# Networked Convergence of Fractional-Order Multiagent Systems with a Leader and Delay

Authors: Yuntao Shi; Junjun Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper investigates the convergence of fractional-order discrete-time multiagent systems with a leader and sampling delay by using Hermite-Biehler theorem and the change of bilinearity. It is shown that such system can achieve convergence depending on the sampling intervalh, the fractional-order α, and the sampling delay τ and its interconnection topology. Finally, some numerical simulations are given to illustrate the results.


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## Body

## 1. Introduction

Recently, more and more scholars focus on the coordinated control [1, 2] of multiagent systems such as the consensus [3–5] and the controllability [6–8]. However, most of the practical distribution systems are fractional order [9–12]. Recently, with the development of society, fractional-order calculus theory [13–16] is widely used to study the signal processing and control, picture processing and artificial intelligence, and so on. The consensus of multiagent systems refers to the fact that agents in the system can transfer information and influence each other according to a certain protocol or algorithm, and eventually agents will tend to the consensus behavior with the evolution of the time in [17]. In fact, for most of multiagent systems, there widely exist time delays as in [18]. So the property of multiagent systems with time delays has always been the hot problem. In [19], the authors studied consensus of multiagent systems with heterogeneous delays and leader-following with integer-order and continuous time. In [20], the paper considered the consensus of fractional-order multiagent systems with sampling delays without the leader.However, for a complex environment, multiagent systems with fractional-order can be better to describe some real natural phenomena. Some basic issues of fractional-order multiagent systems with time delay, such as the convergence, are still lacking in studying. Specially, for a fractional-order multiagent system, which depends crucially on sampling intervalh, the fractional-order α, and its interconnection topology, therefore, it is more difficult to study the convergence of the fractional-order multiagent system.In this paper, we consider the convergence of fractional-order discrete-time multiagent systems with a leader and sampling delay. The leader plays the role of an external input or signal to followers, and the followers update their states based on the information available from their neighbors and the leader. We will establish convergence conditions and discuss relations among sampling intervalh, the fractional-order α, its sampling delay τ, and its interconnection topology of such network.The remainder of this paper is organized as follows. Section2 gives the model and some preliminaries. Section 3 presents the main results, and some simulations are given in Section 4. Finally, Section 5 gives the conclusion.


## 2. Preliminaries and Problem Statement

In this section, we introduce some useful concepts and notations about the definition of fractional derivative [21], graph theory, and convergence of the multiagent systems.Denote a directed graph asG = (V,E,A) consisting of a nonempty set of vertices V and E={(i,j):i,j∈V} is a set of edges, where (i,j) means an arc starts from i and ends by j. If i,j∈V and (i,j)∈E, then we say that i and j are adjacent or j is a neighbor of i. We make Ni={i∈V:(i,j)∈E} be the neighborhood set of node i. A=[aij] is an adjacency matrix of graph G, where aij≥0 is the coupling weight between any two agents. D=diag⁡{d1,d2,…,dn}∈Rn×n is a degree matrix of G; its diagonal elements di=∑j∈Niaij, i=1,2,…,n, for the graph. Then the Laplacian of the weighted graph G is defined as (1)L=D-A∈Rn×n.The agent i is a globally reachable agent if it has paths to all of other agents.Definition 1 (see [17]).
Assume that, for arbitrary given initial values, if(2)limt→∞⁡xit-six0t=0,i∈N, where xi(t)∈Rn is the state value of agent of the multiagent system i (i∈N, N presents an index set (1,2,…,N)), x0(t)∈Rn, and si is a constant which is changed with different i. Then we have that the multiagent system is convergence.Definition 2 (see [21] (Grunwald-Letnikov)).
For any real numberα, the integer part written for α is [α]. If the function f(t) has continuous (m+1)-order derivative in the interval [α,t] and m equals [α] at last when α>0, then let α-order derivative be (3)fαt=limh→0⁡h-α∑i=0t-α/h-1i-αift-ih.Consider a multiagent system is composed ofN+1 agents, where the first N (labeled from 1 to N) are followers and the remainder agent N+1 (labeled 0) is leader. The fractional-order discrete-time multiagent system with a leader and sampling time is described by (4)xik+1=αxik+hαuik,x0k+1=x0k,where(5)uik=∑j∈Niaijxjk-1-xik-1+bi0x0k-1-xik-1,t∈kh,kh+τ,∑j∈Niaijxjk-xik+bi0x0k-xik,t∈kh+τk+1h+τ,α∈(0,1), xi∈Rn is the state of follower i (i∈N, N presents an index set (1,2,…,N)), and x0∈Rn is the state of the leader. Ni is the neighbor set of agent i. aij≥0, bi0≥0 represent the coupling information between followers and from the leader to the followers, respectively; otherwise, aij=0 and bi0=0; h>0 is the sampling interval and the sampling interval is h and the sampling delay is 0<τ<h.


