# Distributed Consensus Tracking for Second-Order Nonlinear Multiagent Systems with a Specified Reference State

Authors: Guoguang Wen; Yongguang Yu; Zhaoxia Peng; Ahmed Rahmani
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper mainly addresses the distributed consensus tracking problem for second-order nonlinear multiagent systems with a specified reference trajectory. The dynamics of each follower consists of two terms: nonlinear inherent dynamics and a simple communication protocol relying only on the position and velocity information of its neighbors. The consensus reference is taken as a virtual leader, whose output is only its position and velocity information that is available to only a subset of a group of followers. To achieve consensus tracking, a class of nonsmooth control protocols is proposed which reply on the relative information among the neighboring agents. Then some corresponding sufficient conditions are derived. It is shown that if the communication graph associated with the virtual leader and followers is connected at each time instant, the consensus can be achieved at least globally exponentially with the proposed protocol. Rigorous proofs are given by using graph theory, matrix theory, and Lyapunov theory. Finally, numerical examples are presented to illustrate the theoretical analysis.


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## Body

## 1. Introduction

Consensus problem, one of the most important and fundamental issues in the cooperative control, has attracted great attention from researchers in recent years because of broad applications in various real-world multiagent systems, such as cooperative control of unmanned (air) vehicles, formation control of robots and aircrafts, and design of sensor networks, to name a few (see the survey papers [1–7] and extensive references therein). The consensus of multiagent systems means to design a network distributed control policy based on the local information obtained by each agent, such that all agents reach an agreement on certain quantities of interest by negotiating with their neighbors. The quantities of interest might represent attitude, position, velocity, temperature, voltage, and so on, depending on different applications.Numerous interesting results for consensus problem have been obtained in the past decade. In 1986, Reynolds [8] fist proposed a computer animation model to simulate collective behaviors of multiple agents. In [9], Vicsek et al. proposed a discrete-time distributed model to simulate a group of autonomous agents moving in the plane with the same speed but different heading. In [10], Jadbabaie et al. gave a theoretical explanation for the consensus behavior of Vicsek’s model based on graph and matrix theories. Ren and Beard [11] extend the work of Jadbabaie et al. [10] to the case of directed graphs and gave some more relaxed conditions. Olfati-Saber and Murray [1] investigated a systematical framework of consensus problems with directed interconnection graphs or time-delay by Lyapunov-based approach.Note from the literatures which were concerned with consensus problem that the multiagent systems can usually be classified into leaderless and leader-follower systems. In a leaderless consensus problem [12–14], there does not exist a (virtual) leader, while in a leader-following consensus problem [15–19], there exists a (virtual) leader. The (virtual) leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones. Therefore, in recent literatures, a (virtual) leader-follower approach has been widely used to the consensus problem [20–25]. Such a consensus problem with a dynamic (virtual) leader is commonly called consensus tracking problem. Most of the above works are based on the first-order control protocols that control the velocities of agents rather than their accelerations (which are commonly easier to control and are especially meaningful for analysis of dynamical system in the real world). Hence, it is necessary to study the consensus tracking problem of second-order multiagent systems. In [26], the consensus tracking problem of multiagent systems with double-integrator dynamics was studied. However, [26] requires the availability of the virtual leader's acceleration input to all followers. Compared with previous works of [26]. Reference [20] studied a distributed leader-following consensus problem with single-integrator dynamics and double-integrator dynamics under fixed and switching communication topologies. It was shown that the acceleration measurements of the virtual leader and followers are not required, and the consensus tracking can be achieved in finite time. In [27], the authors studied the consensus in multiagent systems with second-order dynamics and sampled data. Note also from the above literatures that most of common models used to study the consensus tracking of the second-order multiagent systems are double integrator models. However, in reality, mobile agents may be governed by more commonly inherent nonlinear dynamics [15, 28–31]. References [15, 28] studied the leaderless consensus problems for second-order multiagent systems with nonlinear dynamics under fixed network topology. Reference [32] investigated second-order leader-follower consensus of nonlinear multiagent systems via pinning control. Reference [33] investigated leader-following consensus for second-order multiagent systems with nonlinear inherent dynamics. However, in the above literatures each agent's state only depends on their common inherent dynamics. It is difficult to track a time-varying desired reference state for all agents.Motivated by the above discussions, the main contributions of this paper lie in that we deal with the consensus tracking for second-order nonlinear multiagent systems with a specified reference state. The dynamics of each follower consists of two terms: nonlinear inherent dynamics and a simple communication protocol relying only on the position and velocity information of its neighbors. The consensus reference is taken as a virtual leader, whose output is only its position and velocity information that is available to only a subset of a group of followers. To achieve consensus tracking, a class of nonsmooth control protocols is proposed which reply on the relative information among the neighboring agents. Then some corresponding sufficient conditions are derived. It is shown that if the communication graph associated with the virtual leader and followers is connected at each time instant, the consensus can be achieved at least globally exponentially with the proposed protocol. Rigorous proofs are given by using graph theory, matrix theory, and Lyapunov theory. Compared with the existing results, this paper has the following advantages. Firstly, with contrast of consensus for multiagent systems with double-integrator dynamics [19, 20, 25–27], we investigate the consensus tracking problems for multiagent systems with nonlinear inherent dynamics. Secondly, in contrast to the existing results in [15, 28, 32, 33], where each agent's state only depends on their common inherent dynamics, in this paper all agents can well track a time varying desired reference state. Thirdly, in this paper the consensus can be achieved globally exponentially with the proposed protocol, while in [15, 27–31] the consensus can be archived asymptotically. Finally, it is well known that most of real-world systems, for example, biological systems, autonomous vehicles systems, complex systems, and so on, are time-varying systems [34, 35]. Therefore, in this paper, we consider the consensus tracking problems for time-varying multiagent systems, where the states of all agents are time varying.The rest of this paper is organized as follows. In Section2, the relevant notations and preliminaries are presented. In Section 3, the consensus tracking problem to be solved in this paper is described. The main results are presented in Section 4. Numerical examples are shown in Section 5. Finally, some conclusion remakes are given in Section 6.


