$$

(4)

The modified Laplacian matrix $\mathcal{L}$ plays an important role in the consensus analysis of group-bipartite networks [33]. As a special case of cluster consensus, the collective behavior of such networks is termed group-bipartite consensus, and specifies multiple dual cluster consensus behaviors.

Figure 4 depicts the concept of group-bipartite networks. The node $\mathcal{V}$ in a group-bipartite network topology $\mathcal{G}$ is partitioned into three clusters ${\mathcal{G}}_1$, ${\mathcal{G}}_2$, and ${\mathcal{G}}_3$ with ${\mathcal{V}}_1=\{1,2,3\}$, ${\mathcal{V}}_2=\{4,5,6,7\}$, and ${\mathcal{V}}_3=\{8,9,10,11\}$. It can be seen that coopetition exists within and between clusters. Clearly, each subcluster ${\mathcal{G}}_i,i=(1,2,3)$ is satisfied with the structural balance condition. Then, it can be obtained that ${\varphi }_i=\left\{\begin{array}{cc}1,\hfill & i=1,3,4,5,8,9\hfill \\ -1,\hfill & i=2,6,7,10,11\hfill \end{array}\right.$.

According to the definition of (4), $l_{44}=0,l_{ii}=1,i=(1,2,3,5,6,7,8,9,10,11)$. Then, the corresponding $\mathcal{L}$ is

$$\mathcal{L}=\left(\begin{array}{ccccccccccc}1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& 0& 1& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& -1& -1& -1& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& 0\\ -1& -1& 0& 0& 0& 0& 0& 1& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1\\ 0& 0& 0& 0& -1& 0& -1& 1& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 1\end{array}\right)$$

(5)

2.2. Cooperative Networks

In the subsection, an intermittent clustered network characterized only by cooperative connections is described. This network utilizes a combination of continuous and discrete communication mechanisms to facilitate interaction among agents. Specifically, continuous communication is used within clusters, and inter-cluster communication is discrete. Examples encompass spatially clustered robots, which exhibit a unique set of characteristics [48]. These robots are organized into clusters, wherein continuous interactions persist among the members of each cluster. However, owing to limitations posed by energy and communication resources, long-distance interactions are restricted to discrete exchanges.

Consider a clustered network $\mathcal{G}=(\mathcal{V},\mathcal{E})$ modeled as a non-empty union consisting of several independently connected subnetworks or clusters ${\mathcal{G}}_k=({\mathcal{V}}_k,{\mathcal{E}}_k)$ such that ${\bigcup }_{k=1}^m{\mathcal{V}}_k=\mathcal{V}$, ${\mathcal{V}}_k\bigcap {\mathcal{V}}_{\tau }=\varnothing$ for all $k,\tau \in \{1,\cdots ,m\},\tau \ne k$. Each subcluster ${\mathcal{G}}_k,k\in \{1,\cdots ,m\}$ contains a specific agent called a leader ${\ell }_k\in {\mathcal{V}}_k$, and the remaining ones are called followers $f_{\tau }\in {\mathcal{V}}_k/{\ell }_k$. The instantaneous communication between clusters is executed by the leader, with its associated communication network represented as ${\mathcal{G}}_{\ell }=(\mathcal{I},{\mathcal{E}}_{\ell })$, where $\mathcal{I}=\{{\ell }_1,{\ell }_2,\cdots ,{\ell }_m\}$ and ${\mathcal{E}}_{\ell }=\{e_{\ell \left(ij\right)}\mid (i,j)\in \mathcal{I}\times \mathcal{I}\}$. Correspondingly, the Laplacian matrices of ${\mathcal{G}}_k$ and ${\mathcal{G}}_{\ell }$ are denoted by ${\mathcal{L}}_k$ and ${\mathcal{L}}_{\ell }$, respectively. For simplicity, the first agent in each cluster is the leader. Thus, the node set of the cluster ${\mathcal{G}}_k,k\in \{1,\cdots ,m\}$ is given by

$${\mathcal{V}}_k=\{{\ell }_k,f_{o_{k-1}+2},\cdots ,f_{o_k}\},$$

(6)

where $o_0=0$ and $o_m=N$. The cardinality of ${\mathcal{G}}_k$ is represented by $∥{\mathcal{V}}_k∥=n_k=o_k-o_{k-1},\forall k\ge 1$, and ${\sum }_{k=1}^m{n}_k=N$.

Under this network topology, a global Laplace matrix is defined as follows:

$$\mathcal{L}=\left[\begin{array}{ccc}{\mathcal{L}}_1& \cdots & {\mathbf{0}}_{n_1\times n_k}\\ ⋮& \ddots & ⋮\\ {\mathbf{0}}_{n_k\times n_1}& \cdots & {\mathcal{L}}_k\end{array}\right]$$

(7)

The objective of such cluster networks with continuous–discrete communications is synchronization. The following assumptions are required for intermittent clustered networks:

Assumption 4.

The subcluster ${\mathcal{G}}_k,k\in \{1,\cdots ,m\}$ is strongly connected.

Assumption 5.

The communication topology ${\mathcal{G}}_{\ell }$ formed between the leaders contains a directed spanning tree.

To illustrate the notation (6), Figure 6 illustrates a CMAS that consists of seven agents grouped into two clusters. Agents 1 and 5 are the leaders in each cluster. Moreover, the nodes in ${\mathcal{G}}_k$ and ${\mathcal{G}}_{\ell }$ are represented as ${\mathcal{V}}_1=\{{\ell }_1,f_2,f_3,f_4\}$, ${\mathcal{V}}_2=\{{\ell }_2,f_6,f_7\}$, and ${\mathcal{G}}_{\ell }=\{{\ell }_1,{\ell }_2\}$. Then, the corresponding $\mathcal{L}$ is

$$\mathcal{L}=\left(\begin{array}{ccccccc}2& 0& -1& -1& 0& 0& 0\\ -1& 2& -1& 0& 0& 0& 0\\ -1& -1& 2& 0& 0& 0& 0\\ -1& 0& -1& 2& 0& 0& 0\\ 0& 0& 0& 0& 2& -1& -1\\ 0& 0& 0& 0& -1& 1& 0\\ 0& 0& 0& 0& -1& -1& 2\end{array}\right)$$

(8)

3. Distributed Control of Coopetitive Networks

This section focuses on the development of collective emergent behavior in coopetitive networks, including bipartite consensus, cluster consensus, and group-bipartite consensus. We review the subjects and issues pertaining to these consensuses that have been explored in recent years.

