Amir Amini, Student Member, IEEE, Amir Asif, Senior Member, IEEE, and Arash Mohammadi, Senior Member, IEEE

Abstract—The paper proposes a novel approach for formationcontainment control based on a dynamic event-triggering mechanism for multi-agent systems. The leader-leader and follower-follower communications are reduced by utilizing the distributed dynamic event-triggered framework. We consider two separate sets of design parameters: one set comprising control and dynamic event-triggering parameters for the leaders and a second set similar to the first one with different values for the followers. The proposed algorithm includes two novel stages of codesign optimization to simultaneously compute the two sets of parameters. The design optimizations are convex and use the weighted sum approach to enable a structured trade-off between the formation-containment convergence rate and associated communications. Simulations based on non-holonomic mobile robot multi-agent systems quantify the effectiveness of the proposed approach.

I. Introduction

COOPERATIVE behaviours have attracted considerable attention in a variety of multi-agent system (MAS)applications, including leaderless consensus [1], leaderfollowing consensus [2], [3], containment control [4]–[7], and formation control [8], [9]. Recently, the formationcontainment control (FCC) framework, which can be regarded as the combined problem of formation and containment for multi-agent systems, has arisen in several engineering applications [10]–[20]. In FCC, the leaders converge to a desired geometric formation. Simultaneously, the followers merge within the convex hull spanned by the leaders. As compared to solitary containment [4]–[7] and solitary formation [8], [9], FCC is more complex and a topic of increasing interest in the control and signal processing community. A related application for FCC is the mixed containment-sensing problem [21] where the objective is to have a group of mobile agents (followers) cover and provide surveillance sequentially from one region of interest to another. In this application, the leaders steer the followers from one operational region (formation) to another and coordinate the sensing task for the followers. This paper considers the FCC problem for general linear multi-agent systems using a comprehensive event-triggering scheme,usually known as the dynamic event-triggering mechanism.

Formation-containment has been studied for agents with different dynamics, including second-order linear agents [10],[11], general linear agents [12]–[14], heterogeneous agents[15], [16], Euler–Lagrange systems [17], [18], and a class of nonlinear agents [19]. All of these implementations impose the strict condition of real-time data transmissions between the agents. To preserve the limited energy allocated to each agent,event-triggered mechanisms [22]–[28] that reduce communications are of great interest in FCC applications.

The implementation of an event-triggered scheme often requires a design step for computing parameters associated with the control protocol and event-triggering scheme. In many event-triggered implementations used in cooperative control of networked systems, control gains are either assumed as a priori information [22]–[24] or designed as a separate step based on the Hurwitz stability of the closed-loop systems [25], [26]. In such emulation-based approaches, the design of event-triggering thresholds is based on a preselected value for the control gain. The operational regions obtained for the event-triggered parameters are, therefore,conditioned on the control gains and may not be the best choices. Alternatively, the control gain and event-triggering parameters are designed simultaneously through a unified optimization framework. In this co-design approach, all parameters are computed together based on a predefined objective, such as H∞optimization [27] or inter-event interval maximization [28].

As one of the most advanced event-triggered schemes,dynamic event-triggered mechanism (DEM) have recently been proposed in [29]–[31]. In DEM, an internal dynamic variable is included as an additional threshold to the eventtriggering parameters. One interesting feature of the DEMs is that their inter-event interval can be longer than the so-called static event-triggered schemes. At the same time, the desired cooperative objectives (such as formation and containment)can still be reached using DEM without introducing steadystate errors. This is in contrast to some other implementations[32] where the event-triggered scheme reduces the number of transmissions at the expense of a bounded error for the desired cooperative behaviour.

Motivated by the aforementioned limitations in the existing FCC approaches, the paper proposes a formation-containment control approach using a dynamic event-triggered mechanism(FCC/DEME) that offers optimality for design parameters,namely the control gains and event-triggering parameters. The main features of the proposed FCC/DEME are listed below:

1) To the best of our knowledge, FCC/DEME is the first implementation for formation-containment that utilizes the dynamic event-triggered mechanism. This leads to considerable energy and communication savings for the multi-agent systems.

2) Two different sets of control and dynamic eventtriggering parameters are introduced for: i) formation of the leaders; and ii) containment of the followers. To design these parameters, FCC/DEME utilizes two convex optimizations based on enabling a trade-off between the rate of convergence for formation-containment and the frequency of the events.

3) The design approaches [30], [31], derive some bounded regions for the DEM design parameters. It should be noted that even when the regions for design parameters are known,selecting the operating values that efficiently save transmissions is still difficult and requires some trial and error.Instead, in FCC/DEME the co-design optimization computes the exact values of the design parameters based on one proposed objective function.

Perhaps, the closest work to FCC/DEME is [20], where an event-triggered formation-containment implementation is proposed. Unlike [20], the DEM used in this paper is more general and adds additional degrees of freedom. As another difference, FCC/DEME (unlike [20]) is based on an optimization framework to develop a structured trade-off between the formation-containment convergence rate and frequency of the transmissions.

The remaining paper is organized as follows. Section II introduces notation and preliminary concepts. Section III formulates the formation-containment problem. Section IV develops two unified optimizations (one for the leaders and one for the followers) for parameter design. Simulation examples are included in Section V. Finally, Section VI concludes the paper.

II. Preliminaries