Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 957028, 7 pages
http://dx.doi.org/10.1155/2015/957028

Research Article

Agreement of Networks of Discrete-Time Agents with Mixed Dynamics and Time Delays

Xue Li, Xueer Chen, and Yingchun Xie

School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu 611731, China

Received 17 October 2015; Accepted 29 November 2015

Academic Editor: Peng Lin

Copyright © 2015 Xue Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper considers agreement problems of networks of discrete-time agents with mixed dynamics and arbitrary bounded time delays, and networks consist of first-order agents and second-order agents. By using the properties of nonnegative matrices and model transformations, we derive sufficient conditions for stationary agreement of networks with bounded time delays. It is shown that stationary agreement can be achieved with arbitrary bounded time delays, if and only if fixed topology has a spanning tree and the union of the dynamically changing topologies has a spanning tree. Simulation results are also given to demonstrate the effectiveness of our theoretical results.

1. Introduction

Agreement problems for multiagent networks have attracted great attention in recent years. Agreement means that multiple agents can reach a common value with time going, which might be attitude in multiple spacecraft alignment, heading direction in flocking behaviour, or average in distributed computation. In several decades, numerous literatures have studied the agreement problems for multiagent networks [1–17]. In [1], by using Lyapunov function method and LaSalle’s invariance principle, the agreement problems can be solved when the communication topologies were undirected connected graphs and leader-following network. And in [2], the authors studied the stability properties of linear time-varying networks in continuous time whose network matrix was Mezter with zero row sums and provided that the delay only affects the off-diagonal terms in the differential equation.

Because of the noise, packet loss, and limited communication bandwidth, time delays are inevitable when agents send and receive the information from their neighbors. Agreement problems for the continuous-time multiagent networks with time delays have attracted great attention in the past decades [2–8]. In [3], the authors addressed a coordination problem of a multiagent network with jointly connected interconnection topologies, a sufficient condition to make all the agents converge to a common value based on Lyapunov-based approach and related space decomposition technique. In [4], based on reduced-order system, the authors presented conditions under which all agents reach consensus with the desired performance. In addition, agreement problems for the discrete-time multiagent networks with time delays have also attracted great attention in the past decades [9–12]. In [9], the authors investigated the agreement problem of second-order discrete-time multiagent networks with nonuniform time delays and changing communication topologies, and by using the properties of nonnegative matrices, all agents reached agreement with arbitrary bounded nonuniform time delays. In [10], the authors extended the results of [9, 11] by proposing a new method based on an undelayed equivalent network which has two parts: the linear main body and the error auxiliary, and it was shown that the network can reach agreement with arbitrary communication delays. In [18], the authors investigated a systematical framework of agreement problems with directed interconnection graphs or time delays by a Lyapunov-based approach. In this section, we want to pay more attention to mixed multiagent networks with time delays [19, 20]. In [19], the author obtained two agreement conditions by using the properties of nonnegative matrix. In [20], by using frequency domain method and Gershgorin disk theorem, agreement of the protocol with time delay was only dependent on network coupling strength and each input time delay, but independent of communication delay.

The purpose of this work is to extend the method based on the nonnegative matrices [9, 11, 19, 21, 22] to the discrete-time agents with mixed dynamics and mixed orders. We first introduce two different linear agreement protocols of second-order network and first-order network, respectively. Then, by model transformations, we turn the original network into an equivalent undelayed network whose network matrix is stochastic. Compared with the works [9, 19], we use much easier dynamics and two different protocols with varying control parameters. Under some restrictions on the sampling interval and the coupling weights, we obtain important results. Agreement of two different protocols with time delays is only dependent on the connectedness of the interconnection topology, but independent of communication time delays.

In this paper, the following notation will be used. denotes the set of all -dimensional real column vectors; represents the set of nonnegative integers; represents with dimension; denotes identity matrix; 0 denotes a zero value or zero matrix with an appropriate dimension; represents the set of all real matrices.