## 2. Preliminaries

### 2.1. Notations

The following notations will be used throughout this paper: LetIn denote the n×n identity matrix, let 0m×n denote the m×n zero matrix, and let 1n=[1,1,…,1]T∈Rn (1 for short, when there is no confusion). λmin⁡(A) and λmax⁡(A) are the smallest and the largest eigenvalues of the matrix A, respectively.


### 2.2. Graph Theory Notions

Using graph theory, we can model the interaction topology in multiagent systems consisting ofn agents. Let G=(V,E,A) be a weighted graph of order n with the finite nonempty set of nodes V={v1,…,vn}, the set of edges E⊆V×V, and a weighted adjacency matrix A=(aij)n×n. Here, each node vi in V corresponds to an agent i, and each edge (vi,vj)∈E in a weighted directed graph corresponds to an information link from agent j to agent i, which means that agent i can receive information from agent j. In contrast, the pairs of nodes in weighted undirected graph are unordered, where an edge (vj,vi)∈E denotes that agent i and j can receive information from each other. The weighted adjacency matrix A of a weighted directed graph is defined such that aii=0 for any vi∈V, aij>0 if (vj,vi)∈E, and aij=0 otherwise. The weighted adjacency matrix A of a weighted undirected graph is defined analogously except that aij=aji, for all i≠j, since (vi,vj)∈E implies (vj,vi)∈E. We can say vi is a neighbor vertex of vj, if (vi,vj)∈E.The Laplacian matrixL=(lij)n×n of graph G is defined by lij=-aij for i≠j, and lii=∑j=1,j≠in‍aij, i,j∈{1,…,n}. For an undirected graph, L is symmetric positive semidefinite. However, L is not necessarily symmetric for a directed graph.Lemma 1 (see [36]).
AssumeG is a weighted undirected graph with Laplacian matrix L; then the following two statements are equivalent: (i)
the matrixL has an eigenvalue zero with multiplicity 1 and corresponding eigenvector 1, and all other eigenvalues are positive;(ii)
G is connected.


### 2.3. Nonsmooth Analysis Background

Consider the following vector differential equation:(1)x˙(t)=f(x(t)),
where x(t)=[x1(t),x2(t),…,xn(t)]T and f(x(t)) is not necessarily continuous. A Filippov solution of (1) on [t0,t1], where t1 could be ∞, is absolutely continuous function x(t) satisfying the following differential inclusion:
(2)x˙(t)∈K[f](x(t)),
where K[f](x(t))≜⋂δ>0⋂μ(N)=0co⁡¯f(t,B(x,δ)∖N), ⋂μ(N)=0 denotes the intersection over all sets of Lebesgue measure zeroes, B(x,δ) denotes the open ball of radius δ centered at x, and co⁡¯(Ω) denotes the convex closure of the convex hull of the set Ω.Definition 2 (see [37]).
For a locally Lipschitz continuous functionV, define the Clarke's generalized gradient of V by ∂V≜co⁡{lim⁡∇V(x)∣xi→x,xi∈Ωv⋃‍N-}, where co⁡ denotes the convex hull, Ωv is the set of Lebesgue measure zero, where ∇V does not exist, and N- is an arbitrary set of zero measure. The set-valued Lie derivative of V with respect to (1) is defined as
(3)LKV≜⋂ξ∈∂V‍ξTK[f](x(t)).In the following, a Lyapunov stability theorem in terms of the set-valued mapLKV is stated.Lemma 3 (see [38]).
Given (1), let f(x(t)) be locally essentially bounded and 0∈K[f](0) in a region Q⊃{x∈Rm∣∥x∥<r}×t∣t0≤t<∞, where r>0. Also, let V:Rd→R be a regular function satisfying V(0,t)=0 and 0<V1(∥x∥)≤V≤V2(∥x∥), for z≠0, in Q for some V1 and V2 belonging to class K. If there exists a class K function w(·) in Q such that the set-valued Lie derivative of V(x,t) satisfies
(4)max⁡LKV≤-w(x)<0,forx≠0,
then the solution x(t)≡0 is asymptotically stable.


