Ordered-Bipartite Consensus of Multi-Agent Systems under Finite Time Control

Abstract

Since group consensus (GC) control accounts for one-half of the multi-robot coordination domain, as does complete consensus control, this paper will formulate a certain kind of GC and group bipartite consensus (GBC) in finite time for multi-agent systems (MAS). It should be noted that the key to this work is that information transfer under acyclic partition networks is zero-sum between groups and the cooperative–competitive interaction. The general GC will firstly be achieved under acyclic partition, which reflects the fact that such topology allows the MAS to reach agreement in an ordered and hierarchical process. Further, the information communication is zero-sum within each group under the acyclic partition signed digraph. Thus, the group–bipartite agreement for MAS is given under acyclic bipartite topology. These results display not only an ordered, hierarchical state, but also an intra-group symmetric state. In addition, simulation examples are presented in parallel with the theoretical results.

1. Introduction

No matter what the background of the times is, the exploration of unknown regions and the monitoring of dangerous areas always stand at the forefront, which is why research work in the field of robotics is keeping up with the times. In the current period, intelligent agents have been born along with rapid development. Some intelligent agents are already utilized in civilian life and military reserves. Compared to well-designed robots, multi-agent systems are adaptive and self-organizing because they are formed by a considerable number of simple individuals that can perform distributed perception and behavior. Therefore, the failure of individual behavior does not cause the overall behavior of this multi-agent system to fail. This reflects that the multi-agent system has better fault tolerance and higher robustness. In addition, the systems respond to complex tasks by choosing the number and the type of individuals. The control approach of the multi-agent system is to implement the code with a central processor. Therefore, the combination of information technologies makes it more advantageous in terms of implementation benefits. So, as far as the present day is concerned, the study of MAS is a task that must be carried out in the course of human development [1,2,3,4]. According to the different task objectives of group systems, collaborative control problems are mainly classified into three types of problems, namely, consensus problems, swarm control, and formation control. Nonetheless, research on consensus control problems has always been fundamental to solving other collaborative control problems. For example, the formation control of a MAS is to achieve consensus of relative positions. Therefore, this paper discusses problems that belong to the field of consensus control of the MAS.

Reviewing the scientific process at the end of the 20th century, the convergence and rapid development of scientific fields such as computer science and asynchronous computing have given rise to the concept of consensus control [5]. Olfati developed a general framework for the consensus control problem with respect to integrator networks [6]. Stepping into the era of the 21st century, the field of MAS has seen a boom in research on networks. Consensus control algorithms based on graph theory have received increasing attention and have a relatively mature theory. Liu extended the consensus problem to second-order systems, and obtained sufficient and necessary conditions for the stability of MAS in directed networks [7]. Rezaee et al. proposed a consensus protocol for higher-order multi-agent systems, illustrating the way in which agents achieve consensus using the relative position information shared by their surrounding neighbors [8].

With the increasing theoretical research and the increased requirements of modern industrial production in terms of scale and refinement, complex behavior control has become an important aspect of MAS’s theory research. Among the methods to portray the coordinated control of multi-task objectives for networked MAS, GC and bipartite consensus are in irreplaceable positions. As a result, various approaches for GC have been under development, with studies represented by [9,10,11] among others. Since Altafini pioneered a bipartite consensus model for networked systems in a structurally balanced network topology in 2013 [12], various sequential works have been carried out. Recent research in [13,14,15] gave discussions on the consensus problems for cooperative–competitive networks. Inspired by these works, Zhang provided a novel symmetric positive definite matrix to derive the gain selection criteria in the leader-following case for networked Lagrangian systems [16]. Zhang further introduced an acyclic partition and obtained the explicit expressions of the final states in the leaderless case [17].

However, once the MAS’s consensus problems are mentioned, its convergence rate is further considered. On the one hand, infinite time convergence is not feasible in practical problems. On the other hand, a predictable convergence time is expected for MAS with multi-task objectives. Therefore, finite-time control methods and fixed-time control methods are proposed to increase the convergence rate of MAS. The finite-time/fixed-time control of MAS uses symbolic functions to solve the consensus problem. Due to the advantage of the high convergence rate of finite-time protocols, finite-time control of the MAS has evolved since its introduction. In recent related research, finite-time control ideas have been applied to various types of control for MAS. The work in [18] derived a criterion and provided a sufficient condition to ensure fixed-time stabilization of switched positive nonlinear systems; the article [19] solved finite-time leader–follower output consensus problems by using the mismatched disturbance observer. By using hierarchical non-singular terminal sliding mode, finite-time observer-based consensus was designed for underactuated USVs to deal with the trajectory tracking tasks [20]. This method was also applicable to earth-observation tasks for multiple airships [21].

The existing studies on the single group or single bipartite consensus will not be suitable for a MAS to complete complex behavior goals. For such problems, GC control of a MAS is feasible. However, those complex behaviors require massive information interaction in complete consensus control. Thus, the control protocols for MAS reaching complex behavior goals should consider the problem of handling redundant information, which is also an essential concern for GC. Thus, this paper proposes the acyclic GC and acyclic GBC control protocols for a MAS. Featured by graded (or step-wise) multi-objective tasks, cooperative–competitive relations are required in hierarchical situations. For instance, the unmanned management system perceives order as a fundamental requirement. In addition to achieving complex behavior consensus, the MAS needs to improve its convergence rate. From this point, both the acyclic GC and acyclic GBC control protocols are given based on finite-time control.