3.1. Bipartite Consensus

In 2012, Altafini et al. [49] first proposed a dynamic model of coopetitive networks based on a linear Laplace feedback design. It demonstrated that under the structural balance assumption, individuals form two groups with diametrically opposite final states, i.e., the absolute values of the individuals’ final states are the same, but the signs are opposite. Such a dynamical behavior is referred to as bipartite consensus.

Hu et al. [45] investigated the modeling of the coexistence of cooperation and competition in social networks, along with the collective behaviors, under this modeling framework. Considering the influence of structural balance properties, This paper employed directed signed graphs to describe the interaction network of coopetitive relationships. The dynamics of each agent is modeled as

$${\dot{x}}_i\left(t\right)=\sum _{j\in N_i}{a}_{ij}\left[x_j\left(t\right)-sgn\left(a_{ij}\right)x_i\left(t\right)\right],i=1,\dots n$$

(9)

where the sign function $sign\left(a_{ij}\right)$ equals 1 or −1, indicating the coopetitive relationship between agents. The above equation can be written in the following equivalent form:

$$\dot{x}\left(t\right)=-\mathcal{L}x\left(t\right)$$

(10)

where $x\left(t\right)=col(x_1\left(t\right),\cdots ,x_N\left(t\right)$ is the state vector of all individuals, and $\mathcal{L}$ is the Laplacian matrix defined in (1). Three emergent collective behaviors (consensus, polarization or bipartite consensus, and fragmentation) were explored.

Case 1 (consensus): When the network is a purely cooperative network containing a spanning tree in Figure 3a, the state limits of all individuals satisfy ${lim}_{t\to \infty }∥x_i\left(t\right)-x_j\left(t\right)∥=0$. Figure 7 depicts the state evolution on this network.

Case 2 (bipartite consensus): When the network is Figure 3b, Figure 3c, or Figure 3f, i.e., the network is a vacuum-balanced or structurally balanced graph containing spanning trees, the final state of all individuals exhibits a bipartite consensus, that is, ${lim}_{t\to \infty }∥x_i\left(t\right)-sgn\left(a_{ij}\right)x_j\left(t\right)∥=0.$ In Figure 8, the seven individuals eventually appear polarized, forming two limit states, 1 and −1.

Case 3 (fragmentation): When the coopetitive network is a structurally unbalanced network, as shown in Figure 3d or Figure 3e, then the state of groups eventually splits, i.e., the states of all individuals converge to more than two limit states. Figure 9 depicts the state evolution on a structurally unbalanced network, where the blue solid lines and the red dotted lines respectively denote the trajectories of agents from two competitive subgroups. The three limit states are 1, 0, and −1.

From the above simulation results, it can be observed that for a coopetitive network, the structural balance condition is a necessary requirement to achieve bipartite consensus.

Immediately afterward, Proskurnikov et al. [50] extended Altafini’s model for opinion dynamics in social networks and discussed the modulus consensus problem on time-varying directed signed graphs. This paper studied linear and nonlinear consensus protocols within a common framework, relaxing the constraints of strong connectivity and structural balance.

Ma et al. [51] investigated the bipartite consensus problem of first-order MASs on fixed directed graphs in the presence of measurement noise. This paper uncovered that even in the presence of measurement noise, achieving bipartite consensus on a signed graph with structural balance and a spanning tree was the minimum connectivity assumption required. Moreover, when the signed graph was structurally unbalanced, the state of the closed-loop system can achieve mean square convergence to zero as long as certain mild conditions are satisfied.

Guo et al. [52] discussed the bipartite consensus problem in MASs on directed signed graphs. This paper demonstrates the achievability of bipartite consensus on directed graphs with communication delays and strongly connected structurally balanced conditions. Additionally, the paper considered the case of structural balance, and proposed a method to pin a specific agent so that the system can achieve pinning bipartite consensus.

Shao et al. [53] investigated the bipartite consensus problem of MASs with second-order discrete-time dynamics under asynchronous settings. This paper took into account switching topologies and established an asynchronous distributed control protocol, while sufficient conditions were provided for achieving asynchronous bipartite consensus.

Recently, Wu et al. have worked constructively on the bipartite consensus of coopetitive networks with unknown disturbances or communication noise in [54,55,56].

In [54], the interventional bipartite consensus was addressed for high-order MASs with nonlinear unknown time-varying disturbances. Several adaptive estimators based on neural networks were proposed to estimate the nonlinear disturbances. The convergence of bipartite consensus was analyzed using the Lyapunov function method.

In [55], the mean square bipartite consensus problem was investigated in high-order MASs, incorporating modeling of coopetition interactions and communication noise. This paper proposed a distributed control strategy that was independent of the agent state matrix, and developed a novel randomized approximation strategy utilizing relative state information to suppress communication noise.

In [56], the bipartite consensus problem was explored for high-order MASs, considering scenarios with and without external systems (i.e., leaders). By designing distributed adaptive control laws and utilizing the linear parameterization approach to describe the time-varying characteristics of unknown disturbances, two fully distributed control strategies were proposed that did not rely on any global information. These strategies guaranteed the achievement of bipartite consensus MASs.

In addition, the output bipartite consensus of heterogeneous MASs was also discussed in [57,58].

Liang et al. [57] investigated the bipartite output synchronization problem for heterogeneous MASs with time-varying communication networks. This paper proposed a novel edge-based adaptive output feedback control strategy. By utilizing a sophisticated Lyapunov function, the convergence of the closed-loop system was analyzed.

