Applied Sciences, Vol. 12, Pages 12337: Ordered-Bipartite Consensus of Multi-Agent Systems under Finite Time Control

1. IntroductionNo 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 R be the domain of real numbers and RN be the N-dimensional vector space defined on R. By convention, 1N=(1,1,…,1)T∈RN; diag(v)=diag(v1,v2,…,vn) if v is a column vector v1,v2,…,vnT (or a row vector v1,v2,…,vn) in Rn. Based on these, the following is a brief description of the required basic knowledge.

2.1. Basic Graph TheoryG=(V,E) is a weighted directed graph in the communication network, with the node set V= and the edge set E=, and A=[aij]∈RN×N is the adjacency matrix. In this case, the i-th agent is presented by node vi in the topology network. This means that the i-th agent and node vi are different representations of the same object in its appropriate context. It is always assumed that aii=0, which implies that the given graph G does not include any self-loops. For the directed graph G, aij≠0 if and only if there exists a direct edge from vj to vi, and call G a signed graph if and only if the weights of the edge are either negative or positive. A signed digraph G is defined as structurally balanced if there exists two subsets V1 and V2 such that the node set V can be decomposed by V1∪V2=V and V1∩V2=⊘, and the weights of the edges have the relation 1):

x˙i(t)=ui(t),i=1,2…,N,

(1)

where x∈Rn and u∈Rn denote the state vector and the input vector of agent vi, i=1,2,…,N, respectively.Assumption 1.

Graph G has a acyclic partition .

Under this assumption, Gi is a subgraph of G made up of all the points in Vi 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=A110⋯0A21A22⋯0⋮⋮⋱⋮Ap1Ap2⋯AppN×N,

(2)

Aii represents the adjacency matrix of Gi, and Aij represents the information transmission from Gi to Gj.Remark 1.

It is worth noting that A is the upper triangular block matrix, and Aij=0,i<j means that the information transmission from topology Gi to topology Gj 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 Gi has a spanning tree.

If Gi is only a digraph with non-negative (or non-negative) weights, the eigenvalue 0 of Gi has its right eigenvector 1ni and its left eigenvector ωi=[ωni−1+1,i,ωni−1+2,i,…,ωni−1+ni,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 Gi has a bipartite structure, gauge transformation is needed before solving the problem.Assumption 3.

The row sum of each block ΦiLijΦj 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(Aii),i=1,…,p, L(A) by the way defined in the last subsection. Note that L(Aii) is not necessarily equal to L(A)ii. However, under Assumption 3, there is L(Aii)=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 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.

0isasimpleeigenvalueofLwithmultiplicityp,

(3)

therealpartsofthenonzeroeigenvaluesofLareallpositive.

(4)

Moreover, the left eigenvector belonging to eigenvalue 0 is

π1=(μ1[1],…,μki[1],0,…,0︸n−k1),⋮πi=(ρ1,1[i],…,ρ1,k1[i],…,ρi−1,1[i],…,ρi−1,ki−1[i],μ1[i],…,μki[i],0,…,0︸n−∑j=1ikj),⋮πp=(ρ1,1[p],…,ρ1,k1[p],…,ρp−1,1[p],…,ρp−1,ki−1[p],μ1[p],…,μkp[p]),

where ∑ℓ=1kiμℓ[i]=1 with μℓ[i]≠0 and ∑l=1kjρj,l[i]=0 for j=1,3,…,i−1, and i = 1, 2, …, p.Furthermore, Figure 1 presents a certain acyclic partition of 18 agents, which are divided into 4 groups, G1 (green agents ), G2 (pink agents ), G3 (yellow agents ), and G4 (purple agents ). This clearly describes the interaction communication among the groups—that is, the first group G1 only sends state information to groups G2, G3, and G4, and group G2 just receives information from group G1 and sends information to the other groups G3 and G4, etc.Assumption 4.

Every Gi is structurally balanced.

By Lemma 1, Assumption 4 can be stated equivalently that there exists a block matrix Φ such that ΦiAiiΦi 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 Φ 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=[lij]N×N associated with G is required to be updated as

lij=V and any t≥T.

Furthermore, if there exists one non-trivial Φk(1≤k≤p), i.e., there exists a Φk(1≤k≤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., Φ≠IN),

V i to Vj 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 G1 achieves agreement in finite time. This is immediately followed by proving that both the first group G1 and the second group G2 achieve finite time agreement. Finally, the result of the second step of the proof is extended to G2,…,Gp.

Now, for simplicity, denote that U is a bounded closed set; therefore, the minimal value

M1=minv∈U(vTdiag(ω)L+LTdiag(ω)v2)≠0

exists. Then, for y belonging to U, there is

βsigα(y)+γyTdiag(ω1,…,ωn2)Lβsigα(y)+γyβsigα(y)+γyTβsigα(y)+γy≥M1.

(14)

This leads to

−V˙2(t)≥M1∑j=1n2βsigα(yj)+γyj2.

(15)

Consider the case when 0<|y|≤1, there are

V22αα+1(t)≤maxℓ∈I(ωℓ2αα+1)(β1+α)2αα+1+(γ2)2αα+1∑j=1n2|yj|2α≤maxℓ∈I(ωℓ2αα+1)(β1+α)2αα+1+(γ2)2αα+1

(16)

−V˙2(t)≥M1(β+γ).

(17)

Thus,

−V˙2(t)V22αα+1(t)≥M1(β+γ)maxℓ∈I(ωℓ2αα+1)(β1+α)2αα+1+(γ2)2αα+1≐C1.

(18)

For the other case 1<|y|≤δ, consider the function ∑j=1n2∑ℓ=1n2ljℓωjβsigα(vℓ)+γvℓβsigα(vj)+γvj∑j=1n2ωjβ1+α|vj|1+α+γ2vj22αα+1 with respect to the variable v=v1,…,vn2, which is continuous on the compact set O=; therefore, there exists a minimal value

C2=min1<|v|≤δ∑j=1n2∑ℓ=1n2ljℓωjβsigα(vℓ)+γvℓβsigα(vj)+γvj∑j=1n2ωjβ1+α|vj|1+α+γ2vj22αα+1 such that

−V˙2(t)V22αα+1(t)≥C2.

(19)

As a result of the above, we can obtain −V˙2(t)V22αα+1(t)≥C, where C=max.

Based on the Differential Comparison Principle, V2(t) will reach 0 in finite time t=(1+α)V21−α1+α(0)(1−α)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 G(1,…,p−1); then, G(1,…,p−1), Gp is reduced to the same case as the previous G1, G2. Moreover, it is noted that the non-negative vector ω is in Rπ1⨁Rπ2⨁…⨁Rπp−1. Finally, the complete proof of this theorem is given by mathematical induction.   □

Theorem 7.

Consider the MAS described in Theorem 6; if G is a signed directed graph sati

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