# Multiconsensus of Nonlinear Multiagent Systems with Intermittent Communication

Authors: Jie Chen; Guang-Hui Xu; Liang Geng
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Compared with single consensus, the multiconsensus of multiagent systems with nonlinear dynamics can reflect some real-world cases. This paper proposes a novel distributed law based only on intermittent relative information to achieve the multiconsensus. By constructing an appropriate Lyapunov function, sufficient conditions on control parameters are derived to undertake the reliability of closed-loop dynamics. Ultimately, the availability of results is completely validated by these numerical examples.


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## Body

## 1. Introduction

Multiagent systems have attracted much attention in the field as computer science, vehicle systems, unmanned aerial vehicles, or formation flight of spacecraft since 2009. In most of studies on consensus problem, researchers always adopt the same method that makes all agents finally reach the same value in systems [1–5]. Due to various parameters as sudden changes on environment or cooperative tasks in the reality, the purpose of our research also becomes multiple.In [6], the consensus problem from asynchronous group of the discrete-time heterogeneous multiagent system under dynamic-change interaction topology is discussed and studied briefly. For different agents, two asynchronous consensus protocols are given in this paper. Based on fixed and switching topology, the multiconsensus of first-order multiagent system is discussed. And we assumed that interactions could reach balance between two subnetworks in [6, 7]. The relationship between multiconsensus ability and the underlying digraph topology is discussed in [8]. For fixed communication networks, the author proposes a consensus protocol which can be applied to two different second-order multiagent systems under the same assumption [9]. By using pinning control method in [10–12], we can obtain some criteria on multiconsensus of networks without assuming the balance of network topology. For multiconsensus problem from discrete-time multiagent systems with stochastic and fixed topologies, the author performs a professional study in [13]. In [14], the author performs the research on multiconsensus control of switching-directed interaction and fixed topology in common linear multiagent systems, based on pinning control techniques, matrix analysis theory, and Lyapunov stability theory. The cluster consensus of multiagent dynamical systems with impulsive effects and coupling delays was investigated in [15], where interactions among agents were uncertain.In other words, above conclusion on the consensus of multiagent system with nonlinear dynamics is mostly based on common assumption that information is transmitted continuously between all agents, which means that each single agent shares the information with its neighbors without any communication constraints. However, this is not the fact in reality. For instance, all agents can only get the information from their neighbors during certain disconnected time intervals, as a reason of communication restrictions. In [16–23], intermittent control has attracted more attention. For distributed consensus problem from intermittent control of a time-invariant undirected communication topology in the linear multiagent system, the author designs a type of distributed-observer protocols. In [24], the consensus problem on periodical-intermittent control of second-order agent networks, which is based on matrix theory, Lyapunov control method, and algebraic graph theory, is discussed. Based on the above discussion and research, this paper mainly studies on some characteristics of the second-order multiagent system, such as multiconsensus. In the same subnet, nonlinear dynamics of all agents are the same, while all agents in different subnet have different dynamics. Multiconsensus indicates that all agents in every subgroup can be consistent. Between different groups, there is no consistent value. The research performed in this paper can be summarized as three points: firstly, the multiconsensus of the multiagent system with nonlinear dynamics is studied. Secondly, a novel multiconsensus law, which is devised through intermittent and relative state information, is more general than other second-order multiconsensus protocol. And, in such a protocol, all agents always need to communicate with their neighbors.In Section2 of this paper, the research model has been designed. In Section 3, we perform the study and discussion on the multiconsensus problem of second-order multiagent systems with nonlinear dynamics. In Section 4, two numerical examples are given to prove the effectiveness on the designed protocol. And the conclusion is summarized in Section 5.


## 2. Preliminary

### 2.1. Algebraic Graph Theory

In general, the communication topology between agents in a multiagent system is described by a directed graph. LetG=V,ε,A be a system communication topology diagram consisting of N nodes, the vertex set V=ν1,ν2,L,νn is nonempty finite, the edge set εÍV×V, and a nonsymmetric A=aijn×n is nonnegative weighted adjacency matrix. A=aijn×n is defined as aij≠0 if εij∈ε and aij=0 otherwise. There are no self-loops, i.e., aii=0. The set of neighbors of agent i is denoted by Ni=νjεijÎε. A directed path is a sequence of distinct vertices 1,2,…,r such that νi,νj∈ε, i=1,2,…,r−1.