2. Preliminaries

In this section, we give some preliminary knowledge about matrix theory and graph theory. Let be a weighted digraph, where is the set of nodes and is the set of edges. is the set of node indexes. An edge of is denoted by , where the first element of is said to be the tail of the edge and the other to be the head. is a weighted adjacency matrix, where denotes the weight of to . If , this means that the node can obtain information from the node; if , this means that the node cannot obtain information from the node. Define the set of neighbors of node as . Let the matrix of , where denotes the sum of the values in th row of matrix. The matrix of presents the Laplacian matrix of graph. For the arbitrary node and node , the graph is undirected graph when . A directed graph is said to be strongly connected, if there is a directed path from every node to every other node. A digraph is said to have a spanning tree, if there exists a node such that there is a directed path from this node to every other node. The union of a collection of directed graphs with the same node set is a directed graph with node set and the edge set equal to the union of the edge sets of all of the graphs in the collection.

Given , it is said that ( is nonnegative) if all its elements are nonnegative. If a nonnegative matrix satisfies , then it is said to be (row) stochastic. In addition, a stochastic matrix is said to be indecomposable and aperiodic (SIA) if , where .

3. Networks of Discrete-Time Agents with Mixed Dynamics

In this section, we analyse networks of discrete-time multiagent with mixed dynamics, and networks consist of the first-order agents and the second-order agents. Let the total number of agents be . Each agent is regarded as a node in the communication directed graph . Each edge corresponds to an available information channel from agent to at time , where is nonnegative integer and is the sample time. Suppose the number of second-order agents is and the number of first-order agents is . To simplify the notation, we replace all “” by “”. Then, the dynamics of the th second-order agent are given as follows:

where is the position, is the velocity, and is control input. For the second-order multiagent network, it is assumed that and for . Then, the dynamics of the th first-order agent are given as follows:

where is the position and is the control input. For the first-order multiagent network, we assume that for .

The discrete-time multiagent networks with mixed dynamics (1) and (2) are said to reach agreement, if and only if any initial condition satisfies

To solve the agreement problems, the protocol with time delays is proposed for the second-order agents as follows:

for the second-order multiagent network, is control parameter, is the communication time delay from to , and is the coupling weight chosen from any finite set. Define , , and we can rewrite the dynamics of th second-order agent with algorithm (4) as follows:

Then, we give the agreement algorithm of the first-order agents:

for the first-order multiagent network, is the communication time delay from to and is the edge weight chosen from any finite set. With (6), we can rewrite the dynamics of first-order agents:

In the multiagent network composed of first-order agents and second-order agents, the neighbors of each second-order agent include first-order and second-order agents, denoted by , where and are agent ’s second-order and fist-order neighboring agents, respectively. Similarly, the neighbors of each first-order agent are denoted by . And we can denote the Laplacian matrix as follows:

where , is the Laplacian matrix of second-order agents, , and denotes the adjacency relations of second-order agents to first-order agents. Meanwhile, is the Laplacian matrix of first-order agents, , and denotes the adjacency relations of first-order agents to second-order agents.

4. Main Results

In this section, we will solve the agreement problems of networks of discrete-time agents with mixed dynamics and arbitrary bounded time delays. To analyse the stability of such multiagent networks, there are mainly three approaches: the Lyapunov-based approach, the frequency domain approach, and the approach based on the properties of nonnegative matrices. However, the frequency domain approach is limited to the fixed topology case and invalid when the topologies dynamically change, whereas the Lyapunov-based approach is hard to apply to the case of general directed graphs with time delay and switching topologies, especially when the communication graphs have no spanning trees. In this section, to use the approach based on the properties of nonnegative matrices, we performed a model transformation already (see (5)). Then, we can transform the network into an equivalent undelayed one whose network matrix is stochastic. Based on this obtained equivalent network, we present sufficient conditions under which all agents reach agreement with arbitrary bounded time delays under fixed topology and dynamically changing topologies.

