2,256 research outputs found
Synchronization under matrix-weighted Laplacian
Synchronization in a group of linear time-invariant systems is studied where
the coupling between each pair of systems is characterized by a different
output matrix. Simple methods are proposed to generate a (separate) linear
coupling gain for each pair of systems, which ensures that all the solutions
converge to a common trajectory. Both continuous-time and discrete-time cases
are considered.Comment: 21 page
Synchronization of small oscillations
Synchronization is studied in an array of identical oscillators undergoing
small vibrations. The overall coupling is described by a pair of
matrix-weighted Laplacian matrices; one representing the dissipative, the other
the restorative connectors. A construction is proposed to combine these two
real matrices in a single complex matrix. It is shown that whether the
oscillators synchronize in the steady state or not depends on the number of
eigenvalues of this complex matrix on the imaginary axis. Certain refinements
of this condition for the special cases, where the restorative coupling is
either weak or absent, are also presented.Comment: 16 pages, 6 figure
Conditions for synchronizability in arrays of coupled linear systems
Synchronization control in arrays of identical output-coupled continuous-time
linear systems is studied. Sufficiency of new conditions for the existence of a
synchronizing feedback law are analyzed. It is shown that for neutrally stable
systems that are detectable form their outputs, a linear feedback law exists
under which any number of coupled systems synchronize provided that the
(directed, weighted) graph describing the interconnection is fixed and
connected. An algorithm generating one such feedback law is presented. It is
also shown that for critically unstable systems detectability is not
sufficient, whereas full-state coupling is, for the existence of a linear
feedback law that is synchronizing for all connected coupling configurations
Synchronization of harmonic oscillators under restorative coupling with applications in electrical networks
The role of restorative coupling on synchronization of coupled identical
harmonic oscillators is studied. Necessary and sufficient conditions, under
which the individual systems' solutions converge to a common trajectory, are
presented. Through simple physical examples, the meaning and limitations of the
theorems are expounded. Also, to demonstrate their versatility, the results are
extended to cover LTI passive electrical networks. One of the extensions
generalizes the well-known link between the asymptotic stability of the
synchronization subspace and the second smallest eigenvalue of the Laplacian
matrix.Comment: 13 pages, 8 figure
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