# Robust Consensus for Nonlinear Multiagent Systems with Uncertainty and Disturbance

Authors: Zhiqiang Li; Chengjie Xu; Chen Liu; Haichuan Xu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper investigates robust consensus for nonlinear multiagent systems with uncertainty and disturbance. The consensus evolution behavior is studied under general consensus protocol when each node is disturbed by the relative states between the node and its neighbors. At first, the robust consensus condition is obtained and the convergency analysis is given by using Lyapunov stability theory and matrix theory. Then, the practical consensus is investigated and the bound of the error states is presented. Finally, two numerical simulation examples are given to illustrate the proposed theoretical results.


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## Body

## 1. Introduction

In the past few years, increasing attention has been devoted to the coordination of multiagent systems due to its wide applications for sensor network and multirobot systems [1]. As one of the most important problems in the issue of distributed coordination, consensus has even attracted much more interest from researchers. And many profound results are reported [2–6].Consensus means that the states of the agents reach an agreement on a common physical quantity by implementing an appropriate consensus protocol based on the information from local neighbors [7], which is mainly influenced by the topology of the network and the dynamics of each node. Thereby, on one side, many works about consensus of multiagent systems under switching topology are reported [8–10]. On the other side, consensus of multiagent systems with nonlinear dynamics or disturbances is intensively investigated [11–15]. External disturbance widely exists in real processes and is a main source of instability and poor performance. When any of the nodes is disturbed, coordinated behavior will be destroyed. Thereby, it is of great significance to investigate distributed coordination for nonlinear multiagent systems with bounded disturbances or stochastic disturbances. Mean square average consensus is investigated for multiagent systems with noisy measurement under directed topology. Necessary and sufficient consensus conditions are established [16]. Mean square leader-following consensus for multiagent systems with noisy channels under directed switching topology is investigated [17]. Necessary and sufficient average consensus conditions are established for multiple double-integrator systems with noisy measurement. It is proven that mean square average consensus can be realized if and only if the topology is balanced and strongly connected [18]. Using disturbance observer method, robust leader-following consensus was investigated for nonlinear coupled multiagent systems with external disturbance [19]. Containment control problem is studied for general linear multiagent systems with exogenous disturbances. Both the state feedback and output feedback containment protocol are proposed by using the disturbance observer approach [20]. Using sliding-mode control method, finite consensus and containment were investigated for second-order nonlinear multiagent systems with disturbances under directed topology [21]. Bounded consensus tracking is investigated for linear multiagent systems with actuator saturation and input additive uncertainties and disturbances based on low-and-high gain feedback approach [22].However, few works investigate the consensus behavior when the relative information of the subsystem is disturbed, which widely exists in the real world. Therefore, in this paper, the consensus evolution behavior is investigated for nonlinear multiagent systems with uncertainty and disturbance. The main contribution in this paper is that the disturbance considered is dependent on the addition of the relative states between each node and their neighbors, and the disturbance satisfies a very mild condition, which is more relaxed than the condition in [21, 22]. The uncertainty and disturbance function is described as a piecewise continuous function g(x,t), satisfying g(x,t)2≤g0(x2)+D0. Both robust consensus and practical consensus condition are obtained for D0=0 and D0≠0, respectively. This leads to the fact that the problems investigated in this paper are theoretically challenging and practically important.The rest of the paper is organized as follows. Section2 states the model considered in the paper and gives some basic definitions, lemmas, and assumptions. In Section 3, complete consensus protocol is proposed and the convergency analysis is given and practical consensus protocol is obtained in Section 4. In Section 5, two numerical simulation examples are given for illustrating the theoretical results. Finally, Section 6 concludes the paper.


