# Finite-Time Consensus with a Time-Varying Reference State and Switching Topology

Authors: Jian-Yong Wang; Zhen Tian; Xu Zhu; Naif D. Alotaibi
Journal: Complexity
 (2017)

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## Abstract

The finite-time consensus problem in the networks of multiple mobile agents is comprehensively investigated. In order to resolve this problem, a novel nonlinear information exchange protocol is proposed. The proposed protocol ensures that the states of the agents are converged to a weighted-average consensus in finite time if the communication topology is a weighted directed graph with a spanning tree and each strongly connected component is detail-balanced. Furthermore, the proposed protocol is also able to solve the finite-time consensus problem of networks with a switching topology. Finally, computer simulations are presented to demonstrate and validate the effectiveness of the theoretical analysis under the proposed protocol.


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## Body

## 1. Introduction

In recent years, the cooperative control has received considerable attention mainly due to its broad applications to multiagent systems such as the state consensus seeking of multiple mobile vehicles [1–3], flocking, rendezvous, and containment in natural and social systems [4–9], analysis and control of cyberphysical networks [10], and distributed control and dispatching in smart grids [11–17]. The cooperative control is an interdisciplinary subject and several researchers from different scientific communities have conducted various research studies concerning the abovementioned applications [2, 18]. Among the numerous research topics in cooperative control, consensus problem is one of the most important issues [19, 20]. The consensus problem can be generally described as how to design appropriate protocols based on the local information under some communication topology to ensure that the concerned agents reach an agreement on certain quantities of interest.The main concern of this paper is the consensus seeking problem in the distributed control systems. Consensus seeking problem is a long-standing issue in computer science and sometimes called agreement problem. The communication topology and the consensus protocol are crucial for the consensus of multiagent systems. The researchers have mainly focused on the following two issues: one is the determination of the weakest and the simplest communication topology under which the given protocol can ensure the agents reach an agreement; the second is the design of a protocol to ensure that the states of the agents are converged to consensus under the given interaction topology. The multiagent consensus problem has been addressed by using the graph theory and the matrix analysis methods under various interaction topologies with given information exchange protocols. Vicsek et al. [4] have introduced an interesting discrete-time model of mobile agents. Each agent’s motion is updated according to a local rule based on its own and its neighbors states. The theoretical explanation of the consensus property of the Vicsek mode has been introduced in [5], where each agent’s set of neighbors should change over time with evolution of the system. Later, a general framework of the consensus problem for networks of dynamics agents with fixed or switching topology and communication time-delays has been established [19]. The consensus conditions derived in [19] have been further relaxed in [20]. Lin et al. [21, 22] have studied the consensus problem in the context of formation control of autonomous vehicles and have demonstrated that formation stabilization to a certain point is only feasible if the sensor digraph has a globally reachable node. The consensus problem has also been studied for switching communication topology [23, 24], asynchronous consensus [25, 26], high-dimensional consensus [27], consensus with sampled communication [28], and consensus with external disturbances and model uncertainties [29]. The research so far has focused more on the determination of the weakest and the simplest communication topology than the design of the information exchange protocol to achieve consensus under the given topology. The convergence rate is a very important performance index to evaluate the protocols. Although the enlargement of the algebraic connectivity can increase the convergence rate but still the protocols are unable to ensure that the states of the agents will reach a consensus in finite time. Finite-time consensus is desirable in many real-time scenarios. Recently, Wang and Xiao have studied the finite-time consensus problems and provided several finite-time consensus protocols to ensure that the multiagents reach a consensus in finite time with appropriate communication topology [30–32]. Finite-time consensus in multiple Euler-Lagrange systems over undirected topology has been studied in [33]. More recently, finite-time consensus tracking problem of multiple high-order systems has been considered in [34, 35]. In this paper, a new nonlinear protocol based on the abovementioned works is proposed. The proposed protocol ensures that the agents reach the finite-time consensus with a better convergence rate than the typical linear protocol [19] and the nonlinear protocol proposed by Feng and Long [30]. Furthermore, a number of sufficient consensus conditions are derived and discussed using the tools from Lyapunov stability analysis. Finally, numerical examples are provided to validate the analytical results.The remainder of the paper is organized as follows: in Section2, some preliminary notions in graph theory are provided. Then, the problem is formulated in Section 3. Theoretical analysis results are given in Section 4. In Section 5, simulation results are presented followed by concluding remarks in Section 6.


