A Distributed Observer for a Time-Invariant Linear System

A time-invariant, linear, distributed observer is described for estimating the state of an $m>0$ channel, $n$-dimensional continuous-time linear system of the form $\dot{x} = Ax,\ y_i = C_i x,\ i \in \{1,2,\cdots, m\}$. The state $x$ is simultaneously estimated by $m$ agents assuming each agent $i$ senses $y_i$ and receives the state $z_j$ of each of its neighbors' estimators. Neighbor relations are characterized by a constant directed graph $\mathbb{N}$ whose vertices correspond to agents and whose arcs depict neighbor relations. The overall distributed observer consists of $m$ linear estimators, one for each agent; $m-1$ of the estimators are of dimension $n$ and one estimator is of dimension $n+m-1$. Using results from classical decentralized control theory, it is shown that subject to the assumptions that (i) none of the $C_i$ are zero, (ii) the neighbor graph $\mathbb{N}$ is strongly connected, (iii) the system whose state is to be estimated is jointly observable, and nothing more, it is possible to freely assign the spectrum of the overall distributed observer.