Some approaches on how to use the delta operator when identifying continuous-time processes

When identifying a continuous-time process from discrete-time data, an obvious approach is to replace the derivative operator in a continuous-time model by an approximation such as the delta operator. In some cases, a linear regression model can then be formulated. The well-known least squares method, or alternatively an instrumental variable estimator, would be very desirable to apply, since these estimators enjoy good numerical properties and low computational complexity, in particular for fast or nonuniform sampling. The methods work well for deterministic data, while stochastic disturbances bring additional difficulties. The focus of this paper is therefore on the effect of disturbances. More precisely, it is examined under what conditions the least-squares or the instrumental variables method can be applied successfully. The case of an autoregressive process is studied, and techniques to handle the corresponding linear regression are described. The precise choice of derivative approximation is crucial. Standard delta operator approximations cannot be used directly, but some simple modifications can cure the situation. We review several such modifications in the paper