Second-Order Consensus in Multiagent Systems via Nonlinear Protocol

Abstract

This paper focuses on theoretical analysis of second-order consensus in multiagent system. As an extension of the general linear protocol, a nonlinear protocol is designed for multiagent system with undirected communication topology. The nonlinear protocol is also applied to achieve reference velocity consensus. Through choosing the appropriate Lyapunov functions and using LaSalle’s invariance principle, some consensus conditions are derived. Simulation examples are provided to demonstrate the effectiveness of the proposed results.

1. Introduction

Recently, consensus problems have been investigated extensively in the content of distributed coordinated control of dynamic agents, partly owning to the potential broad application in various areas. A basic and fundamental question in consensus is how to design the effective protocols (or algorithms) such that each agent can achieve a common target. Most of the consensus literatures mainly considered general linear protocols for multiagent systems. However, the relation between agents is usually not linear. In some situations, the states of multiagent systems cannot be observed directly, and some nonlinear functions about the states can be observed. Therefore, it is necessary to design nonlinear protocols for multiagent systems.

Nonlinear functions are always used to describe the coupling relations of nodes in complex networks [1–6]. On the other hand, some scholars have paid attentions to nonlinear protocols for multiagent systems recently. Bauso et al. [7] designed the distributed nonlinear protocols and proposed a game theoretical approach to solve consensus problems. Xiao et al. [8] presented a new nonlinear protocol for state consensus of multiagent system which provides faster convergence rate than the typical linear protocol [9]. Liu et al. [10] discussed the consensus problem under two nonlinear protocols with directed topology. Hui and Haddad [11] developed a thermodynamic framework for addressing consensus for nonlinear multiagent dynamical systems. Shang [12] proposed some sufficient criteria guaranteeing multiagent systems to reach a consensus in finite time under nonlinear protocols. Li and Guan [13] investigated nonlinear consensus protocols for dynamic directed networks of multiagent systems based on the central manifold reduction technique. Shi and Hong [14] considered a group of continuous-time agents with nonlinear agent dynamics and concluded that the agents can flock to a convex target set. Sepulchre [15] introduced consensus problems whose underlying state space is not a linear space but a nonlinear space. Yu et al. [16] studied the consensus problem for cooperative agents with nonlinear dynamics in a directed network. Zhou and Wang [17] derived some sufficient conditions for (global/exponential) semistability for general discrete-time nonlinear protocols.

However, it is worthwhile to note that the above-mentioned nonlinear protocols [7–13, 17] are all proposed for the first-order multiagent systems. As we all know, first-order consensus problems are mainly relative to communication topologies, but, for second-order consensus, both the interaction graph and the coupling strength affect the convergence result. Therefore, the consensus of second-order multiagent systems is more challenging and interesting and attracts researchers’ broad attention. For example, Yu et al. [18] considered a second-order consensus problem for multiagent systems with nonlinear dynamics. Furthermore, Song et al. [19] studied the second-order leader-following consensus problem of nonlinear multiagent systems. Ren [20] proposed and analyzed a consensus algorithm for double-integrator dynamics with a bounded control input under an undirected interaction graph. Though [20] extended some existing results in consensus algorithms to account for actuator saturation, saturation function which is assumed to be the hyperbolic tangent function is concrete and simple. So far as we know, there is no result about designing some nonlinear protocols for second-order multiagent systems. For this purpose, this paper mainly designs a nonlinear protocol for second-order dynamics with an undirected communication topology as an extension of the linear protocol. Then, the consensus protocol for second-order multiagent system with a group reference velocity available to each individuality is proposed.

The rest of this paper is organized as follows. Section 2 provides some preliminary graph theory, the proposed nonlinear protocol, and two useful lemmas. In Section 3, consensus problem and reference velocity consensus of a second-order multiagent system with an undirected communication topology are analyzed, respectively. Some numerical examples are given to show the effectiveness and advantage of the theoretical results in Section 4. Finally, a conclusion is provided in Section 5.

