In order to avoid a potential waste of energy during consensus controls in the case where there exist measurement uncertainties, a nonlinear protocol is proposed for multiagent systems under a fixed connected undirected communication topology and extended to both the cases with full and partial access a reference. Distributed estimators are utilized to help all agents agree on the understandings of the reference, even though there may be some agents which cannot access to the reference directly. An additional condition is also considered, where self-known configuration offsets are desired. Theoretical analyses of stability are given. Finally, simulations are performed, and results show that the proposed protocols can lead agents to achieve loose consensus and work effectively with less energy cost to keep the formation, which have illustrated the theoretical results.

One of the most attractive benefits of multiagent systems is that many fractional, inexpensive, and simple agents working together can achieve the same objective as a monolithic, expensive, and complicated agent. Since certain formation configurations are often indispensable for such systems, the coordination problem in the area of formation control has attracted compelling interest in recent years. One critical issue arising from the coordination problem is to develop distributed control laws based on local information that enable all agents to reach an agreement on certain quantities of interest, which is known as the consensus problem.

In the context of multiagent systems, consensus problems have been studied widely in recent years. Reference [

Most results of the aforementioned research are obtained in case where dynamics are linear. And in case where the dynamic agents are physical models, the design and analysis of nonlinear consensus protocols have also been studied, with input constraints of such systems being taken into account for more adaptive designs. Reference [

However, sometimes, energy means a lot for some particular applications, such as certain applications for space, for example, formation keeping and attitude alignment. It is worthwhile to note that, although the aforementioned protocols are able to deal with the measurement uncertainties, they may not perform well in saving energy (which will be shown through comparisons in the later simulations). It can be known that, in the typical tight spacecraft formation keeping missions, the relative position measurements are of precision of centimeter number grade [

The outline of this paper is as follows. Some backgrounds on algebraic graph theory and linear protocol are given in Section

The

A weighted undirected graph

The linear consensus protocol can be given by (see, e.g., [

It is clear to note that protocol (

Intuitively thinking, when the configuration errors are of the same precision as those of the measurements, it is not convincing to control based on the measurements, and, instead, a conservative control strategy is considered. Introduce two nonlinear functions

Then the nonlinear consensus protocol is given by

Before going on, a lemma and its proof are given as follows.

Suppose that

Considered

Using protocol (

Denote

Introduce a Lyapunov function as

Then the time derivative of

Let

It can be seen from (

For the sake of being referred conveniently, if there exists a group reference which is available to all agents, the group reference can be labeled as a virtual agent 0, with

Suppose that a group reference labeled as agent 0 is available to all agents of multiagent system (

Denote

Note that (

Then the time derivative of

Note that

In practice, it is more common that only a portion of agents can access to the reference, while others cannot. Generally in this case, the linear consensus problem often used the same form of protocol as what is used in the case where the group reference is available to all agents (see, e.g., [

In order to better borrow the form of protocol (

It is worthwhile to note that estimators (

Suppose that

the communication topology graph of (

arguments satisfy that

With

Equation (

Note that the fixed undirected graph

Consider the Lyapunov function candidate

Here, we use the fact that

Note that

After some manipulation, we get that

Therefore, there exists a finite time, which equals

After the design of

In this subsection, the configuration offset of each agent, which is denoted as

After using (

If both the conditions in Theorem

Thus, for all

Note that, if one let all

In the simulations, we will show that the proposed protocol (

A multiagent system which contains 6 agents is taken as an example, and they are considered in 1 dimension for simplicity, although the analyses throughout the paper are discussed in any dimensions of order

Communication links of a multiagent system.

When a group reference does not exist, employing protocol (

Position trajectories of all agents with (

Velocity trajectories of all agents with (

When the reference exists, after using (

Position trajectories of all agents with (

Velocity trajectories of all agents with (

Then we compare linear protocol (

Comparison of energy costs.

In this paper, in order to avoid a potential waste of energy in the case where there exist measurement uncertainties, a nonlinear protocol is proposed with a connected undirected communication topology. The protocol is then extended to both the cases where agents can have full or partial access to a group reference. Especially in the case of partial access, a set of distributed estimators is utilized to help all agents agree on the understandings of the reference, even though some agents cannot access to the reference directly. An additional condition where self-known configuration offsets are desired is also considered. Theoretical analyses of stability are given for the proposed protocols. Finally, simulations are performed by MATLAB, and results show that these protocols can lead agents to loose consensus and work effectively with less energy cost to keep formation, while there is little enhancement by using such protocols when agents are trying to form the formation.