## 2. Preliminary Notations

Graph plays an important role in representing the communication topology among multiagents. Therefore, some basic notions and results in algebraic graph theory [36, 37] and a algebraic inequality [38] are provided.LetG(V,E,A) be a directed graph of order n, where V={υ1,…,υn} is the set of nodes, E⊆V×V is the set of edges, and A=aijn×n is a weighted adjacency matrix such that (υi,υj)∈E(A)⇔aji>0. An edge of G(V,E,A) is denoted by εij=(υi,υj), where the first element υi and the other element υj are called the root and child vertices, respectively. The set of neighbors of vertex υi is defined as N(G(V,E,A),υi)=υj:υj,υi∈E. The corresponding index set is denoted by N(G(V,E,A),i)=j:υj∈NG,υi. The Laplacian of the graph G(V,E,A) is defined as L=D-A, where D=dijn×n is a diagonal matrix with dii=∑i=1naij. A path of graph G(V,E,A) is a sequence υ1,…,υk of vertices satisfying υij,υij+1∈E. A directed graph is said to be strongly connected, if there exists a directed path between each pair of distinct vertices. A directed tree is a directed graph such that each vertex has only one parent vertex, except one special vertex (root vertex) without any parent vertex.Directed graphGsVs,Es,As is asubgraph of G(V,E,A) such that Vs⊆V,Es⊆E. If Vs=V,Es⊆E, Gs(Vs,Es,As) is called a spanning subgraph. We say Gs(Vs,Es,As) is induced by Vs, if, for any (υi,υj)∈Es, (υi,υj)∈Es⇔(υi,υj)∈E. A spanning tree of G(V,E,A) is a directed tree as well as a spanning subgraph of G(V,E,A). A strongly connected component of G(V,E,A) is an induced subgraph and is subjected to be strongly connected. Suppose that graph G(V,E,A) has k strongly connected components, denoted by G1cV1c,E1c,A1c,…,GkcVkc,Ekc,Akc. According to the above definitions, each vertex of directed graph G(V,E,A) lies exactly in one strongly connected component and the strongly connected component of G(V,E,A) partitions its vertices. Another directed graph Gc(Vc,Ec,Ac) is presented in [37], where Vc consists of all strongly connected components G1cV1c,E1c,A1c,…,Gkc(Vkc,Ekc,Akc) of G(V,E,A), and, for any i,j∈{1,…,k}, (Gic(Vic,Eic,Ajc),Gjc(Vjc,Ejc,Ajc))∈Ec if only if there exist υm∈Vic and υn∈Vjc such that (υm,υn)∈E.Without causing any confusion,G(A) can be used to denote the directed graph G(V,E,A).Definition 1.
For directed graphG(A), if there exist some scalars ωi>0, i∈In, yields ωiaij=ωjaji, and then it can be said that G satisfies the detailed balance condition or G(A) is a detail-balanced graph.Definition 2.
Letx=x1,…,xn∈Rn, and then x is called a positive vector if xi>0, i∈In.Definition 3.
Forx∈R,(1)sign⁡x=1x>00x=0-1x<0.Lemma 4 (Cr-inequality [38]).
For anya=[a1,…,an]∈Rn,p>0, and then(2)∑i=1naip≤C∑i=1naip,C=1,0<p≤1,C=np-1,p>1.Lemma 5 (see [39]).
Suppose that there exists a positive definite continuous functionV:D→R such that the following condition holds.There exist real numbersc>0 and ρ∈(0,1) and an open neighborhood V⊆D of the origin such that(3)V˙x+cVρx≤0,x∈V∖0.Then, the origin is finite-time stable and, depending on the initial sate x(0)=x0, the setting time satisfies the following:(4)Tx0≤Vx0c1-ρ.


## 3. Problem Formulation

The multiple mobile agent system studied in this paper consists ofn autonomous agents, for example, particles or robots, labeled 1-n. All these agents share a common state space R. For convenience, In is used to denote the set {1,2,…,n}. The state of agent i is denoted by xi, i∈In, and let x=x1,x2,…,xnT.Suppose that the communication topology isG(A) and the agent i,i∈In, is with the following dynamics:(5)x˙it=uit,where ui(t) is the protocol to be designed.The following information exchange protocol is used to solve a weighted-average finite-time consensus problem:(6)uit=α∑j∈NGA,iaijsign⁡xj-xixj-xiβ+γ∑j∈NGA,iaijxj-xi,where α>0, 0<β<1, γ≥0.Particularly, the following cases can be obtained:(i)
Forα=0, γ=1, then(7)uit=∑j∈NGA,iaijxj-xi,and the nonlinear protocol (7) reduces to the typical linear protocol proposed by Olfati-Saber and Murray [19]. The dynamical behaviors of the multiagents have received considerable attention under this linear protocol and significant results have been established. Further details can be found in the survey papers [2, 18].(ii)
α>0, 0<β<1, γ=0, and the consensus transforms into the nonlinear protocol proposed by Feng and Long [30]. Feng and Long have proved that their nonlinear protocol can solve the finite-time average-agreement problem and can be applied to the systems with switching topologies assuming that the communication topology is represented by a connected undirected graph.(iii)
α≠0, β=0, γ=0, and in this case the protocol becomes discontinuous with respect to the state variables. The research on this topic is still under development. It is worth mentioning that the case when α≠0, β=0, γ=0, and the adjacency matrix A of the communication topology GA is a 0-1 symmetric matrix has been addressed in [40].