Wu et al. [58] discussed the output bipartite consensus problem for heterogeneous MASs and further extended the results of [57]. In light of the unavailability of state information for leaders and unknown system matrices, new distributed estimators were designed for each of them.

The existing research works on bipartite consensus, along with additional details, are summarized in

Table 3. A survey of the research undertaken on the bipartite consensus problem.

Reference	Year	Dynamics	Network Topology	Contribution
[49]	2012	First-order integrator	Static signed graphs	Giving the concept of bipartite consensus
[45]	2014	General linear dynamics	Directed signed graphs	Delving into three emergent collective behaviors (consensus, bipartite consensus, and fragmentation)
[50]	2015	First-order integrator	Time-varying topology	Extending Altafini’s model into time-varying directed signed graphs
[51]	2017	First-order integrator	Fixed signed digraphs	Considering bipartite consensus under measurement noise
[52]	2018	First-order integrator	Directed signed graphs	Investigating pinning bipartite consensus under communication delays
[53]	2018	Second-order integrator	Switching topologies	Studying bipartite consensus under switching topologies
[54]	2016	High-order dynamics	Directed signed graphs	Addressing the interventional bipartite consensus with unknown disturbances
[55]	2018	General linear dynamics	Directed signed graphs	Investigating the mean square bipartite consensus with communication noise
[56]	2019	High-order dynamics	Directed signed graphs	Proposing fully distributed adaptive control laws
[57]	2020	Heterogeneous dynamics	Directed signed graphs	Proposing a novel edge-based adaptive output feedback control strategy
[58]	2023	Heterogeneous dynamics	Directed signed graphs	Designing the new distributed estimators for leader’s unavailable information and unknown system matrix

. Currently, there are some constraints on bipartite consensus, such as structural balance, spanning tree, or joint spanning tree. However, in the real world, the relationship between cooperation and competition can be very complex, and the network topology may not necessarily meet these conditions. Therefore, studying the dynamics of MASs on coopetition networks without structural constraints is of significant practical and theoretical importance.

3.2. Cluster Consensus

In numerous engineering applications, it is often necessary to classify individuals within MASs into multiple groups based on physical attributes or assigned tasks. Each of these groups is commonly referred to as a cluster. In response to this scenario, researchers have proposed a broader concept known as cluster consensus. In cluster consensus, individuals belonging to the same cluster are required to converge to the same constant value, while different clusters may not coincide with each other.

In 2009, Yu and Wang [59] studied the cluster consensus problem of a first-order continuous MAS with two groups. In the scenario where information exchange was undirected, a novel consensus protocol was proposed to tackle the challenge of achieving group consensus. Several convergence conditions were established by leveraging principles from graph theory and matrix theory.

Subsequently, Yu et al. [60] incorporated topology switching and communication delays into a directed network and examined the problem of group consensus in MASs by introducing a double tree-form transformation. Certain necessary and/or sufficient conditions were derived for achieving group consensus.

Chen et al. [61] studied a first-order discrete-time CMAS with fixed and switching topologies, presenting a partitioning algorithm for strongly connected directed graphs. Then, by relying on the non-negative matrix analysis and Markov chain, two necessary conditions were presented for achieving cluster consensus. This paper can provide a response to the issues of determining clustering and ensuring consensus in MASs.

Thereafter, the group consensus problem with two groups was extended to a more general group consensus problem in [62]. The concepts of in-degree balance, out-degree balance, and balance pair were first proposed, and the condition of balance pair was used to restrict the coupling interaction between the clusters. These pioneering research works have attracted the attention of, and sparked discussions among, numerous scholars regarding cluster consensus with multiple clusters.

In [46], Qin et al. addressed the cluster consensus problem for linear MASs using an acyclic partitioning approach to rearrange all directed edges in acyclic directed graphs, and gave the following distributed feedback control protocol:

$$u_i\left(t\right)=K\left[\sum _{j\in N_i}{a}_{ij}\left(x_j\left(t\right)-x_i\left(t\right)\right)+d_i\left(s_{\overline{i}}\left(t\right)-x_i\left(t\right)\right)\right]$$

(11)

where K is the feedback matrix that needs to be designed, and $s_i,i=1,\cdots ,p$ are p particular solutions of a homogeneous system $\dot{s}\left(t\right)=As\left(t\right),$ such that ${lim}_{t\to \infty }∥s_i\left(t\right)-s_j\left(t\right)∥\ne 0$, where $d_i>0$ when agent i is pinned by $s_i$, otherwise $d_i=0$. The results showed that for acyclic directed graphs, regardless of the intra- and inter-cluster coupling strength, the cluster consensus for general linear MASs can be achieved by designing the feedback control matrix K.

Later on, Yu et al. [63] introduced another partition structure for coopetitive networks, i.e., balanced coupled partition, to investigate the cluster consensus problem for linear MASs under pinning control. The results indicated that for fixed topology networks, if each cluster contained a directed spanning tree and the intra-cluster coupling strength was strong enough compared to the inter-cluster coupling strength, it was easy to design a feedback controller to make the system achieve cluster consensus; for switching topology networks, the necessary conditions to achieve cluster consensus were given.

Ma et al. [64] examined the consensus problem in first-order multi-agent systems (MASs) with fixed and directed topologies, taking into account time delays. This study introduced the novel concept of cluster-delay consensus. By employing graph theory, Lyapunov stability analysis, and matrix theory, the authors derived sufficient conditions to ensure the maintenance of cluster-delay consensus in MASs.

Chen et al. [65] studied the cluster consensus problem of MASs with heterogeneous dynamics. The issue was addressed by employing output regulation techniques and constructing state feedback controllers. This paper can be considered to extend the previous research that focused on the standard consensus problem in homogeneous MASs, allowing for different dynamics of agents in heterogeneous MASs.

Dong et al. [66] investigated the problem of cluster consensus for general linear MASs and proposed a novel intermittent output control strategy that enables effective cluster consensus control under nonperiodic operation. This paper provided a new approach and direction for addressing the problem of cluster consensus in MASs.