## 2. Preliminaries and Model Description

A networked multiagent system under undirected topology consists of many agents and their communication, which can be described as an undirected graphG=(V,E,A). The set of nodes V=(υ1,υ2,…,υN) denotes the agents. The set of edges E⊆V×V denotes communication relation of the agents. A=[aij] is the adjacent matrix of the nodes, which is defined as follows: aij=1⇔(υj,υi)∈E; otherwise aij=0. A path is a sequence of edges in a directed graph of the form (ϑ1,ϑ2),(ϑ2,ϑ3),…, where ϑj∈V. An undirected graph is connected if there is a path between any two nodes. The Laplacian matrix L=[lij]∈RN×N is defined as lii=∑j=1,j≠iNaij and lij=-aij, i≠j.Assume that there areN follower agents, labeled as 1,2,…,N. The dynamics of the ith agent are described as(1)x˙it=ft,xi+uit,i=1,2,…,N,where xi,ui∈Rn are the position and the input vector of the ith follower, respectively. f:R×Rn→Rn, i=1,2,…,k, is a continuous vector-value function, which describes the intrinsic local dynamics of the ith node.The following assumption is necessary for our main results.Assumption 1.
Assume thatG is connected.Assumption 2.
Suppose thatf(t,x) satisfies the Lipschitz condition; that is, for arbitrary x1,x2, there exists a constant η such that(2)ft,x1-ft,x22≤ηx1-x22,∀x1,x2∈RN.Lemma 3 (see [2]).
IfL=(lij)∈RN×N is a symmetric irreducible matrix with lii=-∑j=1,j≠iNlij, lij=lji≤0(i≠j), then L is semipositive definite, and for any matrix H=diag(h,0,⋯,0) with h>0, all eigenvalues of the matrix (L+H) are positive.Lemma 4 (see [23]).
For an undirected connected graphG with the Laplacian matrix L and the vector x satisfying 1Tx=0,(3)minx≠0⁡xTLxxTx=λ2L.


## 3. Robust Leader-Following Consensus for Nonlinear Multiagent Systems with Uncertainty and Disturbance

In this section, the robust leader-following consensus of nonlinear multiagent systems with bounded channel disturbance is investigated. Suppose that there is a virtual leader in the network, whose dynamics are described as(4)x˙0t=ft,x0.Consider the following consensus protocol with the channel disturbance:(5)ui=∑j∈Niaijαxj-xi+gxj-xi,t+αhix0-xi,i=1,2,…,N,where α>0 denotes the control gain to be determined; the nonlinear function g:Rn×R→Rn denotes the actuator input additive uncertainty and disturbance, satisfying g(xj-xi,t)=-g(xi-xj,t).Assumption 5.
The channel disturbance functiong(x,t) is piecewise continuous in t and locally Lipschitz in x and its norm is bounded by a known function.(6)gx,t2≤g0x2+D0,∀x,t∈Rn×R+,where g0:R+→R+ is locally Lipschitz and satisfies g0(0)=0 and D0 is a positive constant.Remark 6.
For a given positive constantδ, define(7)Mδ=sup⁡g0ss∣s∈0,δ.Under protocol (5), (1) can be rewritten as(8)x˙it=ft,xi+∑j∈Niaijαxj-xi+gxj-xi,t+αhix0-xi,i=1,2,…,N.Let ei=xi-x0; then(9)e˙it=ft,xi-ft,x0+ui=ft,xi-ft,x0+αhix0-xi+∑j∈Niaijαxj-xi+gxj-xi,t=ft,xi-ft,x0-∑j∈Niαlijej-αhiei+∑j∈Niaijgxj-xi,t,i=1,2,…,N.Lemma 7 (see [22]).
For anyei,ej∈Rn,i,j=1,2,…,N,(10)12∑i=1N∑j=1NaijeiTgxj-xi,t=-∑i=1N∑j=1Naijej-eiTgxj-xi,t.Lemma 8 (see [24]).
For anyei,ej∈Rn, i,j=1,2,…,N,(11)12∑i=1N∑j=1Naijej-eiTej-ei=∑i=1N∑j=1NlijeiTej.