Denote , where , , and ; we can rewrite the networks with mixed dynamics (5) and (7):

where and is either zero or equal to the weight of the edge if . Obviously, are matrices satisfying , and is given by

where is a diagonal matrix.

Because of the existence of time delays, it is still hard to perform analysis on network (9). We need to introduce an equivalent augmented network (9). Define a new state variable ; then network (9) can be rewritten as an undelayed network given by

where is defined as

We know that ; then it is easy to see . This property is important and will be used to study the stability of the networks. Before presenting the main theorem, we first introduce some necessary lemmas.

Lemma 1 (Wolfowitz, 1963 [23]). Let be a finite set of SIA matrices with the property that, for each sequence with positive length, the matrix product is SIA. Then, for each infinite sequence , there exists a vector such that

Lemma 2 (Ren and Beard, 2005 [11]). Given a matrix , where , , for all , and for each , then has at least one zero eigenvalue and all of the nonzero eigenvalues are in the open left half plane. Furthermore, has exactly one zero eigenvalue if and only if the directed graph associated with has a spanning tree.

Lemma 3 (Ren and Beard, 2005 [11]). Let be a stochastic matrix. If has an eigenvalue with algebraic multiplicity equal to one, and all the other eigenvalues satisfy , then is SIA; that is, , where satisfies and . Furthermore, each element of is nonnegative.

To lead to the following lemmas and theorems and to be satisfied with some properties of matrix, we give an assumption that

for and

Lemma 4. Under assumption (14), we can know that is a stochastic matrix and is also a stochastic matrix.

Proof. It is clear that all elements of are nonnegative under assumption (14), and . Thus, is a stochastic matrix. Similarly, we can know that defined in (12) is also a stochastic matrix.

Lemma 5. With assumption (14), if the interconnection topology of agents (5) and (7) has a spanning tree, the matrix defined in (12) is SIA.

Proof. Taking the elementary column transform of by adding all the other columns to the last column,

we can know that , where is defined in (10). Similarly, take the elementary column and row transforms of . First, exchange the first and the second column. Then, add the first column to the second column. Finally, the second row divides and the third row divides . We can obtain as follows:

so . Thus, if and only if , has an eigenvalue with the algebraic multiplicity equal to one; that is, the topology of agents (5) and (7) has a spanning tree from Lemma 2. Similarly, by taking the elementary column transform, we can prove that is full rank, which holds under arbitrary topology with assumption (14); that is, it has . Based on the properties of stochastic matrix and Gershgorin disk theorem, except for one eigenvalue , all the other eigenvalues satisfy . Thus, is SIA based on Lemma 3.

Theorem 6. Consider the discrete-time multiagent networks with mixed dynamics (5) and (7) and bounded time delays under fixed topology. In addition, and do not change with time. Under assumption (14), if the digraph of agents has a spanning tree, the agents in (5) and (7) can reach stationary agreement with protocols (4) and (6).

Proof. We can rewrite network (11) as . Because fixed topology has a spanning tree, we can obtain that is SIA from Lemma 5. Then, from Lemma 1, where some vector . Thus, , ; that is, the agents in (5) and (7) reach stationary agreement.

Lemma 7. Under assumption (14), if the union of the digraphs of agents (5) and (7) has spanning trees for the positive integers and with , then is SIA.

Proof. We define the union of the digraphs associated with as . Under assumption (14), if the union of the digraphs of agents (5) and (7) has a spanning tree, then the matrix has an eigenvalue with the algebraic multiplicity equal to one from the proof of Lemma 5; that is, the rank of matrix is . Then, we can obtain that the digraph of the Laplacian matrix has a spanning tree from Lemma 2; that is, the union of the digraphs has a spanning tree. Then, by using a similar proof for Lemma () in Xiao and Wang (2006) [21], Lemma 7 can be proved.