## 2.1. Notations

The following notations will be used throughout this paper: LetIn denote the n×n identity matrix, let 0m×n denote the m×n zero matrix, and let 1n=[1,1,…,1]T∈Rn (1 for short, when there is no confusion). λmin⁡(A) and λmax⁡(A) are the smallest and the largest eigenvalues of the matrix A, respectively.


## 2.2. Graph Theory Notions

Using graph theory, we can model the interaction topology in multiagent systems consisting ofn agents. Let G=(V,E,A) be a weighted graph of order n with the finite nonempty set of nodes V={v1,…,vn}, the set of edges E⊆V×V, and a weighted adjacency matrix A=(aij)n×n. Here, each node vi in V corresponds to an agent i, and each edge (vi,vj)∈E in a weighted directed graph corresponds to an information link from agent j to agent i, which means that agent i can receive information from agent j. In contrast, the pairs of nodes in weighted undirected graph are unordered, where an edge (vj,vi)∈E denotes that agent i and j can receive information from each other. The weighted adjacency matrix A of a weighted directed graph is defined such that aii=0 for any vi∈V, aij>0 if (vj,vi)∈E, and aij=0 otherwise. The weighted adjacency matrix A of a weighted undirected graph is defined analogously except that aij=aji, for all i≠j, since (vi,vj)∈E implies (vj,vi)∈E. We can say vi is a neighbor vertex of vj, if (vi,vj)∈E.The Laplacian matrixL=(lij)n×n of graph G is defined by lij=-aij for i≠j, and lii=∑j=1,j≠in‍aij, i,j∈{1,…,n}. For an undirected graph, L is symmetric positive semidefinite. However, L is not necessarily symmetric for a directed graph.Lemma 1 (see [36]).
AssumeG is a weighted undirected graph with Laplacian matrix L; then the following two statements are equivalent: (i)
the matrixL has an eigenvalue zero with multiplicity 1 and corresponding eigenvector 1, and all other eigenvalues are positive;(ii)
G is connected.


## 2.3. Nonsmooth Analysis Background

Consider the following vector differential equation:(1)x˙(t)=f(x(t)),
where x(t)=[x1(t),x2(t),…,xn(t)]T and f(x(t)) is not necessarily continuous. A Filippov solution of (1) on [t0,t1], where t1 could be ∞, is absolutely continuous function x(t) satisfying the following differential inclusion:
(2)x˙(t)∈K[f](x(t)),
where K[f](x(t))≜⋂δ>0⋂μ(N)=0co⁡¯f(t,B(x,δ)∖N), ⋂μ(N)=0 denotes the intersection over all sets of Lebesgue measure zeroes, B(x,δ) denotes the open ball of radius δ centered at x, and co⁡¯(Ω) denotes the convex closure of the convex hull of the set Ω.Definition 2 (see [37]).
For a locally Lipschitz continuous functionV, define the Clarke's generalized gradient of V by ∂V≜co⁡{lim⁡∇V(x)∣xi→x,xi∈Ωv⋃‍N-}, where co⁡ denotes the convex hull, Ωv is the set of Lebesgue measure zero, where ∇V does not exist, and N- is an arbitrary set of zero measure. The set-valued Lie derivative of V with respect to (1) is defined as
(3)LKV≜⋂ξ∈∂V‍ξTK[f](x(t)).In the following, a Lyapunov stability theorem in terms of the set-valued mapLKV is stated.Lemma 3 (see [38]).
Given (1), let f(x(t)) be locally essentially bounded and 0∈K[f](0) in a region Q⊃{x∈Rm∣∥x∥<r}×t∣t0≤t<∞, where r>0. Also, let V:Rd→R be a regular function satisfying V(0,t)=0 and 0<V1(∥x∥)≤V≤V2(∥x∥), for z≠0, in Q for some V1 and V2 belonging to class K. If there exists a class K function w(·) in Q such that the set-valued Lie derivative of V(x,t) satisfies
(4)max⁡LKV≤-w(x)<0,forx≠0,
then the solution x(t)≡0 is asymptotically stable.


