Layered event-triggered control for group consensus with both competition and cooperation interconnections

Abstract

In this paper, the problem of group consensus of multi-agent systems that exchange information through two-layer directed communication network is investigated. Both competition and cooperation interconnections are considered, which is more in line with the reality. Based on layered event-triggered control strategy, agents only update control input and broadcast information to their neighborhood in discrete time instants according to the designed triggering conditions. Since the triggering conditions within two layers are different, the whole consensus process is asynchronous. We prove that the agents can achieve group consensus with the proposed control protocol and there is no Zeno behavior by choosing appropriate parameters.

Introduction

In recent years, with extensive applications of multi-agent systems, the problem of coordination control of multi-agent systems has been widely discussed, particularly the consensus issue. This kind of subjects has been studied deeply and there are a great deal of results such as [1], [2], [3], [4], [5], [6]. In the cooperative control of multi-agent systems, each agent needs to communicate with others by receiving and broadcasting information among their neighbors. By this way, all the agents are able to reach the same agreement, consensus is thus achieved. Essentially, the completion of tasks depends on the realization of consensus control for multi-agent systems. For instance, formation control, tracking control and attitude alignment, etc. When the consensus issue of multi-agent systems is investigated, the communication among agents is regarded as a network topology. And consensus control can be implemented under the appropriate protocols and algorithms. By making use of graph theory, matrix theory and Lyapunov stability theory, the convergence of agents in a network can be verified.

However, since the interconnections between agents are more and more complex, and in order to meet different task requirements or adapt to the changes of the environment, it is more general to prompt agents in different subnetworks to reach different goals instead of the same goal by dividing the network of multi-agent systems into multiple subnetworks. This is the idea of group consensus which attracted a good deal of attention in the field of control and automation. Refs. [7], [8], [9], [10], [11] elaborate the analysis of group consensus in detail. What they concerned is that agents in a sub-network share a consistent value, while no agreement between any two sub-networks is required.

There are many kinds of multilayer networks such as leader-following networks, coupled complex networks and multi-agent systems with a bipartite graph. They can be treated as multiplex networks, so does group consensus of multi-agent systems. Solribalta et al. [12] reveal the spectral properties of the Laplacian of multiplex networks, while Gmez et al. [13] focus on diffusion dynamics on multiplex networks. In [14], inner and outer synchronization between two coupled networks is investigated and the output synchronization problem is studied for a heterogeneous network in [15]. Synchronization analysis and network design of multilayer networks are explored by decoupling in [16]. Additionally, Xu et al. [17] address the problem of state estimation for stochastic complex networks which varying coupling is governed by a Markov chain.

With the development of wireless network, it is a challenging task to utilize the limited communication resources to control the network efficiently. Traditionally, updating information periodically is adopted. But it is well known that unnecessary communication in a periodic manner will lead to the waste of energy. To address this problem, the event-triggered strategy arises. In event-triggered control of multi-agent systems, sporadic transmissions are generated that agents need to judge on event condition to determine whether to change and broadcast their information or not. Tabuada [18] proposes the event-triggered real-time scheduling on embedded processors which guarantees performance and relaxes the periodic execution requirements. In lieu of periodic approaches, Dimarogonas et al. [19], [20], [21], [22], [23], [24] employ event-triggered communication scheme where agent sends its local state to the network only when it is necessary, i.e., only when a function of state error exceeds a specific threshold. By this way, the consensus problem is solved. In [25], group consensus is eventually achieved by the centralized and decentralized event-triggered control scheme with undirected communication topology.

In reality, there is not only cooperation among individuals, but also competition. Such phenomenon exists extensively, for example the pursuer–invader problem, competition among several species and cannibalism. They can all be modeled as a structured competitive system. As early as 1986, Diekmann et al. [26] proposed simple mathematical models of cannibalism. Then, many kinds of competitive models concerning on stage structure, maturation delay and harvesting, are presented. And the stability of these models was investigated in [27], [28], [29]. Accordingly, there is a competitive relationship among agents in multi-agent systems. Motivated by these references, we consider both competition and cooperation in the problem of group consensus for multi-agent systems. Since there are both competition and cooperation, the relationship among agents is not cooperative but competitive with each other or partly cooperative with some agents and partly competitive against other agents, see [3], [30] and references therein. To the best of our knowledge, there is no result of group consensus for multi-agent systems with both competition and cooperation interconnections.

Compared to the existing work [19], [20], [21], [22], [23], [24], in this paper, the layered event-triggered controller is adopted to achieve group consensus on account of the structural characteristics of the two-layer multi-agent systems. The main contributions of this paper include: (1) We design a new control protocol involving both competition and cooperation effects for the layered multi-agent systems. This is more consistent with the actual situation that a more comprehensive description of the relationship among agents is provided, while no previous work takes the competitive relationship into account in the problem of group consensus for multi-agent systems. (2) We extend the event-triggered consensus results in [25] from undirected graph to directed one. By designing a layered event-triggered control protocol, a class of layered multi-agent systems that agents are connected by a directed communication network can achieve group consensus. (3) This method has wider application prospects compared with [7], [9], [10], [25] since some assumptions are removed in this paper.

Notations: Let ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times n}$ respectively be sets of real vectors and real matrices. For a vector $x={(x_1,x_2,\dots ,x_n)}^{\mathrm{T}}\in {\mathbb{R}}^n,$ let ‖x‖ denote the 2-norm of x. The matrix inequality $A-B>(\ge )0$ means that $A-B$ is positive (semi-) definite.