Third-Order Leader-Following Consensus in a Nonlinear Multiagent Network via Impulsive Control

Abstract

Many facts indicate that the impulsive control method is a finer method, which is simple, efficient, and low in cost, for achieving consensus. In this paper, based on graph theory, Lyapunov stability theory, and matrix theory, a novel impulsive control protocol is given to realize the consensus of the multiagent network. Numerical simulations are performed to verify the theoretical results.

1. Introduction

In the past few years, consensus of multiagent networks has been intensively studied in many fields, such as biological, social, mathematical, and physical sciences ones [1–5]. Generally speaking, consensus refers to designing a system algorithm or protocol such that all agents asymptotically reach an agreement on their states. In particular, leader-following consensus means that there exists a virtual leader which specifies an objective for all agents to follow. Recently, some first-order and second-order leader-following consensus problems were discussed by lots of researchers [6–10], and then some novel system algorithms were given via some different control methods, such as pinning control, delay coupling control, adaptive control, and impulsive control [9–14]. In addition, Qin et al. considered consensus in the second-order multiagent system with communication delay in [15, 16]. Particularly, some multiagent networks cannot be controlled continuously. At this time, the impulsive control becomes a more desirable alternative. The impulsive control is low in cost and then has been widely applied in many fields, such as information science, system control, life science, communication security, and space techniques [17–19]. In the above senses, the impulsive control is very effective for achieving consensus of a multiagent network.

In some real networks, the connections between part nodes are sometimes a failure, and then the network topology may dynamically change over time. Therefore, it is indispensable to consider the case that the network topology is switching. As much as we know, most of the relevant studies focus on second-order consensus for multiagent networks [11, 12]. When the agent states are influenced by speeds, positions, and accelerations, it is necessary and significative to research the third-order consensus problem of a multiagent network with switching topology. At present, just few works considered the third-order consensus problem. In [20], adaptive third-order leader-following consensus of a nonlinear multiagent network with perturbations was addressed, without using the impulsive control method. In [11], impulsive consensus problem of second-order multiagent network with switching topologies was investigated, without considering its own dynamics. In this paper, we consider the third-order consensus problem in a multiagent network with the aforementioned four characters, that is, leader-following, own dynamics, switching topology, and impulsive control. By using the graph theory, Lyapunov stability theory, and matrix theory, some sufficient conditions are obtained to realize the third-order leader-following consensus.

The rest of this paper is designed as follows. Some necessary preliminaries are stated in Section 2. The consensus of a multiagent network is discussed in Section 3. Numerical examples are given to verify the theoretical results in Section 4. Finally, in Section 5, conclusions are presented.

2. Preliminaries

2.1. Multiagent Network

Information exchange among agents can be modeled by an interaction graph. Let 𝐺={𝑉,𝐸,𝐴} be a weighted diagraph with a node set 𝑉={1,…,𝑁}, an edge set 𝐸⊆𝑉×𝑉, and a weight adjacency matrix 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁. A directed edge denoted by (𝑖,𝑗) means that 𝑗 has access to node 𝑖; that is, node 𝑗 can receive information from node 𝑖. The elements of matrix 𝐴 are defined such that 𝑎_𝑖𝑗>0 for (𝑗,𝑖)∈𝐸, while 𝑎_𝑖𝑗=0 for (𝑗,𝑖)∉𝐸. Let 𝑎_𝑖𝑖=0 for 𝑖∈𝑉. The set ℵ_𝑖={𝑗∈𝑉∣(𝑗,𝑖)∈𝐸} is used as the neighbor set of node 𝑖. When the communication topology is switching, the neighbor set is time-varying, and then ℵ_𝑖(𝑡)={𝑗∈𝑉∶𝑎_𝑖𝑗(𝑡)>0}. Let 𝑎_𝑖𝑗(𝑡)=1,𝑗∈ℵ_𝑖(𝑡),0,otherwise. (1) The out-degree of node 𝑖 is defined by deg(𝑖)=∑^𝑁_𝑗=1𝑎_𝑖𝑗=∑_{𝑗∈ℵ𝑖}𝑎_𝑖𝑗=𝑑_𝑖. If the degree matrix of digraph 𝐺 is 𝐷=diag(𝑑₁,…,𝑑_𝑁), then the Laplacian matrix of digraph 𝐺 is 𝐿=𝐷−𝐴.