### 2.2. Problem Description

Consider a multiagent system withn agents, ν=1,2,…,N. Suppose the multiagent system composed with p subgroup, νp is a set of p subgroup. Note that the corresponding subtopology graph of each subgroup is Gp, and the topology diagram of the whole system is G. The corresponding numbering sets for each subgroup are ν1=1,…,l1, ν2=l1,…,l1+l2, νp=l1+l2+⋯+lp−1,…,N, and l1+l2+⋯+lp−1+lp=N. νi≠ϕ,∪i=1pνi=ν, and νi∩νj≠ϕ for i≠j. For i∈ν, let i¯ denote the subscript of the subset to which the integer i belongs. It is assumed in what follows that each agent knows which cluster it belongs to.The dynamics of systems is described as follows:(1)q¨it=fi¯qi,q˙i,t+uit,where qit∈Rn, q˙it∈Rn, and uit∈Rn are the position, velocity, and control input of agent i, respectively. The function fi¯qi,q˙i,t∈Rn, describing the intrinsic dynamics of agent i, is continuously differentiable.In this work, the leader in each group is described by(2)q¨i¯t=fi¯qi¯,q˙i¯,t.Definition 1.
The multiconsensus control for second-order multiagent systems is said to be achieved if(3)limt⟶∞qit−qi¯t=0,limt⟶∞q˙it−q˙i¯t=0.
Firstly, some basic assumptions and lemmas are given as follows:Lemma 1.
There is a constantω∈R, and P,Q,M,N are matrices with suitable dimensions. Then, the Kronecker product has the following properties:(4)1P⊗QT=PT⊗QT,2ωP⊗Q=P⊗ωQ,3P+Q⊗M=P⊗M+Q⊗M,4P⊗QM⊗N=PM⊗QN.Lemma 2.
The linear matrix inequality(5)AsBsBTsCs>0,where As=ATs,Cs=CTs, and Bs depend affinely on s, which is equivalent to Cs>0,As−BsC−1sBTs>0.Lemma 3.
Suppose thatS∈Rn×n is positive definite and D∈Rn×n is symmetric. Then, for ∀x∈Rn, the following inequality holds:(6)λminS−1DxTSx≤xTDx≤λmaxS−1DxTSx.Assumption 1.
(1)
∑j=N+1N+Maij=0 for all i∈l1, ∑j=1Naij=0 for all i∈l2(2)
The subgraphG1 and G2 have a directed spanning tree, respectivelyAssumption 2.
There exist nonnegative constantsp and q such that nonlinear function satisfies the following equality:(7)fi¯xi,vi,t−fi¯yi,zi,t≤pi¯xi−yi2+qi¯vi−zi2,where x,y,v,z∈Rn,∀t≥0.


## 2.1. Algebraic Graph Theory

In general, the communication topology between agents in a multiagent system is described by a directed graph. LetG=V,ε,A be a system communication topology diagram consisting of N nodes, the vertex set V=ν1,ν2,L,νn is nonempty finite, the edge set εÍV×V, and a nonsymmetric A=aijn×n is nonnegative weighted adjacency matrix. A=aijn×n is defined as aij≠0 if εij∈ε and aij=0 otherwise. There are no self-loops, i.e., aii=0. The set of neighbors of agent i is denoted by Ni=νjεijÎε. A directed path is a sequence of distinct vertices 1,2,…,r such that νi,νj∈ε, i=1,2,…,r−1.


