# Consensus of Fractional-Order Multiagent Systems with Double Integrator under Switching Topologies

Authors: Shiyun Shen; Wenjing Li; Wei Zhu
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

Due to the complexity of the practical environments, many systems can only be described with the fractional-order dynamics. In this paper, the consensus of fractional-order multiagent systems with double integrator under switching topologies is investigated. By applying Mittag-Leffler function, Laplace transform, and dwell time technique, a sufficient condition on consensus is obtained. Finally, a numerical simulation is presented to illustrate the effectiveness of the theoretical result.


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## Body

## 1. Introduction

In recent years, cooperative control of multiagent systems has received compelling attention due to its wide applications such as minisatellites [1], unmanned vehicle [2], deployment [3], and rendezvous [4]. A significant and fundamental problem for cooperative control of multiagent systems (MASs) is consensus, which has broad applications in so many fields such as multivehicle systems [5], swarms, and flocks [6]. It was found that many MASs can be explained in the framework of fractional-order dynamics and the distributed control of fractional-order multiagent systems is becoming an important issue.Consensus of FMASs was early discussed by Cao and Ren in [7–9] and then attracted many researchers’ attention. In [10], by using the matrix theory, graph theory and the frequency-domain analysis method, the control algorithm with absolute damping and communication delay was proposed to discuss the distributed formation control of FMASs. In [11], a distributed control law with a constant reference state was given, which ensured the consensus of FMASs with a time-varying reference state. In [12], the fractional Lyapunov direct method and Mittag-Leffler stability were used to study the consensus of nonlinear FMASs with directed topology, some sufficient conditions were given to guarantee the consensus of nonlinear FMASs with/without a virtual leader. In [13], based on the connectivity of the graph and Riccati equation, the control gain matrix was designed and a sufficient condition on leader-following consensus of FMASs with general linear models was obtained. In [14, 15], the authors studied the leader-following consensus of FMASs with nonlinear dynamics by Lyapunov direct method, respectively. In [16], based on the properties of Mittag-Leffler function, matrix theory, and stability theory of fractional-order differential equations, some sufficient conditions on consensus were derived to guarantee the consensuses of linear and nonlinear FMASs for any bounded input time delay. In [17], a distributed observer-type protocol was proposed to study the consensus of FMASs by applying the properties of the Kronecker product, Mittag-Leffler function, and Laplace transform.Most of the existing literatures about FMASs consensus focused on the fixed topology. In fact, when the multiagent system moves harmoniously, communications between agents are usually intermittent due to external disturbances, limited communication range, or the instability of wireless communication itself. The consensus of FMASs under switching topologies is a more important and challenging problem. In this paper, the consensus of FMASs with double integrator under switching topologies is investigated. By using Mittag-Leffler function, Laplace transform, and dwell time technique, a sufficient condition on consensus is presented to guarantee the consensus of the system.


## 2. Preliminaries

### 2.1. Graph Theory

To describe the switching topologies, we define a piecewise constant switching signalσ(t):[0,∞)→Λ={1,2,…,N} represented by {p} according to [tq-1,tq)→p∈Λ, where q=1,2,… and N is the total number of all possible interconnection graphs.The switching topologies of information exchange betweenn agents are described by Gp=(V,Ep,Ap), p∈Λ, where V={1,2,…,n} and Ep⊂V×V are the sets of vertices and edges of the graph Gp, respectively. The adjacency matrices Ap=(aijp)n×n, where aijp is the weight of the directed edge (j,i)∈Ep in the graph Gp and j and i are called the initial and terminal nodes, respectively, which means that node i can receive information from node j. aijp>0 if and only if there is a directed edge (j,i) in Gp. Moreover, we assume aiip=0 for all i=1,2,…,n, p∈Λ. The degree matrices Dp=diag{d1p,…,dnp} are diagonal matrices, whose diagonal elements are given by dip=∑j=1n‍aijp. The Laplacian matrix of the weighted digraph is defined as Lp=Dp-Ap for every p. The set of neighbors of node i is denoted by Nip={j∈V:(j,i)∈Ep}.