### 3.1. Robust Leader-Following Consensus

In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case whereD0=0 in Assumption 5.Theorem 9.
Consider a networked multiagent system withN followers and a virtual leader, in which the dynamics are described as (1)-(4). Suppose that Assumptions 1–5 hold and the constant D0=0 in Assumption 5. Under the consensus protocol (5), robust consensus of system (1) can be achieved if the following two conditions hold:
(i) There exists at least one pinned node.
(ii) For a given positive constantδ,(12)α-4Mδλ1>η,where λ1 is the minimal eigenvalue of (L+H).
Furthermore, all the following nodes will track the virtual leader asymptotically.Proof.
Choose the Lyapunov candidate function as(13)V=12∑i=1NeiTei.Differentiating V with respect to t along (9), one can obtain(14)V˙=∑i=1NeiTe˙i=∑i=1NeiTft,xi-ft,x0-∑i=1NeiT∑j=1Nαlijej+αhiei-∑j=1Naijgxj-xi,t≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+∑i=1N∑j=1NeiTaijgej-ei,t.According to Lemma 4 and Assumption 5, one has(15)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei-2∑i=1N∑j=1Naijej-eiTgej-ei,t≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1Naijej-ei2gej-ei,t2≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1Naijej-ei2g0ej-ei2≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2Mδ∑i=1N∑j=1Naijej-ei22.According to Lemma 7, we get(16)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+4Mδ∑i=1N∑j=1NlijeiTej≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+4Mδ∑i=1N∑j=1NlijeiTej+4Mδαhi∑i=1NeiTei=-α-4MδeTL+He+ηeTe.According to (12), one has V˙<0, which means that the error systems (9) are locally asymptotically stable. Then xi(t)→x0(t), for t→∞. This means that Theorem 9 holds.


### 3.2. Robust Leader-Following Practical Consensus

In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case whereD0≠0 in Assumption 5.Theorem 10.
Consider a networked multiagent system withN followers and a virtual leader, in which the dynamics are described as (1)-(4). Suppose that Assumptions 1–5 hold and the constant D0≠0 in Assumption 5. Under the consensus protocol (5), robust practical consensus of system (1) can be achieved if the following two conditions hold:
(i) There exists at least one pinned node.
(ii) For a given positive constantδ,(17)βδ=α-2+4Mδλ1>η,where λ1 is the minimal eigenvalue of (L+H).Proof.
Propose the Lyapunov candidate function as(18)V=12∑i=1NeiTei.According to (15), one can obtain(19)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei-2∑i=1N∑j=1Naijej-eiTgej-ei,t≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+∑i=1N∑j=1Naijej-eiTej-ei+∑i=1N∑j=1Naijgej-ei,tTgej-ei,t.
According to Lemma7 and Assumption 5, we get(20)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1NlijeiTej+∑i=1N∑j=1Naijg0ej-ei2+D02≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1NlijeiTej+2∑i=1N∑j=1Naijg0ej-ei22+2∑i=1N∑j=1ND02.According to Remark 6 and Assumption 5, one has(21)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1NlijeiTej+2Mδ∑i=1N∑j=1Naijej-ei22+2NN-1D02=η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2+4Mδ∑i=1N∑j=1NlijeiTej+2NN-1D02≤eTηI-α-2+4MδL+H⊗Ine+2NN-1D02≤-βδ-ηV+2NN-1D02=-βδ-ηV-2NN-1D02βδ-η.Since βδ>η, V˙<0 if V>2N(N-1)D02/(βδ-η), which means that all the state errors will enter into the set S={V≤2N(N-1)D02/(βδ-η)}. This means that Theorem 10 holds.


## 3.1. Robust Leader-Following Consensus

In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case whereD0=0 in Assumption 5.Theorem 9.
Consider a networked multiagent system withN followers and a virtual leader, in which the dynamics are described as (1)-(4). Suppose that Assumptions 1–5 hold and the constant D0=0 in Assumption 5. Under the consensus protocol (5), robust consensus of system (1) can be achieved if the following two conditions hold:
(i) There exists at least one pinned node.
(ii) For a given positive constantδ,(12)α-4Mδλ1>η,where λ1 is the minimal eigenvalue of (L+H).
Furthermore, all the following nodes will track the virtual leader asymptotically.Proof.
Choose the Lyapunov candidate function as(13)V=12∑i=1NeiTei.Differentiating V with respect to t along (9), one can obtain(14)V˙=∑i=1NeiTe˙i=∑i=1NeiTft,xi-ft,x0-∑i=1NeiT∑j=1Nαlijej+αhiei-∑j=1Naijgxj-xi,t≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+∑i=1N∑j=1NeiTaijgej-ei,t.According to Lemma 4 and Assumption 5, one has(15)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei-2∑i=1N∑j=1Naijej-eiTgej-ei,t≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1Naijej-ei2gej-ei,t2≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1Naijej-ei2g0ej-ei2≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2Mδ∑i=1N∑j=1Naijej-ei22.According to Lemma 7, we get(16)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+4Mδ∑i=1N∑j=1NlijeiTej≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+4Mδ∑i=1N∑j=1NlijeiTej+4Mδαhi∑i=1NeiTei=-α-4MδeTL+He+ηeTe.According to (12), one has V˙<0, which means that the error systems (9) are locally asymptotically stable. Then xi(t)→x0(t), for t→∞. This means that Theorem 9 holds.


