  Interconnected networks describe the dynamics of important systems in a wide
range such as biological systems and electrical power grids. Some important
features of these systems were successfully studied and understood through
simplified model of linear interconnection of linear subsystems, where provably
global properties, e.g. global convergence to a specific state, usually hold
true. However, in severely disturbed conditions many of those systems exhibit
strongly nonlinear behaviour. Particularly, multiple equilibrium points may
coexist and make the dynamical behavior of the system difficult to predict.
Aiming at understanding the fragility of interconnected systems, we will
provide a hierarchical framework to assess the metastability and resilience of
such systems. This framework is based on independently characterizing stability
of individual subsystems when they are uncoupled from the network, and then
enforcing the diagonal dominance property on a structure matrix capturing the
subsystems stability and the input-to-output gains of interconnection network.
Since the subsystems are usually of low order and the structure matrix has size
equal to the number of subsystems, this framework is easy to implement and thus
scalable to large scale interconnected systems. Possible application of this
framework in assessing stability of microgrids will be discussed at the end of
this paper.
