Architecture of Distributed Control Design
Probably the most important question in distributed control design is about controller architecture, e.g. which sensors should communicate with which actuators? how much degradation in performance is incurred when imposing additional link constraints, etc. In general, these questions are notoriously difficult (e.g. decentralized control). However, there are important classes of problems where surprising answers are obtained.

Control design with prespecified information passing structure
The blue layer represents a distributed dynamical system and its interaction links,
while the green nodes and arrows represent a distributed controller with preconstrained
information passing links (between controller sites) and sensing/actuation links.
For many reasons, it is desirable to design optimal
spatially distributed controllers with such presepcified constraints on
their communication requirements.
The exact problem is nonconvex in general, but there are certain important
cases that correspond to convex problem (see below). Interestingly, in some cases
the optimal controller inherits the plant's constraint structure for free (even when not
explicitly imposed in the optimization problem).


Optimal controllers inherit plant's symmetries
When the plant has certain symmetries, i.e. its dynamics are invariant under
the action of some symmetry group, controllers inhert some of that symmetry automatically.
It can be shown that an optimal controller with the same symmetries of the plant can
be designed when the performance index is any closed loop norm that is also symmetric
(see the paper).
The basic argument is a generalization to the spatial case of an
averaging argument used to investigate time varying versus time invariant
controllers in the paper by
Shamma & Dahleh.
This in turn is based on the averaging arguments used to construct the invariant
Haar measure on groups.
Incidentally, it appears that this principle is violated when one considers nonnorm
performance measures, or when apriori structural constraints are placed on the controller.
For an example of this, see the “symmetry breaking” or
“Mistuning” control design for platoons.


Controller information propagating faster than plant dynamics
The “speed” at which information/effects travel in a spatiotemporal system can be visualized
using their spatiotemporal impulse response. The diagram on the left gives an
example of a support
set for a spatiotemporal impulse response.
We call this “funnel causality”. The (t,x) funnel illustrates the
first time t at which dynamical effects arrive at a site a distance x away in a
spatially invariant system. The diagram on the right illustrates an optimal
control design problem with apriori spatiotemporal causality restrictions. The blue
funnel represents the plant, and the green funnel represents the class of controllers.
When information passing in the controller is at least as fast as that in the plant (i.e.
the controller's funnel includes the plant's funnel), the constrained optimal controller
design problem is convex!
This fact was first observed for spatially invariant systems with
distributed delayed measurements, and is actually based
on the earlier work of
Voulgaris. The notion of
Quadratic Invariance is a
generalization to a larger class of problems. The above interpretation in terms of speed of information travel in controller versus that in the plant was first given for the special class of spatially invariant systems. Rotkowitz, Cogill & Lall obtained a similar statement for a more general class of networks in which a natural notion of distance can be formulated.


Quadratically optimal controllers are inherently localized!
For spatially distributed control design, a central question is the nature of the
spatial interconnectedness of optimal controllers. It is perhaps a surprise that
quadratically optimal controllers designed with no apriori constraints
(i.e. the centralized controller), turn out to have
some inherent localization properties. It can be shown that for most spatially invariant
plants, quadratically optimal controllers have state feedback and observer gains that
decay exponentially with distance, making them effectively localized. The example
on the left is the centralized LQR statefeedback
gain for a string network of size 50.
See the
paper
for the spatially invariant case, and the
recent work by Motee & Jadbabaie
for a farreaching generalization of this concept.