## 4. Theoretical Analysis

This section describes some theoretical results of the agents dynamics under the proposed protocol.


### 4.1. Fixed Communication Topology

In this subsection, dynamic behaviors of the multiagents under fixed topology are considered.Theorem 6.
If the communication topologyG(A) has a spanning tree and each strongly connected component is detail-balanced, then the nonlinear protocol (6) can ensure that the states of agents converge to a weighted-average consensus in finite time.Proof.
The proof is similar to that of [37] and is divided into the following three steps.
Step 1. Suppose that G(A) is strongly connected and detail-balanced.
AsG(A) is a strongly connected and detail-balanced graph, according to Definition 1, there exists a positive vector ω=ω1,ω2,…,ωnT∈Rn such that ωiaij=ωjaji for all i,j∈In. Then, the following can be obtained:(8)∑i=1nωix˙it=0.Let(9)Δ=1∑i=1nωi∑i=1nωixit.Then Δ is a constant and can be considered as a weighted-average of the initial sates of the agents. Let δ(t)=δ1t,…,δntT, where δi(t)=xi(t)-Δ. Taking Lyapunov function V1(t)=1/2∑i=1nωiδit2, then δ˙i(t)=x˙it and ωTδt=0. In [19], δt is referred to as thegroup disagreement vector. In the following analysis, δt is assumed to be a nonzero vector.
DifferentiatingV1t with respect to t provides(10)dV1tdt=∑i=1nωiδitδ˙it=α∑i=1nωiδit∑j∈NGA,iaijsign⁡δjt-δitδjt-δitβ+γ∑i=1nωiδit∑j∈NGA,iaijδjt-δit=12α∑i,j=1nωiaijδitsign⁡δjt-δitδjt-δitβ+γ∑i,j=1nωiaijδitδj-δi+α∑i,j=1nωjajiδjtsign⁡δit-δjtδit-δjtβ+γ∑i,j=1nωjajiδjtδi-δj=-12α∑i,j=1nωiaijδj-δi1+β+γ∑i,j=1nωiaijδj-δi2=-α2∑i,j=1nωiaij2/1+βδj-δi21+β/2-γ2∑i,j=1nωiaijδj-δi2.According to Cr-inequality(11)dV1tdt≤-α2∑i,j=1nωiaij2/1+βδj-δi21+β/2-γ2∑i,j=1nωiaijδj-δi2.Let M1δt=∑i,j=1nωiaij2/1+βδj-δi2, N1δt=∑i,j=1nωiaijδj-δi2 and B=bijn×n, C=cijn×n, where bij=ωiaij2/1+β, cij=ωiaij. It is obvious that B=BT, C=CT, L(B), and L(C) denote the Laplacian matrices of the graphs G(B), and G(C), respectively. Let λLB=λ1,λ2,…,λn, where 0=λ1≤λ2≤⋯≤λn, represents the spectrum of matrix LB. It can easily be checked that 0 is an algebraically simple eigenvalue and 1=1,…,1T is the associated eigenvector. As L(B) is a real symmetric matrix, there exists orthogonal matrix Φ=φ1,φ2,…,φnT, such that φ1=1/n1,1,…,1T and ΦLBΦT=diag⁡(λ(B)). Then(12)Mδt=2δtTLBδt=2ΦδtTdiag⁡λLBΦδt.Let δ~(t)=Φδ(t)=δ~1,δ~2,…,δ~nT, and then δ~1t=1/n∑i=1nδit. It is obvious that δ~Ttδ~t=δTtδt. According the above analysis, (12) can be rewritten as(13)Mδt=2λ2LBδ~2t2+λ3LBδ~3t2+⋯+λnLBδ~nt2≥2λ2LBδ~tTδ~t-λ2LBn∑i=1nδit2=2λ2LBδ~tTδ~t1-∑i=1nδit2n∑i=1nδi2t.Recalling the fact ωTδ(t)=0, it can be found that δ1t,δ2t,…,δn(t) do not have the same sign. By using the Cr-inequality, one can obtain(14)1-∑i=1nδit2n∑i=1nδi2t>1-∑i=1nδit2n∑i=1nδi2t≥0.
Similar to [31], let U=ξ:nonzero  elements  ofξ1,…,ξnare  not  with  the  same  sign, and let U0=U⋂{ξ:ξTξ=1}. It is obvious that U0 is compact set. For convenience, construct function(15)Wδ=∑i=1nδit2-∑i=1nδit2n∑i=1nδi2t.Clearly, Wδ>0. The expression can be rewritten as(16)Wδ=1n∑i=1nsign⁡δiδiδ-∑i=1nδiδ2.
LetS(x)=∑i=1nsign⁡(xi)xi and T(x)=∑i=1nxi2, where x=x1,…,xn∈Rn. Obviously, Sx and Tx are continuous functions with respect to x. Then, one can get(17)Wδ=Wζ=1nSζ-Tζ≥K1>0,where ζ=δ1/δ,…,δn/δ and K1=minθ∈U0⁡1/nSζ-Tζ. Hence,(18)Mδt≥2K1λ2LBδTtδt≥2K1λ2LBωmax∑i=1nωiδit2.
Similar to the above analysis, there exists a constantK2>0 such that(19)Nδt=2δtTLCδt≥2K2λ2LCωmax∑i=1nωiδit2,where λ2(L(B)),λ2(L(C)) represents the second largest eigenvalue of the matrixes L(B) and LC, respectively.
Let(20)K3=α22K1λ2LBωmax1+β/2,K4=4γK2λ2LCωmax.Obviously, K3,K4>0 and(21)dV1tdt≤-K3+K4V1t2/1+βV1t1+β/2.The above analysis indicates that protocol (7) is able to solve the weighted-average consensus problem if the interaction topology is represented by a strongly connected detail-balanced graph. The last inequality also indicates that the proposed protocol in this paper is capable of providing a faster convergence rate than the typical linear protocol proposed in [19] and the nonlinear finite-time consensus protocol(22)uit=α∑j∈NGA,iaijsign⁡xj-xixj-xiβ,proposed by Feng and Long in [30].
Step
2. Suppose that GA has a spanning tree and the associated root vertex is υi and aij=0, for any j∈In. Furthermore, assume that the subgraph induced by the child vertices υjj∈In,j≠i is strongly connected and detail-balanced.