Theorem 8. Under assumption (14), if there exists an infinite sequence of uniformly bounded, nonoverlapping time intervals , , , such that the union of the digraphs of n agents across each interval has a spanning tree, then multiagent networks (5) and (7) reach agreement with protocols (4) and (6).

Proof. For each , let be the largest integer such that . Then, network (11) is rewritten as

where . Since the union of agents interconnection topologies across has a spanning tree, is SIA from Lemma 7. Since and the edge weights are chosen from any finite set, then it is easy to see that the set of all possible is finite. Hence, by Lemma 7 again, we know that for some vector . We know that is a stochastic matrix from Lemma 4, so we can obtain

Thus, , . That is, the stationary agreement of agents (5) and (7) is reached.

Remark 9. In this section, with assumption (14), we just give the sufficient agreement criteria in Theorems 6 and 8. Similar to the first-order and the second-order multiagent networks, the multiagent networks with mixed dynamics and bounded time delays can also achieve an asymptotic agreement with fixed topologies or dynamically changing topologies.

Remark 10. Under assumption (14), we obtain two conclusions about fixed topology and dynamically changing topology with bounded time delays. First, the multiagent networks with mixed dynamics (1) and (2) and fixed topology achieve stationary agreement using protocol (4) and (6) if and only if fixed topology has a spanning tree. Second, the multiagent networks with mixed dynamics (1) and (2) and dynamically changing topologies achieve stationary agreement using protocols (4) and (6) that any of the changing topologies may not have spanning trees, but the union of the changing topologies must have a spanning tree.

5. Simulations

We suppose a multiagent network composed of three second-order agents (5) and two first-order agents (7). The interconnection topology of the network has a spanning tree in Figure 1. First, we simulate the network with fixed topology. In fixed topology, the diamonds denote the first-order agents and the circles denote the second-order agents. The sampling time of networks is  s. The weight of each edge in topology is 1 and the control parameters , , and for the second-order dynamic agents. Besides, the communication time delays associated with topology are , , , , and . Then, assumption (14) is satisfied. The initial assumptions are , . Thus, stationary agreement of the agents in (5) and (7) can be achieved (see Figures 2 and 3). Then, we simulate the network with the dynamically changing topologies. In Figure 4, a finite state machine is shown with two states, and , which denote the states of a network with dynamically changing topology and time delay; it starts at and switches every  s to the next state. Note that in each time interval of  s the union of the communication graphs has a spanning tree. Let the other parameters be similar to fixed topology; then assumption (14) is satisfied, and agreement is achieved as shown in Figures 5 and 6, which is consistent with Theorem 8.

Figure 1: Topology.

Figure 2: Position of the dynamic agents.

Figure 3: Velocity of the dynamic agents.

(a) Ga

(b) Gb

Figure 4: Two directed graphs.

Figure 5: Position of the dynamic agents.

Figure 6: Velocity of the dynamic agents.

According to Figure 1, we can obtain the Laplacian matrix

6. Conclusion

This paper studies the agreement problems of networks of discrete-time agents with mixed dynamics and arbitrary bounded time delays under fixed topology and dynamically changing communication topologies. For discrete-time multiagent networks with mixed dynamics, by using model transformations and the properties of nonnegative matrix, if and only if fixed topology has a spanning tree, the agents with some restriction on the coupling weights and the sampling interval can tolerate arbitrary bounded time delays to reach a stationary agreement. However, the multiagent networks with mixed dynamics and arbitrary bounded time delays under dynamically changing topologies achieve stationary agreement that any of the changing topologies may not have spanning trees, but the union of the changing topologies must have a spanning tree.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61203080, 61573082), the Foundation of State Key Laboratory of Networking and Switching Technology Foundation (SKLNST2011105, SKLNST2013109), National Program 863 of China (2014AA4032), National Program 973 of China (613237201506), and State Key Laboratory of Intelligent Control and Decision of Complex Systems.

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