## 3. Problem Description

Consider a multiagent system consisting ofn agents. In what follows, all agents are assumed in one-dimensional space for the simplicity of presentation. However, all results hereafter are still valid for the m-dimensional (m>1) by the Kronecker product [39]. Here, the dynamics of each agent in the group is given by
(5)ξ˙i=vi,v˙i=f(t,ξi,vi)+ui,
where i(i=1,…,n), ξi∈Rm and vi∈Rm denote the position and velocity vectors, f(t,ξi,vi)∈Rm denotes the inherent nonlinear dynamics, and ui(t) denotes the control input. When f(t,ξi,vi)≡0, the multiagent system has double-integrator dynamics.The consensus problem of the multiagent system (5) is to design control inputs ui, i={1,…,n} such that
(6)lim⁡t→∞∥ξi-ξ0∥2=0,lim⁡t→∞∥vi-v0∥2=0,
for any i=1,2,…,n, where ξ0∈Rm and v0∈Rm are, respectively, the position vector and velocity vector of the virtual leader, which does not have to be an actual agent but is specified by
(7)ξ˙0=v0,v˙0=f(t,ξ0,v0)+g(t,ξ0,v0),
where f(t,ξ0,v0)∈Rm describes the nonlinear dynamics of the virtual leader, and g(t,ξ0,v0)∈Rm is responsible for controlling trajectory of the virtual leader. We assume that |g(t,ξ0,v0)|≤C0, where C0 is a positive constant.Hereafter, then agents in systems (5) are called followers, and their communication topology is represented by the graph G. Suppose that the virtual leader and all followers share the same nonlinear inherent dynamics, and these nonlinear inherent dynamics satisfy a Lipchitz-type condition given by Assumption 4 as follows, which is satisfied in many well-known systems.Assumption 4.
There exist two nonnegative constantsl1 and l2 such that
(8)∥f(t,ξ,v)-f(t,ζ,γ)∥2≤l1∥ξ-ζ∥2+l2∥v-γ∥2,∀ξ,v,ζ,γ∈Rm,∀t≥0.Remark 5.
Note from Assumption4 that the Lipchitz constants l1 or l2 may be chosen by zero. In order to satisfy the requirements of designing consensus protocols, in this paper we assume that r1=⌊l1⌋+1 and r2=⌊l2⌋+1, where ⌊·⌋ denotes floor function (i.e., ⌊x⌋=max⁡{m∈Z∣m≤x}, where Z is the set of integers).Remark 6.
In recent reference [40], the robust cooperative tracking for multiple nonidentical second-order nonlinear systems is investigated. The nonlinear inherent dynamics f(t,ξi,vi) in this paper satisfies the Lipchitz condition (8), while the nonlinear function f(t,ξi,vi) in [40] is required to be continuously differentiable (the derivative f˙(t,ξi,vi) exists in its domain and is itself a continuous function).


