Event-triggered consensus control of disturbed multi-agent systems using output feedback

doi:10.1016/j.isatra.2019.02.004

ISA Transactions

Volume 91, August 2019, Pages 166-173

https://doi.org/10.1016/j.isatra.2019.02.004 Get rights and content

Highlights

•
A novel observer-based event-triggered protocol is proposed using only output information, in which a predictive factor is used to reduce the trigger frequency.
•
The consensus can be achieved with the expected disturbance attenuation ability, and the Zeno behavior will never occur.
•
The proposed protocol is implemented in an asynchronous way.

Abstract

In this article, the event-triggered consensus control is investigated for general linear multi-agent systems with external disturbances, by using the accessible measurement outputs. In particular, a novel protocol is proposed using the local observed state variables at event-triggered instants, which are respectively derived by adopting an observer to each agent based on output signal. Then it is proved that under the designed event-triggered control protocol consensus can be achieved with the desired disturbance attenuation ability and no Zeno behavior occurs. A numerical simulation is given to demonstrate the effectiveness of the developed control strategy

Introduction

Consensus is an important and basic issue in the cooperative control of multi-agent systems and has attracted extensive attention during the past decades [1], [2], [3], [4], [5], [6], [7], [8], where it is always assumed that all agents can access to continuous measurements and/or control signals. However, in some practical applications, each agent is equipped with a microprocessor subject to certain resource constraints like energy power and communication bandwidth. In order to reduce the resource consumption while maintaining the desired system performance, the event-triggered strategy has been applied to the consensus control for multiple agents [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].

To be specific, the event-triggered consensus problem has been studied for single- and double-integrator multi-agent systems in [9], [10], [11], [12], [13], [14], [15], where the controller updates only when the designed event-triggered conditions are satisfied. Moreover, in [16], [17], [18], [19], [20], [21], the event-triggered protocols have been proposed to realize the consensus control for networks of multiple agents with general linear dynamics. For example, in [16], an event-triggered controller has been developed and the asynchronous event-triggering conditions have been given, under which the multi-agent system can reach consensus on fixed and switching topologies. In [17], both the event- and self-triggered control schemes have been proposed with the distributed, asynchronous, and independent features, and it has been shown that the consensus performance can be reached asymptotically. In [20], two event-triggered protocols have been proposed respectively for the leader-following and the non-leader cases, where each agent decides when to transmit its state signals to its neighbors based on the triggering condition.

In the literatures mentioned above, consensus protocols and event-triggered conditions are designed using the full state information that is very hard or even impossible to be accurately obtained in some occasions [22]. It therefore is essential to develop the output-based method to solve the event-triggered consensus problem. Besides, the external disturbance usually exists in real systems, whose influence to the consensus performance cannot be neglected. Recently, the $H_{\infty}$ consensus control over a finite horizon has been investigated for discrete-time linear multi-agent systems subject to energy-bounded external disturbances [23]. A decentralized estimation-based output feedback protocol, combining with a novel event-based mechanism for deciding when to broadcast messages and update control input, has been developed such that the consensus can be achieved with a desired disturbance attenuation level. However, to our best knowledge, there is very little work on the output-based event-triggered consensus for disturbed continuous-time agents.

In this paper, the $H_{\infty}$ consensus is studied for continuous-time multi-agent systems with general linear dynamics and subject to external disturbances, via the event-triggered output feedback control. Compared with the state-feedback control method, there are more difficulties in designing the event-triggered consensus scheme, caused by the need of estimating real-time states, and then in analyzing the $H_{\infty}$ performance of the augmented closed-loop multi-agent system. The contributions of our research are as follows. First, a novel event-triggered consensus scheme is developed using the local estimated state variables that are generated from the state observers of agents based on output signals. Meanwhile, a predictive factor is used in both the protocol and event-triggering condition designs to reduce the trigger frequency. Second, it is demonstrated that under the designed output-feedback protocol, consensus can be achieved with the prescribed disturbance attenuation ability, and Zeno behavior will never occur in the closed-loop system. Meanwhile, the method to calculate undetermined matrices in the consensus protocol and state observer is also given. Third, the developed dynamic event-triggered scheme is implemented in an asynchronous way, in which each agent triggers independently by using its own and the neighbors’ information. To summarize, the characteristics of the proposed protocol can be summarized as output feedback, asynchronous triggering, and disturbance attenuation.

The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced and the problem formulation is given. In Section 3, an event-triggered protocol is proposed based on the local output information, and the consensus performance analysis is provided. A simulation is presented in Section 4, and Section 5 concludes the paper.

