Summary. This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with latticetype nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multi-vehicle formations.

Motivated by applications to sensor, peer-to-peer, and ad hoc networks, we study distributed algorithms, also known as gossip algorithms, for exchanging information and for computing in an arbitrarily connected network of nodes. The topology of such networks changes continuously as new nodes join and old nodes leave the network. Algorithms for such networks need to be robust against changes in topology. Additionally, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of “gossip ” algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the gossip constraint for an arbitrary network graph, and find that the averaging time of a gossip algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Designing the fastest gossip algorithm corresponds to minimizing this eigenvalue, which is a semidefinite program (SDP). In general, SDPs cannot be solved in a distributed fashion; however, exploiting problem structure, we propose a distributed subgradient method that solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities derived from the gossip algorithm. We use this connection to study the performance and scaling of gossip algorithms on two popular networks: Wireless Sensor Networks, which are modeled as Geometric Random Graphs, and the Internet graph under the so-called Preferential Connectivity (PC) model.

Abstract—Inspired by the so-called “bugs ” problem from mathematics, we study the geometric formations of multivehicle systems under cyclic pursuit. First, we introduce the notion of cyclic pursuit by examining a system of identical linear agents in the plane. This idea is then extended to a system of wheeled vehicles, each subject to a single nonholonomic constraint (i.e., unicycles), which is the principal focus of this paper. The pursuit framework is particularly simple in that the identical vehicles are ordered such that vehicle pursues vehicle CImodulo. In this paper, we assume each vehicle has the same constant forward speed. We show that the system’s equilibrium formations are generalized regular polygons and it is exposed how the multivehicle system’s global behavior can be shaped through appropriate controller gain assignments. We then study the local stability of these equilibrium polygons, revealing which formations are stable and which are not. Index Terms—Circulant matrices, cooperative control, multiagent systems, pursuit problems. I.

We study the stability properties of linear time-varying systems in continuous time whose system matrix is Metzler with zero row sums. This class of systems arises naturally in the context of distributed decision problems, coordination and rendezvous tasks and synchronization problems. The equilibrium set contains all states with identical state components. We present sufficient conditions guaranteeing uniform exponential stability of this equilibrium set, implying that all state components converge to a common value as time grows unbounded. Furthermore it is shown that this convergence result is robust with respect to an arbitrary delay, provided that the delay affects only the off-diagonal terms in the differential equation.

In a network of dynamical systems, concurrent synchronization is a regime where multiple groups of fully synchronized elements coexist. In the brain, concurrent synchronization may occur at several scales, with multiple “rhythms ” interacting and functional assemblies combining neural oscillators of many different types. Mathematically, stable concurrent synchronization corresponds to convergence to a flow-invariant linear subspace of the global state space. We derive a general condition for such convergence to occur globally and exponentially. We also show that, under mild conditions, global convergence to a concurrently synchronized regime is preserved under basic system combinations such as negative feedback or hierarchies, so that stable concurrently synchronized aggregates of arbitrary size can be constructed. Simple applications of these results to classical questions in systems neuroscience and robotics are discussed. 1

In this paper, we propose discrete-time and continuous-time consensus update schemes motivated by the discrete-time and continuous-time Kalman filters. With certainty information encoded into each agent, the proposed consensus schemes explicitly account for relative confidence / reliability of information states from each agent in the team. We show mild sufficient conditions under which consensus can be achieved using the proposed consensus schemes in the presence of switching interaction topologies.

Abstract—We study the problem of flocking and velocity alignment in a group of kinematic nonholonomic agents in 2 and 3 dimensions. By analyzing the velocity vectors of agents on a circle (for planar motion) or sphere (for 3-D motion), we develop a geodesic control law that minimizes a misalignment potential and results in velocity alignment and flocking. The proposed control laws are distributed and will provably result in flocking when the underlying proximity graph which represents the neighborhood relation among agents is connected. We further show that flocking is possible even when the topology of the proximity graph changes over time, so long as a weaker notion of joint connectivity is preserved. Index Terms—Cooperative control, distributed coordination, flocking, multiagent systems. I.

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Grayson Theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the problem of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain convex, and the perimeter of the polygon is monotonically decreasing. Index Terms Distributed control, curve shortening, mobile autonomous robots.

This paper studies the consensus problem of multi-agent systems in an asynchronous framework. Under certain assumptions, the consensus protocol leads to stable behaviors even if the updating instants and sets of the agents are asynchronously determined. The model of asynchronous multi-agent systems encompasses those synchronous ones with various communication patterns, i.e., issues of directional, delayed, or failed communication can be addressed in the same framework. The asynchronous results in this paper thus shed new light on the synchronous results reported in the literature. In particular, synchronous protocols under dynamically changing interaction topologies can be seen as a special case of the asynchronous protocol where all communication delays are zero.

Consensus is well accepted as being a fundamental paradigm for coordination of groups of autonomous agents. Recently, we casted previous work on multi-agent consensus protocols into an asynchronous framework, where each agent updates on its own pace, and uses the most recently received information from other agents. Asynchronous consensus protocols encompass those synchronous ones with various communication patterns. In this paper, we study via simulation a number of open new problems introduced by the asynchronous operation of multi-agent systems. More interestingly, the existing consensus results are classified by their communication assumptions and future research directions are proposed. To facilitate our study, we develop a multi-vehicle simulator in Java, built on top of JProwler; JProwler is a discrete event simulator for prototyping, verifying and analyzing communication protocols of ad-hoc wireless networks. Implementation issues with their implications are also discussed.