Distributed Consensus for Discrete-Time Directed Networks of Multiagents with Time-Delays and Random Communication Links

This paper is concerned with the leader-following consensus problem in mean-square for a class of discrete-time multiagent systems. The multiagent systems under consideration are the directed and contain arbitrary discrete time-delays. The communication links are assumed to be time-varying and stochastic. It is also assumed that some agents in the network are well informed and act as leaders, and the others are followers. By introducing novel Lyapunov functionals and employing some new analytical techniques, sufficient conditions are derived to guarantee the leader-following consensus in mean-square for the concerned multiagent systems, so that all the agents are steered to an anticipated state target. A numerical example is presented to illustrate the main results.


Introduction
In recent years, the multiagent distributed coordination problem has attracted many researchers since it has broad applications in satellite formation flying, cooperative search of unmanned air vehicles, scheduling of automated highway systems, air traffic control, and distributed optimization of multiple mobile robotic systems.In many applications involving multiagent systems, one of the most fundamental problems is that groups of agents need to agree upon certain quantities of interest, which is called the consensus or agreement problem in the literature.Consensus problems have a long history in the field of computer science [1], many distributed control and estimation strategies are designed based on consensus algorithms [2][3][4][5][6][7][8], and consensus problems are used to model many different phenomena involving information flow among agents, including flocking, swarming, synchronization, distributed decision making, and schooling; see, for example, the survey paper [9].Consensus problems for networked dynamic systems have been extensively studied in the last few years [10][11][12].
Usually, algebraic graph theory [13] acts as a good framework for analyzing consensus problems; see, for example, [10,11,14,15].In this framework, each agent is modeled as a vertex of a graph, and an edge of the graph joins node  to node  if agent  is receiving information from agent .The models and algorithms for consensus have been recently reported by a number of investigators.In [16], Vicsek et al. proposed a simple discrete-time model to simulate a group of autonomous agents moving in the plane with the same speed but different headings.Vicsek's model in essence is a simplified version of the model introduced earlier by Reynolds [17].Based on the algebraic graph theory [18], it has been shown that the network connectivity is a key factor in reaching consensus [11,14,15].It has also been proved that consensus in a network with a dynamically changing topology can be reached if and only if the time-varying network topology contains a spanning tree frequently enough as the network evolves with time [11,14].Recently, stochasticapproximation-type algorithms with a decreasing step size are developed, and almost sure convergence is established for consensus seeking; see, for example, [19] and the references therein.It has been recognized that time-delay is unavoidable in signal transmission and is also one of the main sources for causing instability and poor performances of systems [20][21][22].Recently, the multiagent networks with time-delay have started to receive some initial attention [15,23,24].
On the other hand, in many multiagent systems, some agents are well informed and served as leaders, and the others track the leaders and act as followers.It was reported that the leader-following configuration is an energy saving mechanism [25] which was found in many biological systems, and it can also enhance the communication and orientation of the flock [26].The leader-following consensus has been an active area of research [14,27,28].Such a leader-following consensus problem is considered and proved in [14] that if all the agents were jointly connected with their leader, their states would converge to that of the leader as time goes on.Reference [28] studied a leader-following consensus problem for a multiagent system with a varying-velocity leader and time-varying delays, where the interaction graph among the followers was switching and balanced.Reference [27] investigated the leader-following consensus problem of higher-order multiagent systems.Unfortunately, so far, the delayed networks considered for the leader-following consensus problem are almost continuous-time multiagent systems, and the leader-following consensus problems for discrete-time multiagent systems with time-delay and random communication links have received little research attention.Hence, it is our intention in this paper to tackle such an important yet challenging problem.
In this paper, we will investigate the leader-following consensus problem for the discrete-time directed multiagent systems with time-delay and random communication links.By constructing new Lyapunov functionals and employing some analytical techniques, sufficient conditions for the leader-following consensus in mean-square are established for multiagent system, so that all the agents are steered to an anticipated state target.A numerical example is used to illustrate the proposed theory.

