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In this paper, we consider the consensus control problem for a multiagent system (MAS) consisting of integrator dynamics with input and output time delays. First, we investigate a consensus condition for the MAS with a linear controller and without any delay compensation. We then propose a consensus controller with a state predictor to compensate the effect of time delay. The consensus condition for this controller is derived and investigated. Finally, we present an example of solving the consensus control problem for two-wheel mobile robots with feedback loops that pass through a computer network with time delays. To demonstrate the validity of the predictor-based controller, we conduct experiments with two-wheel mobile robots and present the results.

Achieving cooperative control of robotic systems is of increasing interest and has attracted a great attention in recent years. There are many potential applications for multirobot systems, including unmanned aerial vehicles, satellite clusters, automated highways, and search and rescue operations. Control tasks for robotic systems include consensus [

For nonlinear systems with input delay, Oguchi and Nijmeijer [

This article is organized as follows. In Section

Consider a network that consists of

For this system, the consensus problem is formulated as follows.

For multiagent system (

Following the consensus control protocol proposed by [

Assuming that

The dynamics of the total system can then be derived as

Therefore, from the stability of system (

Assume that each system (

Rewriting

Moreover, if the graph is undirected and connected, [

Based on the MAS (

Anticipating synchronization is a kind of master-slave synchronization. The predictor is constituted by the given system dynamics and coupling of the difference of the system output and delayed predictor states. The dynamics of this predictor can be stated as follows:

Controller (

As we use a predictor to predict the states, it is important to prove that the prediction error converges to

When the prediction error

With the use of (

To derive the necessary and sufficient conditions such that the whole system converges to consensus, we consider the coordinate transformation as follows:

From this equation, the consensus condition is given in the following theorem.

Assume that each agent (

The proof is given for the stability of the total synchronization error dynamics (

To make the equation hold, one of the above determinates should be equal to

For the second equation, we consider the smallest value of

Assuming

The first condition corresponds to the consensus condition for the system without delay, and the second comes from the stability of the prediction error. This discussion means that the synchronization-based predictor is an extension of the full-state observer, and a counterpart of the separation principle holds for the stability of the system with the synchronization-based predictor.

Compared with the consensus condition (

It is known that a directed graph contains a directed spanning tree, if and only if the corresponding graph Laplacian

Concerning the average value of agent states, we have the following results.

Consider that the system with agent (

The total system can be summarised as

Following the method shown in [

Therefore, the functional vector

By using controller (

Network structures for three and four agents.

The graph Laplacian

In this simulation the coupling strength is

Simulation results for three agents with controller (

Prediction error

Synchronization error

For a four-agent system, the graph Laplacian

With more agents and a longer time delay, the MAS satisfies Theorem

Simulation results for four agents with controller (

Prediction error

Synchronization error

In this section, by applying controller (

Consider the two-wheel mobile robot shown in Figure

Kinematic model of a mobile robot.

Applying (

For clarity, we simplify

The schematic for the system in the experiment is depicted in Figure

Schematic of the robot

We assume that there is a unified constant input and output time delay

On the way to convergence consensus, a robot may collide with other robots. Here, we assume for robot

Here, the distance

The consensus problem is to converge the coordinates of point

As shown in Figure

Structure of experimental system.

In the experiment, the radius of each robot is

Figure

Experimental results for three robots with controller (

Positions of robots

Synchronization error

In Figure

Figure

Experimental results for four robots with controller (

Positions of robots

Synchronization error

From the experimental results in Figure

Figure

In real applications, time-delay in network communication may be time-varying and/or unknown. According to the experimental results in [

In this paper, we considered the consensus problem of MAS with input and output time delays. A controller with a state predictor based on anticipating synchronization was proposed for this system. The consensus conditions for the controller were given, and we discussed the average consensus. We concluded that the proposed controller and predictor could cope with longer time delays, since the number of robots increased. We provided numerical simulations to show the validity of the control scheme. Validity was further confirmed in experiments with nonholonomic mobile robots based on the theoretical stability criteria and collision avoidance mechanism. It was shown that the validity of the proposed predictor-based controller could be used in real applications to control multiple mobile robots converging to consensus. In this study, to apply the predictor-based control approach, time-delay is considered as a constant value. Since time-delay is variable in real applications, we would like to discuss this problem in the future study.

The authors declare that they have no competing interests.

This research is an extended version of a conference paper [