# Consensus of Multi-Integral Fractional-Order Multiagent Systems with Nonuniform Time-Delays

Authors: Jun Liu; Wei Chen; Kaiyu Qin; Ping Li
Journal: Complexity
 (2018)

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## Abstract

Consensus of fractional-order multiagent systems (FOMASs) with single integral has been wildly studied. However, the dynamics with multiple integral (especially double integral to sextuple integral) also exist in FOMASs, and they are rarely studied at present. In this paper, consensus problems for multi-integral fractional-order multiagent systems (MIFOMASs) with nonuniform time-delays are addressed. The consensus conditions for MIFOMASs are obtained by a novel frequency-domain method which properly eliminates consensus problems of the systems associated with nonuniform time-delays. Besides, the method revealed in this paper is applicable to classical high-order multiagent systems which is a special case of MIFOMASs. Finally, several numerical simulations with different parameters are performed to validate the correctness of the results.


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## Body

## 1. Introduction

The research related to multiagent systems (MASs) has been going on for decades, due to its many meaningful applications, e.g., sweep coverage control of MASs [1], flocking behavior of mobile robots [2], and coordinated attitude control of a formation of satellites [3]. Consensus is an agreement on the quality of certain concerns about the specific states of all agents, which is one of the most fundamental requirements for the research on MASs.Up to now, numerous studies have been conducted to resolve the problems about consensus of MASs with different dynamics. During the past decades, a lot of results have been accomplished about consensus of first-order MASs [4–11]. In [4], a simple model was presented for the phase transition of a set of self-driven particle, and it was demonstrated that the headings of all agents in MASs converged to a common value by simulation. In [5], authors provided some theoretical explanations for Vicsek’s linearized model and analyzed the alignment of undirected switching topologies of agents that were regularly connected. Based on the research works of [5], more relaxed consensus conditions over dynamic switching topologies were given in [6, 7]. Authors in [8] put forward a framework about consensus theory of MASs with directed information flow, link/node failures, time-delays, and so on. Robust H∞ control about consensus problems for MASs with parameter uncertainties, external disturbances, nonidentical state, and time-delays was discussed in [9]. In recent years, more and more researchers have paid more attention to consensus of second-order MASs [12–16]. For instance, authors in [12] investigated consensus problems for second-order continuous-time MASs in the presence of jointly connected topologies and time-delay. In [13], two types of consensus problems for second-order MASs with and without delay over switching topology and directed topology were studied. In [14], the local consensus problem for second-order MASs whose dynamics were nonlinear dynamics under directed and switching random topology was discussed, and several sufficient conditions were derived to ensure the MASs reach consensus. Furthermore, taking into account the fact that high-order MASs were widely used, consensus problems for high-order MASs have been studied in [17–21]. In particular, the output consensus problem in [17] was addressed for high-order MASs with external disturbances, and some conditions were derived to ensure consensus for the MASs. The consensus problems in [18] were considered for a class of high-order MASs with time-delays and switching networks, and a nearest-neighbour rule was designed and some conditions were derived to guarantee consensus for the systems with time-delays. The conditions of consensus in [21] for high-order MASs with nonuniform time-delays were proposed by a novel frequency-domain approach which properly resolved the challenges associated with multiple time-delays.It is worth noting that many results above about MASs were based on the integer-order dynamics. In fact, many scholars have declared that the essential characteristic or behavior of an object in the complex environment could be better revealed by adopting fractional-order dynamics. Examples include unmanned aerial vehicles operating in an environment with the impacts of rain and wind [22], food searching with the help of the individual secretions and microbial [23], and submarine robots in the bottom of the sea with large amounts of microorganisms and viscous substances [24]. Compared to integer-order dynamics, fractional-order dynamics provided an excellent tool in the description of memory and hereditary properties [25, 26]. Moreover, authors in [27, 28] indicated that the integer-order systems were only the special examples of the fractional-order systems. Based on these facts, the research results on consensus of FOMASs with single integral in [29–34] have been continuously springing up in recent years. As we know, consensus problem of FOMASs was first proposed and investigated by Caoet al. [29]. Next, consensus control of FOMASs with time-delays was studied by Yanget al. [30, 31], where homogeneous dynamics and heterogeneous dynamics were used to illustrate the agent of system. In [32], consensus problem of linear FOMASs with input time-delay and the consensus problem of nonlinear FOMASs with input time-delay were investigated, respectively. In [33], consensus problems were studied for FOMASs with nonuniform time-delays. Meanwhile, by means of matrix theory tool, Laplace transform and graph theory tool, two delay margins were obtained as the consensus conditions. Lately, consensus of FOMASs with double integral was proposed in [35–39]. The consensus problem of FOMASs with double integral over fixed topology was studied in [35]. By applying Mittag-Leffler function, Laplace transform, and dwell time technique, consensus for FOMASs with double integral over switching topology was investigated in [36]. Based on the sliding mode estimator, consensus problem for FOMASs with double integral was studied in [37]. By means of matrix theory tool, Laplace transform, and graph theory tool, consensus problems for a FOMAS with double integral and time-delay were studied in [38]. Nevertheless, the above research results on the consensus problems of FOMASs with or without time-delays were based on the single-integral fractional-order or double-integral fractional-order dynamics. To this day, there is almost no research on consensus problems of MIFOMASs with time-delays, especially nonuniform time-delays.Motivated by above analysis, we extend FOMASs from single-integral fractional-order dynamics to multi-integral fractional-order ones in this paper. Consensus problems of FOMASs with multiple integral in the presence of nonuniform time-delays are studied. The main idea of this paper is to first obtain the characteristic polynomial of a MIFOMAS with imaginary eigenvalues through the model transformation of the system and then determine the stability conditions of the system according to this characteristic polynomial, so as to determine the consensus conditions of the system according to the stability conditions of the system. The consensus conditions of the MIFOMAS with nonuniform time-delays can be obtained by inequalities.The main contributions of this paper are as follows. Firstly, we consider multi-integral fractional-order dynamics. As far as we know, this paper is the first paper that studies consensus of MIFOMASs. Just as integer-order MASs have first-order (single-integral) MASs [4–11], second-order (double-integral) MASs [12–16], and high-order (multi-integral) MASs [17–21], FOMASs also have single-integral FOMASs [29–34], double-integral FOMASs [35–39], and MIFOMASs, which makes the overall theory of FOMASs perfect from single-integral to multi-integral FOMASs. In addition, single-integral and double-integral FOMASs are the special cases of MIFOMASs. Secondly, we consider symmetric and asymmetric time-delays. The symmetric time-delays contain up to n(n-1)/2 different values and the asymmetric time-delays contain up to n(n-1) different values when the MIFOMAS consists of n agents. Thirdly, we consider the dynamics of each agent containing multiple state variables with different fractional orders. The MIFOMAS with nonuniform time-delays consists of some agents, and each agent contains multiple state variables with different fractional orders. Finally, we derive the consensus conditions for a MIFOMAS with nonuniform time-delays.The remainder of this article is organized as follows. In Section2, fractional calculus and its Laplace transform are given. In Section 3, the knowledge about graph theory is shown out. In Sections 4 and 5, consensus algorithms for a MIFOMAS in the presence of nonuniform time-delays are studied. In Section 6, some numerical examples with different parameters are simulated to verify the results. Finally, conclusions are drawn out in Section 7.