2. Notation and Preliminaries

2.1. Graph Theory

In this paper, the communication topology among 𝑛 multiagents is assumed to be an undirected graph 𝒢=(𝒱,ℰ,𝒜), where 𝒱={𝑣₁,…,𝑣_𝑛} is the set of nodes, ℰ⊆𝒱×𝒱 is the set of edges, and 𝒜=[𝑎_𝑖𝑗]∈ℝ^𝑛×𝑛 represents the weighted adjacency matrix with nonnegative adjacency elements 𝑎_𝑖𝑗. The node or multiagent indexes belong to a finite index set ℐ={1,2,…,𝑛}. An edge 𝑒_𝑖𝑗=(𝑣_𝑖,𝑣_𝑗)∈ℰ in weighted undirected graph 𝒢 denotes that node 𝑗 can receive information from 𝑖 each other. The adjacency elements associated with the edges of an undirected graph are positive 𝑎_𝑖𝑗>0 if 𝑒_𝑖𝑗∈ℰ, 𝑎_𝑖𝑗=0 otherwise, and have the property that 𝑎_𝑖𝑗=𝑎_𝑗𝑖, for all 𝑖≠𝑗, since 𝑒_𝑖𝑗 implies 𝑒_𝑗𝑖. A path on 𝒢 from node 𝑖₁ to 𝑖_𝑙 is a sequence of undirected edges in the form of (𝑖_𝑘,𝑖_𝑘+1),𝑘=1,…,𝑙−1. An undirected graph is called connected if there exists a path from any node 𝑖 to any other node 𝑗.

2.2. Nonlinear Protocols for Second-Order Dynamics

Consider a second-order multiagent system with the following dynamics: 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑣̇_𝑖(𝑡)=𝑢_𝑖(𝑡),𝑖∈ℐ, (1) where 𝑥_𝑖(𝑡)∈ℝ is the position state, 𝑣_𝑖(𝑡)∈ℝ is the velocity state, and 𝑢_𝑖(𝑡)∈ℝ is the control input (or protocol). Given the dynamical system (1), it is said that protocol 𝑢_𝑖(𝑡) asymptotically solves the consensus problem, if the states of agents satisfy lim_𝑡→+∞𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)=0,lim_𝑡→+∞𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)=0,∀𝑖≠𝑗, (2) for any 𝑥_𝑖(0) and 𝑣_𝑖(0).

The linear protocol [21, 22] is proposed as follows: 𝑢_𝑖(𝑡)=−^𝑛_𝑗=1𝑎_𝑖𝑗𝛼𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝛽𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡), (3) where 𝑎_𝑖𝑗 is the (𝑖,𝑗) entry of the weighted adjacency matrix 𝒜∈ℝ^𝑛×𝑛, and parameters 𝛼, 𝛽 are the coupling strengths.

It is clear that protocol (3) is based on a linear function of difference between each agent and other ones. Linear protocol (3) is simple and easy to use. However, sometimes in order to improve convergence rate or model a bounded control input, the linear protocol is often needed to be extended. Furthermore, in some cases, the state feedback 𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡) may be unobservable; instead, we can observe a nonlinear function 𝜙(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)) of 𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡). Naturally, a nonlinear protocol for second-order system (1) is designed as follows: 𝑢_𝑖(𝑡)=−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡𝑣)+𝜓_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡), (4) where 𝑎_𝑖𝑗 is the (𝑖,𝑗) entry of the weighted adjacency matrix 𝒜 and 𝜙(⋅) and 𝜓(⋅) are two continuous functions.

In order to ensure existence, uniqueness of solution, and the operative property of nonlinear functions 𝜙(⋅) and 𝜓(⋅), the following sets are defined.

Definition 1. Define two function sets Φ={𝜙(𝑥)∣𝜙(𝑥) is an odd function, 𝜙(𝑥)𝑥>0, 𝜙(𝑥)=0 if and only if 𝑥=0}, and Ψ={𝜓(𝑥)∣𝜓(𝑥) is a continuous function and increasing about 𝑥, 𝜓(𝑥)=𝜓(𝑦) if and only if 𝑥=𝑦}.
At the end of this section, two useful lemmas about 𝜙(⋅) and 𝜓(⋅) are introduced.