Motivated by those points, this paper will continue to explore the GC problems as well as GBC problems for MAS with finite-time control. In particular, the discussion covered in this paper is concerned with cooperative–competitive networks with acyclic partition topology. The main innovations are summarized as follows:

• This work deals with acyclic topology for its inter-group communications problems. Combining with the cooperative–competitive topology further, the MAS performs GC behavior in per-group symmetry. In this way, the inter-group problems of signed, directed interaction are also solved.
• The full discussion revolves around the finite-time control of GC coordination and GBC coordination. With these considerations, the smoothness of finite-time control protocols in signed digraphs obtains novel aspects.
• This paper takes the cooperative–competitive mechanism to the information interactions pattern in the first demonstration. The work is also specialized in its mathematical model by constructing a suitable signed graph with an acyclic partition. At this point, the corresponding conditions on algebraic graph theory are summarized. Indeed, based on these properties, the GC will be verified by the mathematical induction method, and the final GC takes an ordered and symmetric result.

Subject to these three points, this article is arranged as follows: Section 1 (i.e., this section) provides the background and overview of this research work; Section 2 unifies the specific notations of the discussed MAS, which includes the basic mathematical knowledge and relevant theories; Section 3 proves that the models in the second section reach the final agreement—that is, a solution is available that allows a MAS to reach GC and GBC in finite time; Section 4 performs simulations based on the model of Section 2, and the results of the simulations verify the conclusions in Section 3.

2. Preliminaries

Before presenting the main conclusions, some basic concepts and theoretical knowledge need to be understood. Let $\mathbb{R}$ be the domain of real numbers and ${\mathbb{R}}^N$ be the N-dimensional vector space defined on $\mathbb{R}$. By convention, $1_N={(1,1,\dots ,1)}^T\in {\mathbb{R}}^N$; $\mathrm{diag}(v)=\mathrm{diag}(v_1,v_2,\dots ,v_n)$ if v is a column vector ${\left(\begin{array}{c}{v}_1,v_2,\dots ,v_n\end{array}\right)}^T$ (or a row vector $\left(\begin{array}{c}{v}_1,v_2,\dots ,v_n\end{array}\right)$) in ${\mathbb{R}}^n$. Based on these, the following is a brief description of the required basic knowledge.

2.1. Basic Graph Theory

$\mathcal{G}=(\mathcal{V},\mathcal{E})$ is a weighted directed graph in the communication network, with the node set $\mathcal{V}=\{v_1,v_2,\dots ,v_N\}$ and the edge set $\mathcal{E}=\{e_{ij}|v_i,v_j\in V\}$, and $A=[a_{ij}]\in {\mathbb{R}}^{N\times N}$ is the adjacency matrix. In this case, the i-th agent is presented by node $v_i$ in the topology network. This means that the i-th agent and node $v_i$ are different representations of the same object in its appropriate context. It is always assumed that $a_{ii}=0$, which implies that the given graph $\mathcal{G}$ does not include any self-loops. For the directed graph $\mathcal{G}$, $a_{ij}\ne 0$ if and only if there exists a direct edge from $v_j$ to $v_i$, and call $\mathcal{G}$ a signed graph if and only if the weights of the edge are either negative or positive. A signed digraph $\mathcal{G}$ is defined as structurally balanced if there exists two subsets ${\mathcal{V}}_1$ and ${\mathcal{V}}_2$ such that the node set $\mathcal{V}$ can be decomposed by ${\mathcal{V}}_1\cup {\mathcal{V}}_2=\mathcal{V}$ and ${\mathcal{V}}_1\cap {\mathcal{V}}_2=\oslash$, and the weights of the edges have the relation $\left\{\begin{array}{cc}{a}_{ij}\ge 0,\mathrm{if}& v_i,v_j\in {\mathcal{V}}_{\ell },\ell =1,2,\\ a_{ij}\le 0,\mathrm{if}& v_i\in {\mathcal{V}}_h,v_j\in {\mathcal{V}}_{\ell },\ell \ne h.\end{array}\right.$ Under this sense, it can be noted that $\overline{i}$ is the set of nodes that have an edge with nodes $v_i$. Then, it is not difficult to explain that $v_j\in \overline{i}$ if and only if $\overline{i}=\overline{j}$. Next, it is necessary to note that ${\mathcal{P}}_{ij}$ are the (directed) paths from $v_j$ to $v_i$ if and only if there exists some indexes $v_{j_1},v_{j_2},\dots ,v_{j_s}$ such that $e_{j_{1}j},e_{j_2{j}_1},\dots ,e_{i{j}_s}$ in $\mathcal{E}$. Furthermore, the directed graph $\mathcal{G}$ is said to contain a directed spanning tree if and only if there exists at least one node (called root) that has a path to all the agents in $\mathcal{V}$. The Laplacian matrix $L(A)={(l_{ij})}_{N\times N}$ related to a matrix A (or to graph $\mathcal{G}$) is defined by $l_{ij}=\left\{\begin{array}{cc}-a_{ij},& i\ne j,\\ {\displaystyle \sum _{\ell \in \mathcal{I}}}{a}_{i\ell },& i=j.\end{array}\right.$

Finally, it is needed to give the algebraic feature of the structurally balanced graph.

Lemma 1

([12]). $\mathcal{G}$ is structurally balanced if and only if there exists a matrix $\Phi =diag({\varphi }_1,\dots ,{\varphi }_N)$ with ${\varphi }_{\ell }\in \{1,-1\}$ such that the entries of $\Phi A\Phi$ are non-negative.

2.2. Problem Formulation

Consider a MAS consisting of the following N identical agents described in dynamics (1):

$$\begin{array}{c}\hfill {\dot{x}}_i(t)=u_i(t),i=1,2\dots ,N,\end{array}$$

(1)

where $x\in {\mathbb{R}}^n$ and $u\in {\mathbb{R}}^n$ denote the state vector and the input vector of agent $v_i$, $i=1,2,\dots ,N$, respectively.

Assumption 1.

Graph $\mathcal{G}$ has a acyclic partition $\{{\mathcal{V}}_1,{\mathcal{V}}_1,\dots ,{\mathcal{V}}_p\}$.