## 2. Fractional Calculus

In [40], several different definitions of fractional calculus have been proposed, in which the Caputo fractional derivative played an important role in fractional-order systems. Because the initial value of Caputo fractional derivative has practical signification in many problems, which is commonly used in the variety of physical fields. Ergo, this paper will model the system dynamical characteristics by using Caputo derivative which is defined by(1)DtαaCxt=1Γm-α∫atxmηt-η1+α-mdη,where a∈R denotes the initial value, α represents the order of the Caputo derivative, and m-1<α≤m(m∈z+). Γ(·) is given by(2)Γy=∫0+∞e-tty-1dt.IfDtαaCx(t) is replaced by x(α)(t), and the Laplace transform of x(t) is represented by X(s)=L{x(t)}=∫0-∞e-stx(t)dt, then the following equation can be used to denote Laplace transform of the Caputo derivative.(3)Lxαt=sαXs-sα-1x0-,α∈0,1sαXs-sα-1x0--sα-2x′0-,α∈1,2where x(0-)=limt→0-x(t) and x′(0-)=limt→0-x′(t).


## 3. Graph Theory

For a MAS withn agents, the network topology can be denoted by a graph G=(V,E), where V={s1,…,sn} and E⊆V2, respectively, represent the set of nodes and the set of edges. The node indices belong to a finite index set I={1,2,…,n}. The weighted adjacency matrix is denoted by A=[aij]n×n. The element of the i-th row and the j-th column in matrix A indicates the connection state between agents si and sj. If nodes si and sj are connected, i.e., eij∈E, then aij>0, and sj is called a neighbor of node si. Ni={j∈I,j≠i} denotes the index set of all neighbors of agent i. If nodes si and sj are connected and aij=aji, then G is an undirected graph; otherwise the G is a directed graph. In a directed graph, a directed path is a sequence of edges by (s1,s2),(s2,s3),…, where (sj,si)∈E. The directed graph has a directed spanning tree if all other nodes have directional paths from the same node. The Laplacian matrix of the graph G is defined by L=Δ-A∈Rn×n, where Δ≜diagdegout(s1),degout(s2),…,degout(si),…,degout(sn) is a diagonal matrix with degout(si)=∑j=1naij. Supposing some graphs G1,G2,…,GM and graph G consist of the same nodes, and the edge set of graph G is the sum of the edge sets of other graphs G1,G2,…,GM, then there is L=∑m=1MLm, which means the Laplacian matrix of graph G is the sum of other graphs’ Laplacian matrix.


## 4. Problem Statement

There are two lemmas [41] for the later analysis.Lemma 1.
If graphG is an undirected and connected graph, then its Laplacian matrix L has one singleton zero eigenvalue and other eigenvalues are all positive; i.e., 0=λ1<λ2≤λ3≤⋯≤λn.Lemma 2.
If graphG is a directed graph with a spanning tree, then its Laplacian matrix L has one singleton zero eigenvalue and other eigenvalues have a positive real part; i.e., λ1=0,Re(λi)>0(i=2,3,…,n).Consider a MIFOMAS composed ofn agents. Each node in graph G corresponds to each agent of the MIFOMAS. If aij>0, we can think that the i-th agent can get state information from the j-th agent. The dynamics of the i-th agent of the MIFOMAS are represented by(4)xi1α1t=xi2t,xi2α2t=xi3t,⋮xil-1αl-1t=xilt,xilαlt=uit,i∈I,where xi1(t),xi2(t),…,xil(t)∈R, respectively, represent l different states of the i-th agent, xi1(α1)(t) is the α1-order Caputo derivative of xi1(t), xi2(α2)(t) is the α2-order Caputo derivative of xi2(t),…,xil(αl)(t) is the αl-order Caputo derivative of xil(t) (α1,α2,…,αl∈(0,1]), and ui(t)∈R is the control input.Definition 3.
If and only if the states of all agents in MIFOMAS (4) satisfy(5)limt→+∞xi1t-xj1t=0,limt→+∞xi2t-xj2t=0,⋮limt→+∞xilt-xjlt=0,i,j∈I,then MIFOMAS (4) can reach consensus. In (5), xi1(t),xi2(t),…,xil(t)∈R, respectively, represent l different states of the i-th agent, and xj1(t),xj2(t),…,xjl(t)∈R, respectively, represent l different states of the j-th agent.In this paper, the control protocol for MIFOMAS (4) will be given by(6)uit=-∑k=1l-1Pk+1xik+1t+∑j∈Niaijxj1t-τij-xi1t-τij,where Ni denotes the neighbors index collection of the agent i, aij is the (i,j)-th element of A, τij>0 is the time-delay which is from the j-th agent to the i-th agent, and Pk+1>0 are scale coefficients. If all τij=τji, then the time-delays are symmetrical; else the time-delays are asymmetrical. The symmetric time-delays and the asymmetric time-delays are two different forms of nonuniform time-delays. For ease of analysis, we define that τm denote M different time-delays of MIFOMAS (4); i.e., τm∈{τij:i,j∈I}(m=1,2,…,M). Then the following control protocol is provided to resolve consensus problems of MIFOMAS (4):(7)uit=-∑k=1l-1Pk+1xik+1t+∑j∈Niaijxj1t-τm-xi1t-τm.Assume the state vector of thei-th agent is ξi(t)=[xi1(t),xi2(t),…,xil(t)]T∈Rl, and the joint state vector of MIFOMAS (4) consisting of n agents is ψ(t)=[ξ1T(t),ξ2T(t),…,ξnT(t)]T∈Rln.If we define two matrices as follows,(8)A=010⋯00001⋯00⋮⋮⋮⋱⋮⋮000⋯10000⋯010-P2-P3⋯-Pl-1-Pl∈Rl×l,and (9)B=00⋯0⋮⋮⋱⋮00⋯010⋯0∈Rl×l,then under the control protocol given by (7), the closed-loop dynamics of MIFOMAS (4) can be described as(10)x11α1t,x12α2t,…,x1lαlt,x21α1t,x22α2t,…,x2lαlt,…,xn1α1t,xn2α2t,…,xnlαltT=In⊗Aψt-∑m=1MLm⊗B·ψt-τm.


