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

. 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.


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][2][3][4][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, leaderfollowing 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][7][8][9][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][10][11][12][13][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][18][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 thirdorder 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.
A directed path from node  to node  in the directed graph  is a sequence of edges (,  1 ), ( 1 ,  2 ), . . ., (  , ) with distinct nodes   ,  = 1, . . ., .A digraph  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, . . ., .The graph  is consisting of the leader and the followers with communication topology.The connection weight between the th follower and the leader is represented by   ,  ∈ .If the th follower is connected to the leader, then   > 0; otherwise,   = 0. Let  = diag{ 1 ,  2 , . . .,   }.
Following, we address the multiagent network with switching topology.The set Ω = { 1 ,  2 , . . .,   } is used as a set of the graphs with all possible topology, which includes all possible graphs (involving  agents and a leader).We define a switching signal  : [0, +∞) → P = {1, 2, . . ., }, which determines the topology structure that corresponds to the network.When the topology is switching, the Laplacian matrix  and the matrix  are also switching, which are denoted by  () and  () .
The following assumptions are needed to derive our main results.

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 where  and  are the state variable and the output, respectively.U(, ) 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  ̸ =   (), we define the time derivative of the function (, ) with respect to system (3) as 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.

Theorem 4. Under Assumption 1, if there exists
where   and  are the maximum eigenvalues of matrices respectively, then the third-order leader-following consensus in the multiagent network (7) is achieved.
Remark 5.According to the proof of Theorem 4, it is not necessary that all the graphs  1 ,  2 , . . .,   have directed spanning tree.

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  for multiagent network: It is easy to verify that the nonlinear function  in (24) satisfies Assumption 1. (Let  = 5).
(28)  Let the equidistant impulsive interval Δ  =   −  −1 = 0.00005.Then,  = 0.00005.By some calculations, we can know that there exists  = 0.9995, which satisfies the inequality (23) of Corollary 6.From Figures 6, 7, and 8, we can see that the position errors   (), the velocity errors  V (), and the acceleration errors   () 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.

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.