Under this assumption, ${\mathcal{G}}_i$ is a subgraph of $\mathcal{G}$ made up of all the points in ${\mathcal{V}}_i$ and the edges between those points. Finally, the network topology of acyclic-partition is given an intuitive algebraic expression in the form of the following adjacency matrix A.

$$A={\left(\begin{array}{cccccc}\hfill & A_{11}\hfill & \hfill 0& \cdots \hfill & \hfill 0& \\ \hfill & A_{21}\hfill & \hfill A_{22}& \cdots \hfill & \hfill 0& \\ \hfill & ⋮\hfill & \hfill ⋮& \ddots \hfill & \hfill ⋮& \\ \hfill & A_{p1}\hfill & \hfill A_{p2}& \cdots \hfill & \hfill A_{pp}\end{array}\right)}_{N\times N},$$

(2)

$A_{ii}$ represents the adjacency matrix of ${\mathcal{G}}_i$, and $A_{ij}$ represents the information transmission from ${\mathcal{G}}_i$ to ${\mathcal{G}}_j$.

Remark 1.

It is worth noting that A is the upper triangular block matrix, and $A_{ij}=0,i<j$ means that the information transmission from topology ${\mathcal{G}}_i$ to topology ${\mathcal{G}}_j$ can only be carried out when $i<j$. In fact, this algebraic condition comes from the property of the acyclic partition structure itself. Acyclic partition can reduce information redundancy and improve the information transfer efficiency of the multi-agent system because of the properties of directed transmission with respect to inter-group information.

Assumption 2.

Each ${\mathcal{G}}_i$ has a spanning tree.

If ${\mathcal{G}}_i$ is only a digraph with non-negative (or non-negative) weights, the eigenvalue 0 of ${\mathcal{G}}_i$ has its right eigenvector $1_{n_i}$ and its left eigenvector ${\omega }_i={[{\omega }_{n_{i-1}+1,i},{\omega }_{n_{i-1}+2,i},\dots ,{\omega }_{n_{i-1}+n_i,i}]}^T$ under Assumption 2. It is well known that the result of this has been widely used in the MAS. Article [22] discussed the problem of finite-time consensus of first-order systems with (unilateral and ungrouped) directed topology and confirmed that the existence of spanning trees is sufficient to achieve finite-time consensus. If ${\mathcal{G}}_i$ has a bipartite structure, gauge transformation is needed before solving the problem.

Assumption 3.

The row sum of each block ${\Phi }_i{L}_{ij}{\Phi }_j$ is zero.

Assumption 3 shows that the effect of all agents in another different group on a certain agent is trivial to zero in the sense of “group”, and the network topology constructed based on this assumption reflects the internal requirements of GC.

Remark 2.

For a general matrix A, one can define $L(A_{ii}),i=1,\dots ,p$, $L(A)$ by the way defined in the last subsection. Note that $L(A_{ii})$ is not necessarily equal to $L{(A)}_{ii}$. However, under Assumption 3, there is $L(A_{ii})=L{(A)}_{ii}$. In fact, Assumption 3 is, on the one hand, an algebraic description of the network topology for acyclic partition, and on the other hand allows the definition of acyclic partition structure to be well-defined. It is specifically reflected in the proof process in the next section—that is, Assumption 3 resolves the solution of a certain GC control problem into a complete consensus problem. From this aspect, it can be said that the addition of the acyclic partition topology to the finite-time GC problems makes it possible to have a solution for practical problems such as the management of intelligent agents.

Lemma 2

([13]). If $\mathcal{G}$ is an unsigned graph satisfying Assumption 1 and Assumption 3, Assumption 2 holds if and only if the following two statements (3) and (4) hold.

$$\begin{array}{c}\begin{array}{cc}& 0\mathit{is}a\mathit{simple}\mathit{eigenvalue}\mathit{of}L\mathit{with}\mathit{multiplicity}p,\hfill \end{array}\end{array}$$

(3)

$$\begin{array}{c}\begin{array}{ccc}& & \hfill \mathit{the}\mathit{real}\mathit{parts}\mathit{of}\mathit{the}\mathit{nonzero}\mathit{eigenvalues}\mathit{of}L\mathit{are}\mathit{all}\mathit{positive}.\end{array}\end{array}$$

(4)

Moreover, the left eigenvector belonging to eigenvalue 0 is

$$\begin{array}{cc}\hfill & {\pi }_1=({\mu }_1^{[1]},\dots ,{\mu }_{k_i}^{[1]},{\displaystyle \underset{n-k_1}{\underbrace{0,\dots ,0}}}),\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\pi }_i=({\rho }_{1,1}^{[i]},\dots ,{\rho }_{1,k_1}^{[i]},\dots ,{\rho }_{i-1,1}^{[i]},\dots ,{\rho }_{i-1,k_{i-1}}^{[i]},{\mu }_1^{[i]},\dots ,{\mu }_{k_i}^{[i]},{\displaystyle \underset{n-{\displaystyle \sum _{j=1}^i}{k}_j}{\underbrace{0,\dots ,0}}}),\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\pi }_p=({\rho }_{1,1}^{[p]},\dots ,{\rho }_{1,k_1}^{[p]},\dots ,{\rho }_{p-1,1}^{[p]},\dots ,{\rho }_{p-1,k_{i-1}}^{[p]},{\mu }_1^{[p]},\dots ,{\mu }_{k_p}^{[p]}),\hfill \end{array}$$

where ${\displaystyle \sum _{\ell =1}^{k_i}}\mu {}_{\ell }^{[i]}=1$ with $\mu {}_{\ell }^{[i]}\ne 0$ and ${\displaystyle \sum _{l=1}^{k_j}}{\rho }_{j,l}^{[i]}=0$ for $j=1,3,\dots ,i-1$, and i = 1, 2, …, p.

