Weihua Li, Huaguang Zhang,, Yu Zhou, and Yingchun Wang,

Abstract—In this study, the bipartite time-varying output formation tracking problem for heterogeneous multi-agent systems (MASs) with multiple leaders and switching communication networks is considered. Note that the switching communication networks may be connected or disconnected. To address this problem, a novel reduced-dimensional observerbased fully distributed asynchronous dynamic edge-eventtriggered output feedback control protocol is developed, and the Zeno behavior is ruled out. The theoretical analysis gives the admissible switching frequency and switching width under the proposed control protocol. Different from the existing works, the control protocol reduces the dimension of information to be transmitted between neighboring agents. Moreover, since an additional positive internal dynamic variable is introduced into the triggering functions, the control protocol can guarantee a larger inter-event time interval compared with previous results.Finally, a simulation example is given to verify the effectiveness and performance of the theoretical result.

I. INTRODUCTION

OVER the past decade, formation tracking control for multi-agent systems (MASs) has attracted considerable attention from researchers owing to its applications in various fields, such as target enclosing, and oceanic and planetary explorations [1]. The main objective of formation tracking control is to design an appropriate control protocol such that the states/outputs of a group of agents can form a specified formation while tracking a expected reference trajectory.Obviously, consensus problem, consensus tracking problem and formation problem are the special cases of formation tracking problem [2]–[9].

Recently, many results about how to design formation tracking control protocols have been obtained. In [10], a distributed state formation tracking control protocol was given, where the expected formation was specified by constant vectors. However, in some practical application scenarios, such as source seeking and obstacle avoidance, the MASs should change formation in real time to meet the mission requirements and adapt to the external environment.Therefore, the research for time-varying formation tracking problem is meaningful and necessary. Previous results on achieving time-varying formation tracking focused on the MASs with a single leader [11]–[14]. However, the reference trajectory may be composed of multiple targets (considered as leaders) in some cases. This makes this problem relatively more complex and challenging. At present, the time-varying formation tracking problem for MASs with multiple leaders is a relatively unexplored area which requires more effort. There are only a few research results [15]–[19]. It should be pointed out that these control protocols in [11]–[19] are valid only if neighboring agents can communicate continuously. Nevertheless, due to the unreliability of information channels and the restriction on power resources in practice, continuous communication among agents is unrealistic. Thus, it is of vital importance to develop an energy-efficient formation tracking control protocol.

On the other hand, there is an increasing interest in applying event-triggered approach to MASs because of its advantage in avoiding continuous communication among system components. For the MASs using event-triggered control protocol,agents communicate according to the predefined triggering mechanism. In earlier static event-triggered control protocols,the triggering mechanism is fixed [20]. Subsequently, dynamic event-triggered control protocols, where the triggering mechanism was adjusted dynamically, were developed in[21]–[24] by introducing an additional internal dynamic variable. Compared with the static triggering mechanism, the dynamic triggering mechanism could bring a larger interevent time interval due to the addition of positive internal dynamic variable. Note that [20]–[24] required global communication network information. However, global communication network information is not available online in several practical applications. By using the adaptive control technology in [25], fully distributed event-triggered control protocols without any global communication network information were proposed in [26]–[29]. Subsequently, fully distributed asynchronous edge-event-triggered control protocols were constructed in [30]–[33], which further reduces the communication frequency. Moreover, fully distributed dynamic eventtriggered control protocol was also developed in [34].

It is noteworthy that cooperative interaction and antagonistic interaction among agents may exist simultaneously in many circumstances [12]. Besides, the communication network among agents may be unreliable due to link failures or new creations [35]. Meanwhile, the full states of agents may be not available due to the limitation of the sensor [36]. Accordingly, it is more meaningful to study the bipartite formation tracking problem for MASs with switching communication networks using only output-feedback information.

Motivated by aforementioned studies, this paper considers the bipartite time-varying output formation tracking (BTVOFT) problem for heterogeneous MASs with multiple leaders and switching communication networks. The main challenges and difficulties of this paper are summarized as follows:

1) To address the above problem, how to design an accepted control protocol needs to be considered;

2) Considering the limited communication and computation resources in real scenario, how to develop a more energyefficient formation tracking control protocol is a problem to be addressed;

3) For unreliable communication network, how to devise a control protocol with larger switching frequency and switching width is an urgent issue to be solved.

To overcome these challenges and difficulties, a novel reduced-dimensional observer-based (RDO-B) fully distributed asynchronous dynamic edge-event-triggered (FDADEET)output feedback control protocol is proposed, and the Zeno behavior is ruled out.

