Consensus of Heterogeneous Multiagent Systems with Arbitrarily Bounded Communication Delay

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.


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][2][3][4][5][6][7][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][10][11][12][13][14][15][16][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][19][20][21][22][23][24][25].There have been a series of papers highlighting first-order, secondorder, 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 discretetime heterogeneous multiagent systems composed of firstorder 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 highorder 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 let R  and Z + represent dimensional real vector space and the set of nonnegative integers, respectively.  ∈ R × is the identity matrix.

Graph Theory
A directed graph G is composed of a vertex set V = { 1 , . . .,   }, an arc set  ⊆ V × V, and a weighted adjacency matrix A = [  ], denoted by G = (V, , A).The node indexes belong to a finite index set L = {1, 2, . . ., }.If there is a directed edge from node   to node   , then   and   are called the tail and the head of (  ,   ), respectively.The neighbors of agent  are denoted by   = {  ∈ V: (  ,   ) ∈ }.The adjacency elements associated with the edges of the digraph are defined as   = 0 and   > 0 if   ∈ .The Laplacian matrix of G(V, , A) is defined as  = Δ − A, where Δ = [Δ  ] is a diagonal matrix with Δ  = ∑  =1   .By the definition of  we can get 1  = 0.A directed path is a sequence of ordered edges of the form (  1 ,   2 ), (  2 ,   3 ), . .., where    ∈ 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.
Given  = [  ] ∈ R × , when all of its elements   are nonnegative there we say  is nonnegative and  ≥ 0. If  ∈ R × is nonnegative and it satisfies 1 = 1, then  is stochastic.When a stochastic matrix  satisfies lim →+∞   = 1  , where  ∈ R  , then  is stochastic, indecomposable, and aperiodic (SIA).
Suppose the dynamics of the first-order agents are where   (0) 1 ∈ R is the position,   1 ∈ R is the control input, and  > 0 is the sample time.To solve the agreement problems, the protocol with communication delays is proposed for the first-order agents as follows: The communication delay 0 ≤   1  () ≤  max corresponds to the information flow from agent  to agent  1 and   1  > 0,  ∈   1 (), is the coupling weight chosen from any finite set and   1 () is the neighboring agents of agent  1 .Suppose the dynamics of the second-order agents are where 2 ∈ R is the velocity, and   2 ∈ R is the control input.To solve the agreement problems, the protocol with communication delays is proposed for the second-order agents as follows: The communication delay 0 ≤   2  () ≤  max corresponds to the information flow from agent  to agent  2 and   2  > 0,  ∈   2 (), is the coupling weight chosen from any finite set and Similarly, we suppose the dynamics of th-order agents are where ∈ R is the th variable of   ,  = 0, 1, . . ., −1, and    () ∈ R is the control input.To solve the agreement problems, the protocol with communication delays is proposed for th agents as follows: The communication delay 0 ≤     () ≤  max corresponds to the information flow from agent  to agent   and     > 0,  ∈    (), is the coupling weight chosen from any finite set and    () is the neighboring agents of agent   .  is control parameter and   > 0 for all  = 1, 2, . . .,  − 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: where ,  ∈ L, agent  belongs to the th-order agents, and agent  belongs to the th-order agents. is any nonnegative integer which satisfies  < ,  < .
There we assume that   is the largest positive integer which satisfies    ≤  for each  ≥ 0. So ( + 1) = Φ() ⋅ ⋅ ⋅ Φ(   )∏   −1 =0 Δ()(0), and Δ() = Φ( +1 − 1)Φ( +1 − 2) ⋅ ⋅ ⋅ Φ(  ).Note that  +1 −   ≤  and the union of {G(  ), G(  + 1), . . ., G( +1 − 1)} has a spanning tree.Under Assumption 1, similar to the proof of lemma 3 of [18], we can get Δ() is SIA.And we can see the union of {G(  ), G(  + 1), . . ., G( + − 1)} has spanning trees for some positive integer .So it is easy to see that ∏ +−1 = Δ() is also SIA.For 0 <  +1 −   ≤  and   () are chosen from a finite set, the number of all possible Δ() is finite.Using Lemma 2, we can get that ∏ +∞ =0 Δ() = 1  for some vector .And each Φ() is a stochastic matrix.So For lim →+∞ ( + 1) = lim →+∞ (), we can get lim →+∞   () = lim →+∞ Γ    () =   (0) for all  ∈ , which implies that lim →∞  (0)   () =   (0) and lim →∞  ()   () = 0, for all  = 1, 2, . . .,  and  ̸ = 0,  = 1, 2, . . .,  − 1.This completes the proof.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.

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 every 2 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.

Conclusion
In this paper, we investigate the consensus problem for networks of high-order heterogeneous systems with timedelay.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

Figure 2 :
Figure 2: Acceleration trajectories of all agents.

Figure 3 :
Figure 3: Velocity trajectories of all agents.

Figure 4 :
Figure 4: Position trajectories of all agents.