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High-order consensus seeking, in which individual high-order dynamic agents share a consistent view of the objectives and the world in a distributed manner, finds its potential broad applications in the field of cooperative control. This paper presents stability switches analysis of arbitrary high-order consensus in multiagent networks with time delays. By employing a frequency domain method, we explicitly derive analytical equations that clarify a rigorous connection between the stability of general high-order consensus and the system parameters such as the network topology, communication time-delays, and feedback gains. Particularly, our results provide a general and a fairly precise notion of how increasing communication time-delay causes the stability switches of consensus. Furthermore, under communication constraints, the stability and robustness problems of consensus algorithms up to third order are discussed in details to illustrate our central results. Numerical examples and simulation results for fourth-order consensus are provided to demonstrate the effectiveness of our theoretical results.

Our understanding of distributed cooperative control for both natural and engineered dynamic networks has been advanced in the last few years [

It is noticed that most algorithms focus on the consensus problems for the networks of agents with single or double integrator dynamics. In the current paper, we investigate arbitrary high-order consensus networks in presence of communication time-delays, which generalize the aforementioned existing results in the literature. The idea of employing high-order integrator dynamic agents under communication constraints is inspired by the following reasons.

First, from the point of view of system science, nonlinear systems of a broad class (i.e., feedback linearizable systems) can be transformed to linear systems via feedback control and transformation of the state vector by using differential-geometric methods [

In our earlier work [

In this paper, we extend earlier work and introduce scalable arbitrary high-order consensus algorithms with communication time-delays. Via frequency domain analysis, we investigate the stability switches and establish an explicit general connection between the arbitrary high-order consensus and the system parameters, including the Laplacian spectrum of the underlying network topology, the feedback gains of the algorithms, and communication constraints. The main contribution of this paper is to provide a theoretical and computational framework for analysis and synthesis of scalable arbitrary high-order consensus algorithms in presence of communication time-delays.

An outline of this paper is as follows. Section

Consider

Information exchange between agents can be naturally modeled by the weighted undirected graph

We employ the following scalable time-delayed high-order consensus algorithm:

The high-order consensus problem discussed in this paper is defined exactly as follows.

For the network of

In this section, we consider the high-order consensus problems in weighted networks of dynamic agents with

Consider a network of n dynamic agents with dynamics described by (

if all

if there exist equations

Since the graph

Despite the nonzero time-delay

Denote

From (

As communication time-delay

Assume that

Next, assume there exist equations

Therefore,

Now suppose that we have obtained values of

Then, differentiating (

Suppose that

Theorem

We now introduce the following existing results reported in Olfati-Saber and Murray [

Consider a network of first-order integrator agents. Assume that the network

Consider a network of second-order integrator agents. Assume that the network

Now we give the necessary and sufficient condition for the stability of the third-order consensus in multiagent networks with time-delays, with the purpose of showing how Theorem

Consider a network of third-order integrator agents. Assume that the network

For the third-order consensus, we have

Hence, the explicit expression for the tight upper bound on the time-delays is given by

The maximum tolerable communication time-delay is a fundamental performance measure for consensus and hence plays an important role in the design of distributed coordination of multiagent systems. It is shown previously that the communication constraints affect the stability of a high-order consensus process in a rather sophisticated fashion. Our results imply that the optimal (or suboptimal) feedback gains and network topologies can be synthesized such that the general high-order consensus robustness of the dynamic network to the communication time-delays is maximized. Thus, in that sense our results can shed light on the whole distributed cooperative control design.

We stress that the results in this paper characterize the robustness of the distributed algorithms to the communication time-delays existing in the network for the arbitrary high-order consensus. In order to illustrate this point, we consider solving fourth-order consensus problem of the network of four agents, whose communication graph

Undirected communication graph

In the first simulation experiment, we choose the time-delay

Evolutions of the fourth-order consensus with communication time-delays on graph

This paper has presented arbitrary high-order consensus algorithms for information consensus in networks of dynamic agents suffering from communication time-delays. Using frequency domain method, a rigorous and general convergence analysis was given. By taking communication time-delay

This work was supported by the National Natural Science Foundation of China (Grant no. 61203032), the Natural Science Foundation of Hubei Province (Grant no. 2012FFB05007), and the Fundamental Research Funds for the Central Universities (Grant no. 2010-IV-014).