Furthermore, Figure 1 presents a certain acyclic partition of 18 agents, which are divided into 4 groups, ${\mathcal{G}}_1$ (green agents $\{v_1,v_2,v_3,v_4,v_5\}$), ${\mathcal{G}}_2$ (pink agents $\{v_6,v_7,v_8,v_9\}$), ${\mathcal{G}}_3$ (yellow agents $\{v_{10},v_{11},$$v_{12},v_{13},v_{14}\}$), and ${\mathcal{G}}_4$ (purple agents $\{v_{15},v_{16},v_{17},v_{18}\}$). This clearly describes the interaction communication among the groups—that is, the first group ${\mathcal{G}}_1$ only sends state information to groups ${\mathcal{G}}_2$, ${\mathcal{G}}_3$, and ${\mathcal{G}}_4$, and group ${\mathcal{G}}_2$ just receives information from group ${\mathcal{G}}_1$ and sends information to the other groups ${\mathcal{G}}_3$ and ${\mathcal{G}}_4$, etc.

Assumption 4.

Every ${\mathcal{G}}_i$ is structurally balanced.

By Lemma 1, Assumption 4 can be stated equivalently that there exists a block matrix $\Phi$ such that ${\Phi }_i{A}_{ii}{\Phi }_i$ has all non-negative entries. This gauge transformation is an indispensable tool in the discussion of bipartite consensus problems for multi-agent systems. The gauge transformation of $\Phi$ provides the first foundation for reaching consensus with cooperative and competitive topology, and makes the research work on bipartite consensus unified with that on unilateral consensus. Therefore, some of the concepts from the previous basic graph theory should be adapted to both cases. For the sake of uniformity in the full narrative, the definition of the Laplacian matrix $L={[l_{ij}]}_{N\times N}$ associated with $\mathcal{G}$ is required to be updated as

$$l_{ij}=\left\{\begin{array}{cc}{\displaystyle \sum _{k\in \overline{i}}}|a_{ik}|+{\displaystyle \sum _{k\notin \overline{i}}}{\varphi }_k{a}_{ik},& i=j,\\ -a_{ij},& i\ne j.\end{array}\right.$$

(5)

Remark 3.

Obviously, Equation (5) is a definition of the Laplacian matrix that includes the three cases of unilateral partition, bipartition, and group bipartition, as well as a classification of these three types of topologies. It is reasonable to take into account the influence of the group networks and bipartite interaction on agent i, as reflected in the values of the elements on the main diagonal of the matrix L in Equation (5). Then, based on this definition of L, ${\Phi }_i{L}_{ii}{\Phi }_i$ can be said to be a traditional Laplacian matrix of non-negative weighted graphs when Assumption 4 holds. Note that ${\Phi }_i{L}_{ii}{\Phi }_i$ is similar to $L_{ii}$. Thus, the result of Lemma 2 can be implemented onto signed directed graphs.

Lemma 3

([15]). If $\mathcal{G}$ is a signed graph satisfying Assumption 1, Assumption 3, and Assumption 4, Assumption 2 holds if and only if $0$ is the p-multiple eigenvalue of L, while all other eigenvalues of $L$ have positive real parts. Furthermore, the p linearly independent right eigenvectors corresponding to eigenvalue 0 are

$$\begin{array}{c}\hfill {\left({({\Phi }_1{\mathbf{1}}_{k_1})}^T,{\displaystyle \underset{n-k_1}{\underbrace{0,\dots ,0}}}\right)}^T,\dots ,{\left({\displaystyle \underset{{\displaystyle \sum _{j=1}^{i-1}}{k}_1}{\underbrace{0,\dots ,0}}},{({\Phi }_i{\mathbf{1}}_{k_i})}^T,{\displaystyle \underset{n-{\displaystyle \sum _{j=1}^i}{k}_j}{\underbrace{0,\dots ,0}}}\right)}^T,\dots ,{\left({\displaystyle \underset{{\displaystyle \sum _{j=1}^{p-1}}{k}_j}{\underbrace{0,\dots ,0}}},{({\Phi }_i{\mathbf{1}}_{k_i})}^T\right)}^T.\end{array}$$

The p linearly uncorrelated left eigenvectors belonging to eigenvalue 0 are

$$\begin{array}{cc}\hfill & {\varpi }_1=({\nu }_1^{[1]},\dots ,{\nu }_{k_i}^{[1]},{\displaystyle \underset{n-k_1}{\underbrace{0,\dots ,0}}}),\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\varpi }_i=({\varrho }_{1,1}^{[i]},\dots ,{\varrho }_{1,k_1}^{[i]},\dots ,{\varrho }_{i-1,1}^{[i]},\dots ,{\varrho }_{i-1,k_{i-1}}^{[i]},{\nu }_1^{[i]},\dots ,{\nu }_{k_i}^{[i]},{\displaystyle \underset{n-{\displaystyle \sum _{j=1}^i}{k}_j}{\underbrace{0,\dots ,0}}}),\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\varpi }_p=({\varrho }_{1,1}^{[p]},\dots ,{\varrho }_{1,k_1}^{[p]},\dots ,{\varrho }_{p-1,1}^{[p]},\dots ,{\varrho }_{p-1,k_{i-1}}^{[p]},{\nu }_1^{[p]},\dots ,{\nu }_{k_p}^{[p]}),\hfill \end{array}$$

where ${\Phi }_i{\nu }^{[i]}$ are non-negative vector satisfying ${({\Phi }_i{\nu }^{[i]})}^T{\mathbf{1}}_{k_i}=1$ and ${({\Phi }_j{\varrho }_j^{[i]})}^T{\mathbf{1}}_{k_j}=0$ (j = 1, 2, …, i − 1) for i = 1, 2, …, p.

Similar to the previous Lemma 2, the graph associated with this lemma is given by Figure 2. The partition represented by this figure is the same as the previous one, with the difference that the groups contain positive and negative weights and are marked in red.

Definition 5.

The system (1) is said to reach GC in finite time in p groups if there exists $T\in {\mathbb{R}}_{+}$ such that

$$\begin{array}{c}\hfill |{\varphi }_i{x}_i(t)-{\varphi }_j{x}_j(t)|=0\mathit{when}\overline{j}=\overline{i},\end{array}$$

(6)

where $\{\overline{1},\dots ,\overline{p}\}$ is an acyclic partition of $\mathcal{V}$ and any $t\ge T$.