A summary of the existing research work on the cluster consensus problem is presented in

Table 4. A survey of the research undertaken on the cluster consensus problem.

Reference	Year	Dynamics	Conditions	Contribution
[59]	2009	First-order integrator	Undirected topology	Solving the cluster consensus with two groups
[60]	2010	First-order integrator	Directed topology	Examining the problem of group consensus incorporated topology switching and communication delays
[61]	2011	First-order integrator	Fixed and switching topology	Proposing a cluster factorization algorithm for directed graphs
[62]	2012	First-order integrator	Directed topology	Studying the group consensus problem with multiple groups; proposing the concept of in-degree balance
[46]	2013	General linear dynamics	Fixed and switching topology	Proposing a acyclic partition; investigating the correlation between cluster consensus behavior and the coupling strength among agents
[63]	2014	General linear dynamics	Fixed and switching topology	Proposing a balanced coupled partition; giving the necessary conditions for achieving cluster consensus
[64]	2016	First-order integrator	Directed topology	Examining the cluster-delay consensus problem for nonlinear dynamics MASs
[65]	2017	Heterogeneous dynamics	Directed topology	Employing output regulation techniques to solve the cluster consensus problem
[66]	2022	General linear dynamics	Directed topology	Proposing a novel intermittent output control strategy in the cluster consensus problem

. Cluster consensus control can enhance the scalability and availability of a system. However, it may also introduce increased complexity and latency, potentially leading to reduced performance. Therefore, in practical applications, it is necessary to carefully weigh the pros and cons and select an appropriate consensus control method based on specific circumstances.

3.3. Group-Bipartite Consensus

Given the potential for competition between agents of the same group, a special class of cluster consensus, group-bipartite consensus has emerged over the past two years. The group-bipartite consensus is expected to accomplish multi-objective symmetric tasks, such as the formation of multiple symmetric shapes by UAVs simultaneously [67].

Liu et al. [33] combined group consensus and bipartite consensus, introduced the concept of group-bipartite consensus, and proposed a distributed control protocol to solve the group-bipartite consensus problem for first-order continuous MASs. The design protocol was as follows:

$$u_i\left(t\right)=\sum _{j\in {\mathcal{V}}_{\overline{i}}}{a}_{ij}\left[x_j\left(t\right)-sgn\left(a_{ij}\right)x_i\left(t\right)\right]+\sum _{j\notin {\mathcal{V}}_{\overline{i}}}{a}_{ij}\left[x_j\left(t\right)-{\varphi }_j{x}_i\left(t\right)\right]$$

(12)

The above equation can be written in the following equivalent form:

$$\dot{x}\left(t\right)=-\mathcal{L}x\left(t\right)$$

(13)

where $x\left(t\right)=col(x_1\left(t\right),\cdots ,x_N\left(t\right)$ is the state vector of all individuals, and $\mathcal{L}$ is a new Laplacian matrix established to facilitate the control implementation and is defined in (4).

When the coopetitive network is a group-bipartite network, as shown in Figure 4, the trajectories of all agents satisfy the definition of group-bipartite consensus, that is, ${lim}_{t\to \infty }∥x_i\left(t\right)-{\alpha }_{\overline{i}}∥=0$ for $i\in {\mathcal{V}}_{\overline{i}}^{\left(1\right)}$ and ${lim}_{t\to \infty }∥x_i\left(t\right)+{\alpha }_{\overline{i}}∥=0$ for $i\in {\mathcal{V}}_{\overline{i}}^{\left(2\right)}$, where $i=1,\cdots ,N$ and ${\alpha }_{\overline{i}}$ are k constants. Figure 10 depicts that the trajectories of all agents gradually converge to a triple-bipartite final convergence state under the control protocol (12). Compared to Figure 8, Figure 10 introduces bipartite consensus as a foundation for a single group, and incorporates multiple groups to achieve bipartite consensus across multiple groups. This implies that group consensus and bipartite consensus are special cases of group-bipartite consensus.

The group-bipartite consensus problem in a heterogeneous MAS composed of first-order integrators and second-order integrators was explored in [68]. By leveraging graph theory and Lyapunov stability methods, sufficient conditions and corresponding consensus protocols were derived for heterogeneous MASs under undirected communication topologies. In contrast to existing work [33], this paper eliminated the constraint of network topology, such as acyclic partitions. However, it only analyzed bipartite networks with two groups.

Thereafter, Liu et al. [69] discussed the problem of oscillatory group-bipartite consensus control in swarm robots with multiple oscillatory leaders. This paper modeled the robotic cluster using Euler–Lagrange equations and verified the proposed control method through two simulations conducted on two typical group-bipartite network topologies.

Wang et al. [67] investigated the problem of coordinated task control in a swarm of robots by introducing the concept of group-bipartite consensus in networked Euler–Lagrange systems. By utilizing the structure of acyclic partition network topologies, they proposed a static group-bipartite consensus control protocol and established geometric criteria to guarantee the achievement of multiple symmetric consensus in networked robot systems.

Recently, considering that most systems in reality are nonlinear, Lu et al. [70] investigated the finite-time problem of second-order nonlinear MASs with leaders. They generalized the system to a nonlinear form in order to achieve finite-time group-bipartite consensus. This paper constrained the nonlinear functions in the system with a semi-Lipschitz condition, which is derived from fundamental inequalities and the Lipschitz condition.

Table 5. A survey of the research undertaken on the group-bipartite consensus problem.

