Topics on the average consensus problems

The research field of networked systems has burgeoned during the last years attracting the attention of outstanding mathematicians, computer scientists and electrical engineers. This is due in no small part to the recent technological advances in communication and computation and to the miniaturization of electronic components which have made large groups of embedded systems, such as sensors and robotic networks, a reality. Such systems are expected to pervade our technological life during the next years performing several complex tasks, such as search and rescue, health case, environmental monitoring and surveillance. The potential advantages of employing networks of agents are numerous. For instance, certain tasks are difficult, if not impossible, when performed by a single agent. Further, a group of agents inherently provides robustness to failures of single agents or communication links. The existence of such networks raise compelling questions: Can large numbers of such autonomous agents be successfully deployed as a team to cooperatively carry out a prescribed task and to respond as a group to high-level management commands? Can such a group reliably function robustly and adaptively, without a centralized controller, with limited communications between its members? These scenarios motivate the study of algorithms for the autonomous adaptation and coordination of large-scale network systems. So is born the emerging discipline of multi-agent systems. This thesis is related to this context. Its focus is on the consensus problem, which in few words, can be described as the problem of making the individuals agents of a network to reach an agreement on key pieces of information or on a common decision (represented by scalar or vector values) that enable them to cooperate in a coordinate fashion. This problem has been massively analyzed in the last years. Indeed, it can be viewed as an intermediate but fundamental step toward a comprehensive understanding of how to assemble the members of a large-scale system into a coherent whole. Moreover consensus algorithms are starting to be applied in a wide-range of applications. In MultiVehicle cooperative control, large numbers of autonomous vehicles (air, ground and water) are envisioned to have a significant impact in numerous civilian and homeland security applications (e.g., monitoring forest fires, oil fields, pipelines, tracking wildlife, patrolling borders, monitoring the perimeter of nuclear power plants, surveillance, reconnaissance). To enable these applications various cooperative control capabilities need to be developed, including formation control, rendezvous, attitude alignment, flocking and cooperative search. Lately, consensus algorithms have shown their effectiveness in reaching these capabilities. In load balancing consensus algorithms are used to balance as much as possible the tasks among the various processors, computers in order to speed up the whole computation. Then, for several applications of a wireless sensor networks, such as mobile target tracking, event detection, efficient TDMA scheduling, and sleep scheduling with very low duty cycle, it is essential that the nodes act in a coordinated and synchronized fashion. All these applications required global clock synchronization, that is, all the nodes of the network need to refer to a common notion of time. Recently novel consensus-based protocol, for synchronizing a wireless has been elaborated. During the last years distributed estimation problems are attracting a large interest. It is possible to see that the solution of some linear inference problems, as for instance the computation of the best linear unbiased estimator, can be reduced to the applications of two or more consensus algorithms. Consensus algorithms have shown their effectiveness also in an other important distributed inference application: tracking of noisy signals. Moreover interesting consensus ideas have been applied, recently, also in more general distributed computation (in particular distributed optimization) in computer vision and in modeling information and decisions dynamics in social networks (learning, opinion dynamics, rules of thumb). The rest of the present document is organized in the following way. First we briefly review the standard consensus algorithm we consider; then we summarize the main results obtained in each chapter.