# Distributed Functional Observers for LTI Systems

## TL;DR

This paper develops conditions and methods for designing distributed functional observers in LTI systems, enabling multiple nodes to collaboratively estimate specific functions of the system state despite coupling and measurement diversity.

## Contribution

It introduces a novel approach to design distributed functional observers by revealing couplings and identifying locally detectable dynamics, extending classical centralized conditions.

## Key findings

- Distributed observers guarantee asymptotic function estimation
- Transformations reveal couplings between dynamics and measurements
- Sufficient network and update rules are provided

## Abstract

We study the problem of designing distributed functional observers for LTI systems. Specifically, we consider a setting consisting of a state vector that evolves over time according to a dynamical process. A set of nodes distributed over a communication network wish to collaboratively estimate certain functions of the state. We first show that classical existence conditions for the design of centralized functional observers do not directly translate to the distributed setting, due to the coupling that exists between the dynamics of the functions of interest and the diverse measurements at the various nodes. Accordingly, we design transformations that reveal such couplings and identify portions of the corresponding dynamics that are locally detectable at each sensor node. We provide sufficient conditions on the network, along with state estimate update and exchange rules for each node, that guarantee asymptotic reconstruction of the functions at each sensor node.