Optimal linear fixed-interval smoothing for colored noise

A convenient canonical form of the optimal fixed-interval smoother and its error covariance matrix are derived for the continuous linear systems with the colored measurement noise.The technique of state augmentation is not used so that the proposed smoother has the same dimension as that of the original state vector to be estimated, which is especially convenient for the higher dimensional system from the computational aspect.It turns out that in the smoothing solution for colored noise it is required to retain the original measurements in addition to the filtering solution.An interesting special case where the signal and the noise processes have the identical statistical property is discussed. In this case no improved estimate is obtained by smoothing the measurements. This fact is interpreted physically in terms of observability. It is also pointed out that the proposed smoothing solution remains valid when the measurements contain no noise