Kalman Filtering and Smoothing to Estimate Real-Valued States and Integer Constants

New algorithms have been developed that solve linear Kalman filtering and fixed-interval smoothing problems that include dynamic real-valued states and static integer unknowns. These algorithms are useful for estimation problems in which some of the estimated quantities are known a priori to be constant integers, as in double-differenced carrier-phase Global Positioning System relative navigation. The new estimation algorithms solve integer linear least-squares problems to derive integer estimates from square-root information equations. The integer linear least-squares solver makes use of least-squares ambiguity decorrelation adjustment methods. The true optimal solution of each filtering or smoothing problem requires that all integers be included for the entire time interval, even those that apply to measurements that are remote from the current time point of interest. This fact causes the size of the integer linear least-squares problem to grow with the length of the time interval, thereby greatly slowing the execution speed of the solution algorithm. Alternative approximate methods for filtering and smoothing are proposed and tested; these are methods that