Synchronization in networks of linear singularly perturbed systems

This work is motivated by the fact that many real systems are characterized by two features. The first one is that they are obtained by interconnecting a bunch of simpler subsystems that have to synchronize in order to reach a global goal. The second one is that each subsystem presents dynamics that evolves on different time-scales. Taking into account the two features leads to the problem of synchronization in networks of singularly perturbed systems. In this work we are providing a preliminary study that considers the problem where each subsystem is linear and the network topology is represented by a connected undirected graph that is fixed in time. We show that we can proceed to a time-scale separation of the overall network dynamics and design the controls that synchronize the slow dynamics and the fast ones. Applying the joint control actions to the network of singularly perturbed systems we obtain an approximation of the synchronization behavior imposed for each scale. The methodology requires a variable transformation to overcome the fact that we are dealing with non-standard singularly perturbed systems. One example illustrates the synchronization behavior of linear singularly perturbed systems.

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