Discrete-time filtering of noise correlated continuous-time processes: Modelling and derivation of the sampling period sensitivities

The optimal discrete-time state estimation of continuous-time processes whose measurements are corrupted by additive white noise is considered, in the case where the measurements are prefiltered by an integrator between sampling times. A discrete-time equivalent model, in which the measurements are written as a function of the state vector at the same instant, is developed for the general case where the continuous-time measurement and process noise signals are correlated. The equations governing the optimal filter, which is based on the discrete-time equivalent model, are presented. The properties of this filter are investigated, in the case of a short sampling period, by deriving the first coefficients of the Taylor's expansions of the optimal gain and the error covariance matrices in powers of the sampling period. The obtained results are compared with the corresponding expressions that have been previously derived for the sampled-data regulator.