A directed path from node 𝑖 to node 𝑗 in the directed graph 𝐺 is a sequence of edges (𝑖,𝑗₁),(𝑗₁,𝑗₂),…,(𝑗_𝑙,𝑗) with distinct nodes 𝑗_𝑘, 𝑘=1,…,𝑙. A digraph 𝐺 has a directed spanning tree if there exists at least one node called root which has a directed path to all the other nodes.

For a leader-follower multiagent network, suppose that the leader (labeled by 0) is denoted by node 0, and the followers are denoted by the nodes 1,2,…,𝑁. The graph 𝐺 is consisting of the leader and the followers with communication topology. The connection weight between the 𝑖th follower and the leader is represented by 𝑏_𝑖,𝑖∈𝑉. If the 𝑖th follower is connected to the leader, then 𝑏_𝑖>0; otherwise, 𝑏_𝑖=0. Let 𝐵=diag{𝑏₁,𝑏₂,…,𝑏_𝑁}.

Following, we address the multiagent network with switching topology. The set Ω={𝐺₁,𝐺₂,…,𝐺_𝑚} is used as a set of the graphs with all possible topology, which includes all possible graphs (involving 𝑁 agents and a leader). We define a switching signal 𝜏∶[0,+∞)→P={1,2,…,𝑚}, which determines the topology structure that corresponds to the network. When the topology is switching, the Laplacian matrix 𝐿 and the matrix 𝐵 are also switching, which are denoted by 𝐿_𝜏(𝑡) and 𝐵_𝜏(𝑡).

The following assumptions are needed to derive our main results.

Assumption 1 (see [21]). Assume that there exists a constant 𝛾>0, such that the vector-valued function 𝑓 satisfies the condition (𝑢−𝑤)^𝑇𝑓(𝑡,𝑥,𝑣,𝑢)−𝑓𝑡,𝑦,𝑧,𝑤≤𝛾𝑥−𝑦^𝑇𝑥−𝑦+(𝑣−𝑧)^𝑇()+()𝑣−𝑧𝑢−𝑤^𝑇(𝑢−𝑤) (2) for any 𝑥,𝑦,𝑧,𝑢,𝑣,𝑤∈𝑅^𝑛.

2.2. Impulsive Control System

Impulsive control systems can be classified into three types based on the characteristics of plants and control laws [22].

A type-I impulsive control system [22] is given by 𝑋̇=𝐅(𝑡,𝑋),𝑡≠𝜏_𝑘(𝑋),Δ𝑋=𝐔(𝑘,𝑌),𝑡=𝜏_𝑘(𝑋),𝑌=𝐆(𝑡,𝑋), (3) where 𝑋 and 𝑌 are the state variable and the output, respectively. 𝐔(𝑘,𝑌) is the impulsive control law. In this kind of system, the control input is implemented by the “sudden jumps" of some state variables.

Definition 2 (see [22]). For 𝑡≠𝜏_𝑘(𝑋), we define the time derivative of the function 𝑉(𝑡,𝑋) with respect to system (3) as 𝑉̇(𝑡,𝑋)≜𝜕𝑉(𝑡,𝑋)+()𝜕𝑡𝜕𝑉𝑡,𝑋𝐅().𝜕𝑋𝑡,𝑋 (4)

Type-II and type-III impulsive control systems and more theoretical results are present in [22].

In this paper, a type-I impulsive control system is considered.

3. Main Results

Consider that a nonlinear multiagent network consists of 𝑁 agents with third-order dynamics: 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑣_𝑖̇(𝑡)=𝑢_𝑖(𝑡),𝑢̇_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑣_𝑖(𝑡),𝑢_𝑖(𝑡),𝑖=1,2,…,𝑁, (5) where 𝑥_𝑖=(𝑥_𝑖1,…,𝑥^𝑇_𝑖𝑛)∈𝑅^𝑛, 𝑣_𝑖=(𝑣_𝑖1,…,𝑣_𝑖𝑛)^𝑇∈𝑅^𝑛, and 𝑢_𝑖=(𝑢_𝑖1,…,𝑢_𝑖𝑛)^𝑇∈𝑅^𝑛 are the position, velocity, and acceleration states of the 𝑖th agent, respectively, and 𝑓=(𝑓₁,…,𝑓_𝑛)^𝑇∈𝑅^𝑛 is a nonlinear vector-valued continuous function used to describe the self-dynamics of the 𝑖th agent.

