Numerical differentiation and parameter estimation for dth order linear stochastic systems

For a linear time invariant system of order d⩾2 with a white noise disturbance, the input and the output are assumed to be sampled at regular time intervals. Using these observations, some approximate values of the first d-1 derivatives are obtained by numerical differentiation and the unknown system parameters are estimated by a discretization of the continuous time least squares formulae. These parameter estimates have an error which does not approach zero as the sampling interval approaches zero. This error is shown to be associated with the inconsistency of the quadratic variation estimate of the white noise variance based on the sampled observations. The use of an explicit correction term in the least squares estimates or the use of some suitable numerical differentiation formulae eliminates the error in the estimates