Realization theory for poset-causal systems: Controllability, observability and duality

TL;DR

This paper develops realization theory for poset-causal systems, exploring controllability, observability, and duality, extending concepts from coordinated systems to more complex partial order communication structures.

Contribution

It introduces new notions of controllability and observability for poset-causal systems and establishes their duality relations, broadening the theoretical framework beyond coordinated systems.

Findings

Poset-causal systems are closed under duality.

New controllability and observability notions are developed.

Duality relations connect these notions in poset-causal systems.

Abstract

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.