#### Abstract

The consensus problem for discrete time second-order multiagent systems with time delay is studied. Some effective methods are presented to deal with consensus problems in discrete time multiagent systems. A necessary and sufficient condition is established to ensure consensus. The convergence rate for reaching consensus is also estimated. It is shown that arbitrary bounded time delay can safely be tolerated. An example is presented to illustrate the theoretical result.

#### 1. Introduction

The study of information flow and interaction among multiple agents in a group plays an important role in understanding the coordinated movements of these agents. As a result, a critical problem for coordinated control is to design appropriate protocols and algorithms such that the group of agents can reach consensus on the shared information in the presence of limited and unreliable information exchange as well as communication time delays.

In multiagent systems, communication time delays between agents are inevitable due to various reasons. For instance, they may be caused by finite signal transmission speeds, traffic congestions, packet losses, and inaccurate sensor measurements. In addition, in practical engineering applications, the agents in multiagent systems transmit sampled information by using sensors or communication network, and the coordination control algorithms are proposed based on the discrete time sampled data to achieve the whole control object. The typical discrete-time consensus control strategy was provided by Jadbabaie et al. [1], which is a simplified Vicsek model [2]. Recently, the consensus analysis of the discrete time first order multiagent systems with or without communication time delays has been studied extensively, see [3–6], to name a few. While it has been realized that modeling more complex practical processes needs the use of double integrator dynamics, as a result, cooperative control for multiple agents with double-integrator dynamics has become an active area of research. Compared with the first-order consensus, Ren and Atkins [7] show that the existence of a directed spanning tree is a necessary rather than a sufficient condition to reach second-order consensus. Therefore, the extension of consensus algorithms from first order to second order is nontrivial. In recent years, more attention has been paid to the consensus problem of multiagent systems with continuous time second-order systems and much progress has been made, some important works include [8–13]. But there has been little attention to the consensus of discrete time second-order systems. In [14], Lin and Jia investigate the consensus of discrete time second order multiagent systems with nonuniform time delays and dynamically changing topologies. A linear consensus protocol is introduced to realize local control strategies for these second-order discrete-time agents. In [15], by using the generalized Nyquist criterion and the Gerschgorin disc theorem, the consensus algorithm with a static leader is proposed to solve the consensus problem of the discrete time second-order multiagent systems with communication delays. In [16], the mean square consentability problem for a network of double-integrator agents with stochastic switching topology is studied. An LMI approach to the design of the consensus protocol is presented. Hence, the consensus problem for discrete time second-order multiagent systems is more important and challenging. The problem becomes more complicated when consensus protocols are extended to systems with time delay.

Motivated by above discussion, in this paper, we consider the consensus problems for discrete time second-order multiagent systems with time delay and provide some effective methods to deal with consensus problems in discrete time multiagent systems.

#### 2. Problem Statement

Let be a set of agents, where . A directed graph will be used to model the interaction topology among these agents. The th vertex represents the th agents . The set of out-neighbors of vertex is denoted by . A path in a digraph is a sequence of distinct nodes such that , . If there exists a path from node to node , we say that is reachable from . If is reachable from all other agents, is said to be globally reachable. A directed tree is a digraph, where every node has exactly one parent except for one node, called the root, which has no parent, and the root has a directed path to every other node. A directed spanning tree of a digraph is a directed tree formed by graph edges that connect all the nodes of the graph. We say that a graph has (or contains) a directed spanning tree if there exists a directed spanning tree that is a subset of the graph. In a digraph , if is a nonempty subset of and for all and , then is said to be closed.

is the adjacency matrix, where denotes the weight of edge and if and only if . Moreover, we assume that for , that is, every agent can use its own instantaneous state information, the same assumption is also taken by [17]. Diagonal matrix is the degree matrix whose diagonal elements are defined by .

The dynamics of agent is described by where the update time instants will be the form , is the initial moment, , the positive real number is the sampled time or time discretization unit, in this paper, we assume that . , denote the position (or angle), velocity (or angular velocity), and control of agent , respectively.

*Definition 2.1. *Second-order consensus in the multiagent systems (2.1) is said to be achieved if for any initial conditions,

#### 3. Consensus Analysis

To solve the consensus problem, we introduce the following neighbor-based feedback control protocol where is a control parameter, is the time delay.

Theorem 3.1. *Under control protocol (3.1), for any bounded time-delay, there exist some such that the consensus for (2.1) is reached asymptotically if and only if the interconnection graph of agents has a globally reachable node.*

Before proving Theorem 3.1, we first need to do model transformation on systems (2.1) under control protocol (3.1) and give some technical lemmas.

Let , , , where , then by (2.1) and (3.1), we have

Since there has a globally reachable node in graph , without loss of generality, we assume the th node is the globally reachable and set:

Let , , then , and

Define , , , .

Noticing that the every row sum of is 1, we have where , is a matrix formed by the first rows and the first columns of matrix , is a row vector formed by the first elements of the th row of matrix .

Then (3.3) can be decoupled as follows:

Let , one can obtain that where

Therefore, the consensus of (2.1) is achieved if and only if as for any initial condition .

Now, we give some useful lemmas for proving Theorem 3.1.

