On graph theoretic results underlying the analysis of consensus in multi-agent systems (original) (raw)

Comments on "Consensus and cooperation in networked multi-agent systems

Proceedings of the IEEE, 2000

The objective of this note is to give several comments regarding the paper [1] published in the Proceedings of the IEEE and to mention some closely related results published in 2000 and 2001. I will focus on the graph theoretic results underlying the analysis of consensus in multiagent systems. As stated in the Introduction of [1], "Graph Laplacians and their spectral properties […] are important graph-related matrices that play a crucial role in convergence analysis of consensus and alignment algorithms." In particular, the stability properties of the distributed consensus algorithms

On Topology of Networked Multi-Agent Systems for Fast Consensus

2011

The rate of convergence of a consensus algorithm for networked multi-agent systems is determined by the second smallest eigenvalue of the graph Laplacian. In this paper, we consider the problem of finding undirected graphs maximizing the second smallest eigenvalue of the Laplacian for the given number of nodes and edges. We show that under certain conditions the second smallest eigenvalue of the Laplacian is maximized for some wellknown classes of graphs such as the cycle graph, the star graph and the complete bipartite graph.

On the Role of Matrix-Weights Elements in Consensus Algorithms for Multi-Agent Systems

Network

This paper examines the roles of the matrix weight elements in matrix-weighted consensus. The consensus algorithms dictate that all agents reach consensus when the weighted graph is connected. However, it is not always the case for matrix weighted graphs. The conditions leading to different types of consensus have been extensively analysed based on the properties of matrix-weighted Laplacians and graph theoretic methods. However, in practice, there is concern on how to pick matrix-weights to achieve some desired consensus, or how the change of elements in matrix weights affects the consensus algorithm. By selecting the elements in the matrix weights, different clusters may be possible. In this paper, we map the roles of the elements of the matrix weights in the systems consensus algorithm. We explore the choice of matrix weights to achieve different types of consensus and clustering. Our results are demonstrated on a network of three agents where each agent has three states.

Strict Lyapunov functions for consensus under directed connected graphs

2020 European Control Conference (ECC), 2020

It is known that for consensus of systems interconnected under a general directed graph topology a necessary and sufficient condition for consensus is that there exist at least one rooted spanning tree. In this paper we present an original statement of linear algebra that serves to characterise the spanning-tree condition for directed graphs in terms of a Lyapunov equation involving the graph’s Laplacian. Our results apply to the case of systems described by simple first and second order integrators. As a result, we provide strict Lyapunov functions that ensure, via direct constructive proof, global exponential stability of the consensus manifold.

Consensus of multi-agent system under directed network: A matrix analysis approach

2009

This paper investigates the consensus of multiagent system in network (i.e. a swarm). The topological structure of the network is characterized by a digraph. The agents of the network are described by an integrator and distributed in Rm. By means of transforming the Laplacian of the digraph into its Frobenius canonical form the system may be decomposed into one or several minimal-independent subsystems and one or several non-independent subsystems. Each minimal-independent subsystem, which consists of some agents of system, achieves consensus of its own. In other worlds, the agents of the subsystem converge into a state (equilibrium position), which is weighted-average of initial states of agents in the subsystem. Thus, the system may has several local consensus positions. When system consists of one or several non-independent subsystems, we further show that all agents in a non-independent subsystem will converge into a state (aggregation position), which are located inside of a convex-combination set of aggregation positions of minimal-independent subsystems. We study these problem mainly by means of graph theory and matrix theory.

Consensus-based distributed estimation of Laplacian eigenvalues of undirected graphs

2013 European Control Conference (ECC), 2013

In this paper, we present a novel algorithm for estimating eigenvalues of the Laplacian matrix associated with the graph describing the network topology of a multi-agent system or a wireless sensor network. As recently shown, the average consensus matrix can be written as a product of Laplacian based consensus matrices whose stepsizes are given by the inverse of the nonzero Laplacian eigenvalues. Therefore, by solving the factorization of the average consensus matrix, we can infer the Laplacian eigenvalues. We show how solving such a matrix factorization problem in a distributed way. In particular, we formulate the problem as a constrained consensus problem. The proposed algorithm does not require great resources in both computation and storage. This algorithm can also be viewed as a way for decentralizing the design of finite-time average consensus protocol recently proposed in the literature. Eventually, the performance of the proposed algorithm is evaluated by means of simulation results.

Distributed consensus in networks of dynamic agents

Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05, 2005

Stationary and distributed consensus protocols for a network of n dynamic agents under local information is considered. Consensus must be reached on a group decision value returned by a function of the agents' initial state values. As a main contribution we show that the agents can reach consensus if the value of such a function computed over the agents' state trajectories is time invariant. We use this basic result to introduce a protocol design rule allowing consensus on a quite general set of values. Such a set includes, e.g., any generalized mean of order p of the agents' initial states. We demonstrate that the asymptotical consensus is reached via a Lyapunov approach. Finally we perform a simulation study concerning the alignment maneuver of a team of unmanned air vehicles.

Consensus problems in networks of agents with switching topology and time-delays

Automatic Control, IEEE …, 2004

In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results. He has been a Postdoctoral Scholar in the Department of Control and Dynamical System, California Institute of Technology, Pasadena, since 2001. His research interests include distributed control of multiagent systems, formation control, consensus and synchronization problems, flocking/swarming, particle-based modeling and simulation, self-assembly, cooperative robotics, mobile sensor networks, self-organizing networks, folding and unfolding, biomolecular systems, dynamic graph theory, graph rigidity, computational geometry, nonlinear control theory, and control of aerospace vehicles and UAVs.