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Somarakis, Christoforos
Baras, John S.
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.