The virtual leader of the multiagent network (5) is an isolated agent described by 𝑥̇₀(𝑡)=𝑣₀(𝑡),𝑣̇₀(𝑡)=𝑢₀(𝑡),𝑢̇₀(𝑡)=𝑓𝑡,𝑥₀(𝑡),𝑣₀(𝑡),𝑢₀(𝑡), (6) where 𝑥₀∈𝑅^𝑛, 𝑣₀∈𝑅^𝑛, and 𝑢₀∈𝑅^𝑛 are the position, velocity, and acceleration of the virtual leader, respectively.

For 𝑡∈(𝑡_𝑘−1,𝑡_𝑘), the state variables 𝑥_𝑖(𝑡), 𝑣_𝑖(𝑡), 𝑢(𝑡) of the system (5) do dynamically change with ordinary differential equations. And at the moment 𝑡_𝑘, if the 𝑥_𝑖(𝑡), 𝑣_𝑖(𝑡), 𝑢(𝑡) supervise the impulsive control, then it can result in the jump. Based on the above senses and the impulsive controller of [11] and applying impulsive control to (5), we have the following consensus scheme for the multiagent system: 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑣̇_𝑖(𝑡)=𝑢_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑢̇_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑣_𝑖(𝑡),𝑢_𝑖(𝑡),𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘≜𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝐶_{𝑘𝑗∈ℵ𝑖(𝑡𝑘)}𝑎_𝑖𝑗𝑡_𝑘𝑥_𝑖𝑡_𝑘−𝑥_𝑗𝑡_𝑘+𝑏_𝑖𝑡_𝑘𝑥_𝑖𝑡_𝑘−𝑥₀𝑡_𝑘,Δ𝑣_𝑖𝑡_𝑘≜𝑣_𝑖𝑡⁺_𝑘−𝑣_𝑖𝑡⁻_𝑘=𝐶_{𝑘𝑗∈ℵ𝑖(𝑡𝑘)}𝑎_𝑖𝑗𝑡_𝑘𝑣_𝑖𝑡_𝑘−𝑣_𝑗𝑡_𝑘+𝑏_𝑖𝑡_𝑘𝑣_𝑖𝑡_𝑘−𝑣₀𝑡_𝑘,Δ𝑢_𝑖𝑡_𝑘≜𝑢_𝑖𝑡⁺_𝑘−𝑢_𝑖𝑡⁻_𝑘=𝐶_{𝑘𝑗∈ℵ𝑖(𝑡𝑘)}𝑎_𝑖𝑗𝑡_𝑘𝑢_𝑖𝑡_𝑘−𝑢_𝑗𝑡_𝑘+𝑏_𝑖𝑡_𝑘𝑢_𝑖𝑡_𝑘−𝑢₀𝑡_𝑘,𝑥_𝑖𝑡⁺₀=𝑥_𝑖𝑡₀,𝑡₀≥0,𝑥_𝑖𝑡⁻_𝑘=𝑥_𝑖𝑡_𝑘,𝑣_𝑖𝑡⁺₀=𝑣_𝑖𝑡₀,𝑡₀≥0,𝑣_𝑖𝑡⁻_𝑘=𝑣_𝑖𝑡_𝑘,𝑢_𝑖𝑡⁺₀=𝑢_𝑖𝑡₀,𝑡₀≥0,𝑢_𝑖𝑡⁻_𝑘=𝑢_𝑖𝑡_𝑘, (7) where Δ𝑥_𝑖(𝑡_𝑘), Δ𝑣_𝑖(𝑡_𝑘), Δ𝑢_𝑖(𝑡_𝑘) are the jump of the position, velocity, and acceleration of the 𝑖th follower agent at the moment 𝑡_𝑘, respectively, 𝑥_𝑖(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑥_𝑖(𝑡_𝑘+ℎ), 𝑥_𝑖(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑥_𝑖(𝑡_𝑘+ℎ), 𝑣_𝑖(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑣_𝑖(𝑡_𝑘+ℎ), 𝑣_𝑖(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑣_𝑖(𝑡_𝑘+ℎ), 𝑢_𝑖(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑢_𝑖(𝑡_𝑘+ℎ), 𝑢_𝑖(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑢_𝑖(𝑡_𝑘+ℎ), 𝐶_𝑘∈𝑅^𝑛×𝑛 is the impulsive controller gain at the moment 𝑡_𝑘, the moments of impulsive satisfy 0≤𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<𝑡_𝑘+1<⋯, and lim_𝑘→+∞𝑡_𝑘=+∞, Δ𝑡_𝑘=𝑡_𝑘−𝑡_𝑘−1≤𝜌<+∞ is the impulsive interval, where 𝑘=1,2,….

