Towards Asynchronous State Reconstruction via Time Projection. Preliminary Results in the case of two Sensors

In this paper we consider a state estimation problem featuring an autonomous linear system and a set of interconnected sensors, each able to measure, in general, different measures and able to acquire their measurements and interact with each other in an asynchronous fashion. In particular, as a first step in the solution of this problem, in this paper we focus on the case of two agents and we develop a provably convergent asynchronous algorithm. Specifically, assuming the sensors take measurements at distinct time instants, we develop a procedure to merge such an information and reconstruct the value assumed by the full system's state at both time instants. The procedure is based on the idea of repeatedly projecting forward/backward in time the state estimated at one time instant and refine the estimation by comparing with the estimate obtained at the other time instant based on the other measure. As a result, the procedure is shown to converge to the actual systems's state at both considered time instants. Notice that, although addressing a particular case, a step of the proposed algorithm has the potential to represent the basic interaction step among pairs of agents in a gossip-like interaction protocol, and in our experimental analysis we provide numerical evidence that supports this conclusion.