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

Authors: Yurong Liu; Hamid Reza Karimi; Xiaohui Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 1. 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 [2–8], 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 [10–12].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 i to node j if agent j is receiving information from agent i. 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, stochastic-approximation-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 [20–22]. 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 thediscrete-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.


## 2. Problem Formulation

Throughout this paper,ℕ and ℤ+ stand for the natural numbers and the positive integer set, respectively; ℝ, ℝn, and ℝn×m denote, respectively, the set of real numbers, the n dimensional Euclidean space, and the set of all n×m real matrices. The superscript T 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[x∣y] will mean the expectation of x conditional on y.Considern agents distributed according to a directed graph 𝒢=(𝒱,ℰ) with a set of nodes 𝒱={1,2,…,n}, a set of edges ℰ∈𝒱×𝒱, and a weighted adjacency matrix A=[aij] with nonnegative adjacency elements aij. In 𝒢, the ith node represents ith agent, and a directed edge (simply called an edge) from node i to node j denoted as an ordered pair (i,j)∈ℰ represents a unidirectional information exchange link from node i to node j; that is, agent j can receive or obtain information from agent i, but not necessarily vice versa. The set of neighbors of node i is denoted by 𝒩i={j:(j,i)∈ℰ}. A weighted adjacency A=[aij]∈ℝn×n of a weighted directed graph is defined such that aij is a positive weight if only (j,i)∈ℰ (so there is no edge between a node and itself; that is, aii=0, for all i∈𝒱). In other words, aij>0, if j∈𝒩i, otherwise aij=0. A directed path (simply called a path) of length k from vt to vl (t,l∈𝒱) is a sequence of edges (i0,i1),(i1,i2),…,(ik,ik+1) with i0=t, ik+1=l and (is,is+1)∈ℰ for s=0,1,…,k. 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 ofn agents distributed over a directed graph 𝒢. Let xi(k)∈ℝ denote the state of node i at time k, x(k)=[x1(k),x2(k),…,xn(k)]T the state of the system accordingly, and let A=[aij] be the weighted adjacency matrix associated with the graph. In general, the dynamics of discrete-time multiagent network with fixed topology are described by
(1)xi(k+1)=xi(k)+∑j∈𝒩i‍aij(xj(k-τij)-xi(k)),i∈𝒱,
where τij∈ℤ+ is the time-delay of the information transmission from node vj to node vi.Remark 1.
The consensus problem for the multiagent system (1) is considered in [29], and the consensus problem for its continuous-time counterpart (analogue)
(2)x˙i(t)=∑j∈𝒩i‍aij(xj(t-τij)-xi(t)),i∈𝒱
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 A(k)=[aij(k)] is time-varying with aij(k) being random variable. Denote a-ij=𝔼[aij(k)] and σij=Var[aij(k)]. As usual, we assume that (j,i)∈ℰ if and only if a-ij>0, and the set of neighbors of node i is denoted by 𝒩i={j:(j,i)∈ℰ}.Now, the dynamics of discrete-time multiagent network with random communication links are given by(3)xi(k+1)=xi(k)+∑j∈𝒩i‍aij(k)(xj(k-τij)-xi(k)),i∈𝒱,
where, as in the previous discussion, τij∈ℤ+ is the time-delay of the information transmission from node vj to node vi. For convenience, we let τii=0, (i∈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 x* be the anticipated state target. If necessary, we can relabel the agents, and without loss of generality, we assume that the first i0 agents serve as leaders, and the other ones act as followers. Consider the following controlled multiagent network:
(4)xi(k+1)={xi(k)+∑j∈𝒩i‍aij(k)(xj(k-τij)-xi(k))-αi(xi(k)-x*),1≤i≤i0,xi(k)+∑j∈𝒩i‍aij(k)(xj(k-τij)-xi(k)),i0+1≤i≤n,
where αi>0 are given constants.Definition 2.
The multiagent network (3) is said to reach leader-following consensus on a state target x* in mean-square if for any solution x(k)=(x1(k),x2(k),…,xn(k))T of system (4), it always holds that
(5)limk→∞𝔼[xi(k)-x*]2=0.