Let 𝑥_𝑖(𝑡)−𝑥₀(𝑡)=𝑥_𝑖(𝑡), 𝑣_𝑖(𝑡)−𝑥₀(𝑡)=𝑣_𝑖̂(𝑡), 𝑢_𝑖(𝑡)−𝑢₀(𝑡)=𝑢_𝑖(𝑡), 𝑥(𝑡)=(𝑥^𝑇₁(𝑡),𝑥^𝑇₂(𝑡),…,𝑥^𝑇_𝑁(𝑡))^𝑇, 𝑣(̂𝑡)=(𝑣^𝑇₁̂(𝑡),𝑣^𝑇₂̂(𝑡),…,𝑣^𝑇_𝑁̂(𝑡))^𝑇, 𝑢(𝑡)=(𝑢^𝑇₁(𝑡),𝑢^𝑇₂(𝑡),…,𝑢^𝑇_𝑁(𝑡))^𝑇, 𝑒(𝑡)=(𝑥^𝑇(𝑡),𝑣^𝑇̂(𝑡),𝑢^𝑇(𝑡))^𝑇; then the error system with (6) and (7) can be written as 𝑒(̇𝑡)=𝐻𝑒(𝑡)+𝑊,𝑡≠𝑡_𝑘,Δ𝑒𝑡_𝑘=𝐶_𝑘𝑒𝑡_𝑘,𝑒𝑡⁺₀=𝑒𝑡₀, (8) where 𝑂𝐻=^{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}𝑂_{𝑛𝑁×𝑛𝑁}𝑂_{𝑛𝑁×𝑛𝑁}𝑂_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}𝑂_{𝑛𝑁×𝑛𝑁}𝑂_{𝑛𝑁×𝑛𝑁}𝑂_{𝑛𝑁×𝑛𝑁},𝑂𝑊=^𝑛𝑁×1𝑂_𝑛𝑁×1𝐹(𝑡,𝑥(𝑡),𝑣(𝑡),𝑢(𝑡))−1_𝑁⊗𝑓𝑡,𝑥₀(𝑡),𝑣₀(𝑡),𝑢₀(𝑡,𝐹((𝑡)(𝑡)(𝑡=𝑓)𝑡,𝑥,𝑣,𝑢))_𝑇𝑡,𝑥₁(𝑡),𝑣₁(𝑡),𝑢₁(𝑡)𝑓,…,_𝑇𝑡,𝑥_𝑁(𝑡),𝑣_𝑁(𝑡),𝑢_𝑁(𝑡)^𝑇,1_𝑁=(1,…,1)^𝑇∈𝑅^𝑁,𝐶_𝑘=diag𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘)⊗𝐶_𝑘,𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘)⊗𝐶_𝑘,𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘)⊗𝐶_𝑘, (9)𝐿_𝜏(𝑘) and 𝐵_𝜏(𝑘) are associated with the switching interconnection graph at time 𝑡_𝑘,𝜏(𝑘)∈P.

Definition 3. Denote 𝑒_𝑥𝑖(𝑡)=‖𝑥_𝑖(𝑡)−𝑥₀(𝑡)‖, 𝑒_𝑣𝑖(𝑡)=‖𝑣_𝑖(𝑡)−𝑣₀(𝑡)‖ and 𝑒_𝑢𝑖(𝑡)=‖𝑢_𝑖(𝑡)−𝑢₀(𝑡)‖. The multiagent network (7) with the virtual leader (6) is said to achieve third-order leader-following consensus if the solution of (8) satisfies lim_𝑡→+∞𝑒_𝑥𝑖(𝑡)=0, lim_𝑡→+∞𝑒_𝑣𝑖(𝑡)=0, and lim_𝑡→+∞𝑒_𝑢𝑖(𝑡)=0, 𝑖=1,2,…,𝑁 for any initial condition.
Basing on graph theory, Lyapunov function method, matrix theory, and the proof of Theorem 1 in [11], we have the following theorem.