## 3. Main Results

LetX(k)=(x1(k),x2(k),…,xN(k))T and X0(k)=X0(k+1) be the state vectors of all the followers and the leader, respectively. Then such system can be rewritten as(6)Xk+1Xk=UXkXk-1+hαB1X0k0,where (7)U=αIn-hα1-τL+B-hατL+BIn0,IN is the N×N identity matrix, B=diag⁡{b10,b20,…,bN0}∈RN×N, 1=(1,1,…,1)T is the N×1 identity matrix, and L=[lij]∈RN×N is Laplacian matrix with (8)lij=-aij,i≠j,j∈Ni,∑j∈Niaij,i=j,0,otherwise.Lemma 3 ((Hermite-Biehler theorem) [22]).
Assume the polynomialq(s)=ρ0+ρ1s+⋯+ρnsn, marking q(jω)=m(ω)+jn(ω). So q(s) is Hurwitz stable if and only if the roots of m(ω)=0, m1<m2<⋯, and n(ω)=0, n1<n2<⋯ satisfy(1)
m(0)n′(0)-m′(0)n(0)>0;(2)
m1<n1<m2<n2<⋯ or n1<m1<n2<m2<⋯.Lemma 4 (see [23]).
M
2
=
ρ
(
M
), if M∈RN×N is a symmetrical matrix.Theorem 5.
Suppose system (4) is a symmetrical and directly weighted network and the leader is a globally reachable agent; then the state of each agent can converge to the spanning space of the leader’s state, if and only if τ<min⁡{1/2,h} and(9)h<min⁡1+αλn1-2τ1/α,1λnτα,where λn is the biggest eigenvalue of matrix (L+B).Proof.
The fractional-order multiagent systems with sampling delay can be convergence if and only ifU<1 which means the eigenvalues of matrix U are less than 1. Because the symmetrically directed weighted network at least has a globally reachable agent, (L+B) can be orthogonal similar to a diagonal matrix. There exists an orthogonal matrix P which makes (10)L+B=PΛP-1,where Λ=diag⁡{λ1,λ2,…,λn} and the eigenvalues of matrix (L+B) satisfy 0<λ1≤λ2≤⋯≤λn. Assuming that z is the eigenvalue of matrix U, then the characteristic polynomial of U is(11)det⁡zI2n×2n-U=det⁡zIn-αIn+hα1-τL+BhατL+B-InzIn=det⁡z2In-zαIn-hα1-τL+B+hατL+B=det⁡z2In-zαIn+zhα1-τ+hατL+B=∏i∈Nz2-zα+zhα1-τ+hατλi≜az.
Leta(z)=0; applying the double linear change z=(s+1)/(s-1), we can have(12)bs=1-α+hαλis2+2-2hατλis+1+α-hαλi+2hατλi.Since the network is symmetrical and directed, (L+B) can be orthogonally similar to a diagonal matrix, whose eigenvalues are all positive. Let s=jω; then (13)bjω=-1-α+hαλiω2+j2-2hατλiω+1+α-hαλi+2hατλi.Denote b(ω)=m(ω)+jn(ω), where (14)mω=-1-α+hαλiω2+1+α-hαλi+2hατλi,nω=2-2hατλiω.
Using Lemma3, system (4) can be asymptotic convergence if and only if we have the following: (1)
ifm0n′(0)-m′(0)n(0)>0, then 1+α-hαλi+2hατλi>0, 1-hατλi>0, and then (15)hα<1+αλi1-2τ,hα<1τλi;(2)
if1-hατλi≠0, then the roots of n(ω) and m(ω) are the same, where the roots of m(ω) satisfy (16)m2ω=1+α-hαλi+2hατλi1-α+hαλiwith 1+α-hαλi+2hατλi>0. Therefore, system (4) is asymptotic convergence.Theorem 6.
Suppose system (4) is a symmetrical and directly weighted network and the leader is a globally reachable agent; then the state of each agent can converge to the spanning space of the leader’s state, if ρ(A)<1, where A=hατ(L+B) and ρ(A) is spectral radius of matrix A.Proof.
The fractional-order multiagent systems with sampling delay can be convergence if and only ifU<1. Using Lemma 4, (17)U=αIn-hα1-τL+BInIn0In00-hατL+B≤αIn-hα1-τL+BInIn0In00-hατL+B=ραIn-hα1-τL+BInIn0ρIn00-hατL+BρA≤1.Remark 7.
Notice from Theorem6 that · is ·2 and ·2 is 2-norm.Remark 8.
Theorem5 describes the relation of the convergence of such system and time delay, while Theorem 6 describes the relation of the convergence of such system and spectral radius of matrix A.


## 4. Simulations

In this section, we will present numerical simulations to illustrate the theoretical results.Consider a multiagent system with nine agents and a leader, in which agent 0 is the leader and the rest are followers. The coupling matrix is defined as follows:(18)A=00.20.100.50.30.1000.200.300.100.20.100.10.300.40000.60.1000.400.200.30.70.20.50.100.200.70.1000.30000.700.20.10.30.10.200.30.10.200000.10.60.700.1000.2000.10.200.300.20,B=0.10000000000.7000000000000000000000.20000000000000000000.3000000000000000000000000000000.1.By computing, the eigenvalues of matrix L+B are (19)0.1277,0.6451,0.9012,1.0373,1.5498,1.8522,2.1573,2.3954,2.7340,respectively. Figures 1–5 show the trajectories of multiagent systems with random initial states and α=0.2, h=0.4, τ=0.3, α=0.1, h=0.4, τ=0.3, α=0.1, h=0.4, τ=0.2, α=0.5, h=0.6, τ=0.3, and α=0.8, h=0.9, τ=0.6, respectively. Under Theorem 5, Figures 1 and 2 are the convergence situation, and Figure 3 is case of divergence. Under Theorem 6, Figures 4 and 5 are the results of simulations under the conditions of convergence and divergence.Figure 1
The trajectories of nine agents in the dynamical network.Figure 2
The trajectories of nine agents in the dynamical network.Figure 3
The trajectories of nine agents in the dynamical network.Figure 4
The trajectories of nine agents in the dynamical network.Figure 5
The trajectories of nine agents in the dynamical network.


## 5. Conclusion

In this paper, we have investigated the convergence problem of the fractional-order discrete-time multiagent system with a leader and sampling delay. We have obtained the convergence results depending on the sampling intervalh, the fractional-order α, and the sampling delay.



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*Source: 314985-2015-10-19.xml*