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).


## 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{aij(k):(j,i)∈ℰ,k∈ℤ+} are independent with respect to (i,j) and k and also independent of the initial states.LetB(k)=[bij(k)] with bij(k)=aij(k) for j∈𝒩i, bii(k)=1-∑j∈𝒩i‍aij(k), otherwise bij(k)=aij(k)=0. Then, (4) is rewritten as
(6)xi(k+1)={∑j=1n‍bij(k)xj(k-τij)-αi(xi(k)-x*),1≤i≤i0,∑j=1n‍bij(k)xj(k-τij),i0+1≤i≤n.DenoteB-=(b-ij)=def𝔼[B(k)], and ΔB(k)=[b~ij(k)]=defB(k)-B-. Then, all row sums of both B(k) and B- are one, and all row sums of ΔB(k) are zero; namely,
(7)∑j=1n‍bij(k)=1,∑j=1n‍b-ij=1,∑j=1n‍b~ij(k)=0,i∈𝒱.Also denote the variance of random variablebij(k) by σij2, where σij is its standard deviation. Notice that 𝔼[b~ij(k)]2 is the variance of random variable bij(k), and b~ij(k) for j≠i is dependent with respect to (i,j) and k. Therefore, we have
(8)𝔼[b~ij(k)]2=Var[bij(k)]=σij2,j≠i,(9)𝔼[b~ij(k)b~il(k)]=0,l≠j,j≠i,l≠i,(10)𝔼[b~ii(k)b~ij(k)]=𝔼[-∑l≠i‍b~il(k)b~ij(k)]=-𝔼[b~ij(k)]2=-σij2,j≠i,(11)𝔼[b~ii(k)]2=𝔼[-∑j≠i‍b~ij(k)]2=∑l≠i‍𝔼[b~ij(k)]2=∑j≠i‍σij2.For the interaction topology of multiagent system (3), we make the following assumption.Assumption 3.
The graph𝒢=(𝒱,ℰ) is strongly connected; namely, the matrix B- is irreducible.Lemma 4 (see [31]).
LetA=[aij]∈ℝn×n be a nonnegative matrix; that is, aij≥0, and let ρ(A) be the spectral radius (called the Perron root of A). In addition, suppose that A is strongly connected, then there is a positive vector x such that Ax=ρ(A)x.From Lemma4, it follows readily that there exists a positive left eigenvector ξ=[ξ1,ξ2,…,ξn]T of B- such that
(12)ξTB-=ξT,∑j=1n‍ξi=1.
In the sequel, we denote
(13)ξ¯=max{ξi:1≤i≤n},(14)ξ_=min{ξi:1≤i≤n},(15)σ¯=max{σij:(j,i)∈ℰ},(16)α_=min{α-i:1≤i≤i0},(17)γ=min{a-ij:1≤i≤i0,j∈𝒩},(18)𝒩^i=𝒩i∪{i}.
Also, we make the following assumption.Assumption 5.
Assume that(19)⋃i=1n0‍𝒩^i=𝒱,αi≤b-iifor1≤i≤i0.Remark 6.
Notice that in Assumption5 the condition ⋃i=1n0𝒩^i=𝒱 means that the set of the first i0 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 σ-<α_ξ_γ/(4(n-1)ξ¯) holds. Then, the multiagent network (3) reaches leader-following consensus on the state target x* in mean-square.Proof.
LetC=[cij] with
(20)cij={b-ij-αi,for1≤i≤i0,j=i,b-ij,otherwise,
and denote ei(k)=xi(k)-x*. Then, the controlled network (6) can be rewritten as
(21)ei(k+1)=∑j=1n‍cijej(k-τij)+∑j=1n‍b~ij(k)ej(k-τij),i∈𝒱.
Letϕ is the initial value of network (21), and denote by ℱk the σ-algebras consisting of all events induced by the random variables ϕ,aij(s) with 0≤s≤k-1, (j,i)∈ℰ; that is, ℱk=σ(ϕ,aij(s),0≤s≤k-1,(j,i)∈ℰ). Also denote τ=max{τij:(i,j)∈ℰ}, and ek(s)=[e1(k+s),e2(k+s),…,en(k+s)]T, -τ≤s≤0.
