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  • 1.
    Markdahl, J.
    et al.
    Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Belval, Luxembourg.
    Thunberg, Johan
    Halmstad University, School of Information Technology, Halmstad Embedded and Intelligent Systems Research (EIS).
    Goncalves, J.
    Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Belval, Luxembourg.
    High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally2020In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 113, article id 108736Article in journal (Refereed)
    Abstract [en]

    The Kuramoto model of coupled phase oscillators is often used to describe synchronization phenomena in nature. Some applications, e.g., quantum synchronization and rigid-body attitude synchronization, involve high-dimensional Kuramoto models where each oscillator lives on the n-sphere or SO(n). These manifolds are special cases of the compact, real Stiefel manifold St(p,n). Using tools from optimization and control theory, we prove that the generalized Kuramoto model on St(p,n) converges to a synchronized state for any connected graph and from almost all initial conditions provided (p,n) satisfies p≤2/3n−1 and all oscillator frequencies are equal. This result could not have been predicted based on knowledge of the Kuramoto model in complex networks over the circle. In that case, almost global synchronization is graph dependent; it applies if the network is acyclic or sufficiently dense. This paper hence identifies a property that distinguishes many high-dimensional generalizations of the Kuramoto models from the original model. © 2019 Elsevier Ltd

  • 2.
    Thunberg, Johan
    et al.
    Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Luxembourg.
    Markdahl, Johan
    Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Luxembourg.
    Bernard, Florian
    Max-Planck-Institute for Informatics, Saarland Informatics Campus, Germany.
    Goncalves, Jorge
    Systems Biomedicine, University of Luxembourg, Luxembourg.
    A lifting method for analyzing distributed synchronization on the unit sphere2018In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 96, p. 253-258Article in journal (Refereed)
    Abstract [en]

    This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d-dimensional unit sphere are lifted to the (d+1)-dimensional Euclidean space. The consensus protocol on the unit sphere is the classical one, where agents move toward weighted averages of their neighbors in their respective tangent planes. Only local and relative state information is used. The directed interaction graph topologies are allowed to switch as a function of time. The dynamics of the lifted variables are governed by a nonlinear consensus protocol for which the weights contain ratios of the norms of state variables. We generalize previous convergence results for hemispheres. For a large class of consensus protocols defined for switching uniformly quasi-strongly connected time-varying graphs, we show that the consensus manifold is uniformly asymptotically stable relative to closed balls contained in a hemisphere. Compared to earlier projection based approaches used in this context such as the gnomonic projection, which is defined for hemispheres only, the lifting method applies globally. With that, the hope is that this method can be useful for future investigations on global convergence.

  • 3.
    Thunberg, Johan
    et al.
    Halmstad University, School of Information Technology, Halmstad Embedded and Intelligent Systems Research (EIS), Centre for Research on Embedded Systems (CERES).
    Markdahl, Johan
    Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Luxembourg.
    Goncalves, Jorge
    Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Luxembourg.
    Dynamic controllers for column synchronization of rotation matrices: A QR-factorization approach2018In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 93, p. 20-25Article in journal (Refereed)
    Abstract [en]

    In the multi-agent systems setting, this paper addresses continuous-time distributed synchronization of columns of rotation matrices. More precisely, k specific columns shall be synchronized and only the corresponding k columns of the relative rotations between the agents are assumed to be available for the control design. When one specific column is considered, the problem is equivalent to synchronization on the (d−1)-dimensional unit sphere and when all the columns are considered, the problem is equivalent to synchronization on SO(d). We design dynamic control laws for these synchronization problems. The control laws are based on the introduction of auxiliary variables in combination with a QR-factorization approach. The benefit of this QR-factorization approach is that we can decouple the dynamics for the k columns from the remaining d−k ones. Under the control scheme, the closed loop system achieves almost global convergence to synchronization for quasi-strong interaction graph topologies.

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