Furthermore, if there exists one non-trivial ${\Phi }_k(1\le k\le p)$, i.e., there exists a ${\Phi }_k(1\le k\le p)$ with positive and negative entries, the system (1) will be said to reach GBC in finite time in p groups.

Remark 4.

Compared with complete consensus (CC), which refers to a unique convergence state, GC refers to multi-agent systems with multiple convergence states. There is no (directed) spanning tree in the communication topology of MAS, and the systems cannot achieve complete consensus. With this consideration, the logic of GC is to achieve multi-consensus by dividing the communication topology into a partition. The result is that the agents partitioned into the same group have the same convergence state (i.e., CC). Thus, GC is an extension of CC. Further, if the communication networks of each group under such partitioning is a bipartite topology, the GC implemented by the systems is called GBC. In other words, GBC means that the system is characterized by multi-pair convergence states, and each pair of convergence states is symmetric.

Remark 5.

The abovementioned expression (6) will be describe as the following equation if Φ is nontrivial (i.e., $\Phi \ne I_N$),

$$\left\{\begin{array}{l}|x_i(t)-x_j(t)|=0,\mathit{the} \mathit{weight} \mathit{of} {\mathcal{P}}_{ij}\mathit{is} \mathit{positive},\hfill \\ |x_i(t)+x_j(t)|=0,\mathit{the} \mathit{weight} \mathit{of} {\mathcal{P}}_{ij}\mathit{is} \mathit{negative}\hfill \end{array}\right.\overline{j}=\overline{i}\mathit{for} \mathit{any}t\ge T.$$

(7)

The bipartite consensus problems require that the agents on the same side reach consensus and the agents from different sides converge to the same value but in the opposite direction [14].

Consider the control protocol of each i-th agent given by (8). This paper will explore the consensus problem of the MAS based on these control protocols.

$$\begin{array}{cc}\hfill & u_i=\beta si{g}^{\alpha }(y_i)+\gamma y_i,i=1,2,\dots ,N,\hfill \end{array}$$

(8)

where $y_i={\displaystyle \sum _{j\in \overline{i}}}(a_{ij}{x}_j-\left|a_{ij}\right|x_i)+{\displaystyle \sum _{j\notin \overline{i}}}(a_{ij}{x}_j-{\varphi }_j{a}_{ij}{x}_i)$ is generally viewed as the reference velocity of agent i.

Remark 6.

The control protocol (8) is a naturally defined finite-time control protocol based on the network topology mentioned in this subsection. When $\beta =0$, (8) is a general control protocol and its consensus has been relatively well studied. When $\beta \ne 0$, this protocol is a finite-time control law at $0\le \alpha <1$, while it is in the scope of a fixed-time control problem at $1<\alpha ,\gamma =0$ [18]. For finite-time consensus problems, ref. [23] proposed a sliding mode to guarantee the excellent robustness and, thus, developed the research on time-continuous case $0<\alpha <1$; moreover, the article [24] contributed to the work of discrete case $\alpha =1$. Next, this paper will explore the finite-time consensus problems and give the conditions for the directed topology to reach consensus. These research works provide a complete theoretical foundation for this paper. Next, this paper extends the grouping and cooperative–competitive consensus of a MAS under finite-time control.

This section ends with a statement of inequality for finite-time control protocols as follows.

Lemma 4.

Suppose that $\mathcal{J}$ is a countable set and $0<\alpha <1$, ${({\displaystyle \sum _i}{p}_i)}^{\alpha }\le {\displaystyle \sum _i}{p}_i^{\alpha }$ holds if $p_i\ge 0$ for $i=1,2\dots ,N$.

3. Main Results

This section will give rigorous proofs for the MAS (1) controlled by (8) to reach GC.

Theorem 6.

For the MAS (8), consider the unsigned directed graph $\mathcal{G}$ as its network topology. If $\mathcal{G}$ satisfies Assumptions 1–3, then the MAS is divided into p groups to achieve finite time agreement in an orderly manner.

Proof. 

Note that $\mathcal{G}$ is a graph with an acyclic partition $\{{\mathcal{V}}_1,{\mathcal{V}}_2,\dots ,{\mathcal{V}}_p\}$, which means that information is transmitted from ${\mathcal{V}}_i$ to ${\mathcal{V}}_j$ only when $j>i$. Therefore, based on this feature, the proof process in this part is divided into three parts. Firstly, it is proved that the first group ${\mathcal{G}}_1$ achieves agreement in finite time. This is immediately followed by proving that both the first group ${\mathcal{G}}_1$ and the second group ${\mathcal{G}}_2$ achieve finite time agreement. Finally, the result of the second step of the proof is extended to ${\mathcal{G}}_2,\dots ,{\mathcal{G}}_p$.

Now, for simplicity, denote that

$$\left\{\begin{array}{c}\hfill x^{[i]}={(x_{n_{i-1}+1},x_{n_{i-1}+2},\dots ,x_{n_{i-1}+k_i})}^T\\ \hfill y^{[i]}={(y_{n_{i-1}+1},y_{n_{i-1}+2},\dots ,y_{n_{i-1}+k_i})}^T\\ \hfill u^{[i]}={(u_{n_{i-1}+1},u_{n_{i-1}+3},\dots ,u_{n_{i-1}+k_i})}^T\end{array}\right.,$$

(9)

where $n_j={\displaystyle \sum _{\ell =1}^j}{k}_{\ell }$ and $k_{\ell }$ is the cardinality of ${\mathcal{V}}_{\ell }$.