To prove that the multiagent network (3) reaches leader-following consensus on the state target x* in mean-square, it suffices to prove the mean-square stability of (21). To this end, we construct the following Lyapunov functional:
(22)V(ek)=V1(ek)+V2(ek)+V3(ek),
where
(23)V1(ek)=∑i=1n‍ξiei2(k),(24)V2(ek)=∑i=1n‍ξi∑j=1n‍∑l≥j‍cijcilVijl(ek),(25)V3(ek)=∑i=1n‍ξi∑j≠i‍2σij2∑s=k-τijk-1‍ej2(s)
with
(26)Vijl(ek)=∑s=k-τijk-1‍ej2(s)+∑s=k-τilk-1‍el2(s).
Then, for system (21), using (8)–(11), we conduct the following computation:
(27)𝔼[V1(ek+1)∣ℱk]-V1(ek)00=𝔼[∑i=1n‍ξi(∑j=1n‍cijej(k-τij)+∑j=1n‍b~ij(k)ej(k-τij))2]0000-∑i=1n‍ξiei2(k)00=∑i=1n‍ξi𝔼[(∑j=1n‍cijej(k-τij))2]0000+∑i=1n‍ξi𝔼[(∑j=1n‍b~ij(k)ej(k-τij))2]0000-∑i=1n‍ξiei2(k)00=∑i=1n‍ξi[∑j=1n‍cij2ej2(k-τij)0000000000+∑j=1n‍∑l>j‍2cijcilej(k-τij)el(k-τil)∑j=1n‍]0000-∑i=1n‍ξiei2(k)0000+∑i=1n‍ξi[∑j=1n‍𝔼[b~ij(k)]2ej2(k-τij)000000000000+∑j=1n‍∑l>j‍2𝔼[b~ij(k)b~il(k)]ej(k-τij)el(k-τil)∑j=1n‍]00=∑i=1n‍ξi[∑j=1n‍cij2ej2(k-τij)0000000000+∑j=1n‍∑l>j‍2cijcilej(k-τij)el(k-τil)∑j=1n‍]0000-∑i=1n‍ξiei2(k)0000+∑i=1n‍ξi[∑j≠i‍σij2ei2(k)+∑j≠i‍σij2ej2(k-τij)000000000000+∑j≠i‍2𝔼[b~ii(k)b~ij(k)]ei(k-τii)ej(k-τij)]00=∑i=1n‍ξi[∑j=1n‍cij2ej2(k-τij)0000000000+∑j=1n‍∑l>j‍2cijcilej(k-τij)el(k-τil)]0000-∑i=1n‍ξiei2(k)0000+∑i=1n‍ξi[∑j≠i‍σij2ei2(k)+∑j≠i‍σij2ej2(k-τij)0000000000000-∑j≠i‍2σij2ei(k)ej(k-τij)].
It is not difficult to see that
(28)𝔼[V2(ek+1)∣ℱk]-V2(ek)00=∑i=1n‍ξi∑j=1n‍∑l>j‍cijcil(ej2(k)-ej2(k-τij)000000000000000000+el2(k)-el2(k-τil))0000+∑i=1n‍ξi∑j=1n‍cij2(ej2(k)-ej2(k-τij)),(29)𝔼[V3(ek+1)∣ℱk]-V3(ek)0000=∑i=1n‍ξi∑j≠i‍2σij2(ej2(k)-ej2(k-τij)).