Without loss of generality, assume that the root vertexυi is υ1. Therefore, one can obtain that a11=a12=⋯=a1n=0 and a11,a21,…,an-11 are not all zeros. GA~ denotes the subgraph induced by the vertices excluding the vertex υ1. In this case, as u1t≡0, the dynamic behaviors of the other agents may be focused. Then, one can obtain(23)uit=α∑j∈NGA,i∖1aijsign⁡xj-xixj-xiβ+γ∑j∈NGA,i∖1aijxj-xi,i≠1.That is,(24)uit=α∑j=2naijsign⁡xj-xixj-xiβ+γ∑j=2naijxj-xi+α·ai1sign⁡x1-xix1-xiβ+γ·ai1x1-xi,i≠1.Let δi=xi-x1, i∈In. It is obvious that δ1≡0. Recalling that x˙1≡0, one can get(25)δ˙i=α∑j=2naijsign⁡δj-δiδj-δiβ+γ∑j=2naijδj-δi-αai1sign⁡δiδiβ-γai1δi,i≠1.Take the Lyapunov function V2t=1/2∑i=1nωiδi(t)2. Differentiating V2t with respect to t leads to (26)dV2tdt=∑i=1nωiδitδ˙it=∑i=2nωiδitα∑j=2naijsign⁡δj-δiδj-δiβ+γ∑j=2naijδj-δi-α·ai1sign⁡δiδiβ-γ·ai1δi=12α∑i,j=2nωiaijδitsign⁡δjt-δitδjt-δitβ+γ∑i,j=2nωiaijδitδj-δi+α∑i,j=2nωjajiδjtsign⁡δit-δjtδit-δjtβ+γ∑i,j=2nωjajiδjtδi-δj-α∑i=2nωiai1δit1+β-γ∑i=2nωiai1δit2=-12α∑i,j=2nωiaijδj-δi1+β+γ∑i,j=2nωiaijδj-δi2-α∑i=2nωiai12/1+βδit21+β/2-γ∑i=2nωiai1δit2=-α2∑i,j=2nωiaij2/1+βδj-δi21+β/2+2∑i=2nωiai12/1+βδit21+β/2-γ2∑i,j=2nωiaijδj-δi2+2∑i=2nωiai1δit2.Applying the Cr-inequality to the above equation, the following is obtained:(27)dV2tdt≤-α2∑i,j=2nωiaij2/1+βδj-δi2+2∑i=2nωiai12/1+βδit21+β/2-γ2∑i,j=2nωiaijδj-δi2+2∑i=2nωiai1δit2.
For simplicity, let(28)M2δt=∑i,j=2nωiaij2/1+βδj-δi2+2∑i=2nωiai12/1+βδit2,N2δt=∑i,j=2nωiaijδj-δi2+2∑i=2nωiai1δit2.
LetA~=a~ij(n-1)×(n-1), where a~ij=ωi+1ai+1j+12/1+β denoted b~=[ω2a212/1+β,…,ωnan12/1+β]. Then one can get that(29)M2δt=2δTLA~δt+2δTdiag⁡b~δ.Then, B~=LA~+diag⁡(b~) is real symmetric and positive definite. Let the smallest eigenvalue of B~ be λ1B~. Then, λ1B~>0, and, for any nonzero vector η∈Rn-1, ηTB~η≥λ1B~ηTη. Therefore,(30)M2δt≥2λ1B~δTδ≥K5∑i=1nωiδi2,where K5=2λ1B~/ωmax.
Similarly, one can obtain that(31)N2δt≥K6∑i=1nωiδi2.
Then, the following inequality holds:(32)dV2tdt≤-α2K51+β/2+γ2K6V22/1+βtV21+β/2t.The above analysis indicates that protocol (7) is able to solve the weighted-average consensus problem if G(A) has a spanning tree and each strongly connected component is detail-balanced.
Step 3. Suppose that G(A) has a spanning tree and each strongly connected component is detail-balanced. Then, the proposed protocol will solve a finite-time weighted-average consensus problem.
Consider the directed graphGcA consisting of the strongly connected components of GA. Obviously, GcA is a directed tree. From the definition of GcA, the following facts can be obtained:(i)
The dynamic behaviors of agents belonging to the vertex set that corresponds to the root vertex ofGc(A) are not affected by others and the local interaction topology among them is strongly connected and detail-balanced. Their states will reach an agreement in a finite time in accordance with the conclusion of Step 1. Considering Gc(A), its root vertex (corresponding to nonempty subset of V, where V is the vertices set of graph G(A)) is labeled 1 and the final consensus state is denoted by x0.(ii)
Consider the dynamics of agents, denoted byυi1,…,υiki, corresponding to vertex i (not the root vertex) of Gc(A). Their dynamic behaviors are only affected by such agents, where there exists at least one vertex that belongs to set {υl,l=1,…,ki} and they can provide information; that is, there exist directed paths connecting them to υl,l=1,…,ki. Suppose such agents excluding υl,l=1,…,ki, are υj1,υj2,…,υjmi, their sates have already reached consensus, and the consensus state is x0. Then, for any l∈{1,2,…,ki},(33)∑s=1kiail,isxis-xil+∑s=1miail,jsxjs-xil=∑s=1kiail,isxis-xil+∑s=1miail,jsx0-xil.Therefore,υj1,…,υjki can be seen as one (virtual) agent and as the leader of υil,l=1,…,ki. If one relabels the vertices υil,l=1,…,ki, by υl,l=2,…,ki+1, and labels the (virtual) agent by 1, then the communication topology of agents υl,l=1,…,ki+1, is G(A0), where (34)A0=00⋯0∑s=1miai1,jsai1,i2⋯ai1,iki⋮⋮⋮⋮∑s=1miaiki,jsaiki,i2⋯aiki,iki.Recalling the theoretical results produced in Step2, the multiagents υl,l=1,…,ki+1, will reach consensus in finite time, and the consensus state will be x0.(iii)
In accordance with the above presented conclusion, the proposed protocol has solved a finite-time weighted-average consensus problem, and the consensus state isx0.
The proof is completed.


