Elsevier

European Journal of Control

Volume 38, November 2017, Pages 52-62
European Journal of Control

Consensus in networks with arbitrary time invariant linear agents

https://doi.org/10.1016/j.ejcon.2017.08.004Get rights and content

Abstract

In this paper, we study the consensus problem from a general control theoretical perspective. For that we identify three entities: the multiagents network that constitutes the control plant, consensus as a control objective, and the consensus algorithm as a feedback controller for the network. Consensus is redefined through the idea of organization (a linear transformation) to define an error vector that resumes the characteristics of the network. With this formulation, we can translate the general consensus problem into a stability problem and, from there, use classical Control Theory to analyze the case of agents with arbitrary linear time invariant dynamics (and not only integrator dynamics) and Laplacian algorithms. The paper is complemented with numeric examples to illustrate the proposed analysis methodology.

Introduction

The topic of consensus in multi agents systems had gain much attention in the control society during the last decade. The analogy of a swarm of birds is a useful way to explain the main characteristic of the problem: a group of similar systems (or agents) agree to coordinate some important variables through a given information exchange strategy (or algorithm). The publication of books like [14], [18], [19], [28] shows that the topic has already reached an advanced state. However, it remains a popular area of research as shown in the review papers [5], [15] where more than three hundred references are quoted. Most of the work in the area is based on Graph Theoretical approaches to the problem and single or double integrators dynamics. Examples of this are the already quoted publication, and an increasing number of papers such as [1], [9], [10], [11], [17].

Consensus can be understood as a control objective in the same way as stability or robustness in classical control. That is, the definition of consensus is independent of the agent’s dynamics or the methodology that the agents follow to reach this objective. It is however not an exception in the field, e.g. [[8], [15], [19], etc.], to find definitions not only in terms of the output signals but also in terms of specific dynamics (usually integrators) and specific consensus algorithms (usually Laplacian algorithms). That is, not as a control objective used for synthesis of controllers, but as a property of particular control plants with particular controllers. Although some publications, e.g. [12], [13], [21], [22], [23], [26], [27], [29], extend the graph theoretical approaches to systems with more general linear dynamics, the particular cases with which these deal, makes it difficult to extend the results to more general cases.

Furthermore, consensus can be intuitively compared with the equilibrium point of a system that resumes the characteristics of the whole network. However, explicitly reducing consensus to a stability problem, is not typically addressed in the existing works. Nevertheless, in some recent papers, e.g. [3], [20], [24], [25], [29], consensus is studied as the stability of a differences vector between the outputs of one of the agents and the rest of them.

In this paper, this idea is further exploited to formally translate the consensus problem into a classical stability one through the introduction of an analysis tool that we named organization. The idea was partially introduced by the author in a conference publication [16], but here it is enriched to obtain analytical conditions to verify if agents with arbitrary linear time-invariant dynamics can reach consensus over all of its outputs. This is done by extending the notion of weighted graphs to include matrix weights, and by identifying three different entities: A multiagents control plant, consensus as a control objective, and a consensus algorithm as a distributed feedback controller. From here, the problem can be studied by means of standard control theory allowing to drop restrictive assumptions on the dynamics of the agents.

After this introduction, Section 2 presents a summary of Graph Theoretical concepts that are needed to characterize the consensus problem. The following section formally defines the problem, while Section 4 analyzes consensus in three different cases. First the most studied case of integrator systems with coupled dynamics, then with arbitrary linear dynamics, and finally for networks where all agents have identical dynamical behavior. The paper is then complemented with numeric examples that show some of the main characteristics of the proposed approach.

The identity and zero matrices are respectively denoted I and 0. A matrix composed by N identity matrices stacked in a column is denoted 1=col{I}i=1N. If necessary, the dimensions of these matrices will be denoted as an index. For example, Iq is the identity matrix in Rq×q and 0m × n the zero matrix in Rm×n.

Section snippets

Graph theory

Most of the consensus works use intensely graph theoretical methods for description and analysis of networks. In this section, basic notions of the subject are presented based on the quoted works and specialized books as [4], [6], [7].

An undirected graph is a tuple G=(V,E), where V={1,2,N} is a set of N nodes or vertices, and E{(i,j)V×V} is a set of edges.

We interpret that the edge denoted (i,j)E is the same as the edge (j,i)E. This is a slightly abuse of notation as we represent an

Multiagents systems

Even though consensus based control is formulated for Multi Agents Systems, it is not easy to find a general description of such a system in the related works. Typically, the plant over which control is performed is considered to be a set of simple integrators in continuous time. This is a very particular case and therefore it can be difficult to generalized it to other relevant configurations.

In a realistic scenario in a control theoretical framework, the different components of the network

Integrator networks

In the standard consensus setup, it is typically considered that A=0 and CB=I. Accepting CB ≠ I leads to a slightly more general case which we will refer as integrator networks (IN). In this case, considering the described feedback law and (5), it is immediate that the dynamics of the error depend only on the characteristics of the consensus algorithm in the following way: e˙=TCBHeThis simplifies the consensus problem greatly as it can be studied by simply analyzing the eigenvalues of matrix G

Examples

Example 1 Coupled Integrators

Consider N=4 integrator systems with, ∀i ∈ {1, 2, 3, 4}, CiBi=[1.00.50.21.0].That is, coupled integrator systems over two dimensions. The network can be studied through any organization to obtain the same conclusions. For example, considering those described by the directed trees of Fig. 2 (a) and (b); and the corresponding matrices: Ta=[I0I00II000II],Tb=[II000II000II].

The fully connected algorithm derived from the graph in Fig. 2 (c) with the corresponding matrix, L1=L^(G1)=[3IIIII3III

Conclusion

In this paper, we have presented a novel characterization of the consensus problem into a classical stability problem. For that we have identified three separate entities: the control plant as a multiagents system, the control objective as consensus defined through an organization transformation, and a feedback controller that represents the consensus algorithm. These definitions make possible to treat the consensus problem with standard control theoretical tools and allow to generalize the

Acknowledgment

This work is partially funded by the Chilean National Agency of Technology and Scientific Research (CONICYT), grant No. 72130056.

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