Consensus control for nonlinear multi-agent systems with event-triggered communications

https://doi.org/10.1016/j.amc.2021.126341Get rights and content

Highlights

  • Since proposed methods are suitable for systems with one-sided Lipschitz constants and a wide range of nonlinear systems, it has less conservatism and stronger generality.

  • The paper introduces two types of control schemes, that is, static and dynamic event-triggered control schemes, to achieve the consensus problems for leader-follower systems and leaderless systems with nonlinear dynamics. In particular, the dynamic event-triggered control protocol and the corresponding triggering function, the construction of which is independent of the global information of the MASs.

  • Event-triggered mechanisms are adopted to reduce network bandwidth requirements. The designed event-triggered mechanisms do not cause Zeno behavior, which can be proved in detail in this paper.

Abstract

This paper investigates the consensus problems for a class of nonlinear multi-agent systems (MASs) by distributed event-based control protocols. Firstly, we design the static distributed event-triggered control protocols to achieve leader-follower and leaderless consensus problems, respectively, and the systems exclude Zeno behavior under the proposed control schemes. In order to avoid utilizing the global information of the system networks, that is, the minimum eigenvalues of the correlation matrices. Then, the fully distributed event-based control schemes are designed by introducing the time-varying coupling weights to achieve the leader-follower and leaderless consensus problems, respectively. It has been shown that the Zeno behavior is ruled out in the MASs under the fully distributed event-based control protocols. All static and dynamic event-based control schemes are able to be constructed based on the solvability of linear matrix inequalities (LMIs). Finally, the rationality of the theoretical results is illustrated by the example.

Introduction

In recent years, cooperative control of MASs has gradually become a research hotspot in the control field. As far as we know, cooperative control generally includes the consensus problem [1]-[2], containment control [3], formation control [4] and output regulation problem [5]-[6] and so on. By designing the distributed control scheme for each agent, the state trajectory of each agent finally tends to be consistent, which is called the consensus problem. As one of the most basic cooperative control problems, consensus problem has received extensive attention from scholars at home and abroad. It not only reveals many natural phenomena in reality, but also has a wide range of applications in many fields, such as robots, energy internet [7], sensor networks and DC Microgrids [8].

In many practical systems, in order to save system resources and reduce the demand of system network bandwidth, the sampled-data control schemes [8], [9], [9], [10] and event-triggered mechanisms [11]-[12] were introduced. Compared with the sampled-data control, the event-triggered control [13]-[14] performed less information communication if and only if the preset triggering condition is established. At present, the application of event-triggered control to multi-agent synchronization has achieved remarkable results [15], [16], [17], [18], [27].

In actual systems, it is not easy for each agent to obtain the global information of the system networks, such as the minimum or maximum non-zero eigenvalues of Laplace matrix, the number of agents in the MASs and so on. More and more scholars began to explore adaptive control [19], [20], [21], [22], which could avoid the use of global information. For example, [19] considered the consensus problem for linear systems with matched uncertainties by designing fully distributed adaptive control protocols, where the construction of controller and triggering function was independent of any global information of the system networks. [22] investigated the continuous-time optimization problem based on an adaptive event-triggered algorithm, where the proposed algorithm did not rely on the parameters of cost functions. Note that the system models we mentioned above are all linear, which first inspires the research in this paper.

Due to the influence of the system itself or the outside, the actual system has certain nonlinearity characteristics, and most of the time only some parts with small influence are ignored. Therefore, it is essential to research the nonlinear systems [23], [24], [25], [26], [27], [28]. For instance, the distributed optimal cooperative control of nonlinear MASs with unknown dynamics was studied in [28] by using adaptive dynamic programming technology. [29], [30], [31] investigated the consensus problems for nonlinear systems with the nonlinear terms satisfying Lipschitz condition. Compared with the one-sided Lipschitz condition, Lipschitz condition is more conservative. Thus, some scholars in [32], [33], [34], [36] used the more general one-sided Lipschitz condition instead of Lipschitz condition. For instance, [32] addressed the leaderless consensus control for a class of nonlinear MASs with the one-sided Lipschitz and quadratic inner-boundedness conditions. In [35], the LMI method was utilized to handle the observer design issue for one-sided Lipschitz nonlinear systems. [36] addressed consensus control problem for one-sided Lipschitz nonlinear MASs by employing relative state feedback. However, the event-triggered control was not considered in the work and not adaptive control, where the controller design required to use global information about the system networks, which further inspires the research in this paper.

Inspired by the previous works, this paper investigates the consensus problems for a class of nonlinear MASs with the one-sided Lipschitz and quadratic inner-boundedness conditions by distributed static and dynamic event-triggered control strategies. The main significance of this paper is drawn summarization as follows.

