A Simple Approach to Distributed Observer Design for Linear Systems

This note investigates the distributed estimation problem for continuous-time linear time-invariant (LTI) systems observed by a network of observers. Our starting point is a given LTI system whose full state vector we want to estimate, based on the measured output of the system. We will call this the observed system. Each observer in the network has access to only part of the output of the observed system, and communicates with its neighbors according to a given network graph. In this note, we recover the known result that if the observed system is observable and the network graph is a strongly connected digraph, then a distributed observer exists. Moreover, the estimation error can be made to converge to zero at any a priori given decay rate. Our approach leads to a relatively straightforward proof of this result, using the mirror of the balanced graph associated with the original network graph. The numerical design of our distributed observer is reduced to solving linear matrix inequalities. Each observer in the network has state dimension equal to that of the observed plant.