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Considering the constrained communication data rate and compute capability that commonly exists in multiagent systems, this paper modifies a current consensus strategy by introducing a kind of dynamic quantizer in both state feedback and control input and updates dynamic quantizers by employing event-triggered strategies, thus forming a new quantitative consensus strategy. The numerical simulation example is built for state quantization and the results show the consistency with expectation.

Control theories for a single agent have been studied over a century; many strategies have been developed maturely, for example, robust control [

Fortunately, Ceragioli, Persis and Frasca in problem of quantization used in single agent has been studied for a long time, and many kinds of quantizers have been designed and well used to solve problems. In [

What is more, the convergence time of multiagent systems with quantizers is always finite. Motivated by this and the successful application of quantizers in single agent, quantization consensus problems of multiagent systems attract more and more attention recently. In [

The rest of this paper is organized as follows: Section

An undirected graph

The Laplacian matrix

The variable quantized is denoted by

Note that there exists a dead region in some neighbourhood of the origin, where the quantization value is zero. This brings something to a quantization system, for instance, the hysteresis in real-time control system. But this is not what we want.

Here, we introduce the approach that is used in [

Thus the range of the quantizer is

The quantizers (

Consider a system with

The communication topology of the considered multiagent system is presented by an undirected graph

Introducing the agreement control laws in [

To state quantization, the closed-loop equation of the nominal system is

One of the purposes of this paper is to give a practical strategy for updating

States quantization: the control law is

Input quantization: the control law is

Meanwhile, the block diagrams studied by this paper are depicted in Figure

Block diagrams of the problems this paper studies.

State quantization case

Input quantization case

Define two designations so that we can use them in the following sections:

Obviously,

In this section, consensus strategies for multiagent systems with state quantization and input quantization are demonstrated, respectively.

Rewrite the control law as

And it is easy to show that the average of the states remains constant all the time.

For system (

Note that

With arbitrary positive integer

This choice is motivated by the analysis of the subdivision of the dead region above. Thus the quantization distance would tend towards origin.

Consider a multiagent system with the state equation (

It is obvious that

Consider

Instead of

Since

This completes the proof.

Assume that

(i) Search the upper bound: set

(ii) Search the lower bound: once condition

From the above two steps, we can pick a

(iii) Set

Since

For input quantization case, the control law of the closed-loop system is

In this case, we also choose the Lyapunov function as

This implies that if positive integer satisfies

Note that only quantized measurements of the inputs are available. The event should be

Consider a multiagent system with the state equation (

It is obvious that

Consider

Instead of

Since

This completes the proof.

Assume that

(i) Search the upper bound: set

(ii) Search the lower bound: once condition

From the above two steps, we can pick a

(iii) Set

Since

From the deductions of event (

The results of this paper are illustrated by the following example, in which we take state quantization case into consideration.

Consider a first-order system with four agents whose Laplacian matrix is given by

Think of the state quantization case. Four agents start from a random position with control law (

Figure

Four agents evolve under (

For

For

Four agents evolve under (

For

For

Four agents of system (

Note that in the two cases discussed above, the quantization interval

Figures

Comparing Figure

This paper investigates the quantized consensus problem of a first-order multiagent system. Both the state quantization and the input quantization are considered. The convergence steps for both cases are given. A simulation result supports the results of this paper perfectly. Future work may be focused on the application of some other effective quantizers in multiagent systems and the solution of the quantized consensus problems of multiagent systems by distributed strategies. Furthermore, systems with directed topology are more practical. So the application of the proposed strategy in these systems is more interesting but more challenging.

The authors declare that there is no conflict of interests regarding the publication of this paper.