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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.

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. [

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.

Let

The dynamics of agent

Second-order consensus in the multiagent systems (

To solve the consensus problem, we introduce the following neighbor-based feedback control protocol

Under control protocol (

Before proving Theorem

Let

Since there has a globally reachable node in graph

Let

Define

Noticing that the every row sum of

Then (

Let

Therefore, the consensus of (

Now, we give some useful lemmas for proving Theorem

If a nonnegative matrix

Equation (

It suffices to verify that

By (

Since the graph

For

Let

If

Inequality

Let

A digraph

Now, we are in the position to prove Theorem

For

For

In the following, we will show that

It is clear that

Next, we first show for any

If (

By (

By the proof procedure of Lemma

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.

Assume that the interaction digraph of ten agents is depicted in Figure

The directed interaction topology of ten agents.

A globally reachable node can be easily found in the digraph. The initial positions and velocities of the ten agents are chosen as [

Positions and velocities of ten agents under control (

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.

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.