Random Sampling of a Continuous-time Stochastic Dynamical System

We consider a dynamical system where the state equation is given by a linear stochastic differential equation and noisy measurements occur at discrete times, in correspondence of the arrivals of a Poisson process. Such a system models a network of a large number of sensors that are not synchronized with one another, where the waiting time between two measurements is modelled by an exponential random variable. We formulate a Kalman Filter-based state estimation algorithm. The sequence of estimation error covariance matrices is not deterministic as for the ordinary Kalman Filter, but is a stochastic process itself: it is a homogeneous Markov process. In the one-dimensional case we compute a complete statistical description of this process: such a description depends on the Poisson sampling rate (which is proportional to the number of sensors on a network) and on the dynamics of the continuous-time system represented by the state equation. Finally, we have found a lower bound on the sampling rate that makes it possible to keep the estimation error variance below a given threshold with an arbitrary probability.