Control design with pre-specified 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 pre-constrained
information passing links (between controller sites) and sensing/actuation links.
For many reasons, it is desirable to design optimal
spatially distributed controllers with such pre-sepcified constraints on
their communication requirements.
The exact problem is apparently non-convex, so one considers various relaxations, which actually turn out to be not too far from the true optimal. For an example of a class of problems in which the design problem is actually convex see below. |
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.
Interstingly, it appears that this principle is violated when one considers non-norm 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 traveling faster than plant dynamics leads to convex problems
The "speed" at which information travels in a spatio-temporal system can be visualized
using their spatio-temporal impulse response. The diagram on the left gives an
example of a support
set for a spatio-temporal 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 spatio-temporal 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. |
Quadratically optimal controllers are inherently "localized"! Their gains decay exponentially with distance
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, 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.
See the paper for the spatially invariant case, and the recent work by Motee & Jadbabaie for a far-reaching generalization of this concept. |
Related Publications
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