## 5. Main Results

Theorem 4.
Suppose that a FOMAS with multiple integral is given by MIFOMAS (4) whose corresponding network topology G satisfies Lemma 1. Define the following functions:(11)Flω=-∑k=1l-jω∑i=1kαiPk+1,θlω=argFlω=arg-∑k=1l-jω∑i=1kαiPk+1=arctan⁡Rlω,Tlω=1ωθlω,where l∈{1,2,3,4,5,6}, Rl(ω)≜Im[Fl(ω)]/Re[Fl(ω)]=tan[θl(ω)], and Re[Fl(ω)] and Im[Fl(ω)], respectively, denote the real part and the imaginary part of Fl(ω).
If allτm<τ¯=Tl(ω¯)=1/ω¯θl(ω¯) for the MIFOMAS (10) with symmetric time-delays, then the control protocol (7) can resolve the consensus problem of the MIFOMAS (10) with symmetric time-delays, and on the contrary, then the control protocol (7) can not resolve the consensus problem of the MIFOMAS (10) with symmetric time-delays. The value of ω¯ corresponding to l in Tl(ω¯) is determined by the following equation:(12)Flω¯=λn,l∈1,2,3,4,5,6,where Flω is the modulus of Fl(ω) and λn is the maximum eigenvalue of L.Proof.
We shall apply the frequency-domain method to analyze the MIFOMAS (10) with symmetric time-delays, and we can get (13)In⊗sα10⋯000sα2⋯00⋮⋮⋱⋮⋮00⋯sαl-1000⋯0sαlΨs-In⊗sα1-10⋯000sα2-1⋯00⋮⋮⋱⋮⋮00⋯sαl-1-1000⋯0sαl-1ψ0--In⊗AΨs+∑m=1MLm⊗Be-sτmΨs=0,Ψs=Rψ0-Gτm-1s,where Ψ(s) is the Laplace transform of ψ(t), ψ(0-) is the initial value of ψ(t),(14)R=In⊗sα1-10⋯000sα2-1⋯00⋮⋮⋱⋮⋮00⋯sαl-1-1000⋯0sαl-1,and(15)Gτms=In⊗sα10⋯000sα2⋯00⋮⋮⋱⋮⋮00⋯sαl-1000⋯0sαl-010⋯0001⋯0⋮⋮⋮⋱⋮000⋯10-P2-P3⋯-Pl+∑m=1MLm⊗Be-sτm.
Motivated by the stability analysis of a fractional-order system in [42], we can study consensus of the MIFOMAS (10) with symmetric time-delays by analyzing the characteristic eigenvalues’ position of the characteristic polynomial det[Gτm(s)] of the MIFOMAS (10) with symmetric time-delays. Specifically, consensus of the MIFOMAS (10) without delays (all τm=0) is necessary for consensus of the MIFOMAS (10) with symmetric time-delays; that is to say, in this case the characteristic eigenvalues of det[Gτm(s)] of the MIFOMAS (10) with symmetric time-delays are all situated in the left half plane (LHP) of the complex plane, and as τm increases continuously from zero, the characteristic eigenvalue of det[Gτm(s)] of the MIFOMAS (10) with symmetric time-delays will change continuously from the LHP to the right half plane (RHP) of the complex plane. Once the characteristic eigenvalue of det[Gτm(s)] of the MIFOMAS (10) with symmetric time-delays reaches the RHP of the complex plane through the imaginary axis, the MIFOMAS (10) with symmetric time-delays will be unstable and can not achieve consensus. Ergo, we only need to consider the critical time-delay when the nonzero characteristic eigenvalue of det[Gτm(s)] of the MIFOMAS (10) with symmetric time-delays is just situated on the imaginary axis for the first time as τm increases continuously from zero, and the corresponding time-delay is just the delay margin τ¯ of the MIFOMAS (10) with symmetric time-delays.
Assumes=-jω≠0 is the characteristic eigenvalue of det[Gτm(s)] of the MIFOMAS (10) with symmetric time-delays on the imaginary axis, u=u1⊗[1,0,…,0,0]T+u2⊗[0,1,0,…,0]T+⋯+ul-1⊗[0,…,0,1,0]T+ul⊗[0,…,0,0,1]T is the corresponding eigenvector, and u=1, u1,u2,…,ul-1,ul∈Cn; we have the following equations: (16)In⊗-jωα10⋯000-jωα2⋯00⋮⋮⋱⋮⋮00⋯-jωαl-1000⋯0-jωαl-010⋯0001⋯0⋮⋮⋮⋱⋮000⋯10-P2-P3⋯-Pl+∑m=1MLm⊗Be-sτmu=0,In⊗-jωα1-10⋯00-jωα2-1⋯0⋮⋮⋮⋱⋮000⋯-10P2P3⋯Pl+-jωαl+∑m=1MLm⊗Be-sτmu=0,In⊗Ωu︸1+∑m=1MLm⊗Bejωτmu︸2=0,(17)1=In⊗Ωu=Inu1⊗Ω1,0,…,0T+Inu2⊗Ω0,1,0,…,0T+Inu3⊗Ω0,0,1,0,…,0T+⋯+Inul-1⊗Ω0,0,⋯,0,1,0T+Inul⊗Ω0,…,0,1T=Inu1⊗-jωα1,0,…,0T+Inu2⊗-1,-jωα2,0,…,0,P2T+Inu3⊗0,-1,-jωα3,0,…,0,P3T+⋯+Inul-1⊗0,…,0,-1,-jωαl-1,Pl-1T+Inul⊗0,…,0,-1,-jωαl+PlT=In⊗-jωα1u1,0,…,0T+In⊗-u2,-jωα2u2,0,…,0,P2u2T+In⊗0,-u3,-jωα3u3,0,…,0,P3u3T+⋯+In⊗0,…,0,-ul-1,-jωαl-1ul-1,Pl-1ul-1T+In⊗0,…,0,-ul,-jωαl+PlulT=In⊗-jωα1u1-u2,-jωα2u2-u3,…,-jωαl-1ul-1-ul,P2u2+P3u3+⋯+Pl-1ul-1+Plul+-jωαlulT,=In-jωα1u1-u2,In-jωα2u2-u3,…,In-jωαl-1ul-1-ul,InP2u2+P3u3+⋯+Pl-1ul-1+Plul+-jωαlulT,(18)2=∑m=1MLm⊗Bejωτmu=∑m=1MLm⊗Buejωτm=∑m=1MLmu1⊗B1,0,…,0T+Lmu2⊗B0,1,0,…,0T+⋯+Lmul-1⊗B0,0,…,0,1,0T+Lmul⊗B0,…,0,1Tejωτm=∑m=1MLmu1⊗0,0,…,0,1T+Lmu2⊗0,0,0,…,0T+⋯+Lmul-1⊗0,0,…,0,0,0T+Lmul⊗0,…,0,0Tejωτm=∑m=1MLm⊗Ilu1⊗0,0,…,0,1T+u2⊗0,0,…,0,0T+⋯+ul-1⊗0,0,…,0,0T+ul⊗0,0,…,0,0Tejωτm=∑m=1MLm⊗Il0,0,…,0,u1Tejωτm=0,0,…,0,∑m=1MLm⊗Ilu1ejωτmT.