Section snippets

Preliminaries

The communication topology among agents is described by an undirected graph $G = (V, E, A)$ . $V = v_{1}, v_{2}, \dots, v_{n}$ is the set of nodes, where node $v_{i}$ represents the $i$ th agent. $E \subseteq V \times V$ is the set of edges, in which edge $(v_{j}, v_{i})$ indicates the information transmission from agents $j$ to $i$ , and the associated weighting factor is denoted by $a_{i j}$ . Then the neighbor set of node $v_{i}$ is denoted by $N_{i} = {v_{j} \in V | (v_{j}, v_{i}) \in E}$ . For an undirected graph, $(v_{i}, v_{j}) \in E$ if and only if $(v_{j}, v_{i}) \in E$ , and meanwhile $a_{i j} = a_{j i} > 0$ holds. In this case, $A =$

Event-triggered protocol design

Since full states are difficult or even impossible to be detected in many systems, the following event-triggered protocol is developed based on local output information: $\{\begin{matrix} u_{i} (t) = K \sum_{j \in N_{i}} a_{i j} (e^{A (t - t_{k}^{i})} {\tilde{x}}_{i} (t_{k}^{i}) - e^{A (t - t_{k^{'}}^{j})} {\tilde{x}}_{j} (t_{k^{'}}^{j})) \\ {\dot{\tilde{x}}}_{i} (t) = A {\tilde{x}}_{i} (t) + B u_{i} (t) + G ({\tilde{y}}_{i} (t) - y_{i} (t)) \\ {\tilde{y}}_{i} (t) = C {\tilde{x}}_{i} (t) \end{matrix}$ where ${\tilde{x}}_{i} (t) \in R^{m}$ and ${\tilde{y}}_{i} (t) \in R^{p}$ are, respectively, the state and the output of dynamic observer for the ith agent, $K \in R^{m_{1} \times m}$ and $G \in R^{m \times p}$ are gain matrices to be determined, $a_{i j}$ is the interaction strength between agents $i$ and $j$ ,

Simulation

In this section, a simulation example is provided to illustrate the obtained theoretical results. To be specific, a system consisting of four agents is considered, whose dynamic equations are given by (2) with the system matrices $A = [\begin{matrix} 0 & - 2 & 0 \\ 1 & 0 & 0.5 \\ - 0.5 & - 2 & - 1 \end{matrix}], \begin{matrix} \end{matrix} B = [\begin{matrix} 1 & 0 \\ 0 & 2 \\ - 1 & - 2 \end{matrix}], \begin{matrix} \end{matrix} B_{1} = [\begin{matrix} 0 \\ 1 \\ 0.5 \end{matrix}], \begin{matrix} \end{matrix} C = [\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}]$ The communication topology is described by Fig. 1 with $a_{i j} = 0.5$ , and then the Laplacian matrix is $L = [\begin{matrix} 1.5 & - 0.5 & - 0.5 & - 0.5 \\ - 0.5 & 0.5 & 0 & 0 \\ - 0.5 & 0 & 0.5 & 0 \\ - 0.5 & 0 & 0 & 0.5 \end{matrix}]$ In addition, we take $γ = 1$ , $σ = 0.1$ , $α = 10$ . According to Theorem 3, it is

Conclusion

In this paper, we studied the event-triggered consensus problem of linear multi-agent systems subjected to external disturbances. An observer-based control protocol and an asynchronous triggering condition were proposed based on the local output information. It was proved that under the proposed protocol, the multi-agent system could reach consensus with the desired disturbance attenuation ability and exhibit no Zeno behavior. Furthermore, the calculation method of gain matrices in the protocol

Acknowledgments

This work was supported by the NSFC, China (61473015, 61327807, 61520106010, 61503231), and the Fundamental Research Funds for the Central Universities, China .

Conflict of interest

No conflict of interest.

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Some results to design DOF protocols have been proposed for undirected communication graphs considering linear homogeneous agents subjected to external disturbances [8–11], switching networks [8], event or self-triggered mechanisms [9,12], and adaptive schemes [13]. The aforementioned works consider full- [9–12], reduced- [13], and any-order output feedback protocols [8] under the assumption that the communication network has a diagonalizable Laplacian matrix. Therefore, the application of these results for MAS with directed graphs is not straightforward.
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¹: Postal Address: Beihang University, Xueyuan Road No.37, Haidian District, Beijing, China. 100191.

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Research articleEvent-triggered consensus control of disturbed multi-agent systems using output feedback

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Event-triggered protocol design

Simulation

Conclusion

Acknowledgments

Conflict of interest

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Research article
Event-triggered consensus control of disturbed multi-agent systems using output feedback

$H_{\infty}$ Consensus control of multi-agent systems with switching topology: A dynamic output feedback protocol

Consensus problem of high-order multi-agent systems with external disturbances: An $H_{\infty}$ analysis approach