### 4.2. Switching Topology

In practice, the information channel among agents may not always be available due to the physical equipment restrictions or the external interferences in the signal, such as exceeding the sensing range or existence of obstacles between the agents. Therefore, it would be reasonable to assume that the communication topology is dynamically changing.Theorem 7.
Suppose thatG(At) is strongly connected and detail-balanced for all t≥0. Then, protocol (7) will solve a finite-time weighted-average consensus under time-varying topology GAt if there exists a common positive vector ω=ω1,ω2,…,ωnT∈Rn such that ωiaijt=ωjajit for all i,j∈In.Proof.
The proof is similar to that of Theorem6 with the same notations, such as Δ and δ(t). Consider the Lyapunov function V3(δ(t)), then the constants, such as K3, K4, K5, K6 that are derived in proof of Theorem 6, will transform into time-varying parameters K3(t), K4(t), K5(t), K6(t). Take Ki(t)=mini⁡Ki(t),i∈3,4,5,6, and then the multiagents will reach an agreement in finite time.It should be noted that a common Lyapunov can be constructed for the consensus error system ifGAt is strongly connected and detail-balanced for all t≥0. Based on this condition, a preliminary result for finite-time consensus of multiagent system is given in theorem. However, how to reach finite-time consensus in multiagent system with switching topology without the condition that GAt is a strongly connected and detail-balanced for all t≥0 is still an unsolved issue.


