Distributed edge-based event-triggered coordination control for multi-agent systems

doi:10.1016/j.automatica.2021.109797

Automatica

Volume 132, October 2021, 109797

https://doi.org/10.1016/j.automatica.2021.109797 Get rights and content

Abstract

This paper studies edge-based event-triggered strategies for the average consensus problem in multi-agent systems. To this end, a distributed event-triggered algorithm is presented based on edge information rather than neighbor information, where positive minimum inter-event times are guaranteed between all communication links in the network. Moreover, the triggering mechanism is constructed in a fully distributed way which uses only relative state differences based on local neighborhood information without any global graph topology information. This treatment greatly reduces the communication frequency compared with continuous-time average consensus algorithms. We show that the proposed algorithm makes all agents converge to the average of their initial states asymptotically, and the positive minimum inter-event time for each link is controllable by two corresponding agents through their triggering parameters.

Introduction

Multi-agent systems (MASs) consisting of individually controlled agents connected over a wireless network have been widely studied in many fields, including consensus (Jin and Gans, 2017, Li et al., 2013, Olfati-Saber and Murray, 2004, Ren et al., 2007, Zhao, Liu et al., 2020), formations (Dong et al., 2019, Dong et al., 2015, Yu et al., 2020), assignments/coverage (Qie et al., 2019, Rahili et al., 2018, Zhong and Cassandras, 2010), social networks (Wang & Jiang, 2014), and distributed estimations (Khodayi-mehr et al., 2019, Olfati-Saber and Jalalkamali, 2012, Wang et al., 2019). Many efforts have been devoted to the fundamental theory of MASs with the assumption of continuous communication between neighboring agents. In practical applications, agents communicate with limited bandwidth and have limited on-board energy. A great amount of energy is consumed by continuous communication among agents. The construction of event- triggered control strategies to relax the communication burden, therefore, attracts considerable attention (Nowzari et al., 2019, Yi et al., 2019).

Event-triggered strategy for MASs was first proposed in Dimarogonas, Frazzoli, and Johansson (2012) to reduce the number of controller updates. It is shown that there exists at least one agent for which the next inter-event time is strictly positive. In Garcia, Cao, Yu, Antsaklis, and Casbeer (2013), continuous communication is relaxed, and Zeno behavior is avoided. A great deal of effort has been devoted to finding triggering conditions which can both relax continuous communication and admit a positive minimum inter-event time (MIET). Existing methods which solve the above challenge can be classified into two types: the sampled-data event detection (Meng and Chen, 2013, Meng et al., 2017, Nowzari and Cortés, 2016) and the time-dependent threshold (Seyboth, Dimarogonas, & Johansson, 2013). In Meng and Chen (2013), the authors utilized a common sampling period for all agents, but it might be restrictive in distributed networks. In Seyboth et al. (2013), the event-based strategy bounds the measurement error of each agent by a time-dependent threshold, and individual agents no longer rely on continuous or periodic access to their neighbors’ information. However, these two approaches rely on a global clock embedded in each agent and global topology information is still required for the triggering parameter design.

Note that the aforementioned references are all neighbor-based event-triggered control, which in general requires an agent broadcasts its state information to all its neighbors synchronously when an event occurs. Edge-based event-triggered control is an alternative to the neighbor-based one, which has been studied in Cao et al., 2015, Cheng and Li, 2019, Wei et al., 2018 and Xiao, Meng, and Chen (2015). In the edge-based event-triggered control, when an edge event is triggered, two agents activate the mutual data sampling and controller updates rather than broadcasting information to all their neighbors. The communications between an agent and its neighbors are aperiodic and asynchronous. The point-to-point communication feature significantly reduces the communication costs in many practical scenarios.

Motivated by the above discussions, in this paper, we focus on designing a fully distributed edge-based event-triggered mechanism. The communication protocol is designed for each edge in a distributed way based on only local information and relative state information, where the global network topology information and global coordinates are unnecessary. Inspired by the time regularization approach used in Zhao, Hua and Guan (2020), we introduce an inactive time constant for each edge to guarantee a positive MIET. Note that an agent can choose distinct parameters for all its edge triggering functions. Also note that the proposed edge-based communication protocol does not require simultaneous communication to all neighboring agents at event times. Lyapunov-based analysis proves that consensus is achieved under the proposed event-triggered control algorithm.

The main contribution in contrast to existing works on event-triggered control of MASs is a fully distributed solution with the following properties:

P1.
Event detection does not require continuous or periodic communication with neighbors.
P2.
Inter-event times for each edge are lower bounded by a controllable constant, which makes the triggering condition implementable.
P3.
Our design uses only local information without any global network topology information.
P4.
Our design does not use a global time variable $t$ or a common synchronous sampling period $h$ . It does not require clock synchronization among agents.
P5.
Our design does not rely on any global coordinate since it uses only relative edge state information and does not involve any absolute state information.
P6.
Point-to-point communication of the edge-based protocol relaxes simultaneous communications with all neighbors at event times.

Note that very few triggering functions proposed in the literature have all the properties P1–P6. The framework proposed here can be applied to most event-triggered control problems of MASs including agents with general linear dynamics and weighted directed graphs by offering a fully distributed solution.