Consider first the trivial case of $p=1$. From Assumption 2, $\mathcal{G}$ has a directed spanning tree, which means that the first group system solves the consensus problem in finite time. The next discussion is on the case of $p=2$—that is, to show that the agents in ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ reach consensus in finite time, which is a key step in the whole proof process. In this case, $L=\left(\begin{array}{cc}{L}_{11}& 0\\ L_{21}& L_{22}\end{array}\right)$ and $L_{22}=L(A_{11}),L_{22}=L(A_{22})$.

Let $y=Lx$; it is not difficult to obtain $\dot{y}=-Lu$. Thus, the control law (8) has that

$$\begin{array}{c}\hfill \dot{y}=-L\left(\beta si{g}^{\alpha }(y)+\gamma y\right).\end{array}$$

(10)

By Lemma 2, the orthogonal space of the column vector space of vector y is $\mathbb{R}{\pi }_1⨁\mathbb{R}{\pi }_2$. Thus, there exists a positive column vector $\omega \in \mathbb{R}{\pi }_1⨁\mathbb{R}{\pi }_2$ such that $\omega \perp y$. For ease of presentation, let $\omega$ be denoted as $({\omega }_1,\dots ,{\omega }_{n_2})$. Now, consider the Lyapunov candidate

$$\begin{array}{cc}\hfill V_2(t)=& y^T\mathrm{diag}({\omega }_1,\dots ,{\omega }_{n_2})\left(\frac{\beta }{1+\alpha }si{g}^{\alpha }(y)+\frac{\gamma }{2}y\right)\hfill \\ \hfill =& {\displaystyle \sum _{j=1}^{n_2}}{\omega }_j\left(\frac{\beta }{1+\alpha }{|y_j|}^{1+\alpha }+\frac{\gamma }{2}{y}_j^2\right)\ge 0,\hfill \end{array}$$

(11)

for which differentiation at t yields

$$\begin{array}{cc}\hfill {\dot{V}}_2(t)=& {\displaystyle \sum _{j=1}^{n_2}}{\omega }_j\left(\beta si{g}^{\alpha }(y_j)+\gamma y_j\right){\dot{y}}_j\hfill \\ \hfill =& \left(\beta si{g}^{\alpha }(y)+\gamma y\right)\mathrm{diag}({\omega }_1,\dots ,{\omega }_{n_2})\dot{y}\hfill \\ \hfill =& -{\left(\beta si{g}^{\alpha }(y)+\gamma y\right)}^T\mathrm{diag}({\omega }_1,\dots ,{\omega }_{n_2})L\left(\beta si{g}^{\alpha }(y)+\gamma y\right)\hfill \\ \hfill \le & 0.\hfill \end{array}$$

(12)

Since $diag({\omega }_1,\dots ,{\omega }_{n_2})$ is positive definite, $V_2=0$ implies that $y=0$ due to the sign of the values of $\alpha ,\beta ,\gamma$. Therefore, it is reasonable to assume that $V_2\ne 0$; further, the following inequalities follow from Lemma 4,

$$\begin{array}{cc}\hfill V_2^{\frac{2\alpha }{\alpha +1}}(t)=& {\left({\displaystyle \sum _{j=1}^{n_2}}{\omega }_j\left(\frac{\beta }{1+\alpha }{|y_j|}^{1+\alpha }+\frac{\gamma }{2}{y}_j^2\right)\right)}^{\frac{2\alpha }{\alpha +1}}\hfill \\ \hfill \le & {\displaystyle \sum _{j=1}^{n_2}}{\omega }_j^{\frac{2\alpha }{\alpha +1}}{\left(\frac{\beta }{1+\alpha }{|y_j|}^{1+\alpha }+\frac{\gamma }{2}{y}_j^2\right)}^{\frac{2\alpha }{\alpha +1}}\hfill \\ \hfill \le & {\displaystyle \sum _{j=1}^{n_2}}{\omega }_j^{\frac{2\alpha }{\alpha +1}}\left({(\frac{\beta }{1+\alpha })}^{\frac{2\alpha }{\alpha +1}}{|y_j|}^{2\alpha }+{(\frac{\gamma }{2})}^{\frac{2\alpha }{\alpha +1}}|y_j{|}^{\frac{4\alpha }{\alpha +1}}\right).\hfill \end{array}$$

(13)

Consider that $\frac{v^T\left(\mathrm{diag}(\omega )L+L^T\mathrm{diag}(\omega )\right)v}{2}$ is a non-negative continuous function defined on ${\mathbb{R}}^N$, and $\frac{v^T\left(\mathrm{diag}(\omega )L+L^T\mathrm{diag}(\omega )\right)v}{2}\ne 0$ for $v\in \mathcal{U}=\{\beta si{g}^{\alpha }(v)+\gamma v|v\perp \omega ,v\in {\mathbb{R}}^N\}\cap S^{N-1}$. In addition, $\mathcal{U}$ is a bounded closed set; therefore, the minimal value

$$M_1={\displaystyle \underset{v\in \mathcal{U}}{min}}(\frac{v^T\left(\mathrm{diag}(\omega )L+L^T\mathrm{diag}(\omega )\right)v}{2})\ne 0$$

exists. Then, for y belonging to $\mathcal{U}$, there is

$$\begin{array}{c}\hfill \frac{{\left(\beta si{g}^{\alpha }(y)+\gamma y\right)}^{T}diag({\omega }_1,\dots ,{\omega }_{n_2})L\left(\beta si{g}^{\alpha }(y)+\gamma y\right)}{{\left(\beta si{g}^{\alpha }(y)+\gamma y\right)}^T\left(\beta si{g}^{\alpha }(y)+\gamma y\right)}\ge M_1.\end{array}$$

(14)

This leads to

$$\begin{array}{c}\hfill -{\dot{V}}_2(t)\ge M_1{\displaystyle \sum _{j=1}^{n_2}}{\left(\beta si{g}^{\alpha }(y_j)+\gamma y_j\right)}^2.\end{array}$$

(15)

