Optimal estimator for linear distributed parameter systems

An estimator is developed for the class of systems which can be modeled as a partial differential equation with input disturbances having Gaussian statistics. The errors associated with output observations are considered to be additive and Gaussian in nature. The estimator is shown to be a linear distributed dynamic system whose input is the observation process. Under the requirement that the estimate is ´unbiased´ it is seen that the characterisation of the gain matrix completely determines the estimator. The optimal gain matrix is obtained using the maximum principle. The optimal estimator so obtained turns out to be a generalised version of Kalman-Bucy filter.