A novel consensus protocol for two-dimension first-order multiagent systems (MAS) is proposed. After a novel protocol for two-dimension first-order MAS is adopted; consensus and two kinds of possible results are concerned. Then consensus asymptotical conditions and ranges of key parameters are given, and convergence speed is analyzed. Numerical examples illustrate effectiveness of the proposed consensus protocol.
Networks have been concerned for many years, and it has been researched in many areas such as in sensor networks [
Consensus problem is applied in robots, spacecraft, and cooperative multiple vehicles, and it mainly concerns exchange ways of information. For example, reference [
Suppose there are
According to [
The following is based on the fact that the premise is satisfied. In this work, a kind of two-dimension first-order MAS is concerned. Then a novel protocol and its consensus are proposed, and range of key parameter is given. Moreover, convergence speed is analyzed. Several simulations are shown to ensure the results.
Suppose there are
The consensus is that all agents finally converge together, with their positions converging to the same value, or it can be expressed as
Protocol of two-dimension system is introduced as
Let
Suppose
Because
Let
Note that
Let
Then the following lemma is proposed.
When
Let
All of the above show that if
According to [
Then suppose there is at least one nonzero eigenvalue having positive real part. Note that
It can be seen that if there is at least one eigenvalue with positive real part of
So the assumption is wrong.
Then the following lemma is proposed.
When
The proof is omitted for similarity with Lemma
Because
According to the premise,
Suppose there is only one pair of conjugate complex eigenvalues marked as
All possible results of
In Figure
If there are more than one pair of conjugate complex eigenvalues, it is obvious that
Convergence speed of system (
From (
From (
From Figure
In this section, examples are illustrated to prove conclusions drawn in Section
A digraph in [
A digraph in [
It is obvious that
Simulation results with different
Simulation results (
Simulation results (
Simulation results (
Simulation results (
From Figure
Moreover, asymptotical consensus is achieved in less than 10 seconds when
Simulation results with the same system when
Simulation results with different
Simulation results (
Simulation results (
If all eigenvalues of
Next,
Simulation results with different
Simulation results (
Simulation results (
Simulation results (
According to Lemma
A novel consensus protocol is presented for the linear two-dimension first-order multiagent systems (MASs). Numerical examples illustrated the effectiveness of the proposed consensus protocol according to different key parameters. Simulation results ensure the relationship between range of key parameter and convergence speed.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by National Natural Science Foundation of China (nos. 61374083 and 61203177) and Zhejiang Province Key Project of Science and Technology (no. 2014C03027).