## 4. Consensus Tracking Protocols

To satisfy (6), the following control input (9) is proposed for each follower:
(9)ui(t)=-α∑j=0n‍aij[r1(ξi-ξj)+r2(vi-vj)]-βsgn⁡{∑j=0n‍aij[r1(ξi-ξj)+r2(vi-vj)]},
where α is a nonnegative constant, β is a positive constant, aij,i,j=1,…,n, is the (i,j)th entry of the adjacency matrix A associated with G, and sgn⁡(·) is the signum function. In addition, ai0>0(i=1,…,n) if the virtual leader's position is available to follower i, and ai0=0 otherwise.Substituting (9) to (5) gives
(10)ξ˙i=vi,v˙i=f(t,ξi,vi)-α∑j=0n‍aij[r1(ξi-ξj)+r2(vi-vj)]-βsgn⁡{∑j=0n‍aij[r1(ξi-ξj)+r2(vi-vj)]}.LetM=L+diag⁡(a10,…,an0), where L is the Laplacian matrix of G.Lemma 7.
Suppose that the communication graphG is connected and at least one follower can receive information from the virtual leader. Let H=[(αr2/2r1)M2(1/2r2)M(1/2r2)M(1/2r2)M] and Q=[αM2(1/2)αM2(1/2)αM2αM2-(r1/r22)M]. If α>(8r22+2r1+2Ω^)λmax⁡(M)/3r22λmin⁡2(M), then the matrices H and Q are symmetric positive definite, and λmin⁡(Q)>2λmax⁡(M), where Ω^=4r24+2r1r22+r12.Proof.
Since the communication graphG is connected and the virtual leader is a neighbor of at least one follower, it follows that the matrix M is symmetric positive definite. Therefore, the matrix M can be diagonalized as M=T-1ΔT, where Δ={λ1,λ2,…,λn}, λi>0 with λi being the ith eigenvalue of M. Then we can obtain a matrix Π such that
(11)Π=[αr22r1Δ212r2Δ12r2Δ12r2Δ]=[T0n×n0n×nT]H[T-10n×n0n×nT-1],
where 0n×n is the n×n zero matrix. It is easy to see that Π is symmetric and has the same eigenvalues as H.
Letδ be an eigenvalue of Π. Because Δ is a diagonal matrix, it follows from (11) that
(12)δ2-αr22λi2+r1λi2r1r2δ+αr22λi3-r1λi24r1r22=0.
As the matrix Π is symmetric, it follows that the roots of (12) are all real. Therefore, they are positive if and only if the following conditions are satisfied:
(13)αr22λi2+r1λi2r1r2>0,αr22λi3-r1λi24r1r22>0.
And, H is positive definite if
(14)α>r1r22λmin⁡(M).
Next, we consider the matrixQ. Similarly with the analysis of the matrix H, there exists a matrix J such that
(15)J=[αΔ212αΔ212αΔ2αΔ2-r1r22Δ]=[T0n×n0n×nT]Q[T-10n×n0n×nT-1].J is symmetric and has the same eigenvalues as Q.
Letε be an eigenvalue of J. Because Δ is a diagonal matrix, it follows from (15) that ε satisfies
(16)ε2-[2αλi2-r1r22λi]ε+αλi2(αλi2-r1r22λi)-14α2λi4=0.
Therefore, λmin⁡(Q)>2λmax⁡(M) if and only if the following inequations are satisfied:
(17)2αλi2-(r1/r22)λi2>2λmax⁡(M),(18)(2λmax⁡(M))2-[2αλi2-r1r22λi](2λmax⁡(M))+αλi2(αλi2-r1r22λi)-14α2λi4>0.
From (17), it follows that
(19)α>(4r22+r1)λmax⁡(M)2r22λmin⁡2(M)≥4r22λmax⁡(M)+r1λi2r22λi2.
From (18), it follows that
(20)α>8r22λmax⁡(M)+2r1λi+2Ω3r22λi2
or
(21)α<8r22λmax⁡(M)+2r1λi-2Ω3r22λi2,
where Ω=4r24λmax⁡2(M)+2r1r22λiλmax⁡(M)+r12λi2.
Therefore,Q is positive definite and λmin⁡(Q)>2λmax⁡(M) if
(22)α>(8r22+2r1+2Ω^)λmax⁡(M)3r22λmin⁡2(M),
where Ω^=4r24+2r1r22+r12.
Combining the results (14) and (22), the proof is completed.Theorem 8.
Suppose that the communication graphG is connected and the virtual leader is a neighbor of at least one follower. If α>(8r22+2r1+2Ω^)λmax⁡(M)/3r22λmin⁡2(M) and β>C0, then second-order consensus tracking in system (10) is achieved at least globally exponentially, where Ω^=4r24+2r1r22+r12.Proof.
Letξ~i=ξi-ξ0,v~i=vi-v0,i∈{1,…,n}. From (7) and (10),
(23)ξ~˙i=v~i,v~˙i=-α∑j=0n‍aij[r1(ξ~i-ξ~j)+r2(v~i-v~j)]-βsgn⁡{∑j=0n‍aij[r1(ξ~i-ξ~j)+r2(v~i-v~j)]}+f(t,ξi,vi)-f(t,ξ0,v0)+g(t,ξ0,v0).
Let ξ~=[ξ~1T,ξ~2T,…,ξ~nT]T, v~=[v~1T,v~2T,…,v~nT]T and F(t,ξ~,v~)=[(f(t,ξ1,v1)-f(t,ξ0,v0))T,…,(f(t,ξn,vn)-f(t,ξ0,v0))T]T. Rewrite (23) in the matrix form as
(24)ξ~˙=v~,v~˙=F(t,ξ~,v~)-αM(r1ξ~+r2v~)-βsgn⁡(M(r1ξ~+r2v~))+1g(t,ξ0,v0).
Consider the Lyapunov function candidate
(25)V(t)=[r1ξ~Tr2v~T]H[r1ξ~r2v~].
According to Definition2, in the following let us compute the set-valued lie derivative of V as(26)LKV(ξ~,v~)=⋂x∈∂V(ξ~,v~)‍xTK[f](ξ~,v~)=⋂x∈∂V(ξ~,v~)‍xTK[r1v~r2{F(t,ξ~,v~)-αM(r1ξ~+r2v~)-βsgn⁡(M(r1ξ~+r2v~))+1g(t,ξ0,v0)}],
where∂V(ξ~,v~) is the generalized gradient of V. Because V is continuously differentiable, ∂V(ξ~,v~)=H[r1ξ~r2v~]. Hence, we have(27)LKV(ξ~,v~)=[r1ξ~r2v~]THTK[r1v~r2{F(t,ξ~,v~)-αM(r1ξ~+r2v~)-βsgn⁡(M(r1ξ~+r2v~))+1g(t,ξ0,v0)}]=K[-[r1ξ~Tr2v~T]Q[r1ξ~r2v~]-β∥(r1ξ~T+r2v~T)M∥1+(ξ~Tr1+v~Tr2)MF(t,ξ~,v~)+(r1ξ~T+r2v~T)M1g(t,ξ0,v0)[r1ξ~r2v~]]={-[r1ξ~Tr2v~T]Q[r1ξ~r2v~]-β∥(r1ξ~T+r2v~T)M∥1+(ξ~Tr1+v~Tr2)MF(t,ξ~,v~)+(r1ξ~T+r2v~T)M1g(t,ξ0,v0)[r1ξ~r2v~]},
where the fact thatxTK[sign⁡(x)]=K[xTsign⁡(x)]=xTsign⁡(x)=∥x∥1 has been used as the fact xTK[sign⁡(x)] is continuous. It then follows that the set-valued Lie derivative LKV is a singleton, whose only element is actually V˙. Since ∥F(t,ξ~,v~)∥2≤∥r1ξ~∥1+∥r2v~∥2,|g(t,ξ0,v0)|<C0, and ∥(r1ξ~T+r2v~T)M∥1>∥(r1ξ~T+r2v~T)M∥2, it follows that
(28)max⁡LKV=V˙≤-λmin⁡(Q)(∥r1ξ~∥22+∥r2v~∥22)+λmax⁡(M)(∥r1ξ~∥2+∥r2v~∥2)2-β∥(r1ξ~T+r2v~T)M∥2+C0∥(r1ξ~T+r2v~T)M∥2≤-[λmin⁡(Q)-2λmax⁡(M)](∥r1ξ~∥22+∥r2v~∥22)-(β-C0)∥(r1ξ~T+r2v~T)∥2.
Note α>(8r22+2r1+2Ω^)λmax⁡(M)/3r22λmin⁡2(M) and β>C0; it follows from Lemma 7 that max⁡LKV is negative definite. Therefore, ξ~i(t)→0n and v~i(t)→0n as t→∞, where 0n is n×1 zero vector. Equivalently, it follows that ξi(t)→ξ0(t) and vi(t)→v0(t) as t→∞.
Next, we prove that the consensus can be achieved at least globally exponentially. Note that(29)V=[r1ξ~Tr2v~T]H[r1ξ~r2v~]≤λmax⁡(H)(∥r1ξ~∥22+∥r2v~∥22).
From (28), V˙(t) satisfies that
(30)max⁡LKV(t)=V˙(t)≤-(λmin⁡(Q)-2λmax⁡(M))(∥r1ξ~∥22+∥r2v~∥22)≤-(λmin⁡(Q)-2λmax⁡(M))λmax⁡(H)V(t).
Therefore, V(t)≤V(0)e-((λmin⁡(Q)-2λmax⁡(M))/λmax⁡(H))t. The proof is completed.Remark 9.
In this paper, we assume that there exists a virtual leader, which does not have to be an actual agent and can be a specified reference state. In fact, the conclusion obtained in this paper can also be extended to the leader-less case where there exists no virtual leader. Because the proof is similar to that of Theorem8 in this paper, it is therefore omitted here.