Lemma 2. Letting 𝐚=[𝑎₁𝑎₂⋯𝑎_𝑛]^T∈ℝ^𝑛, 𝐛=[𝑏₁𝑏₂⋯𝑏_𝑛]^T∈ℝ^𝑛, and 𝐶=[𝑐_𝑖𝑗]∈ℝ^𝑛×𝑛, if matrix 𝐶 is symmetrical, that is, 𝑐_𝑖𝑗=𝑐_𝑗𝑖, and odd function 𝜙(𝑥)∈Φ, then it is derived that 12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑖−𝑎_𝑗𝜙𝑏_𝑖−𝑏_𝑗=^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑖𝜙𝑏_𝑖−𝑏_𝑗. (5)

Proof. Consider that 12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑖−𝑎_𝑗𝜙𝑏_𝑖−𝑏_𝑗=12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑖𝜙𝑏_𝑖−𝑏_𝑗−12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑗𝜙𝑏_𝑖−𝑏_𝑗=12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑖𝜙𝑏_𝑖−𝑏_𝑗+12^𝑛𝑛_𝑗=1𝑖=1𝑐_𝑗𝑖𝑎_𝑗𝜙𝑏_𝑗−𝑏_𝑖=^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑖𝜙𝑏_𝑖−𝑏_𝑗, (6) where the facts that 𝑐_𝑖𝑗=𝑐_𝑗𝑖 and 𝜙(⋅) is an odd function that is, 𝜙(𝑏_𝑗−𝑏_𝑖)=−𝜙(𝑏_𝑖−𝑏_𝑗), are used to obtain the equalities.

Using the same proof and the fact that 𝜓(𝑏_𝑗)−𝜓(𝑏_𝑖)=−(𝜓(𝑏_𝑖)−𝜓(𝑏_𝑗)), the following lemma is obvious.

Lemma 3. Letting 𝐚=[𝑎₁𝑎₂⋯𝑎_𝑛]^T∈ℝ^𝑛, 𝐛=[𝑏₁𝑏₂⋯𝑏_𝑛]^T∈ℝ^𝑛, and 𝐶=[𝑐_𝑖𝑗]∈ℝ^𝑛×𝑛, if matrix 𝐶 is symmetrical, function 𝜓(𝑥)∈Ψ, then it follows that 12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑖−𝑎_𝑗𝜓𝑏_𝑖−𝜓𝑏_𝑗=^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑎_𝑖𝜓𝑏_𝑖−𝜓𝑏_𝑗. (7)

3. Consensus Analysis

In this section, the consensus and reference velocity consensus of second-order multiagent system (1) with the designed nonlinear protocol will be analyzed.

3.1. Consensus with Nonlinear Protocol

Second-order multiagent system (1) with protocol (4) is written as 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑣̇_𝑖(𝑡)=−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡). (8)

Theorem 4. For nonlinear protocol (4), suppose that functions 𝜙(⋅)∈Φ and 𝜓(⋅)∈Ψ. If information topology graph 𝒢 is undirected and connected, then it is held that 𝑥_𝑖(𝑡)→𝑥_𝑗(𝑡), 𝑣_𝑖(𝑡)→𝑣_𝑗(𝑡), as 𝑡→∞; that is to say, multiagent system (8) achieves consensus asymptotically.