## 2.2. Problem Description

Consider a multiagent system withn agents, ν=1,2,…,N. Suppose the multiagent system composed with p subgroup, νp is a set of p subgroup. Note that the corresponding subtopology graph of each subgroup is Gp, and the topology diagram of the whole system is G. The corresponding numbering sets for each subgroup are ν1=1,…,l1, ν2=l1,…,l1+l2, νp=l1+l2+⋯+lp−1,…,N, and l1+l2+⋯+lp−1+lp=N. νi≠ϕ,∪i=1pνi=ν, and νi∩νj≠ϕ for i≠j. For i∈ν, let i¯ denote the subscript of the subset to which the integer i belongs. It is assumed in what follows that each agent knows which cluster it belongs to.The dynamics of systems is described as follows:(1)q¨it=fi¯qi,q˙i,t+uit,where qit∈Rn, q˙it∈Rn, and uit∈Rn are the position, velocity, and control input of agent i, respectively. The function fi¯qi,q˙i,t∈Rn, describing the intrinsic dynamics of agent i, is continuously differentiable.In this work, the leader in each group is described by(2)q¨i¯t=fi¯qi¯,q˙i¯,t.Definition 1.
The multiconsensus control for second-order multiagent systems is said to be achieved if(3)limt⟶∞qit−qi¯t=0,limt⟶∞q˙it−q˙i¯t=0.
Firstly, some basic assumptions and lemmas are given as follows:Lemma 1.
There is a constantω∈R, and P,Q,M,N are matrices with suitable dimensions. Then, the Kronecker product has the following properties:(4)1P⊗QT=PT⊗QT,2ωP⊗Q=P⊗ωQ,3P+Q⊗M=P⊗M+Q⊗M,4P⊗QM⊗N=PM⊗QN.Lemma 2.
The linear matrix inequality(5)AsBsBTsCs>0,where As=ATs,Cs=CTs, and Bs depend affinely on s, which is equivalent to Cs>0,As−BsC−1sBTs>0.Lemma 3.
Suppose thatS∈Rn×n is positive definite and D∈Rn×n is symmetric. Then, for ∀x∈Rn, the following inequality holds:(6)λminS−1DxTSx≤xTDx≤λmaxS−1DxTSx.Assumption 1.
(1)
∑j=N+1N+Maij=0 for all i∈l1, ∑j=1Naij=0 for all i∈l2(2)
The subgraphG1 and G2 have a directed spanning tree, respectivelyAssumption 2.
There exist nonnegative constantsp and q such that nonlinear function satisfies the following equality:(7)fi¯xi,vi,t−fi¯yi,zi,t≤pi¯xi−yi2+qi¯vi−zi2,where x,y,v,z∈Rn,∀t≥0.


