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This paper concerns the problem of consensus tracking for multiagent systems with a dynamical leader. In particular, it proposes the corresponding explicit control laws for multiple first-order nonlinear systems, second-order nonlinear systems, and quite general nonlinear systems based on the leader-follower and the tree shaped network topologies. Several numerical simulations are given to verify the theoretical results.

There have been a lot of recent researches paying attention to the problem of multiagent cooperative control which means a group of agents working cooperatively to achieve coverage, formation, and consensus [

In the pioneering work on consensus tracking of Ren [

Li et al. [

Nonlinear dynamics are now studied in the consensus problem from various perspectives such as [

In those papers mentioned above, some focus on the problem that the linear follower agents track a leader who is governed by an external input, yet others focus on the problem that the nonlinear follower agents track a leader who has no explicit external input. In the practical network with a linear or a nonlinear leader, the external input is unavoidable or even is important for guiding the group to behave correctly. Thus, the study of consensus tracking for a group of nonlinear agents with the leader having an external input will be significative. In this paper, we consider the problem of consensus tracking for the network of a group of

We have noted that the intrinsic dynamics of the leader in [

Due to the existence of nonlinearity in the agents’ dynamics and the external input of the leader, the existing consensus algorithms are not applicable to our problem. By synthetically using the Lipschitz conditions, the variable structure technique [

The remainder of the paper is organized as follows. In Section

We use

Since graph theory plays an important role in modeling the communication topology of the network of the multiagent systems, some basic concepts in graph theory that will be used in this paper are introduced in the following.

In the problem of nonlinear consensus tracking, a kind of communication topology of

For a directed graph,

We start by considering the first-order nonlinearity case:

As in most existing works on networks of nonlinear agents [

There exists

In order to guarantee these

The undirected graph

To deal with the problem of consensus tracking for the network with the first-order nonlinear agents, we propose a control algorithm for (

Then, the main result on the problem of consensus tracking for the network with first-order nonlinear agents is proposed by the following theorem.

If Assumptions

Let

From Assumption 2 and Lemma 1 in [

Next, we discuss the second-order nonlinearity case. Suppose that each of the

There exist

Compared with Assumption

Similar to the first-order case in Section

Then, we propose the following control algorithm applied for the system (

Given the matrix

Since

Then, the main result follows.

Suppose that Assumptions

Let

In order to deal with the nonlinear term of the agents’ dynamics in Theorem

The result in Theorem

In the general nonlinear case, suppose that a network system with

Note that the differential equation (

Throughout the subsequent analysis we assume that the network topology satisfies the following two assumptions.

The graph of the network topology is tree shaped with the leader as the root node, where the tree shaped graph means each node has only one parent node except the root node.

For the network system, each agent knows the measurement of the control input of its parent agent at the same time.

Assumption

In order to propose the consensus tracking algorithm for agents (

Let

If there exist two numbers

If Assumptions

Let

For an undirected connected graph which contains a tree shaped subgraph, or a directed graph which contains a directed spanning tree, we can choose such a tree as Assumption

It has been shown that systems (

In this section, three numerical simulation examples are given to illustrate the theoretical results. Consider the first example, a network of three followers with a leader shown in Figure

The undirected graph for a group of three followers with a leader. Here

Consensus tracking for the first-order nonlinear systems.

The second example is also given for the graph in Figure

Consensus tracking for the second-order nonlinear systems.

The tree shaped graph for a network of four followers with a leader, where

Now let us see the third example. In the case of the general nonlinear dynamics, the graph of the network topology is shown in Figure

We consider each agent’s dynamics to be a simple nonholonomic system specified by the equations as follows:

The norm of error

Since

In this paper, we studied the problem of nonlinear consensus tracking via the variable structure technique, the feedback linearization technique, and the Lyapunov theory when there is a leader governed by the external input. Suppose that the leader’s external input is upper bounded and a connectivity requirement for the network topology is satisfied; we proposed the consensus tracking algorithms for the followers with the first-order nonlinear dynamics, the second-order nonlinear dynamics, and the general nonlinear dynamics to asymptotically track the corresponding nonlinear leader. And several numerical simulations were given to show the effectiveness of our algorithms. The future works include the study of nonlinear consensus tracking in the general directed network topologies.

The author declares that there is no conflict of interests regarding the publication of this paper.