### 2.2. Caputo Fractional Derivative

The Caputo fractional derivative is defined as follows:(1)DaCtαft=1Γn-α∫atfnτt-τα-n+1dτ,n-1<α≤n,where α>0 is the fractional order, n is any given natural number, and Γ(·) is the Gamma function defined as Γ(τ)=∫0∞‍tτ-1e-tdt. A simple notion Dαf(t) will be used to denote aCDtαf(t) if no confusion has arisen.LetL{·} denote the Laplace transform of a function. Then we have(2)LDαft=sαFs-sα-1ft0,α∈0,1,where F(s) is the Laplace transform of f(t).Similar to the exponential function frequently used in the solutions of integer-order systems, a function frequently used in the solutions of fractional-order systems is the Mittag-Leffler function.Definition 1 (see [18]).
The two-parameter function of the Mittag-Leffler type is defined by(3)Eα,βz=∑k=0∞zkΓkα+β,where α>0, β>0, and z is a complex number.In particular, letβ=1 and α>0; (3) becomes (4)Eα,1z≜Eαz=∑k=0∞zkΓkα+1.


### 2.3. System Model Description

Consider the following fractional-order double-integrator multiagent systems withn agents:(5)Dαxit=vit,Dαvit=uit,t≥t0,i∈Δ,where 0<α≤1, Δ={1,2,…,n}, xi(t),vi(t)∈R, and ui(t)∈R are the states and control input of the ith agent, respectively. Assume that the initial states of each agent i are denoted by xi(t0) and vi(t0).Definition 2.
For multiagent systems (5), the protocols ui(t) are said to solve the consensus if and only if, for any initial values, the states of the agents satisfy (6)limt→∞xjt-xit=0,i,j∈Δ,limt→∞Dαxjt-Dαxit=0,i,j∈Δ.To study the consensus of multiagent systems (5), the following consensus control protocol will be used:(7)uit=-∑j∈Nip,j≠iaijpxit-xjt-kDαxit,i∈Δ,where k>0 is the control parameter to be designed later.Letx(t)=(x1(t),x2(t),…,xn(t))T, v(t)=(v1(t),v2(t),…,vn(t))T. Then (5) can be rewritten in the following compact form:(8)Dαxt=vt,Dαvt=-Lpxt-kvt. Furthermore, let 1n-1T=[1,…,1]︸n-1, x~(t)=Sx(t), v~(t)=Sv(t), where (9)S=10-1n-1In-1, that is, x~(t)=(x1(t),x2(t)-x1(t),…,xn(t)-x1(t))T, v~(t)=(v1(t),v2(t)-v1(t),…,vn(t)-v1(t))T. Then, by (8), we have(10)Dαx~t=v~t,Dαv~t=-SLpS-1x~t-kv~t. Define x^(t)=(x2(t)-x1(t),…,xn(t)-x1(t))T, v^(t)=(v2(t)-v1(t),…,vn(t)-v1(t))T. Then, (10) can be decoupled as follows:(11)Dαx^t=v^t,Dαv^t=-L~px^t-kv^t,where (12)L~p=d2p+a12pa13p-a23p⋯a1np-a2npa12p-a32pd3p+a13p⋯a1np-a3np⋮⋮⋱⋮a12p-an2pa13p-an3p⋯dnp+a1np.Letε(t)=(x^T(t),v^T(t))T; (11) can be rewritten in the following compact form:(13)Dαεt=Cpεt,where (14)Cp=0In-1-L~p-kIn-1.