## 4.1. Fixed Communication Topology

In this subsection, dynamic behaviors of the multiagents under fixed topology are considered.Theorem 6.
If the communication topologyG(A) has a spanning tree and each strongly connected component is detail-balanced, then the nonlinear protocol (6) can ensure that the states of agents converge to a weighted-average consensus in finite time.Proof.
The proof is similar to that of [37] and is divided into the following three steps.
Step 1. Suppose that G(A) is strongly connected and detail-balanced.
AsG(A) is a strongly connected and detail-balanced graph, according to Definition 1, there exists a positive vector ω=ω1,ω2,…,ωnT∈Rn such that ωiaij=ωjaji for all i,j∈In. Then, the following can be obtained:(8)∑i=1nωix˙it=0.Let(9)Δ=1∑i=1nωi∑i=1nωixit.Then Δ is a constant and can be considered as a weighted-average of the initial sates of the agents. Let δ(t)=δ1t,…,δntT, where δi(t)=xi(t)-Δ. Taking Lyapunov function V1(t)=1/2∑i=1nωiδit2, then δ˙i(t)=x˙it and ωTδt=0. In [19], δt is referred to as thegroup disagreement vector. In the following analysis, δt is assumed to be a nonzero vector.
DifferentiatingV1t with respect to t provides(10)dV1tdt=∑i=1nωiδitδ˙it=α∑i=1nωiδit∑j∈NGA,iaijsign⁡δjt-δitδjt-δitβ+γ∑i=1nωiδit∑j∈NGA,iaijδjt-δit=12α∑i,j=1nωiaijδitsign⁡δjt-δitδjt-δitβ+γ∑i,j=1nωiaijδitδj-δi+α∑i,j=1nωjajiδjtsign⁡δit-δjtδit-δjtβ+γ∑i,j=1nωjajiδjtδi-δj=-12α∑i,j=1nωiaijδj-δi1+β+γ∑i,j=1nωiaijδj-δi2=-α2∑i,j=1nωiaij2/1+βδj-δi21+β/2-γ2∑i,j=1nωiaijδj-δi2.According to Cr-inequality(11)dV1tdt≤-α2∑i,j=1nωiaij2/1+βδj-δi21+β/2-γ2∑i,j=1nωiaijδj-δi2.Let M1δt=∑i,j=1nωiaij2/1+βδj-δi2, N1δt=∑i,j=1nωiaijδj-δi2 and B=bijn×n, C=cijn×n, where bij=ωiaij2/1+β, cij=ωiaij. It is obvious that B=BT, C=CT, L(B), and L(C) denote the Laplacian matrices of the graphs G(B), and G(C), respectively. Let λLB=λ1,λ2,…,λn, where 0=λ1≤λ2≤⋯≤λn, represents the spectrum of matrix LB. It can easily be checked that 0 is an algebraically simple eigenvalue and 1=1,…,1T is the associated eigenvector. As L(B) is a real symmetric matrix, there exists orthogonal matrix Φ=φ1,φ2,…,φnT, such that φ1=1/n1,1,…,1T and ΦLBΦT=diag⁡(λ(B)). Then(12)Mδt=2δtTLBδt=2ΦδtTdiag⁡λLBΦδt.Let δ~(t)=Φδ(t)=δ~1,δ~2,…,δ~nT, and then δ~1t=1/n∑i=1nδit. It is obvious that δ~Ttδ~t=δTtδt. According the above analysis, (12) can be rewritten as(13)Mδt=2λ2LBδ~2t2+λ3LBδ~3t2+⋯+λnLBδ~nt2≥2λ2LBδ~tTδ~t-λ2LBn∑i=1nδit2=2λ2LBδ~tTδ~t1-∑i=1nδit2n∑i=1nδi2t.Recalling the fact ωTδ(t)=0, it can be found that δ1t,δ2t,…,δn(t) do not have the same sign. By using the Cr-inequality, one can obtain(14)1-∑i=1nδit2n∑i=1nδi2t>1-∑i=1nδit2n∑i=1nδi2t≥0.
Similar to [31], let U=ξ:nonzero  elements  ofξ1,…,ξnare  not  with  the  same  sign, and let U0=U⋂{ξ:ξTξ=1}. It is obvious that U0 is compact set. For convenience, construct function(15)Wδ=∑i=1nδit2-∑i=1nδit2n∑i=1nδi2t.Clearly, Wδ>0. The expression can be rewritten as(16)Wδ=1n∑i=1nsign⁡δiδiδ-∑i=1nδiδ2.
LetS(x)=∑i=1nsign⁡(xi)xi and T(x)=∑i=1nxi2, where x=x1,…,xn∈Rn. Obviously, Sx and Tx are continuous functions with respect to x. Then, one can get(17)Wδ=Wζ=1nSζ-Tζ≥K1>0,where ζ=δ1/δ,…,δn/δ and K1=minθ∈U0⁡1/nSζ-Tζ. Hence,(18)Mδt≥2K1λ2LBδTtδt≥2K1λ2LBωmax∑i=1nωiδit2.
Similar to the above analysis, there exists a constantK2>0 such that(19)Nδt=2δtTLCδt≥2K2λ2LCωmax∑i=1nωiδit2,where λ2(L(B)),λ2(L(C)) represents the second largest eigenvalue of the matrixes L(B) and LC, respectively.
Let(20)K3=α22K1λ2LBωmax1+β/2,K4=4γK2λ2LCωmax.Obviously, K3,K4>0 and(21)dV1tdt≤-K3+K4V1t2/1+βV1t1+β/2.The above analysis indicates that protocol (7) is able to solve the weighted-average consensus problem if the interaction topology is represented by a strongly connected detail-balanced graph. The last inequality also indicates that the proposed protocol in this paper is capable of providing a faster convergence rate than the typical linear protocol proposed in [19] and the nonlinear finite-time consensus protocol(22)uit=α∑j∈NGA,iaijsign⁡xj-xixj-xiβ,proposed by Feng and Long in [30].
Step
2. Suppose that GA has a spanning tree and the associated root vertex is υi and aij=0, for any j∈In. Furthermore, assume that the subgraph induced by the child vertices υjj∈In,j≠i is strongly connected and detail-balanced.