Proof. Choose the following Lyapunov function: 𝑉(𝑡)=12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗^{𝑥𝑖(𝑡)−𝑥𝑗}₀^(𝑡)𝜙(𝑠)1𝑑𝑠+2^𝑛_𝑖=1𝑣²_𝑖(𝑡). (9) The time derivative of the Lyapunov function (9) along any trajectory of (8) is 𝑉̇(𝑡)=12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−^𝑛_𝑖=1𝑣_𝑖(𝑡)^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡)=^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡)1=−2^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡), (10) where the second equality and the last equality are derived from Lemmas 2 and 3, respectively. Since (𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡))(𝜓(𝑣_𝑖(𝑡))−𝜓(𝑣_𝑗(𝑡)))≥0 and the information topology graph 𝒢 is connected, it is derived that −12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡)≤0; (11) that is to say, 𝑉̇(𝑡)≤0. Letting 𝑉̇(𝑡)≡0, it is easily seen that 𝑣_𝑖(𝑡)≡𝑣_𝑗(𝑡) and 𝑣̇_𝑖(𝑡)≡𝑣̇_𝑗(𝑡). Therefore, it follows that 𝑣(̇𝑡)∈Span(𝟏×𝜍), where 𝜍 is a constant and 𝑣(𝑡)=[𝑣₁(𝑡)𝑣₂(𝑡)⋯𝑣_𝑛(𝑡)]^T. Furthermore, from 𝑣_𝑖(𝑡)≡𝑣_𝑗(𝑡), it is obtained that 𝑣̇_𝑖(𝑡)≡−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡,())𝟏×𝜍^T𝑣̇_𝑖(𝑡)≡−𝜍^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡). (12)(13) Since information topology graph is undirected, that is, 𝑎_𝑖𝑗=𝑎_𝑗𝑖, and 𝜙(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))=−𝜙(𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)), it is held that ^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)≡0, (14) which implies that (𝟏×𝜍)^T𝑣(̇𝑡)≡0. Thus, it means that 𝑣(̇𝑡) is orthogonal to 𝟏×𝜍. From the above discussion, it is concluded that 𝑣(̇𝑡)≡0 which in turn implies that −∑^𝑛_𝑗=1𝑎_𝑖𝑗𝜙(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))≡0 from (12). Furthermore, it is obtained that −∑^𝑛_𝑖=1𝑥_𝑖(𝑡)∑^𝑛_𝑗=1𝑎_𝑖𝑗𝜙(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))≡0 which implies that −12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)≡0, (15) from Lemma 2. For 𝜙(𝑥)𝑥>0 and information topology graph is connected, from (15), it is derived that 𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)≡0,∀𝑖≠𝑗. (16) As a result, it follows that 𝑥_𝑖(𝑡)≡𝑥_𝑗(𝑡), for all 𝑖≠𝑗. By LaSalle’s invariance principle, it is concluded that 𝑥_𝑖(𝑡)→𝑥_𝑗(𝑡), 𝑣_𝑖(𝑡)→𝑣_𝑗(𝑡), for all 𝑖≠𝑗, as t→∞. The proof is ended.

When nonlinear protocol (4) is simplified as follows: 𝑢_𝑖(𝑡)=−𝜓𝑣_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡), (17) and the Lyapunov function (9) is chosen, the following conclusion is easily obtained.

Corollary 5. If the information topology graph 𝒢 is undirected and connected, nonlinear functions 𝜙(⋅)∈Φ and 𝜓∈Ψ, then the multiagent system (1) with protocol (17) can achieve consensus, namely, 𝑥_𝑖(𝑡)→𝑥_𝑗(𝑡), 𝑣_𝑖(𝑡)→𝑣_𝑗(𝑡)→0, asymptotically as 𝑡→∞.
In practice, since the position of each agent is easily measured relatively, nonlinear function 𝜙(𝑥) in protocol (4) can be assumed to be 𝜙(𝑥)=𝑥. Therefore, letting 𝜙(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))=𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡), protocol (4) is reduced to 𝑢_𝑖(𝑡)=−^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡). (18) Applying protocol (18) to system (1), the following corollary is derived from Theorem 4 directly.

Corollary 6. For nonlinear protocol (18), suppose the function 𝜓(⋅)∈Ψ. If the information topology graph 𝒢 is undirected and connected, multiagent system (1) with protocol (18) achieves consensus asymptotically, namely, 𝑥_𝑖(𝑡)→𝑥_𝑗(𝑡), 𝑣_𝑖(𝑡)→𝑣_𝑗(𝑡), as 𝑡→∞.