Theorem 4. Under Assumption 1, if there exists 0<𝜑<1 such that 𝛿_𝑘𝑒^{𝜇(Δ𝑡𝑘)}<𝜑, (10) where 𝛿_𝑘 and 𝜇 are the maximum eigenvalues of matrices 𝐼_{3𝑛𝑁×3𝑛𝑁}+𝐶_𝑘^𝑇𝐼_{3𝑛𝑁×3𝑛𝑁}+𝐶_𝑘,2𝛾𝐼_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}𝑂_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}2𝛾𝐼_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}𝑂_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}2𝛾𝐼_{𝑛𝑁×𝑛𝑁}, (11)(12) respectively, then the third-order leader-following consensus in the multiagent network (7) is achieved.

Proof. Consider the following Lyapunov function: 𝑉(𝑒(𝑡))=𝑒^𝑇(𝑡)𝑒(𝑡). (13) For any 𝑡∈(𝑡_𝑘−1,𝑡_𝑘], the time derivative of 𝑉(𝑒(𝑡)) along the trajectory of (8) is 𝐃𝑉(𝑒(𝑡))=(𝑒(̇𝑡))^𝑇𝑒(𝑡)+𝑒^𝑇(𝑡)𝑒(̇𝑡)=𝑒^𝑇(𝑡)𝐻+𝐻^𝑇𝑒(𝑡)+2𝑊^𝑇𝑒(𝑡)=𝑒^𝑇(𝑡)𝐻+𝐻^𝑇𝑒(𝑡)+2𝑢^𝑇(𝑡)𝐹(𝑡,𝑥(𝑡),𝑣(𝑡),𝑢(𝑡))−1_𝑁⊗𝑓𝑡,𝑥₀(𝑡),𝑣₀(𝑡),𝑢₀(𝑡)=𝑒^𝑇(𝑡)𝐻+𝐻^𝑇𝑒(𝑡)+2^𝑁_𝑖=1𝑢^𝑇_𝑖(𝑡)𝑓𝑡,𝑥_𝑖(𝑡),𝑣_𝑖(𝑡),𝑢_𝑖(𝑡)−𝑓𝑡,𝑥₀(𝑡),𝑣₀(𝑡),𝑢₀(𝑡)≤𝑒^𝑇(𝑡)𝐻+𝐻^𝑇𝑒(𝑡)+2𝛾^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+𝑣^𝑇_𝑖̂(𝑡)𝑣_𝑖̂(𝑡)+𝑢^𝑇_𝑖(𝑡)𝑢_𝑖(𝑡)=𝑒^𝑇(𝑡)𝐻+𝐻^𝑇𝑒(𝑡)+𝑒^𝑇(𝑡)2𝛾𝐼_{𝑛𝑁×𝑛𝑁}0_{𝑛𝑁×𝑛𝑁}0_{𝑛𝑁×𝑛𝑁}0_{𝑛𝑁×𝑛𝑁}2𝛾𝐼_{𝑛𝑁×𝑛𝑁}0_{𝑛𝑁×𝑛𝑁}0_{𝑛𝑁×𝑛𝑁}0_{𝑛𝑁×𝑛𝑁}2𝛾𝐼_{𝑛𝑁×𝑛𝑁}𝑒(𝑡)=𝑒^𝑇(𝑡)2𝛾𝐼_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}0_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}2𝛾𝐼_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}0_{𝑛𝑁×𝑛𝑁}𝐼_{𝑛𝑁×𝑛𝑁}2𝛾𝐼_{𝑛𝑁×𝑛𝑁}𝑒(𝑡)≤𝜇𝑉(𝑒(𝑡)). (14) Moreover, for any 𝑡∈(𝑡_𝑘−1,𝑡_𝑘], 0≤𝑉(𝑒(𝑡))≤𝑉𝑡⁺_𝑘−1𝑒^{𝜇(𝑡−𝑡𝑘−1)}. (15)
For 𝑡=𝑡⁺_𝑘, similar to the proof of Theorem 1 in [11], we have 𝑉𝑒𝑡⁺_𝑘=𝑒^𝑇𝑡⁺_𝑘𝑒𝑡⁺_𝑘=𝑒𝑡_𝑘+Δ𝑒𝑡_𝑘^𝑇𝑒𝑡_𝑘+Δ𝑒𝑡_𝑘=𝑒^𝑇𝑡_𝑘𝐼_{3𝑛𝑁×3𝑛𝑁}+𝐶_𝑘^𝑇𝐼_{3𝑛𝑁×3𝑛𝑁}+𝐶_𝑘𝑒𝑡_𝑘; (16) that is, 𝑉𝑒𝑡⁺_𝑘≤𝛿_𝑘𝑉𝑒𝑡_𝑘. (17) Consequently, according to (10) and (15), we obtain 𝑉𝑒𝑡⁺_𝑘≤𝛿_𝑘𝑉𝑡⁺_𝑘−1𝑒^{𝜇(𝑡𝑘−𝑡𝑘−1)}≤𝜑𝑉𝑡⁺_𝑘−1. (18) By recurrence, we have 𝑉𝑒𝑡⁺_𝑘≤𝜑^𝑘𝑉𝑡₀. (19) Then, lim_𝑘→+∞𝑉𝑒𝑡⁺_𝑘=0. (20) Using Δ𝑡_𝑘=𝑡_𝑘−𝑡_𝑘−1≤𝜌<+∞ and (15), lim_𝑡→+∞𝑉(𝑒(𝑡))=0; (21) that is, lim_𝑡→+∞𝑒(𝑡)=0. (22) Therefore, third-order leader-following consensus in the multiagent network (5) is achieved under the impulsive controllers (7).