Problem Formulation
Throughout this paper, N and Z + stand for the natural numbers and the positive integer set, respectively; R, R  , and R × denote, respectively, the set of real numbers, the  dimensional Euclidean space, and the set of all  ×  real matrices.The superscript  represents the transpose for a matrix, and | ⋅ | may stand for any absolute value of real numbers or the standard Euclidean norm from the context.In an underlying probability space (Ω, F, P), E[⋅] and Var[⋅] denote, respectively, the mean and the variance for a random variable, and E[ | ] will mean the expectation of  conditional on .
Consider  agents distributed according to a directed graph G = (V, E) with a set of nodes V = {1, 2, . . ., }, a set of edges E ∈ V × V, and a weighted adjacency matrix  = [  ] with nonnegative adjacency elements   .In G, the th node represents th agent, and a directed edge (simply called an edge) from node  to node  denoted as an ordered pair (, ) ∈ E represents a unidirectional information exchange link from node  to node ; that is, agent  can receive or obtain information from agent , but not necessarily vice versa.The set of neighbors of node  is denoted by N  = { : (, ) ∈ E}.A weighted adjacency  = [  ] ∈ R × of a weighted directed graph is defined such that   is a positive weight if only (, ) ∈ E (so there is no edge between a node and itself; that is,   = 0, for all  ∈ V).In other words,   > 0, if  ∈ N  , otherwise   = 0.A directed path (simply called a path) of length  from V  to V  (,  ∈ V) is a sequence of edges ( 0 ,  1 ), ( 1 ,  2 ), . . ., (  ,  +1 ) with  0 = ,  +1 =  and (  ,  +1 ) ∈ E for  = 0, 1, . . ., .A graph G is said to be strongly connected if there exists a path between any two distinct nodes in it.For convenience of presentation, the two names, agent and node, will be used interchangeably.Now consider the dynamics of  agents distributed over a directed graph G. Let   () ∈ R denote the state of node  at time , x() = [ 1 (),  2 (), . . .,   ()]  the state of the system accordingly, and let  = [  ] be the weighted adjacency matrix associated with the graph.In general, the dynamics of discrete-time multiagent network with fixed topology are described by where   ∈ Z + is the time-delay of the information transmission from node V  to node V  .
Remark 1.The consensus problem for the multiagent system (1) is considered in [29], and the consensus problem for its continuous-time counterpart (analogue) is investigated in [24], and system (1) without time-delays is also investigated extensively; see, for example, [10,19] and the references therein.
In multiagent network (1), it is assumed that there is no communication failure between agents.However, during signal exchange of the sensor nodes, an important uncertainty feature is signal losses, which may be caused by the temporary extreme deterioration of the link quality, for instance, due to blocking objects traveling between the transmitter or receiver [30].Therefore, we consider the general case where each communication link is subject to some probability distribution.Assume that weighted adjacency matrix  () = [ ()   ] is timevarying with  ()   being random variable.Denote   = E[ ()  ] and   = Var[ ()   ].As usual, we assume that (, ) ∈ E if and only if   > 0, and the set of neighbors of node  is denoted by N  = { : (, ) ∈ E}.Now, the dynamics of discrete-time multiagent network with random communication links are given by where, as in the previous discussion,   ∈ Z + is the timedelay of the information transmission from node V  to node V  .For convenience, we let   = 0, ( ∈ V), hereafter.
In practical applications, it is often important to steer the state of each agent in a network to a fixed objective.In this paper, we will consider the regulation of the multiagent network (3) so that all agents can reach a common objective.Suppose that there are some agents acting as leaders and well informed.Specifically, let  * be the anticipated state target.If necessary, we can relabel the agents, and without loss of generality, we assume that the first  0 agents serve as leaders, and the other ones act as followers.Consider the following controlled multiagent network: where   > 0 are given constants.
Definition 2. The multiagent network (3) is said to reach leader-following consensus on a state target  * in meansquare if for any solution x() = ( 1 (),  2 (), . . .,   ())  of system (4), it always holds that lim In this paper, we will investigate the leader-following consensus problem in mean-square for discrete-time multiagent system (3).By constructing novel Lyapunov functionals and employing some new analytical techniques, sufficient conditions are established to ensure the leader-following consensus in mean-square for multiagent system (3).