Section snippets

Graphs

A graph is formally defined as the pair $G = (V, E)$ , where $V = \{v_{1}, v_{2}, \dots, v_{N}\}$ and $E \subseteq {[V]}^{2}$ which is the 2-element subset of $V$ , are referred to as the vertex set and the edge set, respectively. For undirected graphs, edges are unordered pairs of distinct vertices, that is, edge $(v_{i}, v_{j})$ means that agent $v_{i}$ can exchange information with agent $v_{j}$ . We say that agent $v_{i}$ is a neighbor of agent $v_{j}$ . The set of neighbors of agent $v_{i}$ is denoted as $N_{i}$ . The number of neighbors $d_{i}$ of an agent $v_{i}$ is termed the valency of the

Distributed edge-based event-triggered algorithm design

The design schematic is illustrated in Fig. 1, where each agent consists of four main components: a plant, a detector, a zero order holder (ZOH) and a controller. Each edge $(v_{i}, v_{j})$ associates with a sequence of discrete event-triggering times $T_{i j} ≜ {t_{k}^{i j} | k \in Z_{\geq 0}}$ . At edge event instant $t_{k}^{i j}$ , a point-to-point communication link is established between agent $v_{i}$ and agent $v_{j}$ . These two agents sample the relative state difference and update their controllers. The sampled relative state difference of agent

Positive MIET

To make the proposed event-triggered control scheme practical for real applications, we go beyond Zeno free and show that a positive MIET is guaranteed and is controllable for all $(v_{i}, v_{j}) \in E$ . The result is shown in the theorem below.

Theorem 1

Given the MAS (1) and the triggering mechanism (4), the inter-event times for edge $(v_{i}, v_{j})$ are lower-bounded by $t_{k + 1}^{i j} - t_{k}^{i j} \geq min {τ_{i j}, τ_{j i}} > 0$ for any $k \in Z_{\geq 0}$ , where $τ_{i j}$ and $τ_{j i}$ are defined in (7). Moreover, the result holds for all edges $(v_{i}, v_{j}) \in E$ .

Proof

The proof is straightforward.

Example 1

Consider agents with single integrator dynamics under the undirected and connected topology $G_{e} = (V, E)$ , where $V = \{v_{1}, v_{2}, \dots, v_{5}\}$ and $E = \{(v_{1}, v_{2}), (v_{1}, v_{3}), (v_{1}, v_{4}), (v_{2}, v_{3}), (v_{2}, v_{5}), (v_{3}, v_{5}), (v_{4}, v_{5})\} .$ The initial states of the system are chosen randomly from the uniform distribution on the interval $[- 1, 1]$ . We choose $α_{i}$ and $β_{i j}$ randomly from the uniform distribution on the interval $(0, 1)$ for all agents $v_{i}$ and edges $(v_{i}, v_{j}) \in E$ , respectively. The simulation results are shown in Fig. 2. The evolution of the states of all

Conclusions

This paper presents a fully distributed edge-based event-triggered coordination algorithm for MASs with undirected communication networks. A distributed inactive time constant is introduced for each edge to guarantee a positive MIET. In addition, this triggering mechanism uses only local information and relative state information, which is useful in practical applications. The Lyapunov-based consensus analysis proves that all agents achieve the average consensus under the proposed algorithm.

Hongbo Zhao received B.S. degree in the School of Automation Science and Electrical Engineering at Beihang University, Beijing, China, in 2016, where she is currently pursuing the Ph.D. degree in Control Science and Engineering. She was a Research Associate in the Division of Electrical & Computer Engineering at the Louisiana State University (LSU), United States, between January 2020 and December 2020. Her current research interests include multi-agent systems, event-triggered control,

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Xiangyu Meng is an Assistant Professor with the Division of Electrical & Computer Engineering at Louisiana State University, the United States of America. He received his Ph.D. degree in Control Systems from the University of Alberta, Canada, in 2014. He was a Research Associate in the Department of Mechanical Engineering at the University of Hong Kong between June 2007 and July 2007, and between November 2007 and January 2008. He was a Research Award Recipient in the Department of Electrical and Computer Engineering at the University of Alberta between February 2009 and August 2010. Between December 2014 and December 2016, he was with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, as a Research Fellow. He worked as a Postdoctoral Associate in the Division of Systems Engineering at Boston University, the United States, between January 2017 and December 2018. His research interests include event-triggered control, multi-agent systems, and connected and autonomous vehicles.

Sentang Wu received Ph.D. degree in dynamics, ballistics, and aircraft motion control system from Ukraine National Aviation University, in 1992. He was a postdoctoral researcher in aviation and astronautical science at Beihang University, Beijing, China, between 1992 and 1994, where he is currently a Professor in the School of Automation Science and Electrical Engineering. His research interests include the theory and application of nonlinear stochastic systems, computer information processing and control, and aircraft cooperative control and precision guidance.

^☆: The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Dimos V. Dimarogonas under the direction of Editor Christos G. Cassandras.

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Brief paperDistributed edge-based event-triggered coordination control for multi-agent systems☆

Abstract

Introduction

Section snippets

Graphs

Distributed edge-based event-triggered algorithm design

Positive MIET

Example 1

Conclusions

Robotics and Autonomous Systems

Automatica

Automatica

Automatica

Automatica

Automatica

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IEEE Transactions on Automatic Control

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IEEE Transactions on Automatic Control

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IEEE Transactions on Automatic Control

Theory and experiment on formation-containment control of multiple multirotor unmanned aerial vehicle systems

IEEE Transactions on Automation Science and Engineering

Time-varying formation control for unmanned aerial vehicles: Theories and applications

IEEE Transactions on Control Systems Technology

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Algebraic graph theory, Vol. 207

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Brief paper
Distributed edge-based event-triggered coordination control for multi-agent systems☆