Consider the case when $0<|y|\le 1$, there are

$$\begin{array}{cc}\hfill V_2^{\frac{2\alpha }{\alpha +1}}(t)\le & {\displaystyle \underset{\ell \in \mathcal{I}}{max}}({\omega }_{\ell }^{\frac{2\alpha }{\alpha +1}})\left({(\frac{\beta }{1+\alpha })}^{\frac{2\alpha }{\alpha +1}}+{(\frac{\gamma }{2})}^{\frac{2\alpha }{\alpha +1}}\right){\displaystyle \sum _{j=1}^{n_2}}{|y_j|}^{2\alpha }\hfill \\ \hfill \le & {\displaystyle \underset{\ell \in \mathcal{I}}{max}}({\omega }_{\ell }^{\frac{2\alpha }{\alpha +1}})\left({(\frac{\beta }{1+\alpha })}^{\frac{2\alpha }{\alpha +1}}+{(\frac{\gamma }{2})}^{\frac{2\alpha }{\alpha +1}}\right)\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill -{\dot{V}}_2(t)\ge M_1(\beta +\gamma ).\end{array}$$

(17)

Thus,

$$\begin{array}{cc}\hfill & -\frac{{\dot{V}}_2(t)}{V_2^{\frac{2\alpha }{\alpha +1}}(t)}\ge \frac{M_1(\beta +\gamma )}{{\displaystyle \underset{\ell \in \mathcal{I}}{max}}({\omega }_{\ell }^{\frac{2\alpha }{\alpha +1}})\left({(\frac{\beta }{1+\alpha })}^{\frac{2\alpha }{\alpha +1}}+{(\frac{\gamma }{2})}^{\frac{2\alpha }{\alpha +1}}\right)}\doteq C_1.\hfill \end{array}$$

(18)

For the other case $1<|y|\le \delta$, consider the function $\frac{{\displaystyle \sum _{j=1}^{n_2}}{\displaystyle \sum _{\ell =1}^{n_2}}{l}_{j\ell }{\omega }_j\left(\beta si{g}^{\alpha }(v_{\ell })+\gamma v_{\ell }\right)\left(\beta si{g}^{\alpha }(v_j)+\gamma v_j\right)}{{\left({\displaystyle \sum _{j=1}^{n_2}}{\omega }_j\left(\frac{\beta }{1+\alpha }{|v_j|}^{1+\alpha }+\frac{\gamma }{2}{v}_j^2\right)\right)}^{\frac{2\alpha }{\alpha +1}}}$ with respect to the variable $v=\left(\begin{array}{c}{v}_1,\dots ,v_{n_2}\end{array}\right)$, which is continuous on the compact set $\mathcal{O}=\{v:1\le v^{T}v\le \delta \}$; therefore, there exists a minimal value

$C_2={\displaystyle \underset{1<|v|\le \delta }{min}}\left(\frac{{\displaystyle \sum _{j=1}^{n_2}}{\displaystyle \sum _{\ell =1}^{n_2}}{l}_{j\ell }{\omega }_j\left(\beta si{g}^{\alpha }(v_{\ell })+\gamma v_{\ell }\right)\left(\beta si{g}^{\alpha }(v_j)+\gamma v_j\right)}{{\left({\displaystyle \sum _{j=1}^{n_2}}{\omega }_j\left(\frac{\beta }{1+\alpha }{|v_j|}^{1+\alpha }+\frac{\gamma }{2}{v}_j^2\right)\right)}^{\frac{2\alpha }{\alpha +1}}}\right)$ such that

$$\begin{array}{cc}\hfill & -\frac{{\dot{V}}_2(t)}{V_2^{\frac{2\alpha }{\alpha +1}}(t)}\ge C_2.\hfill \end{array}$$

(19)

As a result of the above, we can obtain $-\frac{{\dot{V}}_2(t)}{V_2^{\frac{2\alpha }{\alpha +1}}(t)}\ge C,$ where $C=max\{C_1,C_2\}$.

Based on the Differential Comparison Principle, $V_2(t)$ will reach 0 in finite time $t=\frac{(1+\alpha )V_2^{\frac{1-\alpha }{1+\alpha }}(0)}{(1-\alpha )C}$.

The above proof process is also applicable when the case of $p>2$ is considered. In this case, the first $p-1$ groups are considered as a whole and noted as ${\mathcal{G}}_{(1,\dots ,p-1)}$; then, ${\mathcal{G}}_{(1,\dots ,p-1)}$, ${\mathcal{G}}_p$ is reduced to the same case as the previous ${\mathcal{G}}_1$, ${\mathcal{G}}_2$. Moreover, it is noted that the non-negative vector $\omega$ is in $\mathbb{R}{\pi }_1⨁\mathbb{R}{\pi }_2⨁\dots ⨁\mathbb{R}{\pi }_{p-1}$. Finally, the complete proof of this theorem is given by mathematical induction.   □

Theorem 7.

Consider the MAS described in Theorem 6; if $\mathcal{G}$ is a signed directed graph satisfying Assumption 4, then this MAS reaches bipartite agreement with p groups, respectively, in finite time.

Proof. 

This theorem can be proved similarly by simply taking the Lyapunov function in the proof of Theorem 6 as $V(t)=y^T\mathrm{diag}(\Phi w)(\frac{\beta }{1+\alpha }si{g}^{\alpha }(y)+\frac{\gamma }{2}y).$   □

Remark 7.

The network topology of acyclic partition is more applicable to practical scenes because of its graded characteristics. This includes hierarchical management of agents, production of assembly line, and unmanned vessel formation [25]. Thus, it should be aware of the significance of its related research. This article takes advantage of the algebraic condition of acyclic partition to reduce a kind of GC control problem to a complete consensus problem and, thus, obtains the results of reaching orderly GC in finite time for the research work of MAS.

4. Simulation

This section first gives the ordered performances of the finite time GC control under an acyclic partition topology. Moreover, a set of simulations for the finite time GBC are given under the acyclic partition topology with cooperative–competitive interaction. The third example finally presents comparisons of two cases of the non-acyclic partition with a case of the acyclic partition.

