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This study addresses one of the most essential distributed control problems in multiagent systems, called the average consensus issue, using a new event-triggered sampling control perspective. Although the continuous-time sampling for average consensus has provided good results currently, a systematic investigation into the continuous-time agent dynamics with sampled-data control inputs under an event-triggered mechanism is critically lacking. The problem considered in this paper can be formulated into an average consensus problem of hybrid systems. The method considers three types of control schemes, among which periodic sampling is integrant. The first scheme is a classical sampling controller reinvestigated through a lemma. The second scheme realizes aperiodic control update as well as periodic communication, while the third scheme achieves both aspects aperiodically. Corresponding sufficient conditions of the aforementioned three schemes are derived such that the asymptotic stability of systems is ensured by using algebraic graph theory, matrix analysis, and Lyapunov theory. The constraints for the allowed sampling period, event parameter, and maximum eigenvalue of graph Laplacian are explicitly derived. Moreover, the potential Zeno behavior of agents due to the sampling control theory is avoided. Thus, a digitally implementable technique is provided. Finally, some numerical examples are provided to verify the effectiveness of the proposed theoretical analysis.

Over the past decade, the distributed coordination problems of multiagent systems have attracted immense attention of researchers due to their potential scientific significance and broad engineering application prospects. For instance, multiagent systems will help reveal the working mechanism of brain neurons [

However, in most previous studies, continuous-time sampling, communication, and control update are often assumed for each agent in a distributed consensus protocol. However, continuous sampling and communication are an ideal assumption, so it is more realistic to sample data periodically or aperiodically and exchange information intermittently. To address this contradiction between theory and practice, this study aims to reanalyze one of the most essential problems for multiagent systems, called the average consensus issue, which has core applications in sensor fusion and filters, using a new event-triggered sampling control perspective. Furthermore, consensus theory is the essential principle underlying the collaborative behaviors of bird flocks, fish schools, group of dancers, and any animals or human beings performing similar collective actions. For the sake of simplification of the problem in order to highlight the main concepts of our method, single integrators are chosen as the dynamics of each agent. Nevertheless, the obtained results based on single-order multiagent systems can be potentially extended to more complex agent dynamics. However, the introduction of a communication topology in the system increases the complexity of the control system; therefore, additional constraints need to be further considered with respect to multiagent systems, including constraints of communication delays [

Currently, diverse research exists on periodic event-triggered control or intermittent communication strategies. A continuous sampling method was used to solve the distributed consensus tracking problem for networked Lur’e systems based on an event-triggered mechanism [

Based on the observation of the aforementioned studies, most existing work on the average consensus problem is still limited to continuous sampling, communication, and control update; a systematic investigation of an event-triggered sampling control mechanism for the average consensus problem is critically lacking. When sampling control and event-triggered control are well combined, two fundamental problems associated with the closed-loop systems are (i) how to determine the constraint of the sampling period and (ii) how to synthesize a sampled-data controller and event conditions to guarantee the global stability of multiagent systems without loss of asymptotic stability. The difficulties related to the second problem lie in ensuring that the continuous-time Lyapunov function is negative definite with the discrete control input and in deriving the allowed sampling period, the appropriate parameters of event conditions, and the gains of the controller in order to determine an optimal balance between systematical performance and communication cost.

Based on the above discussion, this paper aims to investigate the average consensus problem for single-order multiagent systems under a fixed topology with periodic or aperiodic communication with respect to an intermittent control update mechanism. The main contributions of this study can be summarized as follows: (1) A consensus problem in demonstrational Lemma

The rest of this paper is organized as follows. In Section

For a vector

We model a group of agents and their interaction relation as a time-invariant undirected graph

Let

Consider a multiagent system having single-integrator agent dynamics, which is the most essential model in nature and engineering. Its state can denote an opinion, the solution of a linear equation, animal population, etc. Based on Newtonian mechanics, the state can be position and speed. All the agents are labeled 1, …,

The control objective in the problem is multivariate, and some of them even cannot be quantitative. Nevertheless, it is most critical to design a distributed event-triggered sampling controller in order to satisfy the following limit:

In addition, our solutions must reduce as much as possible the number of communications and the frequency of control updates, high values of which are detrimental to mechanical components and the limited energy of microembedded systems in most cases. In addition, an event detector is configured at each agent to determine how to use the sampled data and control the communication unit.

The sampling frequency of each agent satisfies the Nyquist–Shannon sampling theorem.

In general, Assumption

As a basic preparation to formally present an event-triggered sampling control in the next section, we first provide the readers with a consensus result by adopting a pure sampling control. Thereafter, we will gradually lead the readers deep into the design of a diverse distributed sampled-data controller combined with an event-triggered mechanism, so as to thoroughly improve the control performance step by step.

Consider the continuous-time first-order multiagent system (

The dynamics of follower

Integrating the above formula from

Denote the state error for each agent by

We construct the following Lyapunov function candidate:

Taking the time derivative of

Then,

In this case, we solve the consensus problem by using the standard sampled-data control scheme. The closed-loop multiagent system is proven to converge asymptotically fast. Meanwhile, we derive the allowed sampling period

In this section, we consider two types of event-triggered mechanisms. The first section mainly aims to reduce the continuous communication into periodic information interaction and reduce the control update frequency. By contrast, the second section attempts to drastically decrease the communication cost and control update simultaneously.

