On optimal linear smoothing theory

The algorithm is developed for generating the optimal smoothed estimate xˆ(t|t+T) of the state x(t) of a continuous linear system, where t is continuous time, T is a positive real constant, and t + T is the time of the most recent measurement. A linear matrix differential equation whose solution gives the covariance matrix of the smoothing error xˆ(t|t+T)=x(T)-xˆ(t|t+T) is then derived. Computational aspects involved in mechanizing the algorithm are discussed in terms of the algorithm's dependence on the solutions of the prediction, filtering, and fixed-point smoothing problems. The relation of the present results to other studies in estimation theory is indicated