#### 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.

#### 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 let and represent -dimensional real vector space and the set of nonnegative integers, respectively. is the identity matrix. . is the Kronecker product.

#### 2. Graph Theory

A directed graph is composed of a vertex set , an arc set , and a weighted adjacency matrix , denoted by . The node indexes belong to a finite index set . 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 . The adjacency elements associated with the edges of the digraph are defined as and if . The Laplacian matrix of is defined as , where is a diagonal matrix with . By the definition of we can get . A directed path is a sequence of ordered edges of the form , where 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 , when all of its elements are nonnegative there we say is nonnegative and . If is nonnegative and it satisfies , then is stochastic. When a stochastic matrix satisfies , where , then 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, until th-order agents and the total number of agents is , where , and denotes the quantity of the th-order agents.

Suppose the dynamics of the first-order agents arewhere is the position, is the control input, and 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 corresponds to the information flow from agent to agent and , is the coupling weight chosen from any finite set and is the neighboring agents of agent .

Suppose the dynamics of the second-order agents arewhere . is the position, is the velocity, and 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 corresponds to the information flow from agent to agent and , is the coupling weight chosen from any finite set and is the neighboring agents of agent . is control parameter.

Similarly, we suppose the dynamics of th-order agents arewhere . is the th variable of , , and is the control input. To solve the agreement problems, the protocol with communication delays is proposed for th agents as follows:The communication delay corresponds to the information flow from agent to agent and , is the coupling weight chosen from any finite set and is the neighboring agents of agent . is control parameter and for all . 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 , agent belongs to the th-order agents, and agent belongs to the th-order agents. is any nonnegative integer which satisfies , .

Let , where , , , andHere we set and , where . So we define :where denotes the Laplacian matrix of the graph , , is the Laplacian matrix of th-order agents, and , . denotes agent ’s neighboring agents except all the agents which are the same order as agent . are the adjacency relations of th-order agents to th-order agents:The network of the multiagent system iswhere and the th element of is equal to zero or the weight of the edge if . And we can see that

#### 4. Main Results

*Assumption 1. *Considerfor , and , where denotes the largest entry of all possible .

Lemma 2 (see [26]). *Let be a finite set of SIA matrices with the property that, for each sequence of positive length, the matrix product is SIA. Then, for each infinite sequence , there exists a vector such that .*

Theorem 3. *Suppose that there exists an infinite strictly increasing with and , . If the union of has a spanning tree, then the agents in systems (1), (3), and (5) converge to a stationary consensus asymptotically.*

*Proof. *Let , and :, and for some positive constants .

LetThen we can transform system (11) into the following system:where and , , and the th element of is either zero or equal to the weight of the element of if : and and for constants .

Define . Then system (15) can be transformed intowhere is defined as follows:We can see , and thusDue to the fact that the elements of the first column of are all 1, and , so we can easily get that only the first row sum of is 1 and the other row sums of are all 0. Thus we can see . Considering that all row sums of are 0, so . Considering that , so . Under Assumption 1, there is no negative element in . So from the above we can get is a stochastic matrix.

There we assume that is the largest positive integer which satisfies for each . So , and . Note that and the union of has a spanning tree. Under Assumption 1, similar to the proof of lemma 3 of [18], we can get is SIA. And we can see the union of has spanning trees for some positive integer . So it is easy to see that is also SIA. For and are chosen from a finite set, the number of all possible is finite. Using Lemma 2, we can get that for some vector . And each is a stochastic matrix. SoFor , we can get for all , which implies that and , for all and . This completes the proof.

*Remark 4. *Similar to Proposition 1 of [18], when all parameters , are in the same value, it is to say that , and we can always find parameters , satisfying Assumption 1 for any given parameters and .

*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 every 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 has a spanning tree. The sample time is . Set , , and , 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.

**(a) Topology Ga**

**(b) Topology Gb**

#### 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.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (61203080, 61573082, and 61304155), the National Program 863 of China (2015AA1528), and the National Program 973 of China (613237201406).