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 (x) over dot = A(x), y(i) = C(i)x, i is an element of {1, 2,..., 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 N 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 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 the classical decentralized control theory, it is shown that subject to the assumptions that none of the C-i are zero, the neighbor graph N 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 N has q > 1 strongly connected components, it is explained how to construct a family of q distributed observers, one for each component, which can estimate x at a preassigned convergence rate.