Because of(1)+(2)=0, we have(19)In-jωα1u1-u2=0,In-jωα2u2-u3=0,⋮In-jωαl-1ul-1-ul=0,InP2u2+P3u3+⋯+Pl-1ul-1+Pl+-jωαlul+∑m=1MLm⊗Ilu1ejωτm=0,and it yields that(20)u2=-jωα1u1,u3=-jωα2u2,⋮ul=-jωαl-1ul-1,∑m=1MLm⊗Ilu1ejωτm=-P2u2+P3u3+⋯+Pl-1ul-1+Pl+-jωαlul.
According to (20), we have(21)∑m=1MLm⊗Ilu1ejωτm=-P2-jωα1+P3-jωα2+α1+⋯+Pl-1-jωαl-2+αl-3+⋯+α1+Pl-jωαl-1+αl-2+⋯+α1+-jωαl+αl-1+⋯+α1u1,then we can multiply both sides of (21) by uH (the conjugate transpose of u); the following equation can be obtained:(22)∑m=1MuHLm⊗Ilu1uHu1ejωτm=-P2-jωα1+P3-jωα2+α1+⋯+Pl-1-jωαl-2+αl-3+⋯+α1+Pl-jωαl-1+αl-2+⋯+α1+-jωαl+αl-1+⋯+α1=-∑k=1l-jω∑i=1kαiPk+1,where l⩾1,Pl+1=1.
Due to(23)u=u1,u2,u3,…,ulT=u1,-jωα1u1,-jωα1+α2u1,…,-jωαl-1+αl-2+⋯+α1u1T,u1=u1,-jωα1,-jωα1+α2,…,-jωαl-1+αl-2+⋯+α1T, (22) can be simplified to(24)∑m=1MuHLm⊗Ilu1uHu1ejωτm=∑m=1MuHLm⊗IluuHuejωτm=-∑k=1l-jω∑i=1kαiPk+1≜Flω,where l⩾1,Pl+1=1.
LetuH(Lm⊗Il)u/(uHu)=am; we take modulus of both sides of (24). According to Lemma 1, we can get the following inequality:(25)Mlω=Flω=∑m=1Mamejωτm≤∑m=1Mam=uHL⊗IluuHu≤Mlω¯=Flω¯=λn.It is obvious that Ml(ω) is an increasing function for ω>0, and if ω≤ω¯, we can get Ml(ω)≤Ml(ω¯)=Fl(ω¯)=λn; that is, inequality (25) is true.
According to (24), we can get(26)argFlω=arg∑m=1Mamejωτm≜θlω,=arctan⁡Rlω,where θl(ω) is the principal value of the argument of Fl(ω), Rl(ω)≜Im[Fl(ω)]/Re[Fl(ω)]=tan[θl(ω)], and Re[Fl(ω)] and Im[Fl(ω)], respectively, denote the real part and the imaginary part of Fl(ω).
According to (26), it is easy to obtain that(27)θlω=arg∑m=1Mamejωτm≤maxωτm.
Consider a FOMAS with single integral; we haveF1(ω)=-(-jω)α1P2=ωα1ejπ(2-α1)/2P2. It is apparent that θ1(ω)=π(2-α1)/2, and M1(ω)=ωα1P2(P2=1). According to (25), we should only consider ω≤ω¯=λn(1/α1), and if all τm<τ¯=T1(ω¯)=T1[λn(1/α1)]=1/ω¯θ1(ω¯)=π(2-α1)/[2λn(1/α1)], then ωτm<λn(1/α1)τ¯=λn(1/α1)T1[λn(1/α1)]=π(2-α1)/2=θ1(ω), which contradicts to (27). Therefore, when all τm<τ¯, the characteristic eigenvalues of det[Gτm(s)] of the MIFOMAS (10) with symmetric time-delays are all situated in the LHP and the FOMAS with single integral will remain stable and can achieve consensus. On the contrary, the FOMAS with single integral will not remain stable and can not achieve consensus. Theorem 4 is proven for l=1.
In the following, the FOMAS with multiple integral (double integral to sextuple integral) shall be analyzed step by step. For convenience of analysis, we first need to define some symbolic parameters:(28)z1=ω2α1P22,z2=ω2α1+α2P32,z3=2ω2α1+α2cosπα22P2P3,z4=2ω2α1+α2+α3cosπα2+α32P2P4,z5=2ω2α1+α2+α3+α4cosπα2+α3+α42P2P5,z6=2ω2α1+α2+α3+α4+α5cosπα2+α3+α4+α52P2P6,z7=2ω2α1+α2+α3+α4+α5+α6cos⁡πα2+α3+α4+α5+α62P2P7,z8=2ω2α1+2α2+α3cosπα32P3P4,z9=2ω2α1+2α2+α3+α4cosπα3+α42P3P5,z10=2ω2α1+2α2+α3+α4+α5cosπα3+α4+α52P3P6,z11=2ω2α1+2α2+α3+α4+α5+α6cosπα3+α4+α5+α62P3P7,z12=ω2α1+α2+α3P42,z13=2ω2α1+2α2+2α3+α4cosπα42P4P5,z14=2ω2α1+2α2+2α3+α4+α5cosπα4+α52P4P6,z15=2ω2α1+2α2+2α3+α4+α5+α6cosπα4+α5+α62P4P7,z16=ω2α1+α2+α3+α4P52,z17=2ω2α1+2α2+2α3+2α4+α5cosπα52P5P6,z18=2ω2α1+2α2+2α3+2α4+α5+α6cosπα5+α62P5P7,z19=ω2α1+α2+α3+α4+α5P62,z20=2ω2α1+2α2+2α3+2α4+2α5+α6cosπα62P6P7,z21=ω2α1+α2+α3+α4+α5+α6P72,and(29)d1=α2ω2α1+α2-1P2P3sin⁡πα22,d2=α2+α3ω2α1+α2+α3-1P2P4sin⁡πα2+α32,d3=α2+α3+α4ω2α1+α2+α3+α4-1P2P5sin⁡πα2+α3+α42,d4=α2+α3+α4+α5ω2α1+α2+α3+α4+α5-1·P2P6sin⁡πα2+α3+α4+α52,d5=α2+α3+α4+α5+α6ω2α1+α2+α3+α4+α5+α6-1·P2P7sin⁡πα2+α3+α4+α5+α62,d6=α3ω2α1+2α2+α3-1P3P4sin⁡πα32,d7=α3+α4ω2α1+2α2+α3+α4-1P3P5sin⁡πα3+α42,d8=α3+α4+α5ω2α1+2α2+α3+α4+α5-1·P3P6sin⁡πα3+α4+α52,d9=α3+α4+α5+α6ω2α1+2α2+α3+α4+α5+α6-1·P3P7sin⁡πα3+α4+α5+α62,d10=α4ω2α1+2α2+2α3+α4-1P4P5sin⁡πα42,d11=α4+α5ω2α1+2α2+2α3+α4+α5-1P4P6sin⁡πα4+α52,d12=α4+α5+α6ω2α1+2α2+2α3+α4+α5+α6-1·P4P7sin⁡πα4+α5+α62,d13=α5ω2α1+2α2+2α3+2α4+α5-1P5P6sin⁡πα52,d14=α5+α6ω2α1+2α2+2α3+2α4+α5+α6-1P5P7sin⁡πα5+α62,d15=α6ω2α1+2α2+2α3+2α4+2α5+α6-1P6P7sin⁡πα62.
For the FOMAS with double integral,(30)F2ω=-∑k=12-jω∑i=1kαiPk+1=--jωα1P2+-jωα1+α2P3=ωα1ejπ2-α1/2P2+ωα1+α2ejπ2-α1-α2/2P3=ωα1P2cos⁡π2-α12+ωα1+α2P3cos⁡π2-α1-α22+jωα1P2sin⁡π2-α12+ωα1+α2P3sin⁡π2-α1-α22=ReF2ω+jIm⁡F2ω.(31)M2ω2=F2ω2=ω2α1P22+ω2α1+α2P32+2ω2α1+α2cosπα22P2P3=z1+z2+z3.Because of R2(ω)=Im[F2(ω)]/Re[F2(ω)], we can get the first derivative of R2(ω):(32)R2′ω=-α2ω2α1+α2-1P2P3sinπα2/2ωα1P2cosπ2-α1/2+ωα1+α2P3cosπ2-α1-α2/22=-d1ReF2ω2<0.