The contributions of the control protocol lie in the following four aspects:

1) Compared with all the aforementioned studies, even [26],the proposed control protocol reduces the dimension of information to be transmitted between neighboring agents.This is the greatest contribution of this paper;

2) Since the proposed control protocol is designed by combining dynamic event-triggered control protocol with fully distributed asynchronous edge-event-triggered control protocol, it has their advantages at the same time. In contrast to [30]–[32], the control protocol can guarantee a larger interevent time interval. Compared with [22], [34], the control protocol can reduce more communication frequency;

3) A reduced-dimensional observer is designed. In contrast to the full-dimensional observer in [36], the design cost of observer is greatly reduced;

4) In contrast to [5], this paper considers the case that the switching communication networks may be disconnected. The theoretical analysis gives the admissible switching frequency and switching width under the proposed control protocol.Compared with [37], the admissible switching frequency and switching width are larger.

II. PRELIMINARIES AND PROBLEM STATEMENT

A. Graph Theory

B. Problem Statement

Consider heterogeneous MASs described by

Algorithm 1 Construction of Parameters in Fully Distributed Dynamic Edge-Event-Triggered Control Protocol (5) and (6)Step 1: Choose an expected output formation . Check the following feasible formation condition:f(t)t→∞(?BiAi ˉCi fi(t)??Bi ˉCi ˙fi(t))=0ni?qi. (9)lim If (9) is satisfied, then continue. Otherwise, is unfeasible under the control protocol and the algorithm stops.zi(t)zi=?ˉBi(Ai ˉCi fi ?ˉCi ˙fi)f(t)Step 2: The formation compensation input is designed as.Hi, H E1i ˇA11 i +Hi ˇA21i , ˇ?A11+H ˇ?A21 Ai+BiE1i E2i E2i=Ψi ?E1iΠi Γ=P?1 ?CT P ∈Rn×n>0 Step 3: Choose and satisfying and are Hurwitz. and Γ are designed as and, where is the solution of the following linear matrix inequality:P ?A+ ?AT P??CT ?C+In<0. (10)Step 4: Select such that. Choose to be any positive constants.ιij, κij, ιi, κi, ?σij, νij, ˉσij, ?σi?, νi?, ˉσi? mini,j∈VF{ιijκi j, ιiκi}<1/λmax(P), 0< ?σij<1, (1??σi j)νij ?ˉσij<0, 0< ?σi? <1, (1??σi?)νi? ?ˉσi? <0 ρij, ?μij, ˉμij, ρi, ?μi?, ˉμi?, ?μ?, ˉμ?

Remark 7:Note that the Lyapunov function candidate selected in this paper is independent of the communication network, which makes it a common Lyapunov function candidate. Thus, the theoretical analysis gets rid of the limitation of average dwell time, and gives a larger admissible switching frequency and switching width compared with [37].

Remark 8:Because many practical MASs have limited running time, it is meaningful to rule out the Zeno behavior for any finite time. The proof of ruling out the Zeno behavior is partly inspired by [27]. In fact, setting μˉi j=0 can guarantee an explicit lower bound. However, this will pay the cost of increasing the upper bound of the output formation tracking errors.

Remark 9:It should be pointed out that the control protocol proposed in this paper has a limitation, that is the output formation tracking errors are asymptotically convergent to a bounded set. However, this control objective is usually expected to be achieved within a fixed time in practice. As a significant performance metric of the control protocol, the convergence rate should be taken into account. In this regard,the fixed-time techniques in [9], [18], [40] are helpful. At present, how to combine these techniques with the proposed control protocol deserves future study.

IV. NUMERICAL SIMULATION

In this section, a simulation example is given to verify the effectiveness and performance of the theoretical result.Consider heterogeneous MASs (1) with three different communication networks as shown in Fig. 1, where the system matrices are

Fig. 1. Possible communication networks. (a) G 1; (b) G 2; (c) G 3.

V. CONCLUSIONS

Fig. 2. Change of switching signal.

Fig. 3. Output trajectories and output snapshots.

Fig. 4. Output formation tracking errors.

Fig. 5. Time-varying coupling weights.

In this paper, the BTVOFT problem for heterogeneous MASs with multiple leaders and switching communication networks has been considered. To address this problem, a novel RDO-B FDADEET output feedback control protocol has been developed, and the Zeno behavior has been ruled out.Compared with previous results, the proposed control protocol can guarantee a larger inter-event time interval. Moreover, the proposed control protocol reduces the dimension of information to be transmitted between neighboring agents and the design cost of observer. The theoretical analysis gives the admissible switching frequency and switching width under the proposed control protocol. Extending the result to directed communication networks is an interesting topic for future study.

Fig. 6. Event-triggered instants.