  We study internal stability in the context of diffusively-coupled control
architectures, common in multi-agent systems (i.e. the celebrated consensus
protocol), for linear time-invariant agents. We derive a condition under which
the system can not be stabilized by any controller from that class. In the
finite-dimensional case the condition states that diffusive controllers cannot
stabilize agents that share common unstable dynamics, directions included. This
class always contains the group of homogeneous unstable agents, like
integrators. We argue that the underlying reason is intrinsic cancellations of
unstable agent dynamics by such controllers, even static ones, where
directional properties play a key role. The intrinsic lack of internal
stability explains the notorious behavior of some distributed control protocols
when affected by measurement noise or exogenous disturbances.