Reference	Year	Dynamics	Typical Structure	Contribution
[33]	2020	First-order integrator	Acyclic partition, sign-balanced couple	Giving the concept of group-bipartite consensus
[68]	2022	Heterogeneous dynamics	Structurally balanced	Solving the group-bipartite consensus problem in heterogeneous CMASs
[69]	2022	Euler–Lagrange systems	Acyclic partition, sign-balanced couple	Applying oscillatory group-bipartite consensus to swarm robots
[67]	2023	Networked robot systems	Acyclic partition	Proposing a static group-bipartite consensus control protocol
[70]	2023	Double integrator	Structurally balanced	Studying the finite-time group-bipartite consensus for nonlinear systems

presents a comprehensive overview of references related to the group-bipartite problem from the preceding three years. Although group-bipartite consensus shows potential in handling multi-objective symmetric tasks, the current research fails to consider the impact of various practical factors, such as communication delays, unobservable states, and unknown perturbations. Therefore, in order to bridge the gap between the inherent properties or constraints of consensus algorithms and practical models, further investigation is needed on consensus problems that involve these practical factors.

4. Cooperative Control over Intermittent Clustered Networks

Although cooperation and competition are prevalent in nature and human society, cooperation is also widespread. Bragagnolo et al. [35] first proposed a cooperative clustered network with continuous intra-cluster communication and discrete inter-cluster communication. Two fundamental challenges can be posed for this kind of intermittent clustered network:

• Problem 1: Design the distributed consensus protocols such that the hybrid continuous–discrete CMASs can achieve global consensus behavior.
• Problem 2: Determine the characterization of the global consensus value.

The single-integrator dynamic model was expressed as (Bragagnolo et al. [35])

$$\left\{\begin{array}{cc}\dot{x}\left(t\right)=-\mathcal{L}x\left(t\right),\hfill & \forall t\in {\mathbb{R}}_{+}\backslash \mathcal{T}\hfill \\ x_l\left(t_k\right)=P_l{x}_l\left(t_k^{-}\right)\hfill & \forall t_k\in \mathcal{T}\hfill \\ x\left(0\right)=x_0\hfill \end{array}\right.$$

(14)

where $x_l\left(t\right)$ is the set of leader states, $P_l\in {\mathbb{R}}^{m\times m}$ is a row stochastic matrix associated with ${\mathcal{G}}_{\ell }$, $\mathcal{T}=\{t_k\in {\mathbb{R}}_{+}|t_k<t_{k+1},\forall k\in \mathbb{N},t_k\mathrm{reset}\mathrm{time}\}$, and $\mathcal{L}$ is the Laplacian matrix defined in (7).

To address the first question, Bragagnolo et al. [35] proposed a quasi-periodic reset strategy and provided LMI conditions to ensure the global consensus index stability of subnetworks represented by directed and strongly connected graphs. As for the second question, the global consensus value depended on the initial conditions and the topology of the networks involved, including networks associated with clusters and networks associated with leaders. It is worth noting that the consensus value is independent of the reset sequence used for the leader state.

The simulation results in Figure 11 demonstrate that the leader’s trajectory was non-smooth while the follower’s trajectory was smooth, and the phenomenon of jumps at the moment of reset had no effect on the calculated consensus.

Morarescu et al. [36] investigated the reset control problem for first-order time-varying CMASs. In contrast to [35], this study also took into account the possibility that the reset time could be triggered by certain events.

The consensus problem in cluster networks with general linear MASs is more challenging than the case of integrators, as pointed out by Pham et al. [37,38] and Wang et al. [40]:

$$\left\{\begin{array}{cc}{\dot{x}}_i=A{x}_i+B{u}_i,\hfill & \\ y_i=C{x}_i,\hfill \end{array}\right.t\in (t_{k-1},t_k)$$

(15)

where $x_i\in {\mathbb{R}}^n$, $u_i\in {\mathbb{R}}^p$, and $y_i\in {\mathbb{R}}^q$ are the state, input, and output of the i-th $(i=1,\cdots ,N)$ agent, respectively.

To achieve global state consensus for (15), Pham et al. [37] proposed a reset state feedback control strategy using relative state measurements. Subsequently, due to physical or economical constraints in real systems, it is not possible to measure all states with sensors, so the states need to be observed. In [38], an output feedback control protocol based on an impulsive observer, more precisely a full-order state observer, was designed for linear CMASs:

$$\left\{\begin{array}{c}\left\{\begin{array}{cc}{\dot{\widehat{x}}}_i=A{\widehat{x}}_i+B{u}_i+H\left({\widehat{y}}_i-y_i\right)+qL{\xi }_i,\hfill & \\ {\widehat{y}}_i=c{\widehat{x}}_i,\hfill \end{array}\right.\hfill \\ u_i=pK\sum _{j=1}^N{a}_{ij}({\widehat{x}}_j-{\widehat{x}}_i),\hfill \end{array}\right.t\in \left(t_{k-1},t_k\right)$$

(16)

where ${\widehat{x}}_i\in {\mathbb{R}}^n$ is the observer state, ${\widehat{y}}_i\in {\mathbb{R}}^n$ is the output of the observer, $H,L\in {\mathbb{R}}^{n\times q},K\in {\mathbb{R}}^{p\times n}$ are the gain matrices, $p>0,q>0$ are the coupling gains, and ${\xi }_i$ is the relative output measurement of the i-th agent defined by ${\xi }_i={\sum }_{j=1}^N{a}_{ij}[({\widehat{y}}_j-{\widehat{y}}_i)-(y_j-y_i)]$.

However, at the reset time, when it came to state updates between clusters, the state information that was not measurable at the prior time was exploited:

$$\left\{\begin{array}{cc}{x}_{{\ell }_i}\left(t_k\right)=\sum _{j=1}^m(P_{l\left(ij\right)}\otimes I_n)x_{{\ell }_j}\left(t_k^{-}\right),x_{f_{\tau }}\left(t_k\right)=x_{f_{\tau }}\left(t_k^{-}\right),\hfill & \\ {\widehat{x}}_{{\ell }_i}\left(t_k\right)=\sum _{j=1}^m(P_{l\left(ij\right)}\otimes I_n){\widehat{x}}_{{\ell }_j}\left(t_k^{-}\right),{\widehat{x}}_{f_{\tau }}\left(t_k\right)={\widehat{x}}_{f_{\tau }}\left(t_k^{-}\right),\hfill \end{array}\right.t=t_k$$

(17)

where $x_{{\ell }_i}\left(t_k\right),x_{f_{\tau }}\left(t_k\right)$ and ${\widehat{x}}_{{\ell }_i}\left(t_k\right),{\widehat{x}}_{f_{\tau }}\left(t_k\right)$ represent the states and observer states of leaders and followers at time $t_k$, respectively.

