A Hybrid Observer for Estimating the State of a Distributed Linear System

Abstract

A hybrid observer is described for estimating the state of an m > 0 channel, n-dimensional, continuous-time, linear system of the form ẋ = Ax, yi = Cix, i ∈ {1, 2, . . . , m}. The system's state x is simultaneously estimated by m agents assuming each agent i senses yi and receives appropriately defined data from each of its current neighbors. Neighbor relations are characterized by a time-varying directed graph N(t) whose vertices correspond to agents and whose arcs depict neighbor relations. Agent i updates its estimate xi of x at "event times" ti1, ti2, ti3, . . . using a local continuous-time linear observer and a local parameter estimator which iterates q times during each event time interval [t i(s−1) , tis), s ≥ 1 to obtain an estimate of x(tis). Subject to the assumptions that none of the Ci's are zero, the neighbor graph N(t) is strongly connected for all time, and the system whose state is to be estimated is jointly observable, it is shown that for any number λ > 0, it is possible to choose q and the local observer gains so that each estimate xi converges to x at least as fast as e −λt does. This result holds whether or not agents communicate synchronously, although in the asynchronous case it is necessary to assume that N(t) changes in a suitably defined sense. Exponential convergence is also assured if the event time sequences of the m agents are slightly different than each other, although in this case only if the system being observed is exponentially stable; this limitation however, is primarily a robustness issue shared by all state estimators, centralized or not, which are operating in "open loop" in the face of small modeling errors. The result also holds facing abrupt changes in the number of vertices and arcs in the inter-agent communication graph upon which the algorithm depends.