Lemma 3.2 (see [18]). *If a nonnegative matrix has the same positive constant row sums given by , then is an eigenvalue of with an associated eigenvector 1 and , where denotes the spectral radius. In addition, the eigenvalue of has algebraic multiplicity equal to one, if and only if the graph associated with has a spanning tree. *

Lemma 3.3. *Equation (3.7) has a unique equilibrium 0 if the interconnection graph of agents has a globally reachable node. *

*Proof. *It suffices to verify that , that is,
has a unique solution 0 if interconnection graph has a globally reachable node.

By (3.9), it is obvious that , then it is equivalent to prove that has a unique solution , namely, has no eigenvalue 1.

Since the graph associated with has a globally reachable node, by the definition of , we know that the graph associated with has a directed spanning tree. By Lemma 3.2, matrix has the eigenvalue 1 with algebraic multiplicity 1. Therefore, by (3.5), matrix has no eigenvalue 1, that is, 1 is not the eigenvalue of matrix . The proof is completed.

Lemma 3.4. *For , if , then , where represents the spectral radius of matrix . *

*Proof. *Let be any eigenvalue of , that is,
*Case I*. If , then it follows from the Laplace theorem for a partitioned matrix that , that is , which is a contradiction.*Case II*. For , using the Laplace theorem for a partitioned matrix again, one can derive from (3.10) that , then
Therefore, we have and . One can easily verify that , that is, , for , . The proof is completed.

Lemma 3.5. *If , then there exist positive constants and such that , . *

Lemma 3.6. *Inequality has at least one solution if .*

*Proof. *Let , then . Set , then , so for . Thus is monotonically increasing for . Since , so there exists at least a such that .

Lemma 3.7 (see [19]). *A digraph with has no globally reachable node if and only if it has at least two disjoint closed subsets of .*

Now, we are in the position to prove Theorem 3.1.

* Proof. **Sufficiency*. By (3.5), one derives that
where , .

For , by Lemma 3.4, . Noticing that Lemma 3.5, there exist constants and such that . Therefore by (3.12), we have

For , by Lemma 3.6, there exists a positive constant satisfying such that .

In the following, we will show that
where

It is clear that , for .

Next, we first show for any ,

If (3.15) is not true, then there must exists a such that

By (3.13), one can obtain that
which is a contradiction. Thus, for any , (3.15) holds, let , (3.14) holds. Since there has a globally reachable node in graph , for , as for any initial condition invoking Lemma 3.3, that is, there exist some such that the consensus for (2.1) is reached asymptotically.*Necessity.* The consensus of (2.1) can be reached asymptotically, that is, for any initial position and velocity and any bounded time delay , and as . By way of contradiction, suppose that the graph has no globally reachable node. Then it follows from Lemma 3.7 that there are at least two disjoint closed sets of nodes in graph . Without loss of generality, we consider the following special case, that is, and there are exactly two disjoint closed sets in graph , say and , that is, there is only one node in and , respectively, (if there are more disjoint closed sets or there are more nodes in each disjoint closed sets, it can be proved by a similar argument only with more complex computation). Given the initial condition satisfying and , by a direct computation, we have , , , , where , . Hence, if , the consensus cannot be reached, a contradiction. The proof of Theorem 3.1 is completed.

*Remark 3.8. *By the proof procedure of Lemma 3.4 and Theorem 3.1, one can conclude that Lemma 3.4 and Theorem 3.1 are also true for . When and there is no communication time delay between agents, that is, , Theorem 3.1 is consist with Theorem 2 in [16].

#### 4. A Simulation Example

In this section, an example is given to demonstrate the efficiency and applicability of the proposed method and to validate the theoretical analysis. For simplicity, we suppose that all the edge weights are 1 in the following example.

*Example 4.1. *Assume that the interaction digraph of ten agents is depicted in Figure 1.

A globally reachable node can be easily found in the digraph. The initial positions and velocities of the ten agents are chosen as [, , , , , , , , , and [, , , , , , , , , , , respectively. Select control parameter , transmission time delay , and . The simulation result under the control protocol (3.1) is shown in Figure 2, which illustrates that consensus has been achieved within about 150 seconds. Change transmission time delay as , by simulation experiments, we also find that consensus is accomplished within about 120, 200, 250, and 700 seconds, respectively. This shows that our designed algorithms can effectively tolerate arbitrary bounded time delays. But when the time delay increased, the convergence rate will decrease.

**(a)**

**(b)**

#### 5. Conclusion

Based on algebraic graph theory, matrix theory, and stability theory of difference equation, the consensus problem of discrete time second-order multiagent systems with time delay is investigated. A necessary and sufficient condition for achieving consensus is presented. Furthermore, the convergence rate for consensus is given. The main results presented in this work are delay-independent (i.e., the results are valid for arbitrary bounded time delay). In addition, the present paper applies graph theoretic tools to explore explicit graphical conditions of the information exchange topologies under which consensus can be achieved. Since the interagent connection structures may vary over time, the consensus of discrete time second-order multiagent systems with time delays and switching topologies is also very interesting to us; this case will be investigated in future research.

#### Acknowledgments

The work is supported jointly by National Natural Science Foundation of China under Grant 61004042, Construction Project of Engineering and Technology Research Center of Chongqing (cstc2011pt-gc40006), and Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ100513.