Remark 5. According to the proof of Theorem 4, it is not necessary that all the graphs 𝐺₁,𝐺₂,…,𝐺_𝑚 have directed spanning tree.

On the basis of Gersgorin disk theorem [23], we get 𝜇≤2(𝛾+1). For (𝐼_{3𝑛𝑁×3𝑛𝑁}+𝐶_𝑘)^𝑇(𝐼_{3𝑛𝑁×3𝑛𝑁}+𝐶_𝑘)=𝐼_3×3⊗[(𝐼_{𝑛𝑁×𝑛𝑁}+(𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘))⊗𝐶_𝑘)^𝑇(𝐼_{𝑛𝑁×𝑛𝑁}+(𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘))⊗𝐶_𝑘)], accordingly, 𝛿_𝑘 is equal to the maximum eigenvalue of (𝐼_{𝑛𝑁×𝑛𝑁}+(𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘))⊗𝐶_𝑘)^𝑇(𝐼_{𝑛𝑁×𝑛𝑁}+(𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘))⊗𝐶_𝑘). Therefore, we have the following Corollary 6 (recalling Δ𝑡_𝑘=𝑡_𝑘−𝑡_𝑘−1≤𝜌 and P={1,2,…,𝑚}).

Corollary 6. Under Assumption 1, if there exists 0<𝜑<1 such that 𝛿𝑒^{2(𝛾+1)𝜌}<𝜑, (23) where 𝛿=sup{𝛿_𝑘}, 𝛿_𝑘 is the maximum eigenvalue of the matrix (𝐼_{𝑛𝑁×𝑛𝑁}+(𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘))⊗𝐶_𝑘)^𝑇(𝐼_{𝑛𝑁×𝑛𝑁}+(𝐿_𝜏(𝑘)+𝐵_𝜏(𝑘))⊗𝐶_𝑘); then, the third-order leader-following consensus in the multiagent network (7) is achieved.

4. Numerical Simulations

In this section, we give some numerical examples to verify the theory results given in the previous section.

Consider the following nonlinear function 𝑓 for multiagent network: 𝑓𝑥_𝑖(𝑡),𝑣_𝑖(𝑡),𝑎_𝑖(𝑡)=2𝑎_𝑖2+𝑎_𝑖1+1−𝑎_𝑖1𝑎−1_𝑖1−𝑎_𝑖2+𝑎_𝑖3−2cos𝑥_𝑖1+2cos𝑣_𝑖1−𝑎_𝑖2−𝑎_𝑖3. (24) It is easy to verify that the nonlinear function 𝑓 in (24) satisfies Assumption 1. (Let 𝛾=5).

Following, we consider that the topology of the multiagent network is switching (P={1,2}).