## 2.1. Graph Theory

To describe the switching topologies, we define a piecewise constant switching signalσ(t):[0,∞)→Λ={1,2,…,N} represented by {p} according to [tq-1,tq)→p∈Λ, where q=1,2,… and N is the total number of all possible interconnection graphs.The switching topologies of information exchange betweenn agents are described by Gp=(V,Ep,Ap), p∈Λ, where V={1,2,…,n} and Ep⊂V×V are the sets of vertices and edges of the graph Gp, respectively. The adjacency matrices Ap=(aijp)n×n, where aijp is the weight of the directed edge (j,i)∈Ep in the graph Gp and j and i are called the initial and terminal nodes, respectively, which means that node i can receive information from node j. aijp>0 if and only if there is a directed edge (j,i) in Gp. Moreover, we assume aiip=0 for all i=1,2,…,n, p∈Λ. The degree matrices Dp=diag{d1p,…,dnp} are diagonal matrices, whose diagonal elements are given by dip=∑j=1n‍aijp. The Laplacian matrix of the weighted digraph is defined as Lp=Dp-Ap for every p. The set of neighbors of node i is denoted by Nip={j∈V:(j,i)∈Ep}.


## 2.2. Caputo Fractional Derivative

The Caputo fractional derivative is defined as follows:(1)DaCtαft=1Γn-α∫atfnτt-τα-n+1dτ,n-1<α≤n,where α>0 is the fractional order, n is any given natural number, and Γ(·) is the Gamma function defined as Γ(τ)=∫0∞‍tτ-1e-tdt. A simple notion Dαf(t) will be used to denote aCDtαf(t) if no confusion has arisen.LetL{·} denote the Laplace transform of a function. Then we have(2)LDαft=sαFs-sα-1ft0,α∈0,1,where F(s) is the Laplace transform of f(t).Similar to the exponential function frequently used in the solutions of integer-order systems, a function frequently used in the solutions of fractional-order systems is the Mittag-Leffler function.Definition 1 (see [18]).
The two-parameter function of the Mittag-Leffler type is defined by(3)Eα,βz=∑k=0∞zkΓkα+β,where α>0, β>0, and z is a complex number.In particular, letβ=1 and α>0; (3) becomes (4)Eα,1z≜Eαz=∑k=0∞zkΓkα+1.


## 2.3. System Model Description

Consider the following fractional-order double-integrator multiagent systems withn agents:(5)Dαxit=vit,Dαvit=uit,t≥t0,i∈Δ,where 0<α≤1, Δ={1,2,…,n}, xi(t),vi(t)∈R, and ui(t)∈R are the states and control input of the ith agent, respectively. Assume that the initial states of each agent i are denoted by xi(t0) and vi(t0).Definition 2.
For multiagent systems (5), the protocols ui(t) are said to solve the consensus if and only if, for any initial values, the states of the agents satisfy (6)limt→∞xjt-xit=0,i,j∈Δ,limt→∞Dαxjt-Dαxit=0,i,j∈Δ.To study the consensus of multiagent systems (5), the following consensus control protocol will be used:(7)uit=-∑j∈Nip,j≠iaijpxit-xjt-kDαxit,i∈Δ,where k>0 is the control parameter to be designed later.Letx(t)=(x1(t),x2(t),…,xn(t))T, v(t)=(v1(t),v2(t),…,vn(t))T. Then (5) can be rewritten in the following compact form:(8)Dαxt=vt,Dαvt=-Lpxt-kvt. Furthermore, let 1n-1T=[1,…,1]︸n-1, x~(t)=Sx(t), v~(t)=Sv(t), where (9)S=10-1n-1In-1, that is, x~(t)=(x1(t),x2(t)-x1(t),…,xn(t)-x1(t))T, v~(t)=(v1(t),v2(t)-v1(t),…,vn(t)-v1(t))T. Then, by (8), we have(10)Dαx~t=v~t,Dαv~t=-SLpS-1x~t-kv~t. Define x^(t)=(x2(t)-x1(t),…,xn(t)-x1(t))T, v^(t)=(v2(t)-v1(t),…,vn(t)-v1(t))T. Then, (10) can be decoupled as follows:(11)Dαx^t=v^t,Dαv^t=-L~px^t-kv^t,where (12)L~p=d2p+a12pa13p-a23p⋯a1np-a2npa12p-a32pd3p+a13p⋯a1np-a3np⋮⋮⋱⋮a12p-an2pa13p-an3p⋯dnp+a1np.Letε(t)=(x^T(t),v^T(t))T; (11) can be rewritten in the following compact form:(13)Dαεt=Cpεt,where (14)Cp=0In-1-L~p-kIn-1.