From (27)–(29), it follows that
(30)𝔼[V(ek+1)∣ℱk]-V(ek)00=∑i=1n‍ξi[∑j=1n‍cij2ej2(k)0000000000+∑j=1n‍∑l>j‍cijcil(ej2(k)-ej2(k-τij)00000000000000000000000+el2(k)-el2(k-τil)00000000000000000000000+2ej(k-τij)el(k-τil))∑j=1n‍]0000-∑i=1n‍ξiei2(k)0000+∑i=1n‍ξi[∑j≠i‍σij2ei2(k)+∑j≠i‍σij2ej2(k)000000000000+∑j≠i‍σij2(ej2(k)-ej2(k-τij)00000000000000000000-2ei(k)ej(k-τij))∑j≠i‍]00=∑i=1n‍ξi[∑j=1n‍cij2ej2(k)00000000000+∑j=1n‍∑l>j‍cijcil((el(k-τil)ej(k-τij))2ej2(k)+el2(k)000000000000000000000000-(ej(k-τij)000000000000000000000000000-el(k-τil)ej(k-τij))2)∑j=1n‍]0000-∑i=1n‍ξiei2(k)0000+∑i=1n‍ξi[∑j≠i‍σij2ei2(k)+∑j≠i‍σij2ej2(k)0000000000000+∑j≠i‍σij2((ei(k)+ej(k-τij))2ei2(k)+ej2(k)000000000000000000000-(ei(k)+ej(k-τij))2)∑j≠i‍]00≤∑i=1n‍ξi[∑j=1n‍cij2ej2(k)00000000000+∑j=1n‍∑l>j‍cijcil(ej2(k)+el2(k))]0000-∑i=1n‍ξiei2(k)0000+2∑i=1n‍ξi[∑j≠i‍σij2ei2(k)+∑j≠i‍σij2ej2(k)].
A straightforward computation yields that
(31)∑i=1n‍ξi[∑j=1n‍cij2ej2(k)+∑j=1n‍∑l>j‍cijcil(ej2(k)+el2(k))]0000=∑i=1n‍ξi∑j=1n‍∑l=1n‍cijcilej2(k).
Noticing the equality∑j=1n‍b-ij=1 (see (7)), it follows readily that
(32)∑i=1n‍ξi∑j=1n‍∑l=1n‍cijcilej2(k)0000=∑i=1i0‍ξi∑j=1n‍∑l=1n‍cijcilej2(k)000000+∑i=i0+1n‍ξi∑j=1n‍∑l=1n‍cijcilej2(k)0000=∑i=1i0‍ξi∑j=1n‍∑l=1n‍b-ijb-ilej2(k)000000+∑i=i0+1n‍ξi∑j=1n‍∑l=1n‍b-ijc-ilej2(k)000000-∑i=1i0‍αiξi[∑j=1n‍b-ijej2(k)+(1-αi)ei2(k)]0000=∑i=1n‍ξi∑j=1n‍∑l=1n‍b-ijb-ilej2(k)000000-∑i=1i0‍αiξi[∑j=1n‍b-ijej2(k)+(1-αi)ei2(k)]0000=∑i=1n‍ξi∑j=1n‍b-ijej2(k)000000-∑i=1i0‍αiξi[∑j=1n‍b-ijej2(k)+(1-αi)ei2(k)]0000=∑i=1n‍ξiei2(k)000000-∑i=1i0‍αiξi[∑j=1n‍b-ijej2(k)+(1-αi)ei2(k)].
Substituting (31) and (32) into (30) yields that
(33)𝔼[V(ek+1)∣ℱk]-V(ek)0000≤2∑i=1n‍ξi[∑j≠i‍σij2ei2(k)+∑j≠i‍σij2ej2(k)]000000-∑i=1i0‍αiξi[∑j=1n‍b-ijej2(k)+(1-αi)ei2(k)].
It is easy to see that
(34)∑i=1n‍ξi[∑j≠i‍σij2ei2(k)+∑j≠i‍σij2ej2(k)]≤ξ-∑i=1n‍∑j≠i‍σij2ei2(k)+ξ-∑i=1n‍∑j≠i‍σij2ej2(k)≤ξ-∑i=1n‍∑j≠i‍σij2ei2(k)+ξ-∑j=1n‍∑i≠j‍σij2ej2(k)≤2(n-1)ξ-σ-2∑i=1n‍ei2(k),
and
(35)∑i=1i0‍αiξi[∑j=1n‍b-ijej2(k)+(1-αi)ei2(k)]=∑i=1i0‍αiξi[∑j∈𝒩i‍b-ijej2(k)0000000000000+(1+b-ii-αi)ei2(k)∑j∈Ni‍]≥α_ξ_γ∑i=1i0‍∑j∈𝒩^i‍ej2(k)≥α_ξ_γ∑i=1n‍ei2(k).
