# Consensus of Heterogeneous Multiagent Systems with Arbitrarily Bounded Communication Delay

Authors: Xue Li; Huai Wu; Yikang Yang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper focuses on the consensus problem of high-order heterogeneous multiagent systems with arbitrarily bounded communication delays. Through the method of nonnegative matrices, we get a sufficient consensus condition for the systems with dynamically changing topology. The results of this paper show, even when there are arbitrarily bounded communication delays in the systems, all agents can reach a consensus no matter whether there are spanning trees for the corresponding communication graphs at any time.


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## Body

## 1. Introduction

In the past few years, consensus problems for multiagent systems have had a significant impact on many fields, including wireless sensor networks, mobile robot formation mission, and formation flying for satellite. Consensus means the outputs of the agents that are spatially distributed can reach a common value. The consensus problems have attracted much attention from academia, and there have been a great number of results investigating the consensus problems for networks of dynamic agents [1–8].In reality, for the influences of the finite speeds of transmission and spreading, multiagent systems are often restricted to communication delays. There have been many works dealing with the study of consensus problems with time-delays [9–17]. For example, [9] investigated the consensus problem with time-delay, obtained the sufficient and necessary condition, and gave the largest tolerable input delay to guarantee the consensus. In [10], Lin and Jia extended the results in [9] to second-order system through the method of linear matrix inequalities. In [11], an upper bound for delay tolerance is obtained for the high-order system when all eigenvalues of each agent are in the closed left half plane. In [12], Zhou and Lin used output feedback protocols to investigate the consensus problem when the time-delays are constant and exactly known even if arbitrarily bounded. The approaches to analyse the consensus problems for multiagent systems fall into three major groups: the Lyapunov functions, the frequency-domain analysis, and the method based on the properties of nonnegative matrices [3, 6, 18–25]. There have been a series of papers highlighting first-order, second-order, high-order, and mixed-order multiagent systems, and great progress has been made in this field. In [3], C.-L. Liu and F. Liu researched the consensus problem of discrete-time heterogeneous multiagent systems composed of first-order agents and second-order agents through the method of nonnegative matrices. In [24], Zheng and Wang proposed two classes of consensus protocols for heterogeneous multiagent systems with and without velocity measurements. However, no clear advancement has so far been seen in the field of high-order heterogeneous multiagent systems.In this paper, we investigate the consensus problem of high-order heterogeneous multiagent systems with arbitrarily bounded time-delays through the method of nonnegative matrices. Under some assumptions, we get a sufficient consensus condition for the multiagent systems with dynamically changing topology and arbitrarily bounded delays.Through this article, we letRm and Z+ represent m-dimensional real vector space and the set of nonnegative integers, respectively. In∈Rn×n is the identity matrix. 1n=[1,1,…,1]T∈Rn. ⊗ is the Kronecker product.


## 2. Graph Theory

A directed graphG is composed of a vertex set V={s1,…,sn}, an arc set ε⊆V×V, and a weighted adjacency matrix A=[aij], denoted by G=(V,ε,A). The node indexes belong to a finite index set L={1,2,…,n}. If there is a directed edge from node si to node sj, then si and sj are called the tail and the head of (si,sj), respectively. The neighbors of agent i are denoted by Ni={sj∈V:si,sj∈ε}. The adjacency elements associated with the edges of the digraph are defined as aii=0 and aij>0 if eij∈ε. The Laplacian matrix of G(V,ε,A) is defined as L=Δ-A, where Δ=[Δij] is a diagonal matrix with Δii=∑j=1naij. By the definition of L we can get L1n=0. A directed path is a sequence of ordered edges of the form (si1,si2),(si2,si3),…, where sij∈v in a directed graph. A directed graph is said to be strongly connected if and only if there is a directed path from every node to every other node. A directed graph has a spanning tree if there is a node such that there exists a directed path from every other node to this node.GivenQ=[qij]∈Rn×r, when all of its elements qij are nonnegative there we say Q is nonnegative and Q≥0. If Q∈Rn×n is nonnegative and it satisfies Q1=1, then Q is stochastic. When a stochastic matrix Q satisfies limk→+∞Qk=1fT, where f∈Rn, then Q is stochastic, indecomposable, and aperiodic (SIA).