## 5. Numerical Results

In order to demonstrate the effectiveness of the theoretic results, some simulations are given in this section.


### 5.1. Leader-Follower Case

#### 5.1.1. Dynamics of Agents

Consider a second-order multiagent system consisting of one virtual leader indexed by0 and six followers indexed by 1 to 6, respectively. The communication topology is given in Figure 1(a). The nonlinear inherent dynamics of each follower is given as follows:
(31)f(t,ξi,vi)=[sin(ξix)cos⁡(t)+cos⁡(vix)sin(t)cos⁡(ξiy)sin(t)+sin(viy)cos⁡(t)]∈R2.
It is easy to verify that f(t,ξi,vi) satisfies Assumption 4. Here, the Lipschitz constants are chosen as r1=3 and r2=3. The trajectory of virtual leader is chosen as ξ0(t)=[t,sin(t)]T∈R2. It follows that the dynamics of virtual leader is given as
(32)f(t,ξ0,v0)+g(t,ξ0,v0)=[0-sin(t)]∈R2.
Furthermore,
(33)g(t,ξ0,v0)=[-sin⁡(ξix)cos⁡⁡(t)-cos⁡⁡(vix)sin⁡(t)-cos⁡⁡(ξiy)sin⁡(t)-sin⁡(viy)cos⁡⁡(t)-sin⁡(t)].
From (33), let us take C0=3.The communication topologies for (a) a group of six followers and one virtual, and (b) a group of six agents without virtual leader.
(a)
(b)


#### 5.1.2. Determinations ofα and β

In this simulation, we suppose thataij=1 if agent i can receive information from agent j, aij=0 otherwise, i∈{1,…,n} and j∈{0,1,…,n}. Therefore, the matrix M can be derived from the topology given in Figure 1(a):
(34)M=[2-10-100-14-10-100-1200-1-1001000-1002-100-10-12].
It is easy to obtain that λmin⁡(M)=0.1284. By Theorem 8, when α≤((8r22+2r1)+2(r1+r22)2+3r24)/3r22λmin⁡(M)=31.4657, and β>C0=3, the consensus tracking can be achieved. Here we choose α=32, β=4.