Main Results and Proofs
This section is devoted to the leader-following consensus analysis for system (3), and let us make some necessary preparations before introducing our main results.
Assume that { ()  : (, ) ∈ E,  ∈ Z + } are independent with respect to (, ) and  and also independent of the initial states. Let , otherwise  ()  =  ()  = 0.Then, (4) is rewritten as Denote  = (  ) Then, all row sums of both  () and  are one, and all row sums of Δ () are zero; namely, Also denote the variance of random variable  ()  by  2  , where   is its standard deviation.Notice that 2 is the variance of random variable  ()  , and b()  for  ̸ =  is dependent with respect to (, ) and .Therefore, we have For the interaction topology of multiagent system (3), we make the following assumption.Assumption 3. The graph G = (V, E) is strongly connected; namely, the matrix  is irreducible.
Lemma 4 (see [31]).Let  = [  ] ∈ R × be a nonnegative matrix; that is,   ≥ 0, and let () be the spectral radius (called the Perron root of ).In addition, suppose that  is strongly connected, then there is a positive vector  such that  = ().
From Lemma 4, it follows readily that there exists a positive left eigenvector  = [ 1 ,  2 , . . .,   ]  of  such that In the sequel, we denote Also, we make the following assumption.
Assumption 5. Assume that Remark 6.Notice that in Assumption 5 the condition ⋃  0 =1 N = V means that the set of the first  0 nodes and their neighbors contains all the nodes of the network.
We are now in a position to introduce the main results of this paper.Theorem 7. Consider the multiagent systems (3) and (4).Suppose that Assumptions 3 and 5 are satisfied, and assume that  < √/(4( − 1)) holds.Then, the multiagent network (3) reaches leader-following consensus on the state target  * in mean-square.
To prove that the multiagent network (3) reaches leaderfollowing consensus on the state target  * in mean-square, it suffices to prove the mean-square stability of (21).To this end, we construct the following Lyapunov functional: where with Then, for system (21), using ( 8)-( 11), we conduct the following computation: It is not difficult to see that From ( 27)-( 29), it follows that A straightforward computation yields that Noticing the equality ∑  =1   = 1 (see (7)), it follows readily that Substituting ( 31) and ( 32) into (30) yields that It is easy to see that and Employing the Lyapunov stability theory, we can deduce that lim  → ∞ E[(e  )] = 0.This completes the proof of the theorem.
Remark 8.In Theorem 7, the condition  < √/4( − 1) always holds when  is sufficiently small.In particular, when the interaction topology of multiagent system is deterministic, the system (3) and the controlled network (4) are reduced, respectively, to In this case,  = 0; accordingly, the condition  < √/4( − 1) is always satisfied, and from Theorem 7, we have the following corollary.
In the previous discussion, we only consider scalar individual states, and it is easy to extend them to the case where the individual states are vectors.Consider the following multiagent system of  nodes with vector-valued states: and the controlled network is given by where () ∈ R  .We have the following results.
Proof.The proof of this theorem is similar to that of Theorem 7. The minor modification is to replace some scalar multiplication operations by the Kronecker product of matrices, and we omit the details here.

A Numerical Example
In this section, we present a numerical example to illustrate the proposed methods.
Example 1.Consider the multiagent networks (3) and ( 4), and for simplicity, we take  = 5.The interaction topology between the agents is shown in Figure 1 Clearly, the network topology is strongly connected, and it is also obvious that N1 ⋃ N2 = V.Therefore, we can choose

Conclusions
We have investigated the leader-following consensus problem in mean-square for a class of discrete-time multiagent systems.The network under study is bidirectional and contains arbitrary time-delays and the random communication links.
Some agents in the network are well informed and serve as leaders.By employing novel Lyapunov functionals and analytical skills, sufficient conditions are established to ensure the leader-following consensus in mean-square for multiagent system.A numerical example is given to demonstrate the proposed approach.
(a), and other parameters are taken as follows: = [  ]