A Distributed Observer for a Time-Invariant Linear System
This work provides a theoretical framework for distributed state estimation in linear systems, addressing the problem of decentralized observer design for multi-agent systems.
The paper presents a distributed observer for continuous-time linear systems, enabling each agent to estimate the full state using local measurements and neighbor information. It shows that under joint observability and strong connectivity of the neighbor graph, the observer's spectrum can be arbitrarily assigned.
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.