The most important feature of the technique in this section is that the communication is periodic, whereas the control update is aperiodic and piecewise constant. We define the sum of the relative state errors for each agent as one part of event conditions:

Herein, the event condition of agent

Based on the consensus literature, let the periodic state tracking error be defined as follows:

Equation (

Furthermore, the quadratic form is given by

Then, we have

With the above-distributed event condition (

In the periodic communication case, each pair of neighboring agents exchanges information with each other at each synchronized sampling time instant, while checking the respective event conditions. When the event condition of an agent is violated, it can be said that its event is triggered. Then, this triggered agent transmits its state to the neighbors, and the state is updated in its controller and its measurement error

Consider the multiagent system (

Under the above event-triggered control framework, the dynamical system for agent

Because

We choose the Lyapunov function candidate as

Differentiating (

Now, consider the time evolution of the function

Since

By using the inequality

By invoking inequality (

To obtain the sufficient condition that guarantees the Lyapunov function is negative definite, force

So far,

From Theorem

This can potentially relax the constraint.

In this case, the frequency of the control input update has been decreased dramatically. However, because the communication in each sampling period is required for the event detection, when this value is sufficiently small, the communication cost is still considerably high.

In this section, we consider another strategy, namely, aperiodic communication, to check the event condition, rather than the previous periodic information exchange between neighboring agents. The underlying idea is to maintain the structure of the distributed controller, but modify only the sum of relative state errors in the event condition. To this end, variable

Consider the multiagent system (

The proof follows the same line as controller (

Then, the quadratic form is obtained as follows:

Using the inequality

Finally,

We choose the same Lyapunov function candidate as in the proof of Theorem

Combining inequality (

To make the entire formula (

Hereto,

In the scheme of Section

The core difference between periodic and aperiodic event-triggered control strategies lies in the fact that the former needs communication of an agent with its neighbors at each sampling time instant in order to judge whether an event condition is violated. Nevertheless, the latter overcomes this shortcoming and realizes communication only at an event-triggered time instant. In general, the aperiodic event-triggered control strategy allows longer communication pause time, which enables other agents to use the saved signal channels; this considerably decreases the communication cost and energy consumption. However, in both strategies, sampling of an agent’s own states is performed in a fixed period. Moreover, the control updates of the two strategies happen at a time-triggered instant, which yields a piecewise constant signal. This class of control signals contributes to the reduction of mechanical wear and energy consumption.

In practice, formation reconfiguration might occur, changing the topology pattern. Therefore, we will consider multiagent systems under a switching topology in this section. First, the following assumption is adopted.

The communication topology

In the case of switching topologies, the controllers and event conditions can be designed in the similar method as those in Section

Then, the following theorem can be given.

Consider the multiagent system (

Following the same line of Theorems

To verify the theoretical results obtained in this paper, three contrastive computer simulations are provided to demonstrate the effectiveness of distributed periodic event-triggered controllers in solving the consensus problem while reducing the communication cost and control update frequency. Consider a scenario where six agents intend to make a rendezvous. Figure

Communication topology.

Based on the communication topology, the adjacency matrix

Thus, the Laplacian matrix is given by

By calculation,

Following Lemma

Evolution of each agent.

Sampling and communication time instant of each agent.

Control input of each agent.

This case adopts controller (

Evolution of each agent.

Event-triggered time instant of each agent.

Control input of each agent.

This case adopts controller (

Evolution of each agent.

Event-triggered time instant of each agent.

Control input of each agent.

In addition, some simulation data are reported in Table

Comparison of Theorems

Case | 1 | 2 |
---|---|---|

0.005 | 0.005 | |

0.01 | 0.01 | |

Events of agent 1 | 6 | 6 |

Events of agent 2 | 8 | 5 |

Events of agent 3 | 10 | 4 |

Events of agent 4 | 10 | 6 |

Events of agent 5 | 8 | 7 |

Events of agent 6 | 9 | 5 |

Total number of events triggered | 51 | 33 |

The results for the switching topologies are analog to fixed graph; those simulations are omitted here due to space limitation.

In this study, we proposed an event-triggered sampling control method to drive multiagent systems with fixed and switching topologies contrastive in the sense of average consensus. This method combines the benefits offered by both periodic sampled-data control and event-triggered control. In particular, unlike most existing continuous-time event-triggered control schemes, the underlying idea in this method is to adopt an event-triggering condition that is verified periodically or aperiodically. While periodic and aperiodic communication occurs, control update occurs aperiodically. Corresponding sufficient conditions are derived by ensuring that the continuous-time derivative of the Lyapunov function is negative definite by choosing an appropriate event parameter and sampling period based on graph Laplacian. Moreover, the Zeno behavior for each agent due to the characteristic of periodical sampling is avoided. Our ongoing work is devoted to extending the present control framework to directed graphs and situations with leaders. Other interesting topics for further exploration are proving that the current periodic event-triggered controller will not degenerate to the pure sampling controller and relaxing the fixed sampling period [

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.