Use of approximate discrete-time models in filtering and regulation of continuous-time processes

A commonly used approximate discrete-time model for the estimation of the states of linear, continuous-time invariant processes is analysed. This analysis is performed by carefully defining and evaluating the true-error-covariance matrix of the estimate and by comparing it with what has been commonly believed to be the error covariance matrix. It is shown that there exists a class of unstable processes for which the former error-covariance matrix attains an unbounded norm, in spite of the fact that the norm of the other error-covariance matrix remains bounded. It is also shown that for another class of processes, the approximate model provides a particularly good estimation for short sampling periods. Based on the equations that govern the behaviour of the estimation-error covariance matrices, an alternative improved approximation is proposed that always leads to a true-error-covariance matrix with a bounded norm. Finally, following similar lines of analysis, it is shown that the approximate discrete-time models for the dual linear-quadratic regulation problem exhibit entirely different behaviour in that the cost may become unbounded even for stable continuous-time processes.