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The leader-follower consensus problem of second-order multiagent systems with both absent velocity measurement and time delay is considered. First of all, the consensus protocol is designed by introducing an auxiliary system to compensate for the unavailability of the velocity information. Then, time delay is incorporated into the consensus protocol and two cases with, respectively, constant time delay and time-varying delay are investigated. For the case of constant time delay, Lyapunov-Razumikhin theorem is deployed to obtain the sufficient conditions that guarantee the stability of the consensus algorithm. For the case of time-varying delay, the sufficient conditions are also derived by resorting to the Lyapunov-Razumkhin theorem and linear matrix inequalities (LMIs). Various numerical simulations demonstrate the correctness of the theoretical results.

Consensus, which aims to make a group of agents reach an agreement on a common value by means of local interactions with each other, has been recognized as a fundamental issue in the cooperative control of multiple mobile autonomous systems [

Among the present literatures on consensus problem, the leader-follower consensus is the most extensively investigated due to its potential applications in many fields, such as the coordinated path tracking of unmanned aerial vehicles (UAVs) [

However, the existing consensus protocols are basically designed only when the full state of neighboring agents is available; i.e., both the position and the velocity information are required in the consensus of, in particular, second-order multiagent systems. As a matter of fact, the relative velocity measurement between agents is usually more difficult or even impossible to obtain than the relative position measurement in some practical applications [

On the other hand, communication delay, which is usually called time delay [

Unfortunately, the existing works usually treat the above two issues in separated ways. Literatures that both consider the absent velocity measurement and time delay in the consensus of second-order multiagent systems seem very few. Motivated by this fact, the leader-follower consensus of second-order multiagent systems with both absent velocity measurement and time delay is investigated in this paper. An auxiliary system is firstly designed to compensate for the unavailability of the velocity information and a consensus protocol with absent velocity measurement is proposed. Then, time delay is introduced into the consensus protocol and two cases with, respectively, constant time delay and time-varying delay are discussed. By deploying the Lyapunov-Razumkhin theorem and Lyapunov-Krasovskii theorem, respectively, the sufficient conditions for the stability of the consensus protocol with constant time delay and time-varying delay are derived. Numerical simulations demonstrate the correctness of the theoretical results.

The rest of this paper is organized as follows. In Section

In this section, some mathematical backgrounds, including graph theory, matrix theory, and time delay systems are introduced for the theoretical analysis of this paper.

In a multiagent system, if each agent is regarded as a node, then its topological structure can be simply described by a graph. Here, graph

If there exists a path from node

Given vectors

If the symmetric matrix

Consider the following system:

Let

If there exists a continuous function

if there exists a continuous nondecreasing function

then the solution

Usually,

It can be seen in Lemma

Consider the following differential equation with time delay:

Supposing that the mapping

In addition, the solution

Considering a multiagent system with one leader and

The dynamics of the leader is expressed as

For multiagent system (

The leader-follower consensus of second-order multiagent systems is solved, if for any initial conditions, the following hold:

Form (

Therefore, the main objective of this paper is to synthesis the distributed consensus protocol for second-order multiagent systems such that when the velocity information is unavailable and there exist time delays during the position information transmission between agents, the consensus of all agents will asymptotically be achieved.

For the consensus of second-order multiagent systems without velocity measurement, the distributed consensus protocol can be formulated as

If time delay is considered in the position information transmission between agents, the consensus protocol can then be written as

Let

For simplicity, we denote

Then, the error dynamics of multiagent system (

From (

In this section, two time delay cases, including constant time delay and time-varying delay, are considered in the consensus of second-order multiagent systems with absent velocity measurement. Some theoretical results are derived for the stability of the proposed consensus algorithm (

For the case of constant time delay,

Let

Then, the following theorem for the stability of second-order multiagent systems with absent velocity measurement and constant time delay is obtained.

Consider the multiagent system (

Then, the consensus of multiagent system (

Choose the following Lyapunov-Razumikhin function:

Taking the derivative of

According to the

Thus

Then, (

Invoking (

In addition, the following inequalities hold according to Lemma

Therefore, (

Let

Note that (

Finally, invoking (

Thus, we know

Note that the above assumption that the time delay in the position information transmission between agents is a constant value may not be reasonable in some practical applications, as the time delay caused by unreliable communication networks may be time-varying and even a stochastic value [

For the case of time-varying delay,

Let

Then, the following theorem for the stability of second-order multiagent systems with absent velocity measurement and time-varying delay is introduced.

Consider the multiagent system (

Then, the consensus of second-order multiagent system (

Define the following Lyapunov-Krasovskii functional:

Taking the time derivative of

Letting

According to inequality (

Hence,

In this section, numerical simulations are carried out to verify the effectiveness of the proposed consensus algorithm with absent velocity measurement and time delay. Specifically, we choose one leader and four followers in the simulation.

The topological structure of multiagent system is illustrated in Figure

The topological structure of multiagent system.

In the following, the consensus of second-order multiagent systems with constant time delay and time-varying delay is, respectively, performed to verify the correctness of the theoretical results.

In the constant time delay case, let

Based on the above simulation settings, we let

Positions and velocities of the leader and followers when the time delay

Position

Velocity

From Figure

In the time-varying delay case, let

Based on the above simulation settings, we let

Positions and velocities of the leader and followers when the time delay

Position

Velocity

It can be seen from Figure

In order to illustrate the superiority of the proposed consensus algorithm (

Positions and velocities of the leader and followers when velocity is unavailable in [

Position

Velocity

The position curves of agents in Figure

The main reason lies in that the position information can only guarantee the position consensus of multiagent systems, whereas the velocity consensus cannot be ensured. By using the auxiliary system (

In the above simulations, the underlying topology among followers is assumed to be undirected, which easily obtains a symmetric Laplacian matrix and hence facilitates the subsequent theoretical analysis. However, the communication network between agents may be directed as in some cases agents can only receive information from others but not be able to send information to them. Therefore, it is of more significance to investigate the consensus problem of multiagent systems under directed topology.

In [

In general, the consensus of multiagent systems in directed networks requires that the communication graph of agents systems contains a spanning tree, which is more challenging than undirected topology.

This paper investigates the leader-follower consensus problem of second-order multiagent systems, where the absent velocity measurement and time delay are considered simultaneously. For the absence of velocity information, an auxiliary system is designed to compensate for the unavailability of velocity measurement. Then, two time delay cases, including constant time delay and time-varying delay, are respectively, investigated and the sufficient conditions for the stability of the consensus algorithm are derived via Lyapunov-Razumkhin theorem and Lyapunov-Krasoviskii theorem. Simulation results show that the proposed consensus protocol is able to realize the consensus of multiagent systems even in the presence of absent velocity measurement and time delay.

It should be emphasized that the maximum allowable upper bound of time delay is a little conservative as the LMI approach adopted in this paper only derives the sufficient conditions for the stability of second-order multiagent system with time delay. Therefore, how to reduce conservatism on the estimation of the maximum allowable upper bound of time delay is our new concern in the future work.

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

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was partially supported by the National Natural Science Foundation of China (61803040), the Key Science and Technology Program of Shaanxi Province (2017JQ6060, 2018JQ6098), and the Fundamental Research Funds for the Central Universities of China (300102328403, 310832171004).

_{∞}pinning synchronization of directed networks with aperiodic sampled-data communications