Without loss of generality, assume that the root vertexυi is υ1. Therefore, one can obtain that a11=a12=⋯=a1n=0 and a11,a21,…,an-11 are not all zeros. GA~ denotes the subgraph induced by the vertices excluding the vertex υ1. In this case, as u1t≡0, the dynamic behaviors of the other agents may be focused. Then, one can obtain(23)uit=α∑j∈NGA,i∖1aijsign⁡xj-xixj-xiβ+γ∑j∈NGA,i∖1aijxj-xi,i≠1.That is,(24)uit=α∑j=2naijsign⁡xj-xixj-xiβ+γ∑j=2naijxj-xi+α·ai1sign⁡x1-xix1-xiβ+γ·ai1x1-xi,i≠1.Let δi=xi-x1, i∈In. It is obvious that δ1≡0. Recalling that x˙1≡0, one can get(25)δ˙i=α∑j=2naijsign⁡δj-δiδj-δiβ+γ∑j=2naijδj-δi-αai1sign⁡δiδiβ-γai1δi,i≠1.Take the Lyapunov function V2t=1/2∑i=1nωiδi(t)2. Differentiating V2t with respect to t leads to (26)dV2tdt=∑i=1nωiδitδ˙it=∑i=2nωiδitα∑j=2naijsign⁡δj-δiδj-δiβ+γ∑j=2naijδj-δi-α·ai1sign⁡δiδiβ-γ·ai1δi=12α∑i,j=2nωiaijδitsign⁡δjt-δitδjt-δitβ+γ∑i,j=2nωiaijδitδj-δi+α∑i,j=2nωjajiδjtsign⁡δit-δjtδit-δjtβ+γ∑i,j=2nωjajiδjtδi-δj-α∑i=2nωiai1δit1+β-γ∑i=2nωiai1δit2=-12α∑i,j=2nωiaijδj-δi1+β+γ∑i,j=2nωiaijδj-δi2-α∑i=2nωiai12/1+βδit21+β/2-γ∑i=2nωiai1δit2=-α2∑i,j=2nωiaij2/1+βδj-δi21+β/2+2∑i=2nωiai12/1+βδit21+β/2-γ2∑i,j=2nωiaijδj-δi2+2∑i=2nωiai1δit2.Applying the Cr-inequality to the above equation, the following is obtained:(27)dV2tdt≤-α2∑i,j=2nωiaij2/1+βδj-δi2+2∑i=2nωiai12/1+βδit21+β/2-γ2∑i,j=2nωiaijδj-δi2+2∑i=2nωiai1δit2.
For simplicity, let(28)M2δt=∑i,j=2nωiaij2/1+βδj-δi2+2∑i=2nωiai12/1+βδit2,N2δt=∑i,j=2nωiaijδj-δi2+2∑i=2nωiai1δit2.
LetA~=a~ij(n-1)×(n-1), where a~ij=ωi+1ai+1j+12/1+β denoted b~=[ω2a212/1+β,…,ωnan12/1+β]. Then one can get that(29)M2δt=2δTLA~δt+2δTdiag⁡b~δ.Then, B~=LA~+diag⁡(b~) is real symmetric and positive definite. Let the smallest eigenvalue of B~ be λ1B~. Then, λ1B~>0, and, for any nonzero vector η∈Rn-1, ηTB~η≥λ1B~ηTη. Therefore,(30)M2δt≥2λ1B~δTδ≥K5∑i=1nωiδi2,where K5=2λ1B~/ωmax.
Similarly, one can obtain that(31)N2δt≥K6∑i=1nωiδi2.
Then, the following inequality holds:(32)dV2tdt≤-α2K51+β/2+γ2K6V22/1+βtV21+β/2t.The above analysis indicates that protocol (7) is able to solve the weighted-average consensus problem if G(A) has a spanning tree and each strongly connected component is detail-balanced.
Step 3. Suppose that G(A) has a spanning tree and each strongly connected component is detail-balanced. Then, the proposed protocol will solve a finite-time weighted-average consensus problem.
Consider the directed graphGcA consisting of the strongly connected components of GA. Obviously, GcA is a directed tree. From the definition of GcA, the following facts can be obtained:(i)
The dynamic behaviors of agents belonging to the vertex set that corresponds to the root vertex ofGc(A) are not affected by others and the local interaction topology among them is strongly connected and detail-balanced. Their states will reach an agreement in a finite time in accordance with the conclusion of Step 1. Considering Gc(A), its root vertex (corresponding to nonempty subset of V, where V is the vertices set of graph G(A)) is labeled 1 and the final consensus state is denoted by x0.(ii)
Consider the dynamics of agents, denoted byυi1,…,υiki, corresponding to vertex i (not the root vertex) of Gc(A). Their dynamic behaviors are only affected by such agents, where there exists at least one vertex that belongs to set {υl,l=1,…,ki} and they can provide information; that is, there exist directed paths connecting them to υl,l=1,…,ki. Suppose such agents excluding υl,l=1,…,ki, are υj1,υj2,…,υjmi, their sates have already reached consensus, and the consensus state is x0. Then, for any l∈{1,2,…,ki},(33)∑s=1kiail,isxis-xil+∑s=1miail,jsxjs-xil=∑s=1kiail,isxis-xil+∑s=1miail,jsx0-xil.Therefore,υj1,…,υjki can be seen as one (virtual) agent and as the leader of υil,l=1,…,ki. If one relabels the vertices υil,l=1,…,ki, by υl,l=2,…,ki+1, and labels the (virtual) agent by 1, then the communication topology of agents υl,l=1,…,ki+1, is G(A0), where (34)A0=00⋯0∑s=1miai1,jsai1,i2⋯ai1,iki⋮⋮⋮⋮∑s=1miaiki,jsaiki,i2⋯aiki,iki.Recalling the theoretical results produced in Step2, the multiagents υl,l=1,…,ki+1, will reach consensus in finite time, and the consensus state will be x0.(iii)
In accordance with the above presented conclusion, the proposed protocol has solved a finite-time weighted-average consensus problem, and the consensus state isx0.
The proof is completed.