## 3. Main Results

LetQs be the set over which there is a directed spanning tree for any Gp∈Qs and Qu be the set over which there is no directed spanning tree for any Gp∈Qu. Denote the cardinality of Λ by Λ. Without loss of generality, we assume that Qs={Gp,p=1,2,…,v} and Qu={Gp,p=v+1,v+2,…,Λ} for some v∈Λ. Assume that the activating topology on [tq-1,tq), q=1,2,…, is Gp, where p∈Λ and p=q mod Λ, and set p=Λ if p=0. Let tq-tq-1≥T∗, where T∗ is called to be the dwell time of the switching signal σ(t). During each of intervals, the interconnection topology does not change. For any t>t0, let Tp(t0,t) denote the total activation time of Gp, p∈Λ and Ts(t)=∑p=1v‍Tp(t0,t), Tu(t)=∑p=v+1Λ‍Tp(t0,t).Based on Lemma3.1 in [17], with a mild revision, we have the following.Lemma 3.
All the eigenvalues of matrixL~p have positive real parts if and only if Gp∈Qs.Lemma 4 (see [19]).
Let(15)Q=0I-H-kI.
Assume that the control parameterk satisfies k2>(maxμ∈ρ(H){Imμ})2/minμ∈ρ(H)Reμ, (minμ∈ρ(H)Reμ≠0), where ρ(H) denotes the set of all eigenvalues of H. Then maxθ∈ρ(Q)Reθ<0 if and only if all eigenvalues of H have positive real parts.By Lemma2.2 in [20], with a mild revision, we have the following.Lemma 5.
There exist finite real constantsh≥1 such that, for any 0<α<1 and β>0, (16)Eα,βAtα≤heAt. In addition, if A is a stable matrix, then there are some real constants K≥1, λ>0 such that (17)Eα,βAtα≤Ke-λt.Theorem 6.
Consider the fractional-order double-integrator multiagent systems (5) under the control protocol (7) and take k2>max1≤p≤v{(maxμ∈ρ(L~p){Imμ})2/minμ∈ρ(L~p)Reμ}. Then, the consensus can be achieved if the switching signal satisfying the dwell time condition (28) and the following inequality holds:(18)inft>t0TstTut>γη,where η=min1≤p≤v{ηp}>0 and γ satisfies maxv+1≤p≤ΛCp<γ<∞ and γγ-Cp>0,p=v+1,v+2,…,Λ.Proof.
Fort∈(tp-1,tp], taking the Laplace transform on (13) and using the Laplace transform (2) with α∈(0,1], we have(19)Lεt=sα-1εtp-1Isα-Cp.Then, taking the inverse Laplace transform of (19), one can obtain that (20)εt=EαCpt-tp-1αεtp-1.
Forp=1,2,…,v, since the communication graph Gp has a directed spanning tree, by Lemma 3, all the eigenvalues of matrix L~p have positive real parts. It follows from the choice of parameter k and Lemma 4 that maxθ∈ρ(Cp)Reθ<0. Hence, Cp is a stable matrix. It follows from Lemma 5 that there exist constants Kp≥1 and ηp>0 such that(21)εt=EαCpt-tp-1αεtp-1≤Kpe-ηpt-tp-1εtp-1,t∈tp-1,tp.For p=v+1,v+2,…,Λ, since maxv+1≤p≤ΛCp<γ<∞ and γγ-Cp>0, obviously, γ is well defined. It follows from Lemma 5 that there exist constants hp>0 such that(22)εt=EαCpt-tp-1αεtp-1≤hpeCpt-tp-1εtp-1≤hpeCpt-tp-1εtp-1≤hpeγt-tp-1εtp-1,t∈tp-1,tp.Then, for t∈(tv-1,tv], by (21), we obtain that(23)εt≤Kve-ηvt-tv-1εtv-1≤K1K2⋯Kve-η1t1-t0⋯e-ηv-1tv-1-tv-2e-ηvt-tv-1εt0=K1K2⋯Kve-∑p=1v-1ηptp-tp-1-ηvt-tv-1εt0.For t∈(tv,tv+1], by (22), one can derive that(24)εt≤hv+1eγt-tvεtv≤K1K2⋯Kvhv+1e-∑p=1vηptp-tp-1eγt-tvεt0,and, then, for t∈(tΛ-1,tΛ], we obtain that(25)εt≤hΛeγt-tΛ-1εtΛ-1≤K1K2⋯Kvhv+1hv+2⋯hΛe-∑p=1vηptp-tp-1e∑p=v+1Λ-1γtp-tp-1+γt-tΛ-1εt0≤K1K2⋯Kvhv+1hv+2⋯hΛe-∑p=1vηptp-tp-1e∑p=v+1Λγtp-tp-1εt0.In general, noticing that Gp is finite, using the definitions of Tp(t0,t), Ts(t), Tu(t), and (25), it is clear that(26)εt≤ωt-t0/T∗e-∑p=1vηpTpt0,te∑p=v+1ΛγTpt0,tεt0≤et-t0/T∗lnωεt0e-ηTsteγTut≤elnω/T∗t-t0εt0e-ηTst+γTut,t>t0,where ω=maxp∈Λ{Kp,hp}.
According toinft>t0Ts(t)/Tu(t)>γ/η, there exists a constant σ>0 such that (γ+σ)Tu(t)≤(η-σ)Ts(t), so -ηTs(t)+γTu(t)≤-σ(Tu(t)+Ts(t))=-σ(t-t0). By (26), we have(27)εt≤εt0elnω/T∗-σt-t0,t>t0.Thus, for the switching signal satisfying the dwell time condition,(28)lnωσ<T∗≤tq-tq-1,q=1,2,…,we can derive that lnω/T∗-σ<0, which implies that the consensus can be achieved asymptotically.