For the FOMAS with triple integral,(33)F3ω=-∑k=13-jω∑i=1kαiPk+1=--jωα1P2+-jωα1+α2P3+-jωα1+α2+α3P4=ωα1ejπ2-α1/2P2+ωα1+α2ejπ2-α1-α2/2P3+ωα1+α2+α3ejπ2-α1-α2-α3/2P4=ωα1P2cos⁡π2-α12+ωα1+α2P3cos⁡π2-α1-α22+ωα1+α2+α3P4cos⁡π2-α1-α2-α32+jωα1P2sin⁡π2-α12+ωα1+α2P3sin⁡π2-α1-α22+ωα1+α2+α3P4sin⁡π2-α1-α2-α32=ReF3ω+jIm⁡F3ω,(34)M3ω2=F3ω2=ω2α1P22+ω2α1+α2P32+2ω2α1+α2cos⁡πα22P2P3+2ω2α1+α2+α3cos⁡πα2+α32P2P4+2ω2α1+2α2+α3cos⁡πα32P3P4+ω2α1+α2+α3P42=z1+z2+z3+z4+z8+z12.
Because ofR3(ω)=Im[F3(ω)]/Re[F3(ω)], we can get the first derivative of R3(ω):(35)R3′ω=-α2ω2α1+α2-1P2P3sin⁡πα2/2+α2+α3ω2α1+α2+α3-1·P2P4sin⁡πα2+α3/2+α3ω2α1+2α2+α3-1P3P4sin⁡πα3/2ωα1P2·cos⁡π2-α1/2+ωα1+α2P3cos⁡π2-α1-α2/2+ωα1+α2+α3P4·cos⁡π2-α1-α2-α3/22=-d1+d2+d6ReF3ω2<0.
In a similar way, the[Ml(ω)]2 and Rl′(ω) of the FOMASs with quadruple integral to sextuple integral can be, respectively, calculated under the appropriate parameters, and they are as follows:(36)M4ω2=z1+z2+z3+z4+z5+z8+z9+z12+z13+z16,R4′ω=-d1+d2+d3+d6+d7+d10ReF4ω2<0.(37)M5ω2=z1+z2+z3+z4+z5+z6+z8+z9+z10+z12+z13+z14+z16+z17+z19,R5′ω=-d1+d2+d3+d4+d6+d7+d8+d10+d11+d13ReF5ω2<0.(38)M6ω2=z1+z2+z3+z4+z5+z6+z7+z8+z9+z10+z11+z12+z13+z14+z15+z16+z17+z18+z19+z20+z21,R6′ω=-d1+d2+d3+d4+d5+d6+d7+d8+d9+d10+d11+d12+d13+d14+d15ReF6ω2<0.
In summary, we have found that the first derivatives ofRl(ω)(2≤l≤6) listed above are negative values, and Rl′(ω)<0 means that Rl(ω)=tan[θl(ω)] are monotonically decreasing with the growth of ω. Then it can be deduced that the arguments θl(ω) also decrease monotonically and continuously about ω because the values of Fl(ω) vary smoothly. Evidently, we can analyze the features of θl(ω)(2≤l≤6) together.
If0<ω1<ω2, we have ω2/ω1>1, and because the arguments θl(ω) decrease monotonically and continuously about ω, we have θl(ω1)>θl(ω2); i.e., θl(ω1)/θl(ω2)>1. Thus,(39)Tlω1Tlω2=θlω1ω1·ω2θlω2=θlω1θlω2·ω2ω1>1, so we can get Tl(ω1)>Tl(ω2), which means Tl(ω) also decrease monotonically and continuously about ω. When ω≤ω¯, we have(40)Tlω≥Tlω¯=τ¯.
It is worth noting that inequality (40) can be obtained when the characteristic eigenvalue of det[Gτm(s)] of the MIFOMAS (10) is s=-jω≠0. If we let all τm<τ¯, then we can obtain the following inequality:(41)Tlω=θlωω=arg∑m=1Mamejωτmω≤maxωτmω<ωτ¯ω=τ¯.
Inequality (41) is in contradiction with inequality (40). Therefore, as long as all τm<τ¯, we can ensure all the characteristic eigenvalues of det[Gτm(s)] of the MIFOMAS (10) with symmetric time-delays are situated in the LHP, and the MIFOMAS (10) with symmetric time-delays will remain stable and can achieve consensus. On the contrary, the MIFOMAS (10) with symmetric time-delays will not be stable and can not achieve consensus. Theorem 4 is proven for l∈{2,3,4,5,6}.Remark 5.
Consensus of the MIFOMAS (10) without symmetric time-delays is necessary for consensus of this system with symmetric time-delays.Remark 6.
Althoughl∈{1,2,3,4,5,6} in Theorem 4 due to computational complexity, the value of l may be greater than 6 under the appropriate parameters.Corollary 7.
If we suppose that a FOMAS with multiple integral is given by MIFOMAS (4) whose corresponding network topology G satisfies Lemma 1 and α1=α2=⋯=αl=1, then the MIFOMAS (10) with symmetric time-delays can be transformed into high-order MAS with symmetric time-delays whose dynamic model is an integer-order dynamic model and the following functions can be obtained:(42)Flω=-∑k=1l-jωkPk+1,θlω=argFlω=arg-∑k=1l-jωkPk+1=arctan⁡Rlω,Tlω=1ωθlω,where l∈{1,2,3,4,5,6}, Rl(ω)≜Im[Fl(ω)]/Re[Fl(ω)]=tan[θl(ω)], and Re[Fl(ω)] and Im[Fl(ω)], respectively, denote the real part and the imaginary part of Fl(ω).
For the high-order MAS with symmetric time-delays, if allτm satisfy τm<τ¯=Tl(ω¯)=1/ω¯θl(ω¯), then the control protocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays, and on the contrary, then the control protocol (7) can not resolve the consensus problem for the high-order MAS with symmetric time-delays. The value of ω¯ corresponding to l in Tl(ω¯) is determined by the following equation:(43)Flω¯=λn,l∈1,2,3,4,5,6,where |Fl(ω)| is the modulus of Fl(ω) and λn is the maximum eigenvalue of L.Remark 8.
