Distributed Synchronization of Heterogeneous Oscillators on Networks With Arbitrary Topology

Many network applications rely on the synchronization of coupled oscillators. For example, such synchronization can provide networked devices with a common temporal reference necessary for coordinating actions or decoding transmitted messages. In this paper, we study the problem of using distributed control to achieve phase and frequency synchronization of a network of coupled heterogeneous nonlinear oscillators. Not only do our controllers guarantee zero-phase error in steady state under arbitrary frequency heterogeneity, but they also require little knowledge of the oscillator nonlinearities and network topology. Furthermore, we provide a global convergence analysis, in the absence of noise and propagation delay, for the resulting nonlinear system whose phase vector evolves on the n-torus

Synchronization of Heterogeneous Kuramoto Oscillators with Arbitrary Topology

We study synchronization of coupled Kuramoto oscillators with heterogeneous inherent frequencies and general underlying connectivity. We provide conditions on the coupling strength and the initial phases which guarantee the existence of a Positively Invariant Set (PIS) and lead to synchronization. Unlike previous works that focus only on analytical bounds, here we introduce an optimization approach to provide a computational-analytical bound that can further exploit the particular features of each individual system such as topology and frequency distribution. Examples are provided to illustrate our results as well as the improvement over previous existing bounds

Oscillation modes in symmetrical wireless-locked systems

Time synchronization of multiple elements of a wireless network can be achieved through the wireless coupling of their oscillator circuits. Most previous works on wireless locking of oscillators analyze the system in an idealized manner, representing the oscillator elements with phase models and describing the propagation effects with constant scalar coefficients and time delays. Here, a realistic analysis of the wireless system is presented, which relies on the extraction of the oscillator models from harmonic-balance (HB) simulations and takes into account the antenna gains and propagation effects. The most usual network configurations, corresponding to ring, fully connected, and star topologies, are investigated in detail. In symmetric conditions, the oscillation modes are detected through an eigenvalue/eigenvector calculation of an equivalent coupling matrix. For each particular mode, the system is analyzed in the following manners: by means of an analytical formulation, able to provide all the coexistent solutions, and through a circuit-level HB simulation of an equivalent system with a reduced number of oscillator elements. The stability properties are determined by means of a perturbation system of general application to any coupled structure. A specific formulation is also derived to predict the impact of discrepancies between the oscillator elements. All the results have been validated with independent circuit-level simulations and measurements.This work was supported in part by the Spanish Ministry of Economy and Competitiveness under the research project TEC2017-88242-C3-1-R, in part by the European Regional Development Fund (ERDF/FEDER), in part by Juan de la Cierva Research Program under IJCI-2014-19141, and in part by the Parliament of Cantabria under the project Cantabria Explora 12.JP02.64069

Wireless-coupled oscillator systems with an injection-locking signal

A detailed analysis of wireless-coupled oscillator systems under the effect of an injection-locking signal is presented. The injection source of high spectral purity is introduced at a single node and enables a reduction of the phase-noise spectral density. Under this injection source, the behavior of the coupled system is qualitatively different from the one obtained in free-running conditions. Two cases are considered: bilateral synchronization, in which an independent source is connected to a particular system oscillator, coupled to the other oscillator elements, and unilateral synchronization, in which one of these elements is replaced by an independent source that cannot be influenced by the rest. The two cases are illustrated through the analysis of a wireless-coupled system with a star topology, such that the injection signal is introduced at the central node. The investigation involves an insightful analytical calculation of the coexisting steady-state solutions, as well as a determination of their stability and bifurcation properties and phase noise. The injection signal stabilizes the system in a large and continuous distance interval, enabling a more robust operation than in autonomous (noninjected) conditions. A coupled system operating at 2.45 GHz has been manufactured and experimentally characterized, obtaining a very good agreement between simulations and measurements.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research projects TEC2014-60283-C3-1-R and TEC2017-88242-C3-1-R

Characterization and Control of Conservative and Nonconservative Network Dynamics

Diffusion processes are instrumental to describe the movement of a continuous quantity in a generic network of interacting agents. Here, we present a probabilistic framework for diffusion in networks and study in particular two classes of agent interactions depending on whether the total network quantity follows a conservation law. Focusing on asymmetric interactions between agents, we define how the dynamics of conservative and non-conservative networks relate to the weighted in-degree and out-degree Laplacians. For uncontrolled networks, we define the convergence behavior of our framework, including the case of variable network topologies, as a function of the eigenvalues and eigenvectors of the weighted graph Laplacian. In addition, we study the control of the network dynamics by means of external controls and alterations in the network topology. For networks with exogenous controls, we analyze convergence and provide a method to measure the difference between conservative and non-conservative network dynamics based on the comparison of their respective attainability domains. In order to construct a network topology tailored for a desired behavior, we propose a Markov decision process (MDP) that learns specific network adjustments through a reinforcement learning algorithm. The presented network control and design schemes enable the alteration of the dynamic and stationary network behavior in conservative and non-conservative networks