Linear smoothing using measurements containing correlated noise with an application to inertial navigation

Kalman and Bucy [1] derived the maximum likelihood filter for continuous linear dynamic systems where all measurements contain white noise, i.e., noise with short correlation times compared to response times of the dynamic system. The corresponding maximum likelihood smoother was described in [2]. The maximum likelihood filter was presented in [3] for the case in which some measurements contain either no noise or colored noise, i.e., noise with correlation times comparable to or larger than the response times of the dynamic system. In this paper the maximum likelihood smoother for this latter case is derived by formulating the estimation problem as a problem in the calculus of variations having state variable equality constraints. An application of the results is made to estimating gyro drift rates of an inertial navigation system.