To avoid such irrationality, Wang et al. [40] proposed a reset reduced-order observer-based output feedback control strategy as follows:

$$\left\{\begin{array}{c}{\dot{v}}_i=F{v}_i+G{y}_i+H{u}_i,\hfill \\ u_i=cK{Q}_1\sum _{j=1}^N{a}_{ij}(y_i-y_j)+cK{Q}_2\sum _{j=1}^N{a}_{ij}(v_i-v_j),\hfill \end{array}\right.t\in (t_{k-1},t_k)$$

(18)

and

$$\left\{\begin{array}{c}{y}_{{\ell }_i}\left(t_k\right)=\sum _{j=1}^m(P_{l\left(ij\right)}\otimes I_q)y_{{\ell }_j}\left(t_k^{-}\right),y_{f_{\tau }}\left(t_k\right)=y_{f_{\tau }}\left(t_k^{-}\right),\hfill \\ v_{{\ell }_i}\left(t_k\right)=\sum _{j=1}^m(P_{l\left(ij\right)}\otimes I_{n-q})v_{{\ell }_j}\left(t_k^{-}\right),v_{f_{\tau }}\left(t_k\right)=v_{f_{\tau }}\left(t_k^{-}\right),\hfill \end{array}\right.t=t_k$$

(19)

where $v_i\in {\mathbb{R}}^{n-q}$ is the observer state, $c>0$ is the coupling strength, and $P_l\in {\mathbb{R}}^{m\times m}$ is a row stochastic matrix associated with ${\mathcal{G}}_{\ell }$. $F\in {\mathbb{R}}^{(n-q)\times (n-q)}$, $K\in {\mathbb{R}}^{p\times n}$, $G\in {\mathbb{R}}^{(n-q)\times q}$, $H\in {\mathbb{R}}^{(n-q)\times p}$, $Q_1\in {\mathbb{R}}^{n\times q}$, and $Q_2\in {\mathbb{R}}^{n\times (n-q)}$ are constant matrices designed according to Algorithm 1 in [40] below.

Algorithm 1: Output feedback control algorithm of the homogeneous MAS

Step 1: Hurwitz matrix F is selected to make its eigenvalues different from those of $\widehat{A}$, where $\widehat{A}=CA{C}^{+}$. Select the matrix G so that $(F,G)$ is controllable. Step 2: Find the unique solution T to the Sylvester equation $T{C}^{+}CA-FT=GC$, which satisfies that $\left[\begin{array}{c}C\\ T\end{array}\right]$ is non-singular. If $\left[\begin{array}{c}C\\ T\end{array}\right]$ is singular, return to step 1 and select G again, until it is non-singular. Then, $H=T{C}^{+}\widehat{B}$ with $\widehat{B}=CB$. Compute matrices $Q_1$ and $Q_2$ using $\left[\begin{array}{cc}{Q}_1& Q_2\end{array}\right]={\left[\begin{array}{c}C\\ T\end{array}\right]}^{-1}$. Step 3: For a given positive-definite matrix $\widehat{Q}$, solve the Riccati equation ${\widehat{A}}^{T}P+P\widehat{A}-P\widehat{B}{\widehat{B}}^{T}P=-\widehat{Q}$ to obtain a positive-definite matrix P such that $\widehat{K}=-{\widehat{B}}^{T}P$. Step 4: Select the coupling strength $c\ge \frac{1}{2{min}_{{\lambda }_{\kappa ,i}\ne 0}\left\{\mathrm{Re}\left({\lambda }_{\kappa ,i}\right)\right\}}$, where ${\lambda }_{\kappa ,i}$ is the i-th nonzero eigenvalue of the Laplacian matrix $L_{\kappa }$.

In Algorithm 1, it is worth noting that the Hurwitz matrix F and the matrix T satisfying the Sylvester equation play the critical roles in solving the global output consensus value ${\tilde{y}}^{*}\left(t\right)$ in steps 1 and 2, respectively. In addition, the solution of the Riccati equation in step 3 and the determination of the coupling strength c in step 4 are sufficient conditions for the synchronization of the system described by Equation (15).

Regarding the first problem, due to the unavailability of states, the problem addressed by Wang et al. [40] was no longer the state consensus but the output consensus. To emphasize the second problem, as demonstrated by Theorem 3 in [40], the global output consensus value ${\tilde{y}}^{*}\left(t\right)$ was

$${\tilde{y}}^{*}\left(t\right)-{\displaystyle \frac{e^{\widehat{A}t}({\varphi }^{T}Q\otimes I_q)Cx\left(0\right)}{{\mathsf{\Sigma }}_{i=1}^m{\varphi }_i}}=\mathbf{0},ast\to \infty$$

(20)

Based on the simulation results, when the cooperative network is illustrated in Figure 5, the output convergence $y_i\left(t\right),i=1,\cdots ,7$ of CMASs under Algorithm 1 and the reset reduced-order-based protocol (18)–(19) are displayed in Figure 11. It is evident from the results that the evolution of the two leaders abruptly changes at the reset time, whereas the followers exhibit a smooth evolution, ultimately achieving output consensus. Moreover, the consensus value computed by (20) is ${\overline{y}}^{*}=\left[\begin{array}{c}-0.94\\ -0.94\end{array}\right]$. It can be seen that the convergence of the system does not suffer from the chattering phenomenon during the reset time. From Figure 12, it is evident that the reduced-order observer, as compared to the reset full-order observer protocols in [40], exhibits a longer convergence time. This can be attributed to the reduction in computational redundancy and storage capacity in the system, which comes at the expense of a longer convergence time.

