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

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.


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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][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][3][4][5][6][7][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][10][11][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 discretetime agents with mixed dynamics and mixed orders.We first introduce two different linear agreement protocols of secondorder 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; 1  represents [1, 1, . . ., 1]  with  dimension;   denotes  ×  identity matrix; 0 denotes a zero value or zero matrix with an appropriate dimension;   () represents the set of all  ×  real matrices.

Preliminaries
In this section, we give some preliminary knowledge about matrix theory and graph theory.Let  = (, , ) be a weighted digraph, where  = {V 1 , . . ., V  } is the set of  nodes and  ⊆  ×  is the set of edges. = {1, 2, . . ., } is the set of node indexes.An edge of  is denoted by   = (V  , V  ), where the first element V  of   is said to be the tail of the edge and the other V  to be the head. = [  ] is a weighted adjacency matrix, where   ≥ 0 denotes the weight of V  to V  .If   > 0, this means that the node can obtain information from the node; if   = 0, this means that the node cannot obtain information from the node.Define the set of neighbors of node  as   = { :   > 0}.Let the matrix of  = diag{  ,  = 1, 2, . . ., }, where   = ∑  =1   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  1 ,  2 , . . .,   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  ≥ 0 ( is nonnegative) if all its elements are nonnegative.If a nonnegative matrix  ∈  × satisfies 1  = 1  , then it is said to be (row) stochastic.In addition, a stochastic matrix  ∈  × is said to be indecomposable and aperiodic (SIA) if lim  → +∞   = 1    , where  ∈   .

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  > 0 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, V  ∈  is the velocity, and   ∈  is control input.For the second-order multiagent network, it is assumed that   () =   (0) and 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   () =   (0) for  < 0.
The discrete-time multiagent networks with mixed dynamics (1) and ( 2) are said to reach agreement, if and only if any initial condition satisfies lim 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,   > 0 is control parameter,   () ≤  max is the communication time delay from V  to V  , and   () > 0 is the coupling weight chosen from any finite set.Define   () =   () + V  (),  = 1, 2, . . ., , 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,   () ≤  max is the communication time delay from V  to V  and   () > 0 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,   () = diag{∑ ∈     (),  = 1, 2, . . ., }, and   () denotes the adjacency relations of second-order agents to first-order agents.Meanwhile,   () =   () +   () is the Laplacian matrix of first-order agents,   () = diag{∑ ∈     (),  =  + 1,  + 2, . . .,  + }, and   () denotes the adjacency relations of first-order agents to second-order agents.

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.
To lead to the following lemmas and theorems and to be satisfied with some properties of matrix, we give an assumption that for  = 1, . . .,  and  =  + 1, . . .,  + .

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

Simulations
We suppose a multiagent network composed of three secondorder 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  = 0.2 s.The weight of each edge in topology is 1 and the control parameters  1 = 6,  2 = 7, and  3 = 8 for the second-order dynamic agents.Besides, the communication time delays associated with topology are  12 = 3,  51 = 4,  45 = 5,  34 = 6, and  23 = 7.Then, assumption ( 14) is satisfied.The initial assumptions are (0) = [10, 20, 30, −10, −20]  , V(0) = [0, 20, −10]  .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, Ga and Gb, which denote the states of a network with dynamically changing topology and time delay; it starts at Ga and switches every 2 s to the next state.Note that in each time interval of 4 s the union of the communication graphs Ga ∪ Gb 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.
According to Figure 1

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.