In a multiagent system (MAS), communication signals are affected by harsh wireless networks when they are transmitted from an agent to its neighboring agents, leading to the inconsistency of the MAS. In this paper, an average-iterative learning control (average-ILC) method is studied to address the consensus problem of MAS over wireless networks in the presence of channel noise and data dropout. The combined effects of channel noise and data dropout on iterative learning controllers are carefully analyzed. Based on graph theory and mathematical expectation, the corresponding average-iterative learning scheme is proposed. Especially, a sufficient condition is derived for the average-iterative learning scheme. Rigorous theoretical analysis demonstrates that the convergence of the covariance matrix of tracking error can be guaranteed with the help of an average-iterative learning scheme. Finally, simulation results are given to show the effectiveness of the proposed method.

The consensus problem has become one of the most popular research studies in collaborative control of MAS [

The applications of ILC in multiagent frameworks have been developed for many years. ILC was firstly introduced into MAS by Ahn and Chen in [

However, an underlying condition of all the abovementioned results is that the signal transmission is reliable, which means that signals were not polluted when they are transmitted from one agent to another. But for MAS, all agents are usually located in different sites and communicate with others via wireless networks. Then, channel noise and data dropout always exist in channels of wireless networks and cause interference to the transmitted signals. Thus, reliable transmission is hard to be realized in actual networks when there are channel noise and data dropout in the wireless communication channel. This increases the design difficulty of iterative learning controllers. To solve this problem, many references give corresponding solutions on channel noise and data dropout. First of all, Kalman filter [

Nevertheless, consensus problems of MAS with ILC in presence of channel noise or data dropout have not been fully investigated. These abovementioned references only consider the impact of channel noise or only the impact of data dropout. It is well known that channel noise and data dropout are unavoidable in wireless communication channels, and they simultaneously act on transmitted signals when the signals are transmitted from one agent to another. Channel noise leads to an inaccurate signal, and data dropout means that the signal is unable to be received by other agents. In other words, the signal cannot be received if data dropout exists; otherwise, the signal is received by other agents in an imprecise form. Then, these combined effects of channel noise and data dropout act on the iterative learning controller, causing the ILC algorithms to fail to converge. This indicates that the design complexity of the iterative learning controller is greatly increased when channel noise and data dropout coexist. Unfortunately, no literature considers channel noise and data dropout simultaneously for ILC of MAS, and ILC schemes proposed by existing references cannot solve this problem well. Therefore, our aim is to design an iterative learning controller to guarantee the convergence of ILC for MAS with channel noise and data dropout.

Motivated by this, in this paper, channel noise and data dropout are both taken into account for MAS. The mixing error caused by channel noise and data dropout will be introduced into the iterative learning controller and then make the outputs of MAS more difficult to reach the desired output. For dealing with this problem, an average-ILC method is proposed. As a result, rigorous theories demonstrate that the effects of both channel noise and data dropout on iterative learning controllers can be effectively eliminated by the proposed method.

The main contributions of this paper are highlighted as follows.

As we all know, channel noise and data dropout are important factors affecting signal transmission. As far as we know, there is no paper to study the situation with both channel noise and data loss. Thus, unlike the aforementioned references, channel noise and data dropout are both considered for the consensus problem of MAS using ILC, which further promotes the design complexity of the iterative learning controller and then increases the difficulty of the consensus problem for MAS. In addition, the differences between channel noise and data dropout are described, and their combined effects on ILC for MAS are also depicted in this paper.

For channel noise and data dropout introduced by wireless communication channels between neighboring agents, an average-iterative learning controller is designed to eliminate the mixing error caused by them. With the help of the series convergence property, a sufficient condition is proposed. Then, the perfect tracking can be obtained for MAS. In addition, by comparing the method proposed in this article with traditional methods, we verify the effectiveness of the method proposed in this article.

The remainder of this paper is organized as follows. Some preliminaries are made in Section

Notions:

Based on algebraic graph theory, we can change the network topology into mathematics matrices. Let

A path in directed graph

All agents of MAS are located in different positions, and the multiagent consistency is realized through mutual cooperation. This means each agent needs to send signals to others and receive signals from others via wireless networks. However, channel noise and data dropout in wireless networks can cause interference to transmitted signals. Then, the details of channel noise and data dropout are shown in Figure

Channel noise and data dropout between agent

For instance, agent 1 transmits its output

The effects of both channel noise and data dropout on agent

Iterative process within agent

It also assumes that

Consider MAS with

For describing all agents of MAS (

For a given virtual leader

Based on the descriptions of channel noise and data dropout in Section

For the convenience of the following analysis,

We consider that the topology of MAS (

Define

For MAS (

Then, (

Define

Then, considering all the time, we can just go on and derive the following equations from (

The objective in this paper is to ensure that the following convergence principle holds:

It is clear that

The virtual leader

For all agents

For all agents

Assumption

Channel noise and data dropout widely exist in the process of wireless signal transmission, which increases the design difficulty of the iterative learning controller. Hence, the research of channel noise and data dropout on ILC is undoubtedly important. In this section, an iterative learning controller is designed for MAS. An average-iterative learning scheme is utilized to deal with the combined effects of channel noise and data dropout on iterative learning controllers.

For realizing objective (

According to the definition of

Define

For MAS (

According to the expression of (

Then, the compact form of tracking error (

The relation of

Benefitting from above derivations, we can obtain the following theorem and corollary. For the ease of representation, we define

Let us consider MAS (

From (

Noting that

Norm on both sides of (

It is worth noting that

Obviously, there is an iteration number

It means

It seemed that if

It is worth noting that

This completes the proof.

Let consider MAS (

According to the relation of input error

According to Theorem

This completes the proof.

Theorem

For illustration and verification, we consider discrete-time MAS (

Let the desired output is

Topology of MAS with four agents and a virtual leader.

Average-iterative learning scheme (

To show the effectiveness of the proposed method, a benchmark method is introduced, such as

Figure

The outputs of MAS for

To show the effectiveness of the proposed method more intuitively, we compare the convergence of the covariance matrix under the proposed method and benchmark method, such as Figure

Convergence of the covariance matrix of tracking error with the proposed method or benchmark method.

In this paper, channel noise and data dropout are both considered, and the effects of these two factors on the convergence speed of the covariance matrix of tracking error need to be analyzed when the proposed method is applied.

Figure

Convergence speed of the covariance matrix of tracking error with different variances of channel noise.

The effect of different successful transfer rates on the convergence speed of the covariance matrix of tracking error is depicted in Figure

Convergence speed of the covariance matrix of tracking error with different successful transfer rates.

This paper deals with the consensus problem of multiagent systems over wireless networks in the presence of channel noise and data dropout via an average-ILC approach. The gain matrix is designed to satisfy sufficient conditions. Rigorous theories demonstrate that the convergence of the covariance matrix of tracking error can be guaranteed with the help of an average-iterative learning scheme. To highlight the effectiveness of the proposed method, a benchmark method is realized to be a reference in Section

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The work was supported by National Natural Science Foundation of China (NSFC) under Grant nos. 61673253 and 61901254.