## 3.2. Robust Leader-Following Practical Consensus

In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case whereD0≠0 in Assumption 5.Theorem 10.
Consider a networked multiagent system withN followers and a virtual leader, in which the dynamics are described as (1)-(4). Suppose that Assumptions 1–5 hold and the constant D0≠0 in Assumption 5. Under the consensus protocol (5), robust practical consensus of system (1) can be achieved if the following two conditions hold:
(i) There exists at least one pinned node.
(ii) For a given positive constantδ,(17)βδ=α-2+4Mδλ1>η,where λ1 is the minimal eigenvalue of (L+H).Proof.
Propose the Lyapunov candidate function as(18)V=12∑i=1NeiTei.According to (15), one can obtain(19)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei-2∑i=1N∑j=1Naijej-eiTgej-ei,t≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+∑i=1N∑j=1Naijej-eiTej-ei+∑i=1N∑j=1Naijgej-ei,tTgej-ei,t.
According to Lemma7 and Assumption 5, we get(20)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1NlijeiTej+∑i=1N∑j=1Naijg0ej-ei2+D02≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1NlijeiTej+2∑i=1N∑j=1Naijg0ej-ei22+2∑i=1N∑j=1ND02.According to Remark 6 and Assumption 5, one has(21)V˙≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2∑i=1N∑j=1NlijeiTej+2Mδ∑i=1N∑j=1Naijej-ei22+2NN-1D02=η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej-αhi∑i=1NeiTei+2+4Mδ∑i=1N∑j=1NlijeiTej+2NN-1D02≤eTηI-α-2+4MδL+H⊗Ine+2NN-1D02≤-βδ-ηV+2NN-1D02=-βδ-ηV-2NN-1D02βδ-η.Since βδ>η, V˙<0 if V>2N(N-1)D02/(βδ-η), which means that all the state errors will enter into the set S={V≤2N(N-1)D02/(βδ-η)}. This means that Theorem 10 holds.


## 4. Robust Leaderless Consensus for Nonlinear Multiagent Systems with Uncertainty and Disturbance

In this section, the robust leaderless consensus of nonlinear multiagent systems with bounded channel disturbance is investigated. Consider the following consensus protocol:(22)ui=∑j∈Niaijtαxj-xi+gt,xj-xi,i=1,2,…,N,where α>0 and g:R×Rn→Rn are defined the same way as Section 3. Denoting x¯=∑i=1Nxi/N, x¯=(∑i=1Nf(t,xi)+∑i=1Nui)/N. Since the Laplacian matrix L is symmetric and zero-row-sum, one has ∑j=1Naij(t)(xj-xi)=∑j=1Nlij(t)xj=0; then ∑i=1N∑j=1Naij(t)(xj-xi)=-∑i=1N∑j=1Nlij(t)xj=0. Furthermore, noting that g(t,xj-xi)=-g(t,xi-xj), one can conclude that ∑i=1N∑j=1Naij(t)g(t,xj-xi)=0. Then ∑i=1Nui=∑i=1N∑j=1Naij(t)(α(xj-xi)+g(t,xj-xi))=0. It follows that x¯˙=0. Let x~i=xi-x¯; according to (9), the error system can be described as(23)e˙it=ft,xi-∑j=1Nft,xjN-∑j∈Niαlijej+∑j∈Niaijgej-ei,t,i=1,2,…,N.