Remark 7. The second-order multiagent system with linear protocol can be written as compact matrix form, and the consensus conditions can be derived by analysing the eigenvalues of the compact matrix [22]. When the second-order multiagent system achieves consensus, the positions and velocities will, respectively, converge to a constant value which is relative to the initial position and velocity and the left eigenvector of the Laplacian matrix (see [21, 22]).
However, the nonlinear protocol is different from the linear protocol. The second-order multiagent system with nonlinear protocol cannot be written as a compact matrix form. So, the proof of consensus is different from the method used in the linear protocol and is more difficult. When the consensus of the second-order multiagent system with nonlinear protocol is reached, the value which the positions and velocities converge to cannot be decided. The value is relative to the nonlinear function, the communication graph, and the initial position and velocity.

3.2. Consensus with Group Reference Velocity

In practice, given a reference velocity 𝑣^∗(𝑡), the positions and velocities of the agents in multiagent system (1) are often required to achieve consensus and converge to 𝑣^∗(𝑡); that is, 𝑥_𝑖(𝑡)→𝑥_𝑗(𝑡), 𝑣_𝑖(𝑡)→𝑣_𝑗(𝑡)→𝑣^∗(𝑡), for all 𝑖≠𝑗. Hence, the following nonlinear protocol is designed with a group reference velocity 𝑣^∗(𝑡) as 𝑢_𝑖(𝑡)=𝑣̇^∗(𝑡)−𝜓𝑣_𝑖(𝑡)−𝜓𝑣^∗(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡). (19)

Given a group reference velocity 𝑣^∗(𝑡), second-order multiagent system (1) with protocol (19) is expressed as 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑣̇_𝑖(𝑡)=𝑣̇^∗(𝑡)−𝜓𝑣_𝑖(𝑡)−𝜓𝑣^∗(𝑡−_𝑛𝑗=1)𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡). (20)

Theorem 8. For nonlinear protocol (19), consider functions 𝜙(⋅)∈Φ and 𝜓(⋅)∈Ψ. When information topology graph 𝒢 is undirected and connected, multiagent system (20) achieves consensus; that is, 𝑥_𝑖(𝑡)→𝑥_𝑗(𝑡), 𝑣_𝑖(𝑡)→𝑣_𝑗(𝑡)→𝑣^∗(𝑡), as 𝑡→∞.

Proof. Letting 𝑣_𝑖(𝑡)=𝑣_𝑖(𝑡)−𝑣^∗(𝑡), 𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡)−∫^𝑡₀𝑣^∗(𝑠)𝑑𝑠, system (20) is rewritten as ̇𝑥_𝑖(𝑡)=𝑣_𝑖(𝑡),̇𝑣_𝑖(𝑡)=−𝜓𝑣_𝑖(𝑡)−𝜓𝑣^∗(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡). (21) Choose the following Lyapunov function: 𝑉(𝑡)=12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑥_{𝑖(𝑡)−}𝑥_𝑗0(𝑡)𝜙(𝑠)1𝑑𝑠+2^𝑛_𝑖=1𝑣²_𝑖(𝑡). (22) Taking the derivative of the Lyapunov function (22) with respect to 𝑡 along the trajectory of (21) yields 𝑉̇(𝑡)=12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−^𝑛_𝑖=1𝑣_𝑖(𝑡)𝜓𝑣_𝑖(𝑡)−𝜓𝑣^∗(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡+𝜓𝑣_𝑖)(𝑡)−𝜓𝑣_𝑗(𝑡). (23) From Lemma 2, it is obtained that 𝑉̇(𝑡)=12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡−1)2^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−^𝑛_𝑖=1𝑣_𝑖(𝑡)𝜓𝑣_𝑖(𝑡)−𝜓𝑣^∗(𝑡)−12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡×𝜓𝑣_𝑖)(𝑡)−𝜓𝑣_𝑗(𝑡)=−^𝑛_𝑖=1𝑣_𝑖(𝑡)−𝑣^∗(𝑡)𝜓𝑣_𝑖(𝑡)−𝜓𝑣^∗(𝑡)−12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)×𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡). (24) Since function 𝜓(⋅)∈Ψ and the information topology graph 𝒢 is undirected and connected, there exist the following facts that −^𝑛_𝑖=1𝑣_𝑖(𝑡)−𝑣^∗(𝑡)𝜓𝑣_𝑖(𝑡)−𝜓𝑣^∗(𝑡−1)≤0,2^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)𝜓𝑣_𝑖(𝑡)−𝜓𝑣_𝑗(𝑡)≤0. (25) Combing (25), it follows that 𝑉̇(𝑡)≤0. When 𝑉̇(𝑡)≡0, it is clear that 𝑣_𝑖(𝑡)≡𝑣_𝑗(𝑡), 𝑣_𝑖(𝑡)≡𝑣^∗(𝑡). Then, it is derived that −^𝑛_𝑗=1𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡−)≡0,_𝑛𝑖=1𝑥_𝑖(t)^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)≡0. (26)(27) Note that (27) implies that −12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)≡0, (28) from Lemma 2. Due to the connected information graph 𝒢, it is derived from (28) that 𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)𝜙𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)≡0. (29) As a result, it is concluded that 𝑥_𝑖(𝑡)≡𝑥_𝑗(𝑡) which is equivalent to 𝑥_𝑖(𝑡)≡𝑥_𝑗(𝑡). Making use of LaSalle’s invariance principle, it is concluded that 𝑥_𝑖(𝑡)→𝑥_𝑗(𝑡), 𝑣_𝑖(𝑡)→𝑣_𝑗(𝑡)→𝑣^∗(𝑡), as 𝑡→∞. The proof is completed.