Most recently, Wang et al. [42] investigated the output synchronization problem for heterogeneous CMASs with hybrid continuous–discrete dynamics as follows:

$$\left\{\begin{array}{c}\left\{\begin{array}{cc}{\dot{x}}_i=A_i{x}_i+B_i{u}_i,\hfill & \\ y_i=C_i{x}_i\hfill \end{array}\right.t\in \left[t_{k-1},t_k\right)\hfill \\ y_{{\ell }_i}\left(t_k\right)=\sum _{j=1}^m(P_{l\left(ij\right)}\otimes I_q)y_{{\ell }_j}\left(t_k^{-}\right),y_{f_{\tau }}\left(t_k\right)=y_{f_{\tau }}\left(t_k^{-}\right),t=t_k\hfill \end{array}\right.$$

(21)

where $x_i\in {\mathbb{R}}^{n_i}$, $u_i\in {\mathbb{R}}^{p_i}$, and $y_i\in {\mathbb{R}}^q$ are the state, input, and output of the i-th $(i=1,\cdots ,N)$ agent, respectively. Here $y_{{\ell }_i}\left(t_k\right)$, $y_{f_{\tau }}\left(t_k\right)$, and $P_l$ have the same meanings as in (17). $A_i,B_i$, and $C_i$ denote constant matrices with compatible dimensions and satisfy the following assumption.

Assumption 6.

$(A_i,B_i)$ is stabilizable, $(A_i,C_i)$ is detectable, and $C_i$ is full rank with rank q.

To address the output synchronization problem in heterogeneous CMASs, the ideas in [42] were divided into three steps:

The first step was to artificially create an internal reference model for each agent (as shown in Figure 13, with the same physical network structure as the heterogeneous CMASs) and design a reduced-order observer-based reset output feedback controller based on reset internal models as follows:

$$\left\{\begin{array}{c}\left\{\begin{array}{cc}{\dot{\xi }}_i=S{\xi }_i+K\sum _{j=1}^N{a}_{ij}({\xi }_i\left(t\right)-{\xi }_j\left(t\right)),\hfill & \\ {\widehat{y}}_i=D{\xi }_i,\hfill \end{array}\right.\hfill \\ {\dot{v}}_i=F_i{v}_i+G_i{y}_i+H_i{u}_i,\hfill \\ u_i=K_{i1}{Q}_{i1}{y}_i+K_{i1}{Q}_{i2}{v}_i+K_{i2}{\xi }_i\hfill \end{array}\right.t\in \left[t_{k-1},t_k\right)$$

(22)

and at the reset time $t_k$,

$$\left\{\begin{array}{cc}{\xi }_{{\ell }_i}\left(t_k\right)=\sum _{j=1}^m{P}_{l\left(ij\right)}{\xi }_{{\ell }_j}\left(t_k^{-}\right),{\xi }_{f_{\tau }}\left(t_k\right)={\xi }_{f_{\tau }}\left(t_k^{-}\right),\hfill & \\ v_{{\ell }_i}\left(t_k\right)=v_{{\ell }_i}\left(t_k^{-}\right)+T_{{\ell }_i}{C}_{{\ell }_i}^{+}[\sum _{j=1}^m{P}_{l\left(ij\right)}{y}_{{\ell }_j}\left(t_k^{-}\right)-y_{{\ell }_i}\left(t_k^{-}\right)],\hfill & \\ v_{f_{\tau }}\left(t_k\right)=v_{f_{\tau }}\left(t_k^{-}\right),\hfill \end{array}\right.$$

(23)

where ${\xi }_i\left(t\right)\in {\mathbb{R}}^r$ and ${\widehat{y}}_i\left(t\right)\in {\mathbb{R}}^q$ represent the internal reference model states and outputs, respectively. $v_i\left(t\right)\in {\mathbb{R}}^{n_i-q}$ denotes the reduced-order-observer states. $S\in {\mathbb{R}}^{r\times r}$ and $D\in {\mathbb{R}}^{q\times r}$ represent the state and output matrices of the internal reference models, respectively. $K\in {\mathbb{R}}^{r\times q},F_i\in {\mathbb{R}}^{(n_i-q)\times (n_i-q)},G_i\in {\mathbb{R}}^{(n_i-q)\times q},H_i\in {\mathbb{R}}^{(n_i-q)\times p_i}$, and $T_{{\ell }_i}\in {\mathbb{R}}^{r\times q}$ are designed according to Algorithm 2 below.

Algorithm 2: Output feedback control algorithm of the heterogeneous MAS

Step 1: Select the Hurwitz matrix $F_i$ to ensure that its eigenvalues are different from those of ${\widehat{A}}_i$, where ${\widehat{A}}_i=C_i{A}_i{C}_i^{+}$. Select the matrix $G_i$ such that $(F_i,G_i)$ is controllable. Step 2: Determine the unique solution $T_i$ to the Sylvester equation $T_i{C}_i^{+}{C}_i{A}_i-F_i{T}_i=G_i{C}_i$ such that $\left[\begin{array}{c}{C}_i\\ T_i\end{array}\right]$ is non-singular. If $\left[\begin{array}{c}{C}_i\\ T_i\end{array}\right]$ is singular, return to Step 1 and select $G_i$ again until it is non-singular. Compute matrices $Q_{i1}$ and $Q_{i2}$ using $\left[\begin{array}{cc}{Q}_{i1}& Q_{i2}\end{array}\right]={\left[\begin{array}{c}{C}_i\\ T_i\end{array}\right]}^{-1}$ and then select $H_i=T_i{C}_i^{+}{\widehat{B}}_i$, where ${\widehat{B}}_i=C_i{B}_i$. Step 3: Select the gain matrices $K_{i1}$ such that ${\widehat{A}}_i+{\widehat{B}}_i{K}_{i1}{C}_i^{+}$ is Hurwitz, $K_{i2}={\Gamma }_i-K_{i1}{\Pi }_i$, where the solution pairs $({\Pi }_i,{\Gamma }_i)$ with ${\Pi }_i\in {\mathbb{R}}^{n_i\times r}$, ${\Gamma }_i\in {\mathbb{R}}^{p_i\times r}$ depend on the following regulator equation: $$\begin{array}{cc}\hfill {\Pi }_{i}S& =A_i{C}_i^{+}{C}_i{\Pi }_i+B_i{\Gamma }_i,\hfill \\ \hfill {\Pi }_i& =C_i^{+}D,i=1,2,\cdots ,N.\hfill \end{array}$$ (24)

In the implementation of Algorithm 2, all parameters are carefully selected to ensure the synchronization of heterogeneous CMASs (21) and are utilized during the convergence analysis. Additionally, the utilization of the generalized inverse of matrix $C_i$ from [42] leads to the derivation of the solution pair $({\Pi }_i,{\Gamma }_i)$ using the regulator equation (24), which is based on the information provided by S, D, and $(A_i,B_i,C_i)$.

