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We consider consensus problems of first-order multiagent systems with sampled information and noisy measurements. A distributed stochastic approximation type algorithm is employed to attenuate the measurement noises. We provide conditions under which almost sure strong consensus is guaranteed for fixed and switching directed network topologies. Simulation results are provided to illustrate the theoretical results.

In recent years, distributed coordination control of multiagent systems has received compelling attention from the control community. As a critical issue for coordination control, consensus means that the group of agents reach an agreement on a common value via local communication. Consensus problems are closely related to many different problems that involve interconnection of dynamic systems in various fields of science and engineering, such as synchronization of coupled oscillators, flocking theory, fast consensus in small worlds, rendezvous in space, distributed sensor fusion in sensor networks, and distributed formation control (see [

Vicsek et al. [

In many applications envisioned, due to the application of digital sensors and controllers, the information exchange among the group of agents may only occur at sampling instants. Thus, it is more practical to consider the consensus problems with sampled-data information. Moreover, with sampled-data control, many benefits can be achieved such as flexibility, robustness, and low cost. Some results about consensus problems for multi-agent systems via sampled-data control have been reported [

In this paper, we are interested in the consensus problems with sampled information and noisy measurements. That is, we assume that each agent can only obtain the noisy measurements of its neighbors' states at sampling instants. This paper is partly motivated by [

The following notations will be used throughout this paper. Let

Consider a multi-agent system consisting of

The weighted adjacency matrix of the digraph

Below is an important property of Laplacian

Zero is an eigenvalue of

Consider the following first-order integrator system of

In this paper, we assume that each agent can only obtain noisy measurements of the states of its neighbors at sampling instants. Denote the resulting measurement by agent

The agents are said to reach almost sure strong consensus if there exists a random variable

When there is no noisy measurements, Definition

Our objective is to design a distributed protocol so that the

Assumption (A1′) is weaker than assumption (A1). The above assumptions imply that the impact of the noise can be attenuated as time goes on since

Write

(A2)

Note that (A2) contains that

In this subsection, we consider the case of fixed topology. We begin by studying the noise-free case; that is,

We make the following assumption on the network topology.

It is not easy to analyze the convergence of system (

Assuming (A3), denote

Following the notation in Lemma

Let

Apply protocol (

Take a Lyapunov function

Now we investigate the case of having noisy measurements. We can obtain the following system associated with system (

where

Consider system (

Choose a Lyapunov function

By a similar argument to that in the proof of (

We will make use of the following lemma to finish up our proof.

Lemma 9 (see [

Let

Let

Apply protocol (

From (

At the same time, since

In most existing works concerning consensus problems based on sampled-data control, there is a requirement on the sampling period

In this subsection, we consider the case when the network topology changes dynamically. In order to describe the time-varying topology, we define a switching signal

We introduce the following assumption on the network topology.

Under (A4), there exists an orthogonal matrix

Let

Consider system (

Choose a Lyapunov function

Apply protocol (

Note that

The remaining of the proof is similar to that of Theorem

Note that the condition in Theorem

In this section, two examples are provided to illustrate the theoretical results. In the following two examples, the variance of the i.i.d zero mean Gaussian measurement noises is

Consider a multi-agent system consisting of five agents with the interaction topology shown in Figure

Interaction topology

State trajectories under different sampling periods in the case of noise-free and fixed topologies.

State trajectories under different sampling periods with fixed topology and noisy measurements.

Consider a multi-agent system consisting of five agents. The interaction topology is time varying of switching period

Interaction topologies

State trajectories under different sampling periods with noisy measurements and time-varying topology.

In this paper, a consensus problem for a multi-agent system with sampled information and noisy measurements is investigated. Both the case of fixed topology and time-varying topologies are taken into consideration. For the case of fixed topology, we prove that the agents reach almost sure strong consensus as long as the network topology contains a spanning tree. For the case of time-varying topologies, under the assumption that each interaction topology is balanced and contains a spanning tree, we show that the agents reach almost sure strong consensus. Different from the most existing results concerning sampled-data control of multi-agent systems, it is shown that the convergence conditions are independent of the sampling period, which is due to the introduction of the step size.

This research is supported by Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Science & Technology Research Project (2012GZX0080).