#### 5.1.3. Simulation Results

Figure2 shows the position states of the virtual leader and followers. Figures 3(a) and 3(b) show the position tracking errors along x-axis and y-axis. Figures 4(a) and 4(b) show the velocity tracking errors along x-axis and y-axis, respectively. Here, the initial position and velocity states of followers are randomly chosen from the cubes [-3,3]×[-3,3] and [-2,2]×[-2,2], respectively, and the initial position and velocity states of the virtual leader are ξ0(0)=[0,0]T and v0(0)=[1,1]T. It can be seen that all followers ultimately track the virtual leader. Simulation results verify the theoretical analysis very well.Figure 2
Position states of the followers and virtual leader under the communication topology of Figure1(a).(a) Position tracking errors alongx-axis; (b) position tracking errors along y-axis.
(a)
(b)(a) Velocity tracking errors alongx-axis; (b) velocity tracking errors along y-axis.
(a)
(b)


### 5.2. Leaderless Case

In this part, let us consider a group of six agents without virtual leader. The communication topology is given in Figure1(b). In order to compare with leader-follower case, the nonlinear inherent dynamics of each agent is chosen as (31), which is the same as that in leader-follower case. α and β are also chosen as 32 and 4, respectively; the initial position and velocity states of six agents are still randomly chosen from the cubes [-3,3]×[-3,3] and [-2,2]×[-2,2], respectively,Figure5 shows the position states of six agents under communication topology in Figure 1(b). Figures 6(a) and 6(b) show the position tracking errors along x-axis and y-axis. Figures 7(a) and 7(b) show the velocity tracking errors along x-axis and y-axis, respectively. It can be seen from Figures 6 and 7 that the tracking errors ultimately converge to zero. We see from Figure 5 that consensus tracking is achieved.Figure 5
Position states of six agents.(a) Position tracking errors alongx-axis; (b) position tracking errors along y-axis.
(a)
(b)(a) Velocity tracking errors alongx-axis; (b) velocity tracking errors along y-axis.
(a)
(b)Remark 10.
For the leaderless consensus case, the final states of each agent are determined by the communication topology, the control gains, and the initial value of each agent. The final states of each agent cannot be specified. However, for the leader-follower case, there exists a virtual leader that determines the final state, and the control objective is to guarantee that the final states of all agents reach the state of the virtual leader. From the above simulations, the conclusion obtained in this paper can be extended to the leaderless case.


## 5.1. Leader-Follower Case

### 5.1.1. Dynamics of Agents

Consider a second-order multiagent system consisting of one virtual leader indexed by0 and six followers indexed by 1 to 6, respectively. The communication topology is given in Figure 1(a). The nonlinear inherent dynamics of each follower is given as follows:
(31)f(t,ξi,vi)=[sin(ξix)cos⁡(t)+cos⁡(vix)sin(t)cos⁡(ξiy)sin(t)+sin(viy)cos⁡(t)]∈R2.
It is easy to verify that f(t,ξi,vi) satisfies Assumption 4. Here, the Lipschitz constants are chosen as r1=3 and r2=3. The trajectory of virtual leader is chosen as ξ0(t)=[t,sin(t)]T∈R2. It follows that the dynamics of virtual leader is given as
(32)f(t,ξ0,v0)+g(t,ξ0,v0)=[0-sin(t)]∈R2.
Furthermore,
(33)g(t,ξ0,v0)=[-sin⁡(ξix)cos⁡⁡(t)-cos⁡⁡(vix)sin⁡(t)-cos⁡⁡(ξiy)sin⁡(t)-sin⁡(viy)cos⁡⁡(t)-sin⁡(t)].
From (33), let us take C0=3.The communication topologies for (a) a group of six followers and one virtual, and (b) a group of six agents without virtual leader.
(a)
(b)


### 5.1.2. Determinations ofα and β

In this simulation, we suppose thataij=1 if agent i can receive information from agent j, aij=0 otherwise, i∈{1,…,n} and j∈{0,1,…,n}. Therefore, the matrix M can be derived from the topology given in Figure 1(a):
(34)M=[2-10-100-14-10-100-1200-1-1001000-1002-100-10-12].
It is easy to obtain that λmin⁡(M)=0.1284. By Theorem 8, when α≤((8r22+2r1)+2(r1+r22)2+3r24)/3r22λmin⁡(M)=31.4657, and β>C0=3, the consensus tracking can be achieved. Here we choose α=32, β=4.


### 5.1.3. Simulation Results

Figure2 shows the position states of the virtual leader and followers. Figures 3(a) and 3(b) show the position tracking errors along x-axis and y-axis. Figures 4(a) and 4(b) show the velocity tracking errors along x-axis and y-axis, respectively. Here, the initial position and velocity states of followers are randomly chosen from the cubes [-3,3]×[-3,3] and [-2,2]×[-2,2], respectively, and the initial position and velocity states of the virtual leader are ξ0(0)=[0,0]T and v0(0)=[1,1]T. It can be seen that all followers ultimately track the virtual leader. Simulation results verify the theoretical analysis very well.Figure 2
Position states of the followers and virtual leader under the communication topology of Figure1(a).(a) Position tracking errors alongx-axis; (b) position tracking errors along y-axis.
(a)
(b)(a) Velocity tracking errors alongx-axis; (b) velocity tracking errors along y-axis.
(a)
(b)