The dynamic model and the control protocol in Corollary7 were discussed in [21], and the conclusion in Corollary 7 is less conservative than that in [21]. The proof about Corollary 7 is the same as that of Theorem 4.Theorem 9.
Suppose that a FOMAS with multiple integral is given by MIFOMAS (4) whose corresponding network topology G satisfies Lemma 2. Define the following functions:(44)Blω=-∑k=1l-jω∑i=1kαiPk+1,Θlω=argBlω=arg-∑k=1l-jω∑i=1kαiPk+1=arctan⁡Rlω,Γlω=1ωΘlω-argλi, where l∈{1,2,3,4,5,6}, arg(λi)∈(-π/2,π/2), λi is the i-th eigenvalue of L, Rl(ω)≜Im[Bl(ω)]/Re[Bl(ω)]=tan[Θl(ω)], and Im[Bl(ω)] and Re[Bl(ω)] denote the imaginary part and real part of Bl(ω), respectively.
If allτm satisfy τm<τ¯=min|λi|≠0⁡{Γl(ω¯i)} for the MIFOMAS (10) with asymmetric time-delays, then the control protocol (7) can resolve the consensus problem of the MIFOMAS (10) with asymmetric time-delays, and on the contrary, then the control protocol (7) can not resolve the consensus problem of the MIFOMAS (10) with asymmetric time-delays. The value of ω¯i corresponding to l in Γl(ω¯i) is determined by the following equation:(45)Blω¯i=λi,l∈1,2,3,4,5,6, where |Bl(ω¯i)| is the modulus of Bl(ω¯i).Proof.
By adopting the proof method similar to Theorem4, we suppose that s=-jω≠0 is the characteristic eigenvalue of det[Gτm(s)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis, u=u1⊗[1,0,…,0,0]T+u2⊗[0,1,0,…,0]T+⋯+ul-1⊗[0,…,0,1,0]T+ul⊗[0,…,0,0,1]T is the corresponding eigenvector, and u=1, u1,u2,…,ul-1,ul∈Cn. According to Lemma 2, we can get the following equation:(46)Blω≜∑m=1Mamejωτm=-∑k=1l-jω∑i=1kαiPk+1.
Take modulus of both sides of (46); Blω is an increasing function for ω>0; thus ω(|Bl(ω)|) is also an increasing function for Blω.
Calculate the principal value of the argument of (46); we have(47)Θlω≜argBlω=arg-∑k=1l-jω∑i=1kαiPk+1=arctan⁡Rlω,where Rl(ω)≜Im[Bl(ω)]/Re[Bl(ω)]=tan[Θl(ω)] and Im[Bl(ω)] and Re[Bl(ω)] denote the imaginary part and real part of Bl(ω), respectively.
According to the definition ofBl(ω) in (46), we have(48)Θlω≤arg∑m=1Mam+maxωτm, so there is(49)maxωτm≥Θlω-arg∑m=1Mam.
Due to∑m=1Mam=uH(L⊗Il)u/(uHu), the possible values of ∑m=1Mam must be nonzero eigenvalues of L of graph G; i.e., ∑m=1Mam=λi(λi≠0), which makes the delay margin τ¯ minimized. So when |Bl(ω)|≤|λi|, ω|Bl(ω)|≤ω|λi|=ω¯i. If we let all τm<τ¯, there is(50)maxωτm<ω¯iτ¯=ω¯iminλi≠0Γlω¯i=minλi≠0Θlω¯i-argλiω¯iω¯i≤Θlω-arg∑m=1Mam.
Inequality (50) is in contradiction with inequality (49). Therefore, as long as all τm<τ¯, the characteristic eigenvalues of det[Gτm(s)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axis, then the MIFOMAS (10) with asymmetric time-delays will remain stable and can achieve consensus. On the contrary, the MIFOMAS (10) with asymmetric time-delays will not be stable and can not achieve consensus. Theorem 9 is proven.Remark 10.
Consensus of the MIFOMAS (10) without asymmetric time-delays is necessary for consensus of this system with asymmetric time-delays.Remark 11.
Althoughl∈{1,2,3,4,5,6} in Theorem 9 due to computational complexity, the value of l may be greater than 6 under the appropriate parameters.Corollary 12.
If we suppose that a FOMAS with multiple integral is given by MIFOMAS (4) whose corresponding network topology G satisfies Lemma 2 and α1=α2=…=αl=1, then the MIFOMAS (10) with asymmetric time-delays can be transformed into high-order MAS with asymmetric time-delays whose dynamic model is an integer-order dynamic model and the following functions can be obtained:(51)Blω=-∑k=1l-jωkPk+1,Θlω=argBlω=arg-∑k=1l-jωkPk+1=arctan⁡Rlω,Γlω=1ωΘlω-argλi,where l∈{1,2,3,4,5,6}, arg(λi)∈(-π/2,π/2), λi is the i-th eigenvalue of L, Rl(ω)≜Im[Bl(ω)]/Re[Bl(ω)]=tan[Θl(ω)], and Im[Bl(ω)] and Re[Bl(ω)] denote the imaginary part and real part of Bl(ω), respectively.
For the high-order MAS with asymmetric time-delays, if allτm satisfy τm<τ¯=minλi≠0⁡{Γl(ω¯i)}, then the control protocol (7) can resolve the consensus problem for the high-order MAS with asymmetric time-delays, and on the contrary, then the control protocol (7) can not resolve the consensus problem for the high-order MAS with asymmetric time-delays. The value of ω¯i corresponding to l in Γl(ω¯i) is determined by the following equation:(52)Blω¯i=λi,l∈1,2,3,4,5,6, where Blω¯i is the modulus of Bl(ω¯i).


