Consensus of first-order and second-order multiagent systems has been wildly studied. However, the convergence of high-order (especially the third-order to the sixth-order) state variables is also ubiquitous in various fields. The paper handles consensus problems of high-order multiagent systems in the presence of multiple time delays. Obtained by a novel frequency domain approach which properly resolves the challenges associated with nonuniform time delays, the consensus conditions for the first-order and second-order systems are proven to be nonconservative, and those for the third-order to the sixth-order systems are provided in the form of simple inequalities. The method revealed in this article is applicable to arbitrary-order systems, and the results are less conservative than those based on Lyapunov approaches, because it roots in sufficient and necessary criteria of stabilities. Simulations are carried out to validate the theoretical results.
Consensus problems of multiagent systems have found many applications in the fields that hold great promise, including (but not limited to) biosciences, robotics, and computer sciences. Consensus is the agreement regarding a certain quality of interest on specific states of all the agents, which is widely demanded in the engineering applications. The research on consensus problems has lasted for decades. Various techniques are developed to solve consensus problems of numerous multiagent systems [
This paper addresses the consensus control problems of high-order multiagents systems with nonuniform time delays. One motivation for studying high-order systems is to achieve accurate control of complex motion: for example, when performing consensus motion that requires abrupt change of heading, a team of vehicles should maintain consistency of acceleration (as well as position and velocity) among them by controlling the third-order state (acceleration), while lower-order (first-order and second-order) consensus protocols are usually designed for more regular motion (e.g., rectilinear [
A novel frequency-domain-based method is developed to challenge the system complexities and derive the consensus conditions. Comparing to the universal stability analysis tool Lyapunov approaches, frequency domain methods are more possibly conducing to less conservative results as it roots in sufficient and necessary stability criteria. On the other hand, Lyapunov approaches applied in many literatures yield consensus conditions in the form of Linear Matrix Inequalities (LMIs) [
The main idea of the proposed approach is to transform the high-order systems’ dynamics into high-degree polynomials with respect to hypothetically existing imaginary eigenvalues of the systems. By studying the monotonicity of the polynomials and their derivatives, consensus conditions can be figured out in the form of inequalities. The present work first brings out sufficient consensus conditions for the first-order to sixth-order nonuniformly delayed systems which are most likely to apply to practical engineering applications [
Literature [
The remainder of this note is organized as follows: Section
This section starts with some definitions and results in graph theory.
Consider an
If the undirected graph
Consider an
In [
Let
This paper assumes that the graph
The following lemma presents a sufficient condition for the stability of high-degree polynomials given by [
Consider a polynomial
Let
Assume for
Under Assumption
Matrix
According to Lemma
When
Lemma
By analyzing the effect of nonuniform time delays on the stability of the systems, we will give a proof to the ensuing theorem.
Consider
Consider the network of high-order multiagents with nonuniform time delays. Let
Suppose
Rewrite (
Take modulus of both sides of (
As
Consider the first-order system; then, we have
Unlike
By investigating the locus of
For
Investigate the following parabola when
Let
Consider the interval
Taking the fourth-order system into account, we have
For
Comparing
Consider the interval
Likewise, for
The trajectories of
Despite the fact that it is difficult to present a general solution in the form of inequality to consensus problems of all high-order multiagent systems because the monotonicity of each high-degree polynomial that the solution relies on requires specialized derivation to figure out, the process of deriving the consensus conditions for the first-to-sixth-order systems has demonstrated a general approach, with which one can work out consensus conditions for an arbitrary-order system. Moreover, by further calculation, we provide stronger consensus conditions for the first-order and second-order multiagent systems in the following theorem.
Consider the first-order and second-order multiagent system (
For the first-order multiagent system, we have already proven in Theorem
For
In this section, some simulations are provided to illustrate the theoretical results obtained by the previous analysis.
Consider a multiagent system consisting of 4 agents. Figure
Two communication topologies of the 4-agent system.
For a third-order multiagent system with simply connected graph
The trajectories of agents in the third-order system (the topology is
The trajectories of agents in the third-order system (the topology is
For a sixth-order system connected as
The trajectories of agents in the sixth-order system (the topology is
The trajectories of agents in the sixth-order system (the topology is
To examine Theorem
The trajectories of agents in the second-order system (the topology is
The trajectories of agents in the second-order system (the topology is
This paper has studied consensus problem of high-order multiagent systems with nearest-neighbor control rules in the presence of nonuniform time delays. For each delayed
Future research will seek solutions to consensus problems of nonuniformly delayed high-order systems with directed topologies by applying this method. The main challenge is the calculation of
The authors declare that they have no competing interests.
This work has been supported by the Specialized Research Fund for the Doctoral Program of Higher Education (20130185110023).