The second step is to demonstrate that the homogeneous internal reference models achieve synchronization. With Figure 13 as a simulation example, Figure 14 shows the trajectories of state ${\xi }_i,i=1,\cdots ,7$. It is clear that the internal reference model state achieves synchronization under Algorithm 2 and the hybrid communication mechanism (22)–(23).

The third step is to let each agent track the corresponding internal reference model to obtain the output synchronization of the heterogeneous CMAS, that is, ${lim}_{t\to \infty }∥y_i\left(t\right)-y_j\left(t\right)∥$ = 0. The trajectory of the output variable $y_i\left(t\right)$ can be seen to achieve synchronization, as shown in Figure 15.

To comprehensively investigate the disparity between consensus algorithms and the inherent attributes or constraints present in practical models, a thorough exploration of consensus issues incorporating real-world factors is imperative. These real-world factors encompass communication constraints (e.g., time delay and sampling perturbations), quantization, and state saturation, all of which can be considered vital constraint characteristics within practical models.

Recently, Pham et al. [39] focused on investigating the challenge of formation control in state-constrained clustered network systems. To tackle the issue arising from the combination of hybrid communication and state saturation, a highly resilient formation control protocol was proposed.

The output consensus problem for heterogeneous CMASs with disturbances was investigated in [41]. A new output consensus protocol was proposed in which each agent had an observer to reconstruct the state and disturbances, and a virtual reference model to take into account continuous intra-communication and discrete inter-communication.

Based on the above review,

Table 6. Advantages and disadvantages of cooperation control problems over intermittent cluster networks.

Reference	Year	Dynamics	Advantages and Disadvantages
			State Information	Communications Constraint	Reset Instants
[35]	2016	First-order integrator	✓	None	Predetermined
[36]	2016	First-order integrator	✓	Time-varying topology	Event-triggered
[37]	2019	General linear dynamics	✓	None	Predetermined
[38]	2020	General linear dynamics	×	None	Predetermined
[39]	2020	General linear dynamics	✓	State saturation	Predetermined
[40]	2021	General linear dynamics	×	None	Predetermined
[41]	2022	Heterogeneous dynamics	✓	Disturbances	Predetermined
[42]	2023	Heterogeneous dynamics	×	None	Predetermined

If state information can be available, it is marked by ✓, otherwise use ×.

lists the key characteristics of these studies, which explicitly show the advantages and disadvantages of the research questions. From Table 6, it is easy to see that although various cooperation control problems over intermittent cluster networks have been studied, there are still many critical issues to be further addressed in future work, such as the time sequence of inter-cluster communication being determined by event triggers rather than predetermined.

5. Prospects for Future Research

An overview of the mathematical modeling and distributed control of wide-area networks has been presented. While some coordination controls of MASs over wide-area networks have been discussed in the literature, there are still several critical issues that need to be addressed in future research. In the following section, several relevant issues are suggested as potential directions for future investigations.

• Current research on coordination control of MASs has some structural constraints on communication topology. For example, bipartite consensus requires a structurally balanced condition, cluster consensus requires an in-degree balance assumption, and group-bipartite consensus needs to be structurally balanced, and requires acyclic partition, and sign-balanced couple. Furthermore, to achieve such dynamical behaviors, a spanning tree or joint spanning tree in the network topology is necessary. Therefore, investigating the dynamical behavior of MASs on more general unstructured constrained networks would be highly beneficial.
• In the existing results of cooperative control on intermittent clustered networks, the interaction moments between neighboring clusters are artificially predetermined. However, in order to enhance work efficiency and facilitate flexible information interactions, the inter-cluster communication often needs to change based on actual demand. Therefore, considering the incorporation of demand-based event-triggered impulsive control strategies for intermittent clustered networks is of great practical importance.
• Although group-bipartite consensus has the potential to address multi-objective symmetric tasks, it still faces challenges and limitations in practical applications. For instance, designing effective protocols to achieve group-bipartite consensus and addressing noise and faults in the systems that warrant further investigation.

In the last decade, numerous innovative frameworks and models have emerged in the study of wide-area networks for MASs, enhancing the literature on agent-based systems and expanding their potential applications. However, this field is still in its early stages and requires further refinement of theoretical results. In our review, we focus on group-bipartite networks, which have the potential to effectively handle complex tasks involving multiple levels and structures. However, these networks face various constraints related to topology, and communication constraints (e.g., delays, saturation, disturbances) have not been adequately addressed. To make group-bipartite networks more practical, future research should tackle these limitations. Cooperative intermittent clustered networks are particularly suitable for post-disaster communication networks and power grids due to their distinctive communication methods. Nevertheless, there are significant hurdles to overcome, such as predefined long-distance interactions. To address these issues, it is necessary to develop more flexible control algorithms that facilitate sparse communication between clusters.

6. Conclusions

MASs serve as important methodologies and tools for the analysis and modeling of complex systems. In this review, we categorize wide-area networks based on the interaction modes of cooperation or competition among agents. We provide mathematical modeling for each type, review the relevant literature, and summarize the key findings. Finally, we provide future research prospects. Exploring the clustering and sparsity of MASs can help explain real-world clustering phenomena and can be applied to practical engineering.