## 3. Main Results

In this section, the multiconsensus of multiagent system is analyzed.The distributed feedback controller of agenti is designed as(8)uit=α∑j=1Naijqj−qi+q˙j−q˙i−αdiqi−qi¯+q˙i−q˙i¯,t∈kH,kH+δ,uit=0,t∈kH+δ,k+1H,where H>0 is the control period and δ>0 is called the control time width.Letq˜it=qit−qi¯tandq˙˜it=q˙it−q˙i¯t are the measurement error of position and velocity of the ith agent.Note that ifj∈VGk, then j¯=k and q˜jt=qjt−qk¯t. We then observe(9)∑j=1Naijqjt−qit=−∑k=1p∑j∈VGklijqjt=−∑k=1p∑j∈VGklijqjt−qk¯t+qk¯t=−∑j=1Nlijx˜jt−∑k=1p∑j∈VGklijqk¯t∑j=1Naijx˜jt−q˜it.Obviously,∑j=1Naijq˙jt−q˙it=∑j=1Naijq˙˜jt−q˙˜it, which in turn together (1) and (2) yields the conclusion(10)q¨˜it=fi¯qi,q˙i,t−fi¯qi¯,q˙i¯,t,−α∑j∈Vaijq˜jt−q˜it+q˙˜jt−q˙˜it,−αdiq˜it+q˙˜it,t∈kH,kH+δ,q¨˜it=fi¯qi,q˙i,t−fi¯qi¯,q˙i¯,t,t∈kH+δ,k+1T.Defineq˜t=q˜1Tt,q˜2Tt,…,q˜NTtT and q˙˜t=q˙˜1Tt,q˙˜2Tt,…,q˙˜NTtT.Then, system (10) can be rewritten as(11)q¨˜t=Fi¯qt,q˙t,t−F˜i¯qi¯t,q˙i¯t,t,−αL+D⊗Inq˜t+q˙˜t,t∈kH,kH+δ,q¨˜it=Fi¯qt,q˙t,t−F˜i¯qi¯t,q˙i¯t,t,t∈kH+δ,k+1T.where Fqt,q˙t,t=fi¯Hq1t,q˙1t,t,…,fi¯HqNt,q˙Nt,tH,F˜q¯t,q˙¯t,t=fi¯Hq1¯t,q˙1¯t,t,…,fi¯HqN¯t,q˙N¯t,tH,D=diagd1,…,dN.Theorem 1.
Under Assumption1 and Assumption 2, system (1) can reach the multiconsensus if the parameters meet the following conditions:(12)ρIN−α2L¯−αD<0,(13)δ>γγ+ηT,where ρ=maxp,q+1, p=maxpi¯, q=maxqi¯, γ=2λmaxΩ−1M, η=λminQ/λmaxΩ, Ω=αL¯+2αDINININ, Q=αL¯/2+αD−pIN0N0NαL¯/2+αD−q+1IN, M=pINα/2L¯+αDα/2L¯+αDq+1IN, and L¯=L+LT.Proof.
It follows from (12) that(14)αL¯2+αD−ρIN>0,αL¯+2αDINININ>0.
Definite the Lyapunov function for system (12):(15)Vt=12y˜TtΩ⊗Iny˜t,where Ω=αL¯+2αDINININ and y˜t=q˜Tt,q˙˜TtT.
In view of (8), we know that Lyapunov function (15) satisfies Vt≥0 and Vt=0 if and only if q˜t=q˙˜t=0.(16)Vt=12q˜TαL¯+2αD⊗Inq˜+q˜Tq˙˜+12q˙˜Tq˙˜.
Fort∈kH,t0+kH+δ, the time derivative of Vt along the trajectories of system (11) gives(17)V˙t=q˜TαL¯+2αD⊗Inq˜⋅+q˜⋅Tq˙˜+q˜Tq¨˜+q˙˜Tq¨˜=−q˜TαL+αD⊗Inq˜+q˙˜TIN−αL−αD⊗Inq˙˜+q˜+q˙˜Fi¯qt,q˙t,t−F˜i¯q¯t,q˙¯t,t.
Then, by (A1), we get(18)q˜T+q˙˜TFi¯qt,q˙t,t−F˜i¯q¯t,q˙¯t,t=∑i=1Nq˜it+q˙˜itTfi¯qi,q˙i,t−fi¯qi¯,q˙i¯,t=∑i=1Npi¯q˜it2+qi¯q˙˜it2≤q˜TpIN⊗Inq˜+q˙˜TqIN⊗Inq˙˜.
It follows from (17) and (18) that(19)V˙t≤−q˜TαL¯2+αD⊗Inq˜+q˙˜TqIN⊗Inq˙˜q˙˜TIN−αL¯2−αD⊗Inq˙˜+q˜TpIN⊗Inq˜=q˜T−αL¯2−αD+pIN⊗Inq˜+q˙˜T−αL¯2−αD+q+1IN⊗Inq˙˜=−y˜TQ⊗Iny˜,where Q=αL¯/2+αD−pIN0N0NαL¯/2+αD−q+1IN. In view of condition (12), we can get Q>0. It is known that Lyapunov function satisfies that(20)λminΩy˜t2≤Vt≤λmaxΩy˜t2.
Thus, one has(21)V˙t≤−λminQλmaxΩVt=−ηVt,where η=λminQ/λmaxΩ.
Fort∈kH+δ,k+1T, taking the time derivative of Vt along the trajectories of system (11)(22)V˙t=q˜TαL¯+2αD⊗Inq˜⋅+q˜⋅Hq˙˜+q˜q¨˜+q¨˜Tq¨˜≤y˜TpINα2L¯+αDα2L¯+αDq+1IN⊗Iny˜≤y˜TM⊗Iny˜≤2λmaxΩ−1MVt=γVt,where M=pINα/2L¯+αDα/2L¯+αDq+1IN and γ=2λmaxΩ−1M.
Then, by the above differential inequality (21) and (22), we have the following results:(1)
For0<t<δ, Vt≤V0e−ηt and Vδ≤V0e−ηt(2)
Forδ≤t<H, Vt≤Vδeγt−δ≤V0eγt−δ−ηδ and VH≤V0eγH−δ−ηδ(3)
H≤t<H+δ, Vt≤VHe−ηt−H≤V0eγH−δ−ηδ−ηt−H, and VH+δ≤V0eγH−δ−2ηδ(4)
H+δ≤t<2H,Vt≤VH+δeγt−TH−δ≤V0eγH−δ+γt−H−δ−2ηδ and V2H≤V0eγH−δ+γH−δ−2ηδ=V0e2γH−δ−2ηδ
Thus, one has
ForkH≤t≤kH+δ,(23)Vt≤VkHexp−ηt−kH≤V0expkγH−δ−kηδ−ηt−kH≤V0exp−ηδ−γH−δHt−δexpH−δHγδ≤V0exp−ηδ−γH−δHt−δexpγH−δ.
ForkH+δ≤t<k+1H,(24)Vt≤VkH+δeγt−kH−δ≤V0expk+1γH−δ−k+1ηδ≤V0exp−ηδ−γH−δHt−δexpγH−δ.
Ifδ>γ/γ+ηH, then defining ξ=ηδ−H−δγ/H>0. Let K=V0expηδ−γH−δ/HδexpγH−δ, and from the above analysis, we can draw the following conclusions:(25)Vt≤Ke−ηδ−γH−δ/Ht.
Therefore, the multiconsensus can be achieved. The proof is completed.Remark 1.
Under the condition of Theorem1, the multiconsensus of system (1) can be achieved globally exponentially with the presented law.Remark 2.
When each agent has the identical nonlinear function, all the agents have the same virtual leader. The dynamics of the virtual leader can be described as(26)q¨rt=fqr,q˙r,t.
Under the intermittent control, system (1) can reach consensus.Theorem 2.
Under Assumption1 and Assumption 2, system (1) can reach the multiconsensus if the parameter meets the following conditions:(27)1IN−α2L¯−αD<0,2δ>γγ+ηT,where ρ=maxρ1,ρ2+1, γ=2λmaxΩ−1M, η=λminQ/λmaxΩ, L¯=L+LT, Ω=αL¯+2αDINININ, Q=αL¯/2+αD0N0NαL¯/2+αD−IN, and M=0α/2L¯+αDα/2L¯+αDIN.
Whenfqi,q˙i,t=0, the dynamics of the ith agent can be described as follows:(28)q¨it=uit.
Then, (28) shows that the velocity of virtual leader is constant. Under protocol (8), let q˜it=qit−qi¯t,q˙˜it=q˙it−q˙i¯t, then the closed-loop system (28) becomes:(29)q¨˜it=−α∑j∈Vaijq˜jt−q˜it+q˙˜jt−q˙˜it−αdiq˜it+q˙˜it,t∈kH,kH+δ,q¨˜it=0,t∈kH+δ,k+1H.where i=1,2,…,N. System (29) can be rewritten as(30)q¨˜t=−αL+D⊗Inq˜t+q˙˜t,t∈kH,kH+δ,q¨˜t=0,t∈kH+δ,k+1H.
The proof of this part is similar to that of Theorem1, which is omitted here.