  • 1.

    Since proposed methods are suitable for systems with one-sided Lipschitz constants and a wide range of nonlinear systems, it has less conservatism and stronger generality.

  • 2.

    The paper introduces two types of control schemes, that is, static and dynamic event-triggered control schemes, to achieve the consensus problems for leader-follower systems and leaderless systems with nonlinear dynamics. In particular, the dynamic event-triggered control protocol and the corresponding triggering function, the construction of which is independent of the global information of the MASs.

  • 3.

    Event-triggered mechanisms are adopted to reduce network bandwidth requirements. The designed event-triggered mechanisms do not cause Zeno behavior, which can be proved in detail in this paper.

  • 4.

    The main challenge of this paper is how to design adaptive control protocols to avoid using the global information of the MASs.

The general structure of the paper is divided into the following parts: In Section 2, the system model is established and some basic results are given about graph theory. In Section 3, static distributed event-triggered control protocols are constructed to solve the consensus problems for leader-follower and leaderless systems. It has been shown that Zeno behavior is excluded in the MASs under these control protocols. In Section 4, in order to avoid the use of any global information, fully distributed event-triggered adaptive control protocols are designed to implement the consensus problems of the leader-follower and leaderless systems, respectively. In Section 5, an example is introduced to verify the validity of the theoretical results. Finally, the conclusion of the paper and future research direction are given.

Notations: Rn×m expresses the set of n×mdimensional matrices. A>0Rn×n(A0) represents that A is positive definite (semipositive definite) matrix. λmin(A) expresses the smallest eigenvalue of A. A, A1 mean transpose and inverse matrices of A. denotes the Kronecker product. · denotes the Euclidean norm for vectors or the induced norm for matrices. IN means the identity matrix of dimension N, and 1N is an Ndimensional vector with all elements being 1. x1,x2,,xNRn, col(x1,,xN) represents nN dimensional column vector stacked by x1,,xN. diag{·} expresses the diagonal matrix or diagonal block matrix.

Section snippets

Problem formulation

Consider the nonlinear MASs consisting of N agents. The dynamics of agent i are presented byx˙i(t)=Axi(t)+Bui(t)+f(xi),i=1,,Nwhere xiRp, uiRq are the state, control input for agent i, respectively. f(·):RpRp symbolizes nonlinear function. ARp×p,BRp×q express the system matrices.

The information exchange of agents in the MASs is depicted by an undirected graph G=(v,ɛ,w), where v={1,,N} expresses the set of all agents, ɛv×v represents the set of edge consisting of any pair of agents, w=[wij

Leader-follower consensus control by distributed event-triggered control strategy

Suppose the leader is represented by 1, and its control input u1(t)=0. The followers are denoted by 2,,N. For each follower, we first design the following distributed event-triggered control protocol to achieve the leader-follower consensus of the MASs (1).ui(t)=cKj=1Nwij(x˜i(t)x˜j(t)),fori=2,,Nwhere c>0 is a scalar satisfying cχλmin(L1) and χ>0 appears in (10), K is a gain matrix to be specified later. x˜i(t)=eA(ttki)xi(tki), x˜1(t)=eAtx1(0), tki(t0i=0) is the kth triggering time instant

Dynamic fully distributed event-triggered adaptive control protocols

In order to avoid utilizing any global information of the systems, we design the adaptive control protocols to solve the leader-follower consensus and the leaderless consensus problems in this section, respectively.

Simulation examples

In this section, we present an example [32] to demonstrate the rationality of the main results obtained.

Consider the MASs consisting of six agents, and the communication topology is shown in Fig. 1. The dynamics of agent i are given as[x˙i1x˙i2]=[1111][xi1xi2]+[21]ui+[xi1(xi12+xi22)xi2(xi12+xi22)]where i=1,,6. Through verification, it can be known that the nonlinear function f(·) satisfies Assumptions 1–2 with ϱ=0,σ=99,ς=100.

Select c=3, ψi=20,ωi=0.5. According to Algorithm 1, the matrices K,

Conclusions and future work

In this technical note, we have investigeted the consensus issues for a class of nonlinear MASs by event-based control protocols. Firstly, we designed the static distributed event-triggered control protocols to reach leader-follower and leaderless consensus, respectively. It had been shown the Zeno behavior did not exist under the proposed control schemes. In order to avoid utilizing the global information, the fully distributed event-based adaptive control protocols were constructed by

Acknowledgments

This work was supported by National Key R&D Program of China under grant 2018YFA0702200, and National Natural Science Foundation of China (61627809, 61621004), and Liaoning Revitalization Talents Program (XLYC1801005).

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