  We study fundamental performance limitations of distributed feedback control
in large-scale networked dynamical systems. Specifically, we address the
question of whether dynamic feedback controllers perform better than static
(memoryless) ones when subject to locality constraints. We consider distributed
linear consensus and vehicular formation control problems modeled over toric
lattice networks. For the resulting spatially invariant systems we study the
large-scale asymptotics (in network size) of global performance metrics that
quantify the level of network coherence. With static feedback from relative
state measurements, such metrics are known to scale unfavorably in lattices of
low spatial dimensions, preventing, for example, a 1-dimensional string of
vehicles to move like a rigid object. We show that the same limitations in
general apply also to dynamic feedback control that is locally of first order.
This means that the addition of one local state to the controller gives a
similar asymptotic performance to the memoryless case. This holds unless the
controller can access noiseless measurements of its local state with respect to
an absolute reference frame, in which case the addition of controller memory
may fundamentally improve performance. In simulations of platoons with 20-200
vehicles we show that the performance limitations we derive manifest as
unwanted accordion-like motions. Similar behaviors are to be expected in any
network that is embeddable in a low-dimensional toric lattice, and the same
fundamental limitations would apply. To derive our results, we present a
general technical framework for the analysis of stability and performance of
spatially invariant systems in the limit of large networks.