## 4. Numerical Simulations

In this part, the effectiveness of presented theoretical results has been proven by numerical examples. The multiconsensus of second-order agent systems has been studied. Actually, seven agents can form a second-order system which is split into two clustersν1=1,2,3 and ν2=4,5,6,7. If i∈ν1, then i^=1; and if i∈ν2, then i^=2.The Laplacian matrix is given as follows:(31)L=3−30100−1−23−1−1001−10100001−10200−2000−11000000−110−110−20−13,D=diag3,0,0,3,0,0,0.Case 1.
Setf1xi,vi,t=−xi and f2xi,vi,t=−3vi. By Assumption 2, p1=1,q1=0, and p2=0,q2=3. Then, ρ=4. In view of condition (12), we can choose α=15. By condition (13), we choose δ=0.08. The initial position and velocity of the virtual leader are given as follows:(32)q10,q˙10=0,1,q20,q˙20=1,4.
Figure1 shows the states and velocity of the virtual leaders and followers. It is easy to see that the multiconsensus of systems (1) can be received. The error tracks of the agents are depicted in Figure 2.Figure 1
The trajectories of the agents.
(a)(b)Figure 2
The error trajectories of the agents.
(a)(b)Whenf1¯xi,vi,t=f2¯xi,vi,t=−xi, as shown in Figure 3, the consensus problem of system (1) is indeed solved.Figure 3
The trajectories of the agents.
(a)(b)Case 2.
Letf1¯xi,vi,t=0 and f2¯xi,vi,t=0. The initial velocity of the virtual leader are given as follows: v1¯0=2 and v2¯0=5. The multiconsensus of the systems can be achieved. As a special case, when the initial velocity of the virtual leader is same, the system reaches consensus. The simulation result is shown in Figures 4 and 5, which show that the multiconsensus can be achieved.Figure 4
The trajectories of the agents.
(a)(b)Figure 5
The error trajectories of the agents.
(a)(b)


## 5. Conclusion

In this paper, we have studied on multiconsensus of second-order multiagent systems with nonlinear dynamics. For the realization of multiconsensus, a distributed protocol based on intermittent relative information has been proposed. Some of sufficient conditions have been given to ensure that the states of all agents could reach more consistent values. In this paper, two simulation examples are given and applied to the effectiveness of theoretical results.



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