Example 7. Assume that the graphs 𝐺₁, 𝐺₂ have directed spanning trees. The directed spanning trees of 𝐺₁, 𝐺₂ are described in Figures 1 and 2. The corresponding matrices of the graphs 𝐺₁, 𝐺₁, 𝐺₂, 𝐺₂ are 𝐴₁, 𝐵₁, 𝐴₂, 𝐵₂, respectively, where 𝐴₁=,𝐴001100101000010001100010100000001010₂=,𝐵000101100010110000100010011000101000₁=diag(0,1,0,1,0,0),𝐵₂=diag(1,0,0,0,1,0). (25) In time intervals [𝑡_4𝑘−1,𝑡_4𝑘), the corresponding matrices are 𝐴₂,𝐵₂,𝑘=1,2,…. In other time intervals, the corresponding matrices are 𝐴₁, 𝐵₁. We choose the impulsive gains 𝐶_𝑘1, 𝐶_𝑘2 with topology 𝐺₁ and 𝐺₂, respectively, as follows: 𝐶_𝑘1=,𝐶−0.04000−0.04000−0.04_𝑘2=.−0.05000−0.05000−0.05 (26)
Let the equidistant impulsive interval Δ𝑡_𝑘=𝑡_𝑘−𝑡_𝑘−1=0.00005. Then, 𝜌=0.00005. There exists 𝜑=0.9995, which satisfies the inequality (23) of Corollary 6. From Figures 3, 4, and 5, we can see that the position errors 𝑒_𝑥𝑖(𝑡), the velocity errors 𝑒_𝑣𝑖(𝑡), and the acceleration errors 𝑒_𝑎𝑖(𝑡) converge to zero quickly.

Figure 1 

Spanning tree of graph 𝐺1.

Figure 2 

Spanning tree of graph 𝐺2.

Figure 3 

Position errors 𝑒𝑥𝑖(𝑡) of system (7) with a directed spanning tree.

Figure 4 

Velocity errors 𝑒𝑣𝑖(𝑡) of system (7) with a directed spanning tree.

Figure 5 

Acceleration errors 𝑒𝑎𝑖(𝑡) of system (7) with a directed spanning tree.

Example 8. Assume that the graphs 𝐺₁, 𝐺₂ do not contain directed spanning trees. Let the corresponding matrices of the graphs 𝐺₃, 𝐺₃, 𝐺₄, 𝐺₄ be 𝐴₃, 𝐵₃, 𝐴₄, 𝐵₄, respectively, where 𝐴₃=,𝐴010000000000000000101000011000000110₄=,𝐵001000100100010100000000000000100010₃=diag(0,1,1,0,0,0),𝐵₄=diag(0,0,0,1,1,0). (27)
Similar to Example 7, in time intervals [𝑡_4𝑘−1,𝑡_4𝑘), the corresponding matrices are 𝐴₄,𝐵₄,𝑘=1,2,…, while in other time intervals, the corresponding matrices are 𝐴₃, 𝐵₃. We choose the impulsive gain 𝐶_𝑘3, 𝐶_𝑘4 with topology 𝐺₃ and 𝐺₄, respectively, as follows: 𝐶_𝑘3=,𝐶−0.07000−0.07000−0.07_𝑘4=.−0.08000−0.08000−0.08 (28)
Let the equidistant impulsive interval Δ𝑡_𝑘=𝑡_𝑘−𝑡_𝑘−1=0.00005. Then, 𝜌=0.00005. By some calculations, we can know that there exists 𝜑=0.9995, which satisfies the inequality (23) of Corollary 6. From Figures 6, 7, and 8, we can see that the position errors 𝑒_𝑥𝑖(𝑡), the velocity errors 𝑒_𝑣𝑖(𝑡), and the acceleration errors 𝑒_𝑎𝑖(𝑡) converge to zero quickly. It shows that the condition on a directed spanning tree is not necessary to realize consensus of the multiagent network (7) under the impulsive control. In addition, the researchers considered the second-order multiagent system with communication delay in [21, 22]. In our future work, we will study the third-order consensus problem for the multiagent systems with communication delay.

Figure 6 

Position errors 𝑒𝑥𝑖(𝑡) of system (7) without a directed spanning tree.

Figure 7 

Velocity errors 𝑒𝑣𝑖(𝑡) of system (7) without a directed spanning tree.

Figure 8 

Acceleration errors 𝑒𝑎𝑖(𝑡) of system (7) without a directed spanning tree.

5. Conclusion

By using graph theory, Lyapunov stability theory, and matrix theory, third-order leader-following consensus problem of a nonlinear multiagent network is studied in this paper. By designing proper impulsive controllers, a new criterion on realizing consensus in the multiagent network with switching topology is achieved. Finally, numerical simulations are provided to illustrate the theoretical results. In our future work, we will study the third-order consensus problem for the multiagent systems with communication delay.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 60964006, 61004101, and 11162004) and the Science Foundation of Guangxi Province, China (no. 2013GXNSFAA019006).