### 4.1. Robust Leaderless Consensus

In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case whereD0=0 in Assumption 5.Theorem 11.
Consider a networked multiagent system withN follower agents, in which the dynamics are described as (1). Suppose that Assumptions 1–5 hold and the constant D0=0 in Assumption 5. Under the consensus protocol (22), robust consensus of system (1) can be achieved if(24)βδ=α-Mδλ1>5ηwhere λ2(L) is the minimal eigenvalue of L and δ is a given positive constant.Proof.
Choose the Lyapunov candidate function as(25)V=12∑i=1NeiTei.Differentiating V with respect to t along (23), one can obtain(26)V˙=∑i=1NeiTe˙i=∑i=1NeiTft,xi-∑j=1Nft,xjN-∑i=1NeiT∑j=1Nαlijej-∑j=1Naijgxj-xi,t≤η∑i=1Nei2ei-ej2-α∑i=1N∑j=1NeiTlijej+∑i=1N∑j=1NeiTaijgej-ei,t≤5η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej+∑i=1N∑j=1NeiTaijgej-ei,t.According to (15)-(16), one has(27)V˙≤5η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej+2Mδ∑i=1N∑j=1Naijej-ei22≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej+4Mδ∑i=1N∑j=1NlijeiTej=-α-4MδeTLe+5ηeTe.Noting that 1Te~=∑i=1Nxi-∑i=1Nxi/N=0, according to Lemma 4, eTLe≥eTλ2(L)e. Then, following with (24), one has V˙<0, which means that the error systems (23) are locally asymptotically stable. Then xi(t)→x¯(t), for t→∞. This means that Theorem 11 holds.


### 4.2. Robust Leaderless Practical Consensus

In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case whereD0≠0 in Assumption 5.Theorem 12.
Consider a networked multiagent system withN follower agents, in which the dynamics are described as (1). Suppose that Assumptions 1–5 hold and the constant D0≠0 in Assumption 5. Under the consensus protocol (22), robust practical consensus of system (1) can be achieved if(28)βδ=α-2+4Mδλ2>5η,where λ2(L) is the minimal eigenvalue of L and δ is a given positive constant.Proof.
Propose the Lyapunov candidate function as(29)V=12∑i=1NeiTei.According to (26), one can obtain(30)V˙≤∑i=1NeiTft,xi-∑j=1Nft,xjN-α∑i=1N∑j=1NeiTlijej-2∑i=1N∑j=1Naijej-eiTgej-ei,t≤5η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej+∑i=1N∑j=1Naijej-eiTej-ei+∑i=1N∑j=1Naijgej-ei,tTgej-ei,t≤5η∑i=1NeiTei-α-2∑i=1N∑j=1NeiTlijej+∑i=1N∑j=1Naijg0ej-ei2+D02≤5η∑i=1NeiTei-α-2∑i=1N∑j=1NeiTlijej+2∑i=1N∑j=1Naijg0ej-ei22+2∑i=1N∑j=1ND02.According to Remark 6 and Assumption 5, one has(31)V˙≤5η∑i=1NeiTei-α-2∑i=1N∑j=1NeiTlijej+2Mδ∑i=1N∑j=1Naijej-ei22+2NN-1D02=5η∑i=1NeiTei-α-2∑i=1N∑j=1NeiTlijej+4Mδ∑i=1N∑j=1NlijeiTej+2NN-1D02≤eT5ηI-α-2+4MδL⊗Ine+2NN-1D02≤-βδ-ηV+2NN-1D02=-βδ-5ηV-2NN-1D02βδ-5η.Since βδ>5η, V˙<0 if V>2N(N-1)D02/(βδ-5η), which means that all the state errors will enter into the set S={V≤2N(N-1)D02/(βδ-η)}. This means that Theorem 12 holds.


## 4.1. Robust Leaderless Consensus

In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case whereD0=0 in Assumption 5.Theorem 11.
Consider a networked multiagent system withN follower agents, in which the dynamics are described as (1). Suppose that Assumptions 1–5 hold and the constant D0=0 in Assumption 5. Under the consensus protocol (22), robust consensus of system (1) can be achieved if(24)βδ=α-Mδλ1>5ηwhere λ2(L) is the minimal eigenvalue of L and δ is a given positive constant.Proof.
Choose the Lyapunov candidate function as(25)V=12∑i=1NeiTei.Differentiating V with respect to t along (23), one can obtain(26)V˙=∑i=1NeiTe˙i=∑i=1NeiTft,xi-∑j=1Nft,xjN-∑i=1NeiT∑j=1Nαlijej-∑j=1Naijgxj-xi,t≤η∑i=1Nei2ei-ej2-α∑i=1N∑j=1NeiTlijej+∑i=1N∑j=1NeiTaijgej-ei,t≤5η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej+∑i=1N∑j=1NeiTaijgej-ei,t.According to (15)-(16), one has(27)V˙≤5η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej+2Mδ∑i=1N∑j=1Naijej-ei22≤η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej+4Mδ∑i=1N∑j=1NlijeiTej=-α-4MδeTLe+5ηeTe.Noting that 1Te~=∑i=1Nxi-∑i=1Nxi/N=0, according to Lemma 4, eTLe≥eTλ2(L)e. Then, following with (24), one has V˙<0, which means that the error systems (23) are locally asymptotically stable. Then xi(t)→x¯(t), for t→∞. This means that Theorem 11 holds.


