Discrete-time complementary models and smoothing algorithms: The correlated noise case

The concept of complementary models for discrete-time linear finite-dimensional systems with correlated observation and process noise is developed. Using this concept, a new algorithm for the fixed interval smoothing problem is obtained. The new algorithm offers great flexibility with respect to changes in the initial state variance
\Pi_{0}
. Next, the relationship among the new smoothing algorithm, the two-filter smoother, and the reversed-time Kalman filter is explored. It is shown that a similarity transformation on the Hamiltonian system simultaneously produces the new smoothing algorithm, as well as the reversed-time Kalman filter.