Elsevier

Information Sciences

Volume 596, June 2022, Pages 537-550
Information Sciences

Fully distributed event-triggered bipartite formation tracking for multi-agent systems with multiple leaders and matched uncertainties

https://doi.org/10.1016/j.ins.2022.03.033Get rights and content

Abstract

This paper investigates the bipartite time-varying output formation tracking issue for general linear multi-agent systems with multiple leaders and matched uncertainties. The outputs of followers are required to form two antagonistic prescribed time-varying subformations, one of which tracks the convex combination of the outputs of multiple leaders and the other tracks the symmetric convex combination. An independent asynchronous fully distributed event-triggered control protocol is formulated by using relative information between neighboring agents. It is shown that, with the formulated protocol, the bipartite time-varying output formation tracking can be achieved if the feasible formation condition is satisfied. The Zeno behavior is proved to be excluded. Then, we further construct a novel self-triggered control protocol to avoid continuous monitoring of estimate error. Since the protocol incorporates both event-triggered control and adaptive control, it efficiently avoids continuous communication among agents and can be implemented in a fully distributed manner. Moreover, it is noteworthy that in the case where relative information is available while absolute information is not, the protocol is applicable. Finally, the feasibility of the constructed protocols is verified by a numerical example.

Introduction

Over the past two decades, cooperative control for multi-agent systems (MASs), including but not limited to consensus control [1], [2], [3], containment control [4], [5], [6], [7], formation control [8], [9], [10], has received extensive attention from more and more researchers. This is mainly due to its wide practical applications, such as industrial process control, mobile robots formation, fault detection and attitude synchronization [11]. Formation control, as one of the most important topics in cooperative control for MASs, aims to drive a group of agents to form a prescribed formation in the state/output space by constructing an appropriate distributed control protocol using neighboring information. In robotics community, several classical approaches to address the formation issue have been presented, including leader–follower, virtual structure, and behavior-based approaches. However, as described in [12], these three approaches have various disadvantages. For example, it is difficult to establish a quantitative mathematical model and analyze the stability of the whole system theoretically for using the behavior-based approach. Thus, [13] constructed a consensus-based distributed approach and proved that three approaches mentioned above can be unified in the framework of consensus-based distributed approach. By utilizing the consensus-based distributed approach, many interesting studies have been reported on achieving the time-invariant formation for MASs [14], [15], [16], [17]. However, the formation should change dynamically to meet the mission requirements in many practical applications, such as obstacle avoidance and target enclosing. Therefore, the study on time-varying formation is more meaningful. In [18], [19], [20], [21], [22], the time-varying state/output formation analysis and design issue were investigated for MASs with different dynamics, including heterogeneous agents, a class of nonlinear agents, general linear agents and second-order linear agents. In some practical applications, it is not enough to only form a prescribed formation, and the whole formation still needs to track an expected reference trajectory. Thus, the formation tracking issue (FTI) arises, where the states/outputs of followers are required to form the prescribed formation while tracking the expected reference trajectory generated by not less than a leader. In [23], the time-varying output FTI was investigated for heterogeneous general linear MASs with a non-autonomous leader. In [24], necessary and sufficient conditions for general linear MASs with multiple leaders to achieve time-varying state formation tracking, and formation tracking feasibility constraints were given. In [25], the time-varying output FTI for homogeneous and heterogeneous general linear MASs with multiple leaders was investigated, respectively. On top of that, there are a few results on formation tracking [26], [27], [28], [29]. It should be pointed out that most of the works mentioned above only focus on cooperative interaction between neighboring agents. In fact, cooperative interaction and antagonistic interaction often coexist in many real-world scenarios, such as opposing teams in sports. In [30], the bipartite time-varying state FTI was investigated for general linear MASs with a non-autonomous leader. It was shown that structurally balanced communication network was the prerequisite to achieve bipartite control. The bipartite time-varying output FTI for heterogeneous general linear MASs with multiple leaders was studied in [31].

All of the implementations in [23], [24], [25], [26], [27], [28], [29], [30], [31] impose the strict condition of real-time information transmissions between neighboring agents. However, since each agent is usually bandwidth-constrained and energy-constrained, these implementations might become infeasible or impractical in many applications. To deal with the issue, it is of great interest to add the event-triggering mechanism that reduces communication [32], [33], [34], [35], [36], [45]. In [37], a distributed event-triggered control protocol was constructed by using the absolute information about agents. Considering the absolute information about agents cannot be precisely measured, while relative information between neighboring agents can in some applications, such as deep-space exploration, a distributed event-triggered control protocol using only relative information between neighboring agents was formulated in [38]. Note that in most of the aforementioned works, the design of the control protocols depends on global network information. However, this requirement is very difficult to meet for MASs with a large number of agents. To overcome the limitation, fully distributed event-triggered control protocols were constructed in [39], [40], [41]. It should be pointed out that [39], [40], [41] focused on the MASs with accurate dynamics models. Nevertheless, due to the widely existing uncertainties or disturbances, accurate dynamics models are sometimes not available. The MASs with matched uncertainties were further investigated in [42].

Motivated by the aforementioned works, this paper constructs an independent asynchronous fully distributed event-triggered control protocol for the bipartite time-varying output FTI of general linear MASs with multiple leaders and matched uncertainties. It is proved that the protocol does not exhibit the Zeno behavior. Then, a novel self-triggered control protocol is further constructed to avoid continuous monitoring of estimate error. Compared with the relevant results, the main contributions of this paper lie in the following four aspects:

  • (1)

    A formation tracking control protocol based on the event-triggering mechanism is constructed. Compared with the existing formation tracking results in [23], [24], [25], [26], [27], [28], [29], [30], [31], continuous communication among agents is efficiently avoided.