## 6. Simulation Results

The correctness and validity of the theoretical results for Theorems4 and 9 will be verified by some numerical simulations in this section. Under different network topologies, the FOMAS with different multiple integral will be considered.First of all, to validate Theorem4, we consider a FOMAS composed of 4 agents. Figure 1 shows the network topology depicted with a connected and undirected graph G, and Figure 1 has five different time-delays which are symmetric time-delays and it shows full connectivity. All the delays are marked with τij, where i and j are the indexes, which are used to represent the connected agents i and j. If we suppose the weight of each edge of graph G in Figure 1 is 1, then the adjacency matrix and the corresponding Laplacian matrix of G are, respectively, (53)A=2101131101211113,L=2-10-1-13-1-10-12-1-1-1-13,where λn=4 is the maximum eigenvalue of L.Figure 1
The undirected network topology.Example 1.
For a FOMAS with double integral and symmetric time-delays under the undirected graph, let us setα1=0.9,α2=0.8,αl=0(3≤l≤6), and P2=2.5,P3=1,P4=P5=P6=P7=0; thus the delay margin τ¯=1.015s according to Theorem 4. Two groups of symmetric time-delays are set: τ12=τ21=1.01s,τ14=τ41=1.00s,τ23=τ32=0.99s,τ24=τ42=0.98s,τ34=τ43=0.97s, which are bounded by the delay margin τ¯; τ12=τ21=1.02s,τ14=τ41=1.03s,τ23=τ32=1.04s,τ24=τ42=1.05s,τ34=τ43=1.06s, which exceed the delay margin τ¯. The simulation results about Example 1 are displayed in Figures 2 and 3: the two subfigures in Figure 2 show the trajectories of all agents’ states when all symmetric time-delays are less than the delay margin τ¯, which indicates that the FOMAS with double integral and symmetric time-delays is stable and consensus of the FOMAS with double integral and symmetric time-delays can be reached; the two subfigures in Figure 3 show the trajectories of all agents’ states when all symmetric time-delays exceed the delay margin τ¯, which indicates that the FOMAS with double integral and symmetric time-delays is unstable and consensus of the FOMAS with double integral and symmetric time-delays can not be reached.Figure 2
The trajectories of all agents’ states in the FOMAS with double integral and symmetric time-delays when allτm<τ¯ in Example 1.Figure 3
The trajectories of all agents’ states in the FOMAS with double integral and symmetric time-delays when allτm>τ¯ in Example 1.Example 2.
For a FOMAS with triple integral and symmetric time-delays under the undirected graph, let us setα1=0.9,α2=0.8,α3=0.7,αl=0(4≤l≤6), and P2=2.5,P3=9,P4=1,P5=P6=P7=0; thus τ¯=1.398s according to Theorem 4. Two groups of symmetric time-delays are set: τ12=τ21=1.30s,τ14=τ41=1.25s,τ23=τ32=1.20s,τ24=τ42=1.15s,τ34=τ43=1.10s, which are bounded by the delay margin τ¯; τ12=τ21=1.40s,τ14=τ41=1.41s,τ23=τ32=1.42s,τ24=τ42=1.43s,τ34=τ43=1.44s, which exceed the delay margin τ¯. The simulation results about Example 2 are displayed in Figures 4 and 5: the three subfigures in Figure 4 show the trajectories of all agents’ states when all symmetric time-delays are less than the delay margin τ¯, which indicates that the FOMAS with triple integral and symmetric time-delays is stable and consensus of the FOMAS with triple integral and symmetric time-delays can be reached; the three subfigures in Figure 5 show the trajectories of all agents’ states when all symmetric time-delays exceed the delay margin τ¯, which indicates that the FOMAS with triple integral and symmetric time-delays is unstable and consensus of the FOMAS with triple integral and symmetric time-delays can not be reached.Figure 4
The trajectories of all agents’ states in the FOMAS with triple integral and symmetric time-delays when allτm<τ¯ in Example 2.Figure 5
The trajectories of all agents’ states in the FOMAS with triple integral and symmetric time-delays when allτm>τ¯ in Example 2.Example 3.
For a FOMAS with sextuple integral and symmetric time-delays under the undirected graph, let us setα1=0.9,α2=0.8,α3=0.7,α4=0.6,α5=0.5,α6=0.4, and P2=2.5,P3=9,P4=4,P5=8,P6=1.2,P7=1; thus τ¯=0.2515s according to Theorem 4. Two groups of symmetric time-delays are set: τ12=τ21=0.24s,τ14=τ41=0.22s,τ23=τ32=0.20s,τ24=τ42=0.18s,τ34=τ43=0.16s, which are bounded by the delay margin τ¯; τ12=τ21=0.26s,τ14=τ41=0.27s,τ23=τ32=0.28s,τ24=τ42=0.29s,τ34=τ43=0.30s, which exceed the delay margin τ¯. The simulation results about Example 3 are displayed in Figures 6 and 7: the six subfigures in Figure 6 show the trajectories of all agents’ states when all symmetric time-delays are less than the delay margin τ¯, which indicates that the FOMAS with sextuple integral and symmetric time-delays is stable and consensus of the FOMAS with sextuple integral and symmetric time-delays can be reached; the six subfigures in Figure 7 show the trajectories of all agents’ states when all symmetric time-delays exceed the delay margin τ¯, which indicates that the FOMAS with sextuple integral and symmetric time-delays is unstable and consensus of the FOMAS with sextuple integral and symmetric time-delays can not be reached.
Next, to examine Theorem9, we give a network topology described in Figure 8, which is a directed graph G with a spanning tree. It also contains five different time-delays which are asymmetric time-delays and displays full connectivity. If we suppose the weight of each edge of graph G in Figure 8 is 1, then the adjacency matrix and the corresponding Laplacian matrix are (54)A=2101010101100011,L=2-10-1010-10-11000-11,where λ1=2,λ2=0,λ3=1.5+0.866i, and λ4=1.5-0.866i are all eigenvalues of L.Figure 6
