New conditions for synchronization in dynamical communication networks

Abstract

This paper investigates synchronization in a typical multi-agent system in which the communication network changes according to the system state. Through building new relationships between a matrix and its associated graph and estimating the diameter of the communication network, we prove that synchronization can be achieved if the speed of agents is bounded by $O\left(n^{-\beta }\right)$, where $n$ is the number of agents and $\beta$ is bounded by a constant independent of $n$, which is much better than the existing bound $O\left(n^{-n}\right)$. Some simulations are provided to illustrate the theoretical results.

Introduction

The collective behavior of networks of dynamical agents has greatly interested scientists and researchers not only from biology and ethology, but also from physics, mathematics, computer science, and control theory. This interest has inspired a huge amount of literature [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], in which an intrinsic theme underlying these works is that complex collective behavior could arise from simple interactions among agents with only limited information.

Generally, the dynamics of agents will be influenced by the topologies of the communication networks. In many real-world systems, the dynamics of agents would also influence the topologies of networks in turn. A well-known example of such a model is the so-called Vicsek model [1], in which $n$ autonomous agents move in a plane with the same speed $v$ but with different headings. The heading of each agent updates according to the average direction of its neighbors, and, for a given agent, its neighbors are agents surrounding it with a distance less than a fixed interaction radius $r$. In this model, the communication networks are determined by the positions of agents, and thus they are state dependent and dynamic. Vicsek et al. used this model to investigate the emergence of self-ordered motion, and showed through computer simulations that the agents can coordinate with each other to move in the same direction eventually [1].

The theoretical analysis of the Vicsek model, however, is much more difficult due to the strong coupling between the states of the agents and the corresponding dynamical communication networks. In [2], Jadbabaie et al. initiated a theoretical study for the synchronization of a linearized Vicsek model, and pointed out that the system will synchronize if the associated networks are infinitely often jointly connected. Savkin simplified the model with the headings chosen from a discrete set, and showed that the agents always break into several groups; agents in the same group have the same heading [3]. Li and Jiang investigated the synchronization of the Vicsek model, and presented a necessary and sufficient condition under an assumption of connectivity [4].

Moreover, inspired by the Vicsek model, many generalizations and variations have been devoted to similar problems with differences regarding the types of agent dynamics and the properties of the graph. Particularly, Moreau studied a general nonlinear consensus protocol [5]; Ren et al. [6] considered the consensus of agents modeled by high-order integrators, and Seo et al. [7] and Li et al. [8] presented high-order linear consensus protocols using a dynamic compensator; Olfati-Saber [9] and Su et al. [10] investigated the synchronization of agents with cohesion, separation, and alignment rules; Cao et al. [11] used graph-theoretic constructions to investigate directed networks of dynamical agents with time delays or asynchronous events, while Sun et al. [12] employed a linear matrix inequality method to study undirected networks with time-varying delays. Ren and Beard introduced some more relaxable conditions for synchronization, and presented a detailed description in [13].

For the sake of synchronization, most existing results essentially imposed some connected conditions on the evolution of the communication networks [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. However, it is a difficult task to verify when such conditions actually hold. There are of course other methods to guarantee the connectivity by using additional information and/or constraints, such as the periodic boundary condition in [1], [14], the global interactions in [15], the soft control method in [16], and the designed attractive forces in [17].

For the Vicsek model, once given the initial conditions and the system parameters, the dynamical process of the network topology will be driven by the evolution of the agent states. Therefore, what kinds of initial conditions and system parameters can lead to connectivity turns out to be a difficult and challenging issue.

Recently, Tang and Guo studied the model in a random framework, and, under uniformly distributed initial conditions, they proved that for any given speed $v$ and communication radius $r$ the linearized Vicsek model will synchronize with large probability as long as the size of the population is large enough [18]. Liu and Guo obtained the same result for the original Vicsek model in [19]. However, the above results would definitely lose efficiency if the initial states are arbitrary rather than uniformly distributed, and if the number of agents is finite rather than sufficiently large. An intuitive explanation could be that the larger the group is, the more complicated the initial states could be, and the more rigorous the conditions should be for the sake of the efficiency of the worst initial states. In this case, Liu and Guo presented a preliminary theoretical analysis, and developed a sufficient condition for synchronization that the speed $v$ should be sufficiently slow with $v\le \frac{d}{{△}_0}{\left(\frac{cos\bar{\theta }}{n}\right)}^n$, where $n$ is the size of population and $d,{△}_0,\bar{\theta }$ are determined by the initial states [20]. This result implies that the speed $v$ should decrease exponentially as the population increases, which is rather restrictive and tough for large populations.

The aim of this paper is to give a more efficient synchronization condition in a deterministic framework, in which the initial states are arbitrary and the population is finite. In order to obtain a convergence rate, the structural properties of matrices and the connection with the topologies of communication networks are exploited, which provides a perspective on the time needed for the information to spread in a network. Based on the estimation of the diameter of communication networks, the restriction on speed $v$ is improved from $O\left(n^{-n}\right)$ to $O\left(n^{-\beta }\right)$, where $\beta$ is a bounded constant independent of $n$. Here, the synchronization condition is only imposed on the system parameters and the initial states, which would be easy to verify in comparison with the connectivity constraints on the system trajectory.

The rest of the paper is organized as follows. Section 2 presents some preliminaries, the main results are established in Section 3, and some simulations are given in Section 4. Finally, Section 5 draws a conclusion.