This paper mainly investigates the average consensus of multiagent systems with the problem of packet losses when both the first-order neighbors’ information and the second-order neighbors’ information are used. The problem is formulated under the sampled-data framework by discretizing the first-order agent dynamics with a zero-order hold. The communication graph is undirected and the loss of data across each communication link occurs at certain probability, which is governed by a Bernoulli process. It is found that the distributed average consensus speeds up by using the second-order neighbors’ information when packets are lost. Numerical examples are given to demonstrate the effectiveness of the proposed methods.
In recent years, there has been an increasing research in coordination control of multiagent systems. Information consensus has attracted more and more attentions from many engineering application fields, such as formation control, flocking, artificial intelligence, and automatic control [
An excellent protocol can reduce cost, increase efficiency, and can optimize performance. Convergence rate is an important index to evaluate the performance of consensus. There has been much research interest in dealing with this issue. In [
It is noted that the literatures mentioned above mainly focus on consensus problem for agents under first-order dynamics with time delay. In reality, the agents exchange data over fading communication channels instead of ideal ones. In fact, in many practical applications, this data exchange between sensors is done by wireless communication, which has a possibility of packets lost. Thereby, the packet losses should be taken into consideration. Many related works have been reported. Reference [
Inspired by the above references, we consider multiagent systems with the problem of packet losses based on the second-order neighbors’ information. We construct a group of agents, which can communicate with their second-order neighbors and each communication link has a probability of failure. We assume that all channels are independent and subject to a distributed random process. Thereby, they have the same probability of data loss. Each agent is equipped with a sampler and a zero-order hold, which are synchronized in time. Then, by converting the system to the equivalent error dynamics, stochastic stability of the error dynamic system is studied. Here, a Lyapunov function is constructed and a sufficient condition is established to guarantee the average consensus in the form of linear matrix inequality (LMI). We are curious about whether the protocol based on the second-order neighbors’ information can accelerate the convergence speed with the problem of packet losses. Then, a simulation comparison of the convergence rate between the protocol based on the second-order neighbors and the one in general linear is shown. Comparison of the convergence speed between different probabilities of packet losses is also simulated.
The rest of this paper is organized as follows. Section
The set of real numbers is denoted by
In this paper, the interaction among
From the above definitions, we know some facts:
Consider the following first-order dynamics:
Next, we consider the packet losses among agents. The following control protocol is designed:
Furthermore, we assume that the occurrence of packet loss is governed by a Bernoulli process with uniform probability
The undirected topology is coupled; that is, for any pair of agents
Assumption
We define two sets of matrices
So, the system dynamics can be written as
The nominal communication topologies
The above assumption is necessary for consensus because if the undirected graph is not connected, then it does not have a spanning tree. From [
The average states of the agents
For an undirected graph, given the Laplacian matrix
The following theorem gives a sufficient condition on the average consensus of the system (
Given the scalar
Construct the candidate Lyapunov function as
To illustrate the average consensus of the system (
Nominal communication topology and topology based on the second-order neighbors’ information.
The weights are set to unity for simplicity here. We set the corresponding Laplacian matrices
We choose the sampling period as
Time history of the Bernoulli variable
Nominal communication and the topology based on second-order neighbors’ information with
Comparison of the convergence speed based on second-order neighbors’ information with different package loss probability.
In this paper, we have investigated the average consensus in multiagent systems with the problem of packet losses when second-order neighbors’ information was used. The convergence rates of general protocol and second-order neighbor protocol with packet losses have been compared and it is concluded that second-order neighbor protocol speeds up the consensus rate. What is more, we can see the influence of packet losses. Future work will extend the agent dynamics to second-order or higher-order dynamics with data loss and time-varying delay.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation (NNSF) of China under Grants no. 61004031, no. 61174096, and no. 61104141, Beijing Municipal Commission of Education Science and Technology Program (KM201310017006), and the Fundamental Research Funds for the Central Universities.