Christoforos Somarakis

Convergence results for the linear consensus problem under Markovian random graphs

This paper discusses the linear asymptotic consensus problem for a network of dynamic agents whose communication network is modeled by a randomly switching graph. The switching is determined by a finite state Markov process, each topology corresponding to a state of the process. We address the cases where the dynamics of the agents is expressed both in continuous time and in discrete time. We show that, if the consensus matrices are doubly stochastic, average consensus is achieved in the mean square sense and the almost sure sense if and only if the graph resulting from the union of graphs corresponding to the states of the Markov process is strongly connected. The aim of this paper is to show how techniques from the theory of Markovian jump linear systems, in conjunction with results inspired by matrix and graph theory, can be used to prove convergence results for stochastic consensus problems.

A simple proof of the continuous time linear consensus problem with applications in non-linear flocking networks

We revisit the linear distributed consensus problem in continuous time, to provide a simple and elegant proof under very mild assumptions. Our approach is based on a novel extension of the contraction coefficient that can be adapted to the continuous time version of the model. We apply our results in non-linear second order consensus networks of Cucker- Smale type and we obtain new initial conditions for asymptotic flocking.

LTI continuous time consensus dynamics with multiple and distributed delays

We study linear time invariant (LTI) continuous time consensus dynamics in the presence of bounded communication delays. Contrary to traditional Lyapunov based methods, we approach the problem using Fixed Point Theory. This method, allows us to create an appropriate complete functional metric space and through contraction mappings to establish the existence and uniqueness of a solution of this model. We explore the case of constant as well as distributed delays.

A Generalized Gossip Algorithm on Convex Metric Spaces

A 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, where the communication between agents is controlled by a set of Poisson counters. We study the convergence properties of the algorithm using stochastic calculus. In particular, we show that the distances between the states of the agents converge to zero with probability one and in the rth mean sense. In the special case of complete connectivity and uniform Poisson counters, we give upper bounds on the dynamics of the first and second moments of the distances between the states of the agents. In addition, we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations.

Asymptotic consensus solutions in non-linear delayed distributed algorithms under deterministic & stochastic perturbations

We consider a multi-agent non-linear delayed model which sustains consensus type of solutions. We use fixed point theory arguments to establish sufficient conditions for existence and uniqueness of solutions that converge exponentially fast to a common value with prescribed rate. The conditions depend on the communication topology, the nonlinearity of the model as well as the delay in the propagation of information. Furthermore we test the robustness of our results in the presence of independent time varying perturbations both deterministic and stochastic.

A randomized gossip consensus algorithm on convex metric spaces

A 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.

Interplay between performance and communication delay in noisy linear consensus networks

This work investigates performance of noisy time-delayed linear consensus networks from a graph topological point of view. Performance of the network is measured by the square of the H2-norm of the system. The focus of this paper is on noisy consensus networks with homogeneous time delays affecting both the agent and all its neighbors. We derive an exact expression for the performance measure of the network in terms of time delay parameter and Laplacian eigenvalues of the underlying graph of the network. It is shown that the performance measure is a convex and Schur-convex function of Laplacian eigenvalues. We characterize the network topology with optimal performance. Furthermore, we quantify a fundamental limit on the best achievable performance based on performance of the optimal topology.

Koopman performance analysis of a class of nonlinear dynamical networks

This paper builds upon the Koopman spectral analysis tools to develop a method for assessment of the performance of a class of first-order nonlinear consensus networks. This class of networks is defined over an interconnected graph with state-dependent weights that are nonlinear functions of the state of the network. The mean energy of the output of the system with respect to random initial conditions is utilized as the performance measure. We quantify this performance measure in terms of the Koopman eigenfunctions of the nonlinear dynamics, and the eigenvalues of the corresponding linearized system at the equilibrium of the network, where the eigenvalues of the linearized system are indeed Laplacian eigenvalues of the underlying graph with opposite sign. Our results reveal that the performance measure of the nonlinear network depends on the interconnection topology of the underlying graph. We illustrate effectiveness of our results using several examples, including a Cucker-Smale type consensus network and first-order network of identical Kuramoto oscillators.

Distributed solution of the Economic Dispatch Problem in smart grid power systems framework with delays

We consider the Economic Dispatch Problem in power systems in a smart-grid architecture friendly environment. The problem is tackled with the use of multiple decentralized controllers that execute parallel distributed consensus algorithms. The scenario takes into account the presence of multiple time-varying communication delays.

Dynamics of collectives with opinionated agents: The case of scrambling connectivity

We investigate the problem of social/opinion networks of autonomous agents, each of which has a personal view that follows an independent dynamic behavior and it communicates with their neighbors under a cooperative algorithm. We apply mathematics from non-negative matrix theory. We investigate agents with identical and non-identical views and we establish sufficient conditions for stability and convergence to a common synchronized opinion.