## 3. Model

Suppose that discrete-time heterogeneous multiagent system consists of first-order agents, second-order agents, third-order agents, untillth-order agents and the total number of agents is n, where n=n1+n2+⋯+nl, and ni(i=1,2,…,l-1) denotes the quantity of the ith-order agents.Suppose the dynamics of the first-order agents are(1)ξi10k+1=ξi10k+ui1kT,i1=1,2,…,n1,where ξi10∈R is the position, ui1∈R is the control input, and T>0 is the sample time. To solve the agreement problems, the protocol with communication delays is proposed for the first-order agents as follows:(2)ui1k=-∑j∈Ni1ai1jkξi10k-ξj0k-τi1jk.The communication delay 0≤τi1j(k)≤τmax corresponds to the information flow from agent j to agent i1 and ai1j>0,j∈Ni1(k), is the coupling weight chosen from any finite set and Ni1(k) is the neighboring agents of agent i1.Suppose the dynamics of the second-order agents are(3)ξi20k+1=ξi20k+ξi21kT,ξi21k+1=ξi21k+ui2kT,where i2=n1+1,n1+2,…,n1+n2. ξi2(0)∈R is the position, ξi2(1)∈R is the velocity, and ui2∈R is the control input. To solve the agreement problems, the protocol with communication delays is proposed for the second-order agents as follows: (4)ui2k=-p1ξi21k-∑j∈Ni2kai2jkξi20k-ξj0kk-τi2jk.The communication delay 0≤τi2j(k)≤τmax corresponds to the information flow from agent j to agent i2 and ai2j>0,j∈Ni2(k), is the coupling weight chosen from any finite set and Ni2(k) is the neighboring agents of agent i2. p1>0 is control parameter.Similarly, we suppose the dynamics ofmth-order agents are(5)ξim0k+1=ξim0k+ξim1kT⋮ξimm-2k+1=ξimm-2k+ξimm-1kTξimm-1k+1=ξimm-1k+uimkT,where m=3,4,…,l,im=∑z=1m-1nz+1,∑z=1m-1nz+2,…,∑z=1mnz. ξimj∈R is the jth variable of ξim, j=0,1,…,m-1, and uim(k)∈R is the control input. To solve the agreement problems, the protocol with communication delays is proposed for mth agents as follows:(6)uimk=-∑j=1m-1pjξimjk-∑sj∈Nimkaimjkξim0k-ξj0k-τimj.The communication delay 0≤τimj(k)≤τmax corresponds to the information flow from agent j to agent im and aimj>0,j∈Nim(k), is the coupling weight chosen from any finite set and Nim(k) is the neighboring agents of agent im. pj is control parameter and pj>0 for all j=1,2,…,m-1. It is obvious to see that when the states of agents satisfy the following, the high-order heterogeneous multiagent systems in this paper can reach consensus:(7)limk→+∞ξiak-ξjak=0,where i,j∈L, agent i belongs to the mth-order agents, and agent j belongs to the nth-order agents. a is any nonnegative integer which satisfies a<m, a<n.Letξ(k)=[ξ1T(k),ξ2T(k),…,ξlT(k)]T, where ξ1=[ξ1(0),ξ2(0),…,ξn1(0)], ξ2=[ξn1+10,ξn1+11,ξn1+20,ξn1+21,…,ξn1+n20,ξn1+n21],…,ξl=[ξc+10,ξc+11,…,ξc+1l-1,…,ξc+nl0,ξc+nl1,…,ξc+nl(l-1)],  c=∑i=1l-1ni, and(8)A¯k=1T00⋯0001T0⋯00001T⋯00⋮⋮⋮⋮⋱⋮⋮0000⋯1T0-p1T-p2T-p3T⋯-pk-2T1-pk-1T∈Rk×k.Here we set A1=In1 and Ak=diag⁡A¯k,A¯k,…,A¯k∈Rknk×knk, where k=2,3,…,l. So we define A=diag⁡A1,A2,…,Al:(9)Lk=L¯1k-L12k⋯-L1lk-L21kL¯2k⋯-L2lk⋮⋮⋱⋮-Ll1k-Ll2k⋯-L¯lk,where L(k) denotes the Laplacian matrix of the graph G, L¯i(k)=Li(k)+Di(k), Li(k) is the Laplacian matrix of ith-order agents, and Di(k)=diag{∑j∈Nioaij(k),i=1,2,…,c}, c=∑i=1lni. Nio denotes agent i’s neighboring agents except all the agents which are the same order as agent i. Lij(k) are the adjacency relations of ith-order agents to jth-order agents:(10)Bij=00⋯000⋯0⋮⋮⋱⋮T0⋯0∈Ri×j,i,j=1,2,…,l,L¯k=L¯1k⊗B11-L12k⊗B12⋯-L1lk⊗B1l-L21k⊗B21L¯2k⊗B22⋯-L2lk⊗B2l⋮⋮⋱⋮-L1lk⊗Bl1-Ll2k⊗Bl2⋯L¯lk⊗Bll.The network of the multiagent system is(11)ξk+1=A-L0k-C0kξk+C1kξk-1+⋯+Cτmaxkξk-τmax,where L0(k)=diag(L¯(k)) and the ijth element of Cm(k)(m=0,1,…,τmax) is equal to zero or the weight of the edge eij if τij=m. And we can see that L¯k=L0k-∑m=0τmaxCmk.