  • (2)

    A control protocol using only relative information between neighboring agents is designed. Different from [42], in the case where relative information is available while absolute information is not, the protocol is applicable.

  • (3)

    A fully distributed control protocol independent of any global information is developed. Compared with [38], the protocol can be implemented in a fully distributed manner.

  • (4)

    A novel self-triggered control protocol is constructed. In contrast to [38], the protocol enlarges inter-event interval time.

The rest of this paper is organized as follows. In Section 2, the issue statement is provided. In Section 3 and Section 4, the main results of this paper are presented. In Section 5, the numerical simulation is given. Finally, Section 6 concludes this paper.

Notaion 1

N is the set of non-negative integers. · is the 2-norm of vector or the Frobenius-norm of matrix. P>0 stands for matrix P is positive definite. 1 (0) is the column vector with compatible dimensions, where all entries are equal to 1 (0). I is the identity matrix with compatible dimensions. λmin(P) (λmax(P)) is the minimum (maximum) eigenvalue of matrix P. sgn(·) is the sign function. diag{···} is the diagonal matrix with diagonal elements being “···”.

Denote by a signed directed graph G=(V,E) the communication network among M (M<N) followers and N-M leaders, where V={1,···,N} is the union set of follower set VF={1,···,M} and leader set VL={M+1,···,N}, and EV×V is the edge set. If (i,j)E (agent i can receive information from agent j), then agent j is said to be the in-neighbor of agent i. Each edge Eij=(i,j) of the graph G is assigned with a weight wij such that wii=0,wij0 if (i,j)E and jVF,wij=bj>0 if (i,j)E and jVL, and wij=0 otherwise. Denote the Laplacian matrix L=[lij]N×N associated with the graph G by lii=j=1N|wij| and lij=-wij. If (i,j)E(j,i)E, then the graph G is said to be undirected. In the undirected graph G, one has wij=wji. A directed path from Vi1 to Vik is a sequence of ordered edges (Vir,Vir+1),r=1,2,,k-1. Since all leaders can not receive information from other agents, the Laplacian matrix L can be decomposed as L=L1L200, where L1RM×M and L2RM×(N-M).

Definition 1 [43]

The communication network GF among followers is said to be structurally balanced if there exists a bipartition of V1 and V2 satisfying V1V2=VF and V1V2=, such that wij0,i,jVd,d{1,2}and wij0,iVd,jV3-d.

Definition 2 [24]

If the in-neighbor set of a follower contains all leaders, then the follower is called well-informed. If the in-neighbor set of a follower contains no leaders, then the follower is called uninformed.

Section snippets

Issue Statement

Consider the MASs composed of M followers and N-M leaders. The dynamics of the ith follower and the th leader are described byẋi(t)=Axi(t)+Bui(t)+ζi(xi,t),yi(t)=Cxi(t),iVFandẋ(t)=Ax(t),y(t)=Cx(t),VLrespectively, where xi(t)Rn,ui(t)Rp,yi(t)Rq and ζi(xi,t)Rp denote the state, input, output and uncertainty of the ith follower, respectively; x(t)Rn and y(t)Rq denote the state and output of the th leader, respectively; ARn×n,BRn×p and CRq×n are the known system matrices with

Fully Distributed Event-Triggered Control Protocol Design

For each follower, the following fully distributed event-triggered control law is constructed:ui(t)=ρi(t)Γθ̂i(t)-hi(t)+ρi(t)ψiΓθ̂i(t)-hi(t)+σi(t)ρ̇i(t)=κi-ϱiρi(t)+12Γθ̂i(t)-hi(t)2+Γθ̂i(t)-hi(t)where ρi(t) is the time-varying gain with ρi(0)>0; ΓRp×n is the constant feedback matrix; σi(t)Rp is the formation compensation signal; κi and ϱi are the positive constants; θ̂i(t)Rn is the combined state estimate described by θ̂i(t)=eA(t-tki)θi(tki),t[tki,tk+1i),kN with θi(t)=j=1M|wij|xi(t)-sgn

Fully Distributed Self-Triggered Control Protocol Design

Note that during the implement of the event-triggered control protocol constructed in the previous section, each follower must continuously monitor the estimate error. However, such monitoring requires the follower to continuously monitor its in-neighbors. To overcome the limitation, a novel self-triggered control protocol is further constructed. Different from the event-triggered control protocol, the next triggering instant tk+1i in the self-triggered control protocol is determined by the

Numerical Simulation

Consider the MASs composed of 7 followers and 3 leaders. Their dynamics are described by (1), (2) withA=01200000-20010000,B=00100001,C=10000010,ζi=0.01isin(t+xi)0.01icos(t+xi).

The communication network among MASs is depicted as Fig. 1, where follower {1,2,3,4} are assigned to V1 and follower {5,6,7} are assigned to V2. The outputs of follower {1,2,3,4} are required to form a rotating square formation and track the centre of the outputs of leader {8,9,10}, while the outputs of follower {5,6,7

Conclusion

Considering cooperative interaction and antagonistic interaction between neighboring agents may exist simultaneously, this paper investigates the bipartite time-varying output FTI of general linear MASs with multiple leaders and matched uncertainties. An independent asynchronous fully distributed event-triggered control protocol has been constructed, and the Zeno behavior has been excluded. Note that the event-triggered control protocol requires continuous monitoring of estimate error. To

CRediT authorship contribution statement

Weihua Li: Conceptualization, Methodology, Software, Validation, Writing - original draft. Huaguang Zhang: Validation, Software. Yuliang Cai: Writing - review & editing, Supervision, Validation. Yingchun Wang: Writing - review & editing, Supervision, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by National Key Research and Development Program of China under grant 2018YFA0702200, and National Natural Science Foundation of China (61627809, 62173080), and Liaoning Revitalization Talents Program (XLYC1801005).

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