## 4.2. Switching Topology

In practice, the information channel among agents may not always be available due to the physical equipment restrictions or the external interferences in the signal, such as exceeding the sensing range or existence of obstacles between the agents. Therefore, it would be reasonable to assume that the communication topology is dynamically changing.Theorem 7.
Suppose thatG(At) is strongly connected and detail-balanced for all t≥0. Then, protocol (7) will solve a finite-time weighted-average consensus under time-varying topology GAt if there exists a common positive vector ω=ω1,ω2,…,ωnT∈Rn such that ωiaijt=ωjajit for all i,j∈In.Proof.
The proof is similar to that of Theorem6 with the same notations, such as Δ and δ(t). Consider the Lyapunov function V3(δ(t)), then the constants, such as K3, K4, K5, K6 that are derived in proof of Theorem 6, will transform into time-varying parameters K3(t), K4(t), K5(t), K6(t). Take Ki(t)=mini⁡Ki(t),i∈3,4,5,6, and then the multiagents will reach an agreement in finite time.It should be noted that a common Lyapunov can be constructed for the consensus error system ifGAt is strongly connected and detail-balanced for all t≥0. Based on this condition, a preliminary result for finite-time consensus of multiagent system is given in theorem. However, how to reach finite-time consensus in multiagent system with switching topology without the condition that GAt is a strongly connected and detail-balanced for all t≥0 is still an unsolved issue.


## 5. Simulation and Applications

In this section, some simulations are performed by fixingα=4, β=0.5, and γ=3 in order to validate the effectiveness of the theoretical results.Two digraphs withn=6 and n=7 are presented in Figures 1 and 2, respectively. The adjacency matrix of G1 is(35)A=030002203000090300000.7505000010030.80001.20.Figure 1
DigraphG1.Figure 2
DigraphG2.It can be seen thatG1 is detailed balanced graph and ω1=2,3,1,4,2,5T is satisfying ωiaij=ωjaji. The adjacency matrix of G2 is (36)A=00000003003400600000000.60020100102.50025007000000005000.DigraphG2 has a spanning tree and each strongly connected component is detail-balanced. And ω2=2,1,3,5,4,6,2T is satisfying ωiaij=ωjaji.Figures3 and 4 provide the dynamic behaviors of the multiagents under the interaction topology G1 with initial state x(0)=[3,4,7,-6,8,5] and the state trajectories of agents for communication G2 with initial state x(0)=[3,-4,7,-6,2,5,6], respectively. The simulation results demonstrate that the multiagents can reach a finite-time consensus under the proposed protocol.Figure 3
Dynamics of agents under the communication topologyG2.Figure 4
Dynamics of agents under the communication topologyG2.


## 6. Conclusion

In this paper, finite-time consensus problem in multiagent networks with first-order dynamics has been investigated and examined. A new nonlinear information exchange protocol has been developed to ensure the achievement of consensus. The designed protocol ensures that the states of the agents converge to a weighted-average consensus in a finite time, provided that the communication topology is a weighted directed graph with a spanning tree and each strongly connected component is detail-balanced. Furthermore, the theoretical analysis has demonstrated that the proposed protocol is also capable of solving the finite-time consensus problem of networks with a switching topology. Future work will focus on achieving finite-time consensus with nonlinear intrinsic dynamics and time-varying topology.



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*Source: 6038584-2017-06-01.xml*