4.1. Simulation Examples for Reaching GC in Finite Time

This example considers a digraph of 18 nodes divided into four groups, represented by Figure 1. Obviously, there are edges with positive and negative weights connecting the four groups and making Figure 1 connected. Moreover, these edges between groups are directed and no-loop, i.e., there is an acyclic partition of Figure 1. The first simulation example in this section is about the finite time GC agreement of 18 agents in a networked topology relating to Figure 1. To solve for the state volume of the simulated MAS, the initial state volume is now taken to be $x(0)=(11,-2,3,1.5,$$-15,11,-3.5,4,-10.9,-21,17.5,6,-18,22,-27,$$36,-45,35).$ Three different values for parameter $\alpha$ are chosen below to fully outline the reliability of the simulation example.

First, the evolution of the states $x(t)$ for 18 agents at $\alpha =0.25$ is given by Figure 3. From Figure 3, three aspects of messages can be obtained.

One is that the state volume of each agent is finally converged, which means that the MAS reaches the stable state. Secondly, the state volumes of all the agents converge to four different values, i.e., the MAS reaches stability in four groups. Thirdly, the state volumes of four groups converge to each convergence value in partition order, which means that the MAS reaches the groups’ stability in order, and the order is related to the partition.

Second, when $\alpha =0.50$, Figure 4 gives the evolution of the state volumes of the agents in this MAS.

In this case, the convergence process is consistent with the case of $\alpha =0.25$. In addition, it is necessary to further compare the results of Figure 3 with those of Figure 4, despite the convergence values of each group of state volumes in Figure 3 and Figure 4 being equal. However, it is worth noting that each convergence time given in Figure 4 is less than that in Figure 3.

Finally, considering the above case, this example further considers the case of the control parameter $\alpha =0.75$. Correspondingly, Figure 5 represents the evolution process with regard to the state volumes of all the agents. It can be found from these three figures that the change of the finite time control parameter $\alpha$ does not change the final convergence value of the system state volumes. However, by comparing Figure 4 with Figure 5, it can be seen that the convergence time of each group given in Figure 5 is less than that of each group in Figure 4. With these two sets of comparison results, it can be inferred that as the value of parameter $\alpha$ converges to 1, the convergence values of the state volumes remain constant but the convergence time decreases.

4.2. Simulation Examples for Reaching GBC in Finite Time

Consider a directed graph about 18 nodes represented by Figure 2, where all nodes are divided into four groups, and the groups are connected by directed edges with positive and negative weights. Different from the last simulation example, there are positive and negative weights on the edges of each group. Actually, the sub-graph consisting of nodes in each group is a structurally balanced digraph. The second simulation example of this section will be given below, which is the finite-time GBC agreement with the networked topology represented by Figure 2 for the 18 agents. In this simulation, the initial state volumes are set to $x(0)=(11,-2,3,1.5,$$-15,11,-3.5,4,-10.9,-21,17.5,6,-18,22,-27,36,-45,35)$ and the parameter is taken to be $\alpha =0.75$.

Correspondingly, Figure 6 gives the results of this simulation. Observing the convergence process regarding the states of all the agents given in Figure 6, the following observations can be made. Similar to the first simulation example, the state volumes of the systems reach stabilization in sequence according to the order of partition. In fact, the state volumes finally reached by each group are symmetric about the origin, thus making the state volumes of all agents in the four groups converge to eight different values.

4.3. MAS Consensus Is Reachable under the Group Partition Topology

To further discuss the relationship between the agreement of the finite-time control protocols and the group partition topology, three kinds of finite-time control under the group partition topology of MAS will be considered. Therefore, this simulation example considers the networked topology inscribed by Figure 7, Figure 8 and Figure 9. The topology graph given by Figure 7 is an acyclic partition graph about six nodes divided into two groups; $\{v_1,v_2,v_3\}$ is the set of nodes in the first group, and the second group includes $v_4,v_5,v_6$. The information interaction is directed from the first group to the second group. Compared with Figure 7, both Figure 8 and Figure 9 describe a kind of balanced coupling topology with six nodes, All nodes are also divided into two groups and the group of all nodes is the same as in Figure 7. However, note that in both cases, the links between the groups are no longer acyclic. In other words, they have both directed edges from the first group to the second group and directed edges from the second group to the first group.

For the simulation under these three kinds of topology, given that the parameter takes a value of $\alpha =0.25$, the initial state volumes of the agents are $x(0)=(6.8,-2,2.3,-4.5,-1,-5)$. Figure 10 gives the simulation results of six agents under the topology of Figure 7. Figure 11 and Figure 12 show the cases of six first-order linear systems related to Figure 8 and Figure 9, respectively. From the results of Figure 10, six agents reach GC in two groups sequentially in finite time. Figure 11 gives the finite time GC of six agents, in which it is not easy to distinguish the order. However, Figure 12 illustrates that the MAS fails to reach GC. Combining these three examples above, it can be inferred that the consensus of MAS is reachable under the group partition topology, such as for the acyclic partition topology. However, it is not sufficient for the MAS to reach consensus.

5. Conclusions

In this paper, the fact that the acyclic partition topology eliminates the integral influence among the groups of networks was the key to enable the agent systems to achieve consensus in finite time. As an additional conclusion, the finite time control protocols were further applicable to the group-bipartite interaction structure. According to the experimental part of the simulation, the results were consistent with the above theoretical results. In addition, through simulation experiments, this paper finally obtained the performance of MAS with acyclic division topology with ordered group arrivals under the finite-time control protocol. Obviously, these intuitive motion performances of the agents in the MAS made the control protocol given in this paper practically meaningful. With these conclusions, it is not difficult to surmise that fixed-time GC as well as fixed-time GBC are solvable problems. However, in future work, it will be more meaningful to propose well-defined formations of network structures that are close to real problems, such as to the swarm collaboration problem.