A Distributed Observer for a Time-Invariant Linear System

A time-invariant, linear, distributed observer is described for estimating the state of an <inline-formula> <tex-math notation="LaTeX">$m>0$</tex-math></inline-formula> channel, <inline-formula><tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional continuous-time linear system of the form <inline-formula> <tex-math notation="LaTeX">$\dot{x} = Ax,\;y_i = C_ix,\;i\in \lbrace 1,2,\ldots, m\rbrace$</tex-math></inline-formula>. The state <inline-formula><tex-math notation="LaTeX">$x$</tex-math></inline-formula> is simultaneously estimated by <inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula> agents assuming each agent <inline-formula> <tex-math notation="LaTeX">$i$</tex-math></inline-formula> senses <inline-formula><tex-math notation="LaTeX">$y_i$ </tex-math></inline-formula> and receives the state <inline-formula><tex-math notation="LaTeX">$z_j$</tex-math> </inline-formula> of each of its neighbors’ estimators. Neighbor relations are characterized by a constant directed graph <inline-formula><tex-math notation="LaTeX">$\mathbb {N}$</tex-math></inline-formula> whose vertices correspond to agents and whose arcs depict neighbor relations. For the case when the neighbor graph is strongly connected, the overall distributed observer consists of <inline-formula><tex-math notation="LaTeX">$m$</tex-math> </inline-formula> linear estimators, one for each agent; <inline-formula><tex-math notation="LaTeX">$m-1$</tex-math> </inline-formula> of the estimators are of dimension <inline-formula><tex-math notation="LaTeX">$n$</tex-math> </inline-formula> and one estimator is of dimension <inline-formula><tex-math notation="LaTeX">$n+m-1$</tex-math> </inline-formula>. Using results from the classical decentralized control theory, it is shown that subject to the assumptions that none of the <inline-formula><tex-math notation="LaTeX">$C_i$</tex-math></inline-formula> are zero, the neighbor graph <inline-formula><tex-math notation="LaTeX">$\mathbb {N}$</tex-math></inline-formula> is strongly connected, the system whose state to be estimated is jointly observable, and nothing more, it is possible to freely assign the spectrum of the overall distributed observer. For the more general case, when <inline-formula> <tex-math notation="LaTeX">$\mathbb {N}$</tex-math></inline-formula> has <inline-formula><tex-math notation="LaTeX"> $q>1$</tex-math></inline-formula> strongly connected components, it is explained how to construct a family of <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula> distributed observers, one for each component, which can estimate <inline-formula><tex-math notation="LaTeX">$x$</tex-math></inline-formula> at a preassigned convergence rate.