Browsing by Author "Somarakis, Christoforos"
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Item The Dynamics Of A Simple Rational Map(2011-07-17) Somarakis, ChristoforosThe dynamics of the 2-D rational map are studied for various values of it's control parameters. Despite it's simple structure this model is very rich in non-linear phenomena such as, multi-scroll strange attrac- tors, transitions to chaos via period doubling bifurcations, quasi-periodicity as well as intermittency, interior crisis, hyper-chaos etc. In this work, strange attractors, bifurcation diagrams, periodic windows, invariant characteristics are investigated both analytically and numerically.Item A Fixed Point Theory Approach to Multi-Agent Consensus Dynamics With Delays(2013-01-01) Somarakis, Christoforos; Baras, JohnThe classic linear time-invariant multi-agent consensus scheme is revisited in the presence of constant and distributed bounded delays. We create a fixed point argument and prove exponential convergence with specific rate that depends both on the topology of the communication graph and the upper bound of the allowed delay.Item A GENERAL FRAMEWORK FOR CONSENSUS NETWORKS(2015-03-10) Somarakis, Christoforos; Baras, JohnA new framework for the analysis of consensus networks is developed. The theory consists of necessary and sufficient conditions and it is flexible enough to comprise a variety of consensus systems. Under mild connectivity assumptions, the discussion ranges from linear, nonlinear, ordinary, functional and leader-follower models. The establishment of explicit estimates on the rate of convergence is the central objective. Our work extends and unifies past related works in the literature. Illustrative examples and simulations are presented to outline the theoretical results.Item Linear Processes Under Vanishing Communications - The Consensus Problem(2012-04-07) Somarakis, ChristoforosIn this report, we revisit the classical multi-agent distributed consensus problem under the dropping of the general assumption that the existence of a connection between agent implies weights uniformly bounded away from zero. We reformulate and study the problem by establishing global convergence results both in discrete and continuous time, under fixed, switching and random topologies.Item On the dynamics of linear functional differential equations with asymptotically constant solutions(2014-08-22) Somarakis, Christoforos; Baras, John; Paraskevas, Evripidis; Baras, JohnWe discuss the dynamics of general linear functional differential equations with solutions that exhibit asymptotic constancy. We apply fixed point theory methods to study the stability of these solutions and we provide sufficient conditions of asymptotic stability with emphasis on the rate of convergence. Several examples are provided to illustrate the claim that the derived results generalize, unify and in some cases improve the existing ones.Item PROBLEMS IN DISTRIBUTED CONTROL SYSTEMS, CONSENSUS AND FLOCKING NETWORKS(2015) Somarakis, Christoforos; Baras, John S.; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)An important variant of the linear model is the delayed one where it is discussed in great detail under two theoretical frameworks: a variational stability analysis based on fixed point theory arguments and a standard Lyapunov-based analysis. The investigation revisits scalar variation unifying the behavior of old biologically inspired model and extends to the multi-dimensional (consensus) alternatives. We compare the two methods and assess their applicability and the strength of the results they provide whenever this is possible. The obtained results are applied to a number of nonlinear consensus networks. The first class of networks regards couplings of passive nature. The model is considered on its delayed form and the linear theory is directly applied to provide strong convergence results. The second class of networks is a generally nonlinear one and the study is carried through under a number of different conditions. In additions the non-linearity of the models in conjunction with delays, allows for new type of synchronized solutions. We prove the existence and uniqueness of non-trivial periodic solutions and state sufficient conditions for its local stability. The chapter concludes with a third class of nonlinear models. We introduce and study consensus networks of neutral type. We prove the existence and uniqueness of a consensus point and state sufficient conditions for exponential convergence to it. The discussion continues with the study of a second order flocking network of Cucker-Smale or Motsch-Tadmor type. Based on the derived contraction rates in the linear framework, sufficient conditions are established for these systems' solutions to exhibit exponentially fast asymptotic velocity. The network couplings are essentially state-dependent and non-uniform and the model is studied in both the ordinary and the delayed version. The discussion in flocking models concludes with two noisy networks where convergence with probability one and in the r-th square mean is proved under certain smallness conditions. The linear theory is, finally, applied on a classical problem in electrical power networks. This is the economic dispatch problem (EDP) and the tools of the linear theory are used to solve the problem in a distributed manner. Motivated by the emerging field of Smart Grid systems and the distributed control methods that are needed to be developed in order to t their architecture we introduce a distributed optimization algorithm that calculates the optimal point for a network of power generators that are needed to operate at, in order to serve a given load. In particular, the power grid of interconnected generators and loads is to be served at an optimal point based on the cost of power production for every single power machine. The power grid is supervised by a set of controllers that exchange information on a different communication network that suffers from delays. We define a consensus based dynamic algorithm under which the controllers dynamically learn the overall load of the network and adjust the power generator with respect to the optimal operational point.Item A randomized gossip consensus algorithm on convex metric spaces(2012-02-20) Matei, Ion; Somarakis, Christoforos; Baras, JohnA consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest. This problem can be generalized in the context of convex metric spaces that extend the standard notion of convexity. In this paper we introduce and analyze a randomized gossip algorithm for solving the generalized consensus problem on convex metric spaces. We study the convergence properties of the algorithm using stochastic differential equations theory. We show that the dynamics of the distances between the states of the agents can be upper bounded by the dynamics of a stochastic differential equation driven by Poisson counters. In addition, we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations.Item Stability by Fixed Point Theory in Consensus Dynamics(2014-08-22) Somarakis, Christoforos; Baras, John; Paraskevas, Evripidis; Baras, JohnWe study the stability of linear time invariant distributed consensus dynamics in the presence of multiple propagation and processing delays. We employ fixed point theory (FPT) methods and derive sufficient conditions for asymptotic convergence to a common value while the emphasis is given in estimating the rate of convergence. We argue that this approach is novel in the field of networked dynamics as it is also flexible and thus capable of analyzing a wide variety of consensus based algorithms for which conventional Lyapunov methods are either too restrictive or unsuccessful.Item Towards a unified theory of consensus(2014-10-12) Somarakis, Christoforos; Baras, JohnWe revisit the classic multi-agent distributed consensus problem under mild connectivity assumptions and non-uniformly bounded weights. The analysis is based on a novel application of the standard results from the non-negative matrix theory. It is a simple, yet unifying, approach that yields generalized results. We apply these results to a wide variety of linear, non-linear consensus and flocking algorithms proposed in the literature and we obtain new conditions for asymptotic consensus. Our framework is developed in both discrete and continuous time. Furthermore we extend the discussion to stochastic settings.