## 4. Main Results

Assumption 1.
Consider(12)Hm>T,ki1>dmaxT∏n=1i-1Hn,kij>0,1+kii>0,for m=1,2,…,l,i=2,3,…,l, and j=1,2,…,i-1, where dmax denotes the largest entry of all possible L(k).Lemma 2 (see [26]).
LetP1,P2,…,Pk∈Rq×q be a finite set of SIA matrices with the property that, for each sequence Pi1,Pi2,…,Pij of positive length, the matrix product PijPij-1⋯Pi1 is SIA. Then, for each infinite sequence Pi1,Pi2, there exists a vector f∈Rq such that ∏j=1∞Pij=1fT.Theorem 3.
Suppose that there exists an infinite strictly increasingkα with k0=0 and 0<kα+1-kα≤β, α,β∈Z+. If the union of {G(kα),G(kα+1),…,G(kα+1-1)} has a spanning tree, then the agents in systems (1), (3), and (5) converge to a stationary consensus asymptotically.Proof.
Letϕi(k)=Γiξi(k),i=1,2,…,l, and ϕ(k)=[ϕ1T(k),ϕ2T(k),…,ϕnT(k)]:(13)Γii=100⋯01H10⋯01∑j0=12Hj0H1H2⋯0⋮⋮⋮⋱01∑j0=1i-1Hj0∑j0=1l-2∏j1=j0+1i-1Hj0j1⋯∏j0=1i-1Hj0∈Ri×i,i=2,3,…,l.Γ1=In1, and Γi=diag{Γii,Γii,…,Γii}∈R(ni×i)×(ni×i),i=2,3,…,l for some positive constants Hi∈R.
Let(14)Mk=L¯1k⊗F11-L12k⊗F12⋯-L1lk⊗F1l-L21k⊗F21L¯2k⊗F22⋯-A2lk⊗F2l⋮⋮⋱⋮-Ll1k⊗Fl1-Ll2k⊗Fl2⋯L¯lk⊗Fll.Then we can transform system (11) into the following system:(15)ϕk+1=E-M0k-C¯0kϕk+C¯1kϕk-1+⋯+C¯τmaxkϕk-τmax,where Fij=Bij∏k=1i-1Hk,F1j=B1j,i=2,3,…,l and j=1,2,…,l, M0(k)=diag(M(k)), and the ijth element of C¯m(k)(m=0,1,2,…,τmax) is either zero or equal to the weight of the element of mij if τij=m:(16)Eii=1-TH1TH10⋯0001-TH2TH2⋯00⋮⋮⋮⋱⋮⋮000⋯1-THi-1THi-1ki1ki2ki3⋯ki,i-1kii+1∈Ri×i,i=2,3,…,l.E1=In1 and Ei=diag{Eii,Eii,…,Eii}∈R(ni×i)×(ni×i),i=2,3,…,l and E=diag{E1,E2,…,El} for constants kij∈R.
DefineY(k)=[ϕT(k),ϕT(k-1),…,ϕT(k-τmax)]T. Then system (15) can be transformed into(17)Yk+1=ΦkYk,where Φ(k) is defined as follows:(18)Φk=E-M0k+C¯0kC¯1k⋯C¯τmax-1kC¯τmaxkI0⋯000I⋯000⋮⋱⋮⋮00⋯I0.We can see Eii=ΓiiA¯iΓii-1,i=3,4,…,l, and thus(19)ki1,ki2,…,kii+1=1∑j0=1i-1Hj0∑j0=1i-2∏j1=j0+1i-1Hj0j1⋯∏j0=1i-1Hj0A¯i-I+IΓii-1=1∑j0=1i-1Hj0∑j0=1i-2∏j1=j0+1i-1Hj0j1⋯∏j0=1i-1Hj0A¯i-IΓii-1+0i-1T,1=0,1-p1∏m=1i-1HmT,∑m=1i-1Hm-p2∏m=1i-1HmT,∑j0=1i-2∏j1=j0+1i-1Hj0Hj1-p3∏m=1i-1HmT,…,∑j0=1i-1∏m=1,m≠j0i-1Hm-pi-1∏m=1i-1HmTΓii-1+0i-1T,1.
Due to the fact that the elements of the first column ofΓii are all 1, and ΓiiΓii-1=Ii, so we can easily get that only the first row sum of Γii-1 is 1 and the other row sums of Γii-1 are all 0. Thus we can see [ki1,ki2,…,kii]1=0. Considering that all row sums of L(k) are 0, so [E-M(k)]1=1. Considering that M0(k)=M(k)-∑m=0τmaxC¯m(k), so Φ(k)1=1. Under Assumption 1, there is no negative element in Φ(k). So from the above we can get Φ(k) is a stochastic matrix.
There we assume thatαk is the largest positive integer which satisfies kαk≤k for each k≥0. So Yk+1=Φk⋯Φkαk∏α=0αk-1ΔαY0, and Δ(α)=Φ(kα+1-1)Φ(kα+1-2)⋯Φ(kα). Note that kα+1-kα≤η and the union of {G(kα),G(kα+1),…,G(kα+1-1)} has a spanning tree. Under Assumption 1, similar to the proof of lemma 3 of [18], we can get Δ(α) is SIA. And we can see the union of {G(kα),G(kα+1),…,G(kα+j-1)} has spanning trees for some positive integer j. So it is easy to see that ∏l=αα+j-1Δl is also SIA. For 0<kα+1-kα≤η and aij(k) are chosen from a finite set, the number of all possible Δ(j) is finite. Using Lemma 2, we can get that ∏k=0+∞Δk=1fT for some vector f. And each Φ(k) is a stochastic matrix. So(20)limk→+∞Yk+1=limk→∞Φk⋯Φkmk∏m=0mk-1ΔmY0=limk→∞Φk⋯Φkmk1fTY0=1fTY0.For limk→+∞⁡Yk+1=limk→+∞⁡Y(k), we can get limk→+∞⁡ϕi(k)=limk→+∞⁡Γiξik=fTY(0) for all i∈I, which implies that limk→∞⁡ξim(0)(k)=fTY(0) and limk→∞⁡ξimnk=0, for all m=1,2,…,l and n≠0,n=1,2,…,m-1. This completes the proof.Remark 4.
Similar to Proposition 1 of [18], when all parameters Hn,n=1,2,…,l-1, are in the same value, it is to say that Hn=H, and we can always find parameters pm,m=1,2,…,l-1, satisfying Assumption 1 for any given parameters H and kij,i=2,3,…,l,j=1,2,…,l-1.Remark 5.
The existing results on heterogeneous multiagent systems, for example, [3, 24], only considered the systems composed of first-order and second-order agents, while the systems discussed in our paper do not only contain first-order and second-order agents but also contain high-order agents. The agents considered in the existing works contain at most only two variables (position states and velocity states), while the agents in this paper might contain no smaller than three variables (position states, velocity states, etc.). The variables of the agents are coupled in the form of integral. More kinds of agents with more variables might make the complexities of the whole multiagent systems increase in a geometrical rate. The approach of the existing works on the heterogeneous multiagent systems with first-order and second-order agents is based on the decoupling of the two variables and cannot be applied directly to the multiagent systems with high-order agents. Our approach is to introduce a model transformation to decouple different variables in each agent and decouple different kinds of agents so as to use the properties of the nonnegative matrices.