The trajectories of all agents’ states in the FOMAS with sextuple integral and symmetric time-delays when allτm<τ¯ in Example 3.Figure 7
The trajectories of all agents’ states in the FOMAS with sextuple integral and symmetric time-delays when allτm>τ¯ in Example 3.Figure 8
The directed network topology.Example 4.
For a FOMAS with double integral and asymmetric time-delays under the directed graph, let us setα1=0.9,α2=0.8,αl=0(3≤l≤6), and P2=2.5,P3=1,P4=P5=P6=P7=0; thus τ¯=1.649s according to Theorem 9. Two groups of asymmetric time-delays are set: τ12=1.64s,τ14=1.63s,τ24=1.62s,τ32=1.61s,τ43=1.60s, which are bounded by the delay margin τ¯; τ12=1.66s,τ14=1.67s,τ24=1.68s,τ32=1.69s,τ43=1.70s, which exceed the delay margin τ¯. The simulation results about Example 4 are displayed in Figures 9 and 10: the two subfigures in Figure 9 show the trajectories of all agents’ states when all asymmetric time-delays are less than the delay margin τ¯, which indicates that consensus of the FOMAS with double integral and asymmetric time-delays can be reached; the two subfigures in Figure 10 show the trajectories of all agents’ states when all asymmetric time-delays exceed the delay margin τ¯, which indicates that consensus of the FOMAS with double integral and asymmetric time-delays can not be reached.Figure 9
The trajectories of all agents’ states in the FOMAS with double integral and asymmetric time-delays when allτm<τ¯ in Example 4.Figure 10
The trajectories of all agents’ states in the FOMAS with double integral and asymmetric time-delays when allτm>τ¯ in Example 4.Example 5.
For a FOMAS with triple integral and asymmetric time-delays under the directed graph, let us setα1=0.9,α2=0.8,α3=0.7,αl=0(4≤l≤6), and P2=2.5,P3=9,P4=1,P5=P6=P7=0; thus τ¯=1.327s according to Theorem 9. Two groups of asymmetric time-delays are set: τ12=1.32s,τ14=1.29s,τ24=1.26s,τ32=1.23s,τ43=1.21s, which are bounded by the delay margin τ¯; τ12=1.33s,τ14=1.36s,τ24=1.39s,τ32=1.42s,τ43=1.45s, which exceed the delay margin τ¯. The simulation results about Example 5 are displayed in Figures 11 and 12: the three subfigures in Figure 11 show the trajectories of all agents’ states when all asymmetric time-delays are less than the delay margin τ¯, which indicates that consensus of the FOMAS with triple integral and asymmetric time-delays can be reached; the three subfigures in Figure 12 show the trajectories of all agents’ states when all asymmetric time-delays exceed the delay margin τ¯, which indicates that consensus of the FOMAS with triple integral and asymmetric time-delays can not be reached.Figure 11
The trajectories of all agents’ states in the FOMAS with triple integral and asymmetric time-delays when allτm<τ¯ in Example 5.Figure 12
The trajectories of all agents’ states in the FOMAS with triple integral and asymmetric time-delays when allτm>τ¯ in Example 5.Example 6.
For a FOMAS with sextuple integral and asymmetric time-delays under the directed graph, let us setα1=0.9,α2=0.8,α3=0.7,α4=0.6,α5=0.5,α6=0.4, and P2=2.5,P3=9,P4=4,P5=8,P6=1.2,P7=1; thus τ¯=0.4335s according to Theorem 9. Two groups of asymmetric time-delays are set: τ12=0.43s,τ14=0.40s,τ24=0.37s,τ32=0.34s,τ43=0.31s, which are bounded by the delay margin τ¯; τ12=0.44s,τ14=0.47s,τ24=0.50s,τ32=0.53s,τ43=0.57s, which exceed the delay margin τ¯. The simulation results about Example 6 are displayed in Figures 13 and 14: the six subfigures in Figure 13 show the trajectories of all agents’ states when all asymmetric time-delays are less than the delay margin τ¯, which indicates that consensus of the FOMAS with sextuple integral and asymmetric time-delays can be reached; the six subfigures in Figure 14 show the trajectories of all agents’ states when all asymmetric time-delays exceed the delay margin τ¯, which indicates that consensus of the FOMAS with sextuple integral and asymmetric time-delays can not be reached.Figure 13
The trajectories of all agents’ states in the FOMAS with sextuple integral and asymmetric time-delays when allτm<τ¯ in Example 6.Figure 14
The trajectories of all agents’ states in the FOMAS with sextuple integral and asymmetric time-delays when allτm>τ¯ in Example 6.


## 7. Conclusion

The consensus problems of a FOMAS with multiple integral under nonuniform time-delays are studied in this paper. Taking into account two kinds of nonuniform time-delays, the sufficient conditions have been derived in the form of inequalities for the MIFOMAS with nonuniform time-delays. Numerical simulations of the MIFOMAS with nonuniform time-delays over undirected topology and directed topology are performed to verify these theorems. Finally, the simulation results show that the selected examples have achieved the desired results: the MIFOMAS with nonuniform time-delays under given conditions can achieve the consensus. With the help of the above research of this paper, distributed formation control of the MIFOMAS with nonuniform time-delays will be one of the most significant topics, which will be one of our future research tasks.



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*Source: 8154230-2018-11-01.xml*