4. Numerical Examples

In this section, a number of simulations are given to illustrate the effectiveness of the results proposed in the paper.

Example 9. Consider the communication topology graph 𝒢 as described in Figure 1. Graph 𝒢 has eight nodes, and the edges denote the information interchange between agents. For simplicity, it is assumed that graph 𝒢 has 0-1 weights. To simulate numerical examples, choose 𝜙(𝑥)=𝑥²√if𝑥>1,−√𝑥if0<𝑥≤1,−𝑥if−1<𝑥≤0,−𝑥²if𝑥≤−1, (30) and 𝜓(𝑥)=𝑥³ in the nonlinear protocol (4), and it is clear that 𝜙∈Φ and 𝜓∈Ψ. Therefore, the nonlinear protocol (4) can be achieved asymptotically for multiagent system (1) from Theorem 4. Figure 2 shows that the velocity and position trajectories of second-order multiagent system (8) converge to consensus, respectively. In addition, Figure 3 describes that both the velocity and position trajectories of second-order multiagent system (1) with protocol (18) achieve consensus. Choose a reference velocity 𝑣^∗(𝑡)=sin(3∗𝑡)+cos(𝑡), and using protocol (19), multiagent system (1) can achieve consensus asymptotically. Figure 4 describes the trajectories of velocity and position of multiagent system (1) with protocol (19), respectively. It is seen that the trajectories of velocity follow the reference velocity 𝑣^∗(𝑡) successfully.

Figure 1 

Communication topology graph.

Figure 2 

State trajectories of multiagent system (8).

Figure 3 

State trajectories of multiagent system (1) with protocol (18).

Figure 4 

State trajectories of multiagent system (20).

5. Conclusions

This paper provided consensus analysis for the second-order multiagent system with nonlinear protocol. As an extension of linear protocols, a nonlinear protocol was designed to achieve consensus. In addition, the protocol was also used to achieve reference velocity consensus for second-order multiagent system. The analysis mainly relied on some tools from algebraic graph and control theory. Through using the Lyapunov theory and LaSalle’s invariance principle, sufficient conditions for consensus are derived. Simulation examples show that when the nonlinear function satisfies the assumption, the second-order multiagent can achieve consensus and reference velocity consensus.

Acknowledgment

This work was supported by the Research Funds of Ningxia University Introducing Talent (no. BQD2012004).