  Poset-causal systems form a class of decentralized systems introduced by Shah
and Parrilo [32] and studied mainly in the context of optimal decentralized
control. In this paper we develop part of the realization theory for
poset-causal systems. More specifically, we investigate several notions of
controllability and observability, and their relation under duality. These new
notions extend concepts of controllability and observability in the context of
coordinated linear systems [16]. While for coordinated linear systems there is
a clear hierarchical structure with a single (main) coordinator, for
poset-causal systems there need not be a single coordinator and the
communication structure between the decentralized systems allows for more
intricate structures, governed by partial orders. On the other hand, we show
that the class of poset-causal systems is closed under duality, which is not
the case for coordinated linear systems, and that duality relations between the
various notions of observability and controllability exist.