## 5.1.1. Dynamics of Agents

Consider a second-order multiagent system consisting of one virtual leader indexed by0 and six followers indexed by 1 to 6, respectively. The communication topology is given in Figure 1(a). The nonlinear inherent dynamics of each follower is given as follows:
(31)f(t,ξi,vi)=[sin(ξix)cos⁡(t)+cos⁡(vix)sin(t)cos⁡(ξiy)sin(t)+sin(viy)cos⁡(t)]∈R2.
It is easy to verify that f(t,ξi,vi) satisfies Assumption 4. Here, the Lipschitz constants are chosen as r1=3 and r2=3. The trajectory of virtual leader is chosen as ξ0(t)=[t,sin(t)]T∈R2. It follows that the dynamics of virtual leader is given as
(32)f(t,ξ0,v0)+g(t,ξ0,v0)=[0-sin(t)]∈R2.
Furthermore,
(33)g(t,ξ0,v0)=[-sin⁡(ξix)cos⁡⁡(t)-cos⁡⁡(vix)sin⁡(t)-cos⁡⁡(ξiy)sin⁡(t)-sin⁡(viy)cos⁡⁡(t)-sin⁡(t)].
From (33), let us take C0=3.The communication topologies for (a) a group of six followers and one virtual, and (b) a group of six agents without virtual leader.
(a)
(b)


## 5.1.2. Determinations ofα and β

In this simulation, we suppose thataij=1 if agent i can receive information from agent j, aij=0 otherwise, i∈{1,…,n} and j∈{0,1,…,n}. Therefore, the matrix M can be derived from the topology given in Figure 1(a):
(34)M=[2-10-100-14-10-100-1200-1-1001000-1002-100-10-12].
It is easy to obtain that λmin⁡(M)=0.1284. By Theorem 8, when α≤((8r22+2r1)+2(r1+r22)2+3r24)/3r22λmin⁡(M)=31.4657, and β>C0=3, the consensus tracking can be achieved. Here we choose α=32, β=4.


## 5.1.3. Simulation Results

Figure2 shows the position states of the virtual leader and followers. Figures 3(a) and 3(b) show the position tracking errors along x-axis and y-axis. Figures 4(a) and 4(b) show the velocity tracking errors along x-axis and y-axis, respectively. Here, the initial position and velocity states of followers are randomly chosen from the cubes [-3,3]×[-3,3] and [-2,2]×[-2,2], respectively, and the initial position and velocity states of the virtual leader are ξ0(0)=[0,0]T and v0(0)=[1,1]T. It can be seen that all followers ultimately track the virtual leader. Simulation results verify the theoretical analysis very well.Figure 2
Position states of the followers and virtual leader under the communication topology of Figure1(a).(a) Position tracking errors alongx-axis; (b) position tracking errors along y-axis.
(a)
(b)(a) Velocity tracking errors alongx-axis; (b) velocity tracking errors along y-axis.
(a)
(b)


## 5.2. Leaderless Case

In this part, let us consider a group of six agents without virtual leader. The communication topology is given in Figure1(b). In order to compare with leader-follower case, the nonlinear inherent dynamics of each agent is chosen as (31), which is the same as that in leader-follower case. α and β are also chosen as 32 and 4, respectively; the initial position and velocity states of six agents are still randomly chosen from the cubes [-3,3]×[-3,3] and [-2,2]×[-2,2], respectively,Figure5 shows the position states of six agents under communication topology in Figure 1(b). Figures 6(a) and 6(b) show the position tracking errors along x-axis and y-axis. Figures 7(a) and 7(b) show the velocity tracking errors along x-axis and y-axis, respectively. It can be seen from Figures 6 and 7 that the tracking errors ultimately converge to zero. We see from Figure 5 that consensus tracking is achieved.Figure 5
Position states of six agents.(a) Position tracking errors alongx-axis; (b) position tracking errors along y-axis.
(a)
(b)(a) Velocity tracking errors alongx-axis; (b) velocity tracking errors along y-axis.
(a)
(b)Remark 10.
For the leaderless consensus case, the final states of each agent are determined by the communication topology, the control gains, and the initial value of each agent. The final states of each agent cannot be specified. However, for the leader-follower case, there exists a virtual leader that determines the final state, and the control objective is to guarantee that the final states of all agents reach the state of the virtual leader. From the above simulations, the conclusion obtained in this paper can be extended to the leaderless case.


## 6. Conclusion

The consensus tracking problem for second-order multiagent systems with nonlinear inherent dynamics and a time varying reference state has been studied in this paper. A class of nonsmooth control protocols has been proposed, and the corresponding sufficient conditions have been obtained. It is found that if the communication graph associated with the virtual leader and followers is connected, the consensus can be achieved globally exponentially with the proposed protocol.There are still a number of related interesting problems that deserve further infestation, for example, the consensus tracking problem for high-order multiagent systems with nonlinear inherent dynamics, the consensus tracking problem under directed communication topology and with time delays, and so on, some of which will be studied in the near future.



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*Source: 320687-2014-05-28.xml*