Substituting (34) into (33) results in
(36)𝔼[V(ek+1)∣ℱk]-V(ek)≤(4(n-1)ξ-σ-2-α_ξ_γ)|e(k)|2,
which implies that
(37)𝔼[V(ek+1)]-𝔼[V(ek)]≤(4(n-1)ξ-σ-2-α_ξ_γ)𝔼[|e(k)|2].
Employing the Lyapunov stability theory, we can deduce that limk→∞𝔼[V(ek)]=0. This completes the proof of the theorem.Remark 8.
In Theorem7, the condition σ-<α_ξ_γ/4(n-1)ξ¯ always holds when σ- 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
(38)xi(k+1)=xi(k)+∑j∈𝒩i‍aij(xj(k-τij)-xi(k)),i∈𝒱,(39)xi(k+1)={xi(k)+∑j∈𝒩i‍aij(xj(k-τij)-xi(k))-αi(xi(k)-x*),1≤i≤i0,xi(k)+∑j∈𝒩i‍aij(xj(k-τij)-xi(k)),i0+1≤i≤n.
In this case, σ-=0; accordingly, the condition σ-<α_ξ_γ/4(n-1)ξ¯ is always satisfied, and from Theorem 7, we have the following corollary.Corollary 9.
Consider the multiagent systems (38) and (39). Under Assumptions 3 and 5, the multiagent network (38) reaches leader-following consensus on the state target x*.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 ofn nodes with vector-valued states:
(40)xi(k+1)=xi(k)+∑j∈𝒩i‍aij(k)(xj(k-τij)-xi(k)),i∈𝒱,
and the controlled network is given by
(41)xi(k+1)={xi(k)+∑j∈𝒩i‍aij(k)(xj(k-τij)-xi(k))-αi(xi(k)-x*),1≤i≤i0,xi(k)+∑j∈𝒩i‍aij(k)(xj(k-τij)-xi(k)),i0+1≤i≤n,
where x(k)∈ℝn. We have the following results.Theorem 10.
Consider the multiagent systems (40) and (41). Suppose that Assumptions 3 and 5 are satisfied, and assume that σ-<α_ξ_γ/4(n-1)ξ¯ holds. Then, the multiagent network (40) reaches the leader-following consensus on the state target x* in mean-square.Proof.
The proof of this theorem is similar to that of Theorem7. The minor modification is to replace some scalar multiplication operations by the Kronecker product of matrices, and we omit the details here.


## 4. 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 n=5. The interaction topology between the agents is shown in Figure 1(a), and other parameters are taken as follows:
(42)B-=[b-ij]5×5=[0.400.20.20.20.50.500000.50.500000.50.500000.50.5],[τij]5×5=[0051250000030000030000040].
Clearly, the network topology is strongly connected, and it is also obvious that N^1⋃𝒩^2=𝒱. Therefore, we can choose i0=2. Assume that σ13=σ14=σ15=σ21=σ32=σ43=σ54=0.03, α1=α2=0.38, and x*=5. By a straightforward computation, we can get that ξ=[0.1923,0.2885,0.2885,0.1538,0.0769]T, and it is also easy to see that σ-=0.03, α_=0.38, γ=0.2, ξ¯=0.2885, and ξ_=0.0769. In this case, α_ξ_γ/4(n-1)ξ¯=0.0398, and σ-<α_ξ_γ/4(n-1)ξ¯ the multiagent. Therefore, by Theorem 7, network (3) reaches the leader-following consensus on an anticipated state target in mean-square. With the above parameters and a set of initial values produced in a stochastic way, the numerical simulation shown in Figure 1(b) matches well with the theoretical results.Numerical simulation.
(a)
The interaction topology of multiagent system
(b)
Consensus on an anticipated state target


## 5. 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.



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*Source: 158731-2013-06-26.xml*