## 5. Simulation Results

In this section, by presenting some numerical simulations, we will verify the validity and correctness of the theoretical scheme. Considering the system is composed of six agents and the initial conditions of them are set randomly. The hollow circles represent the first-order agents, the squares represent the second-order agents, and the solid circles represent the third-order agents. We use the changing topology composed of Ga and Gb, which starts at Ga and switches every2T s to the next state, which is shown in Figure 1. Obviously, the corresponding graphs Ga and Gb have no spanning tree, and the union of Ga∪Gb has a spanning tree. The sample time is T=0.2s. Set τ12=2,τ23=3,τ34=4,τ45=5,τ56=6,τ61=7, p3=p4=2,p51=p61=18, and p52=p62=6, and then it can satisfy Assumption 1. The simulation results of each agent’s third variables trajectories are shown in Figure 2, each agent’s second variables trajectories are shown in Figure 3, and each agent’s first variables trajectories are shown in Figure 4.Figure 1
Two subfigures.
(a)
Topology Ga
(b)
Topology GbFigure 2
Acceleration trajectories of all agents.Figure 3
Velocity trajectories of all agents.Figure 4
Position trajectories of all agents.


## 6. Conclusion

In this paper, we investigate the consensus problem for networks of high-order heterogeneous systems with time-delay. Through the method of the properties of the nonnegative matrices, we obtain a sufficient condition to guarantee the consensus of the directed heterogeneous network with arbitrarily bounded time-delay. Even for the high-order heterogeneous systems with dynamically changing topologies, it is shown that the output of the agents in the systems can reach consensus no matter whether there are spanning trees for the corresponding graphs.



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