## 4.2. Robust Leaderless Practical Consensus

In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case whereD0≠0 in Assumption 5.Theorem 12.
Consider a networked multiagent system withN follower agents, in which the dynamics are described as (1). Suppose that Assumptions 1–5 hold and the constant D0≠0 in Assumption 5. Under the consensus protocol (22), robust practical consensus of system (1) can be achieved if(28)βδ=α-2+4Mδλ2>5η,where λ2(L) is the minimal eigenvalue of L and δ is a given positive constant.Proof.
Propose the Lyapunov candidate function as(29)V=12∑i=1NeiTei.According to (26), one can obtain(30)V˙≤∑i=1NeiTft,xi-∑j=1Nft,xjN-α∑i=1N∑j=1NeiTlijej-2∑i=1N∑j=1Naijej-eiTgej-ei,t≤5η∑i=1NeiTei-α∑i=1N∑j=1NeiTlijej+∑i=1N∑j=1Naijej-eiTej-ei+∑i=1N∑j=1Naijgej-ei,tTgej-ei,t≤5η∑i=1NeiTei-α-2∑i=1N∑j=1NeiTlijej+∑i=1N∑j=1Naijg0ej-ei2+D02≤5η∑i=1NeiTei-α-2∑i=1N∑j=1NeiTlijej+2∑i=1N∑j=1Naijg0ej-ei22+2∑i=1N∑j=1ND02.According to Remark 6 and Assumption 5, one has(31)V˙≤5η∑i=1NeiTei-α-2∑i=1N∑j=1NeiTlijej+2Mδ∑i=1N∑j=1Naijej-ei22+2NN-1D02=5η∑i=1NeiTei-α-2∑i=1N∑j=1NeiTlijej+4Mδ∑i=1N∑j=1NlijeiTej+2NN-1D02≤eT5ηI-α-2+4MδL⊗Ine+2NN-1D02≤-βδ-ηV+2NN-1D02=-βδ-5ηV-2NN-1D02βδ-5η.Since βδ>5η, V˙<0 if V>2N(N-1)D02/(βδ-5η), which means that all the state errors will enter into the set S={V≤2N(N-1)D02/(βδ-η)}. This means that Theorem 12 holds.


## 5. Simulations

Consider a network with 8 nodes and a virtual leader. The topology is described as Figure1. We consider both complete consensus and practical consensus for nonlinear first-order multiagent systems for the cases where D0=0 and D0≠0, respectively. The dynamics of each agent are described as x˙i=2xisin⁡3t+ui.Figure 1
Topology graphG.For the complete consensus case, choosing the nonlinear disturbance function is chosen asg(x,t)=0.02sin⁡x; according to Figures 2 and 4, we can know that the consensus for the proposed network can be achieved. For the practical consensus case, choosing the nonlinear disturbance function is chosen as g(x,t)=0.02sin⁡x+0.2cos⁡t; according to Figures 3 and 5, it can be found that there is a tiny error in the trajectories, which means that the practical consensus is achieved.Figure 2
Leader-following consensus of state trajectories.Figure 3
Leader-following practical consensus of state trajectories.Figure 4
Leaderless consensus of state trajectories.Figure 5
Leader-following practical consensus of state trajectories.


## 6. Conclusions

In this paper, both the robust consensus and the practical consensus problem are investigated for nonlinear multiagent systems with uncertainty and disturbance. The uncertainty and disturbance function is piecewise continuous and is described asg(x,t), satisfying g(x,t)2≤g0(x2)+D0. Both complete consensus and practical consensus condition are obtained and the analysis is presented by using hybrid tools from matrix theory and Lyapunov stability theory.



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