## 4. A Numerical Example and Simulation

Consider the fractional-order multiagent systems withα=0.8. The switching topologies are shown in Figure 1.Figure 1
Switching topologies:Gi, i=1,2,3,4.For simplicity, we assumeaijp=1 if (j,i) in Gp, p=1,2,3,4. Then, by the definition of L~p, we have (29)L~1=100102-100-1210-102,L~2=1000010-1000000-12,L~3=100112-1010011-102,L~4=1000110-110101-1-12.By simple computation, we have ρ(L~1)=1, 3.3247, 1.3376+0.5623-1, 1.3376-0.5623-1, ρ(L~4)=2.6180, 0.3820, 1, 1; then k2>0.828. It is clear that there is a directed spanning tree in G1 and G4. Then, the consensus can be achieved for some T∗ and switching signal σ(t) satisfying (18).LetT∗=10, k=3, and the initial states be x1(0)=8, x2(0)=-10, x3(0)=-5, x4(0)=16, x5(0)=-7, v1(0)=-5, v2(0)=1, v3(0)=-9, v4(0)=4, and v5(0)=15. The control ui(t) and switching sequence are depicted in Figure 2. The state responses xi(t) and vi(t) are depicted in Figure 3, which shows the consensus is achieved asymptotically.Figure 2
Controlui(t) and switching sequence.Figure 3
State responsesxi(t) and vi(t) of each agent.Remark 7.
From the above simulation, it is found that the condition (18) is conservative. But within this frame of this paper, the condition (18) can hardly be improved. Sharpening the condition may require a new method. We leave this for further investigation.


## 5. Conclusions

The consensus problem of fractional-order multiagent systems with double integrator under switching topologies is investigated, where the fractional order satisfies0<α≤1. Consensus algorithms with a control parameter are designed. Based on the Mittag-Leffler function, Laplace transform, and dwell time technique, a sufficient condition is obtained, which ensures that the consensus of fractional-order multiagent systems can achieved asymptotically. The consensus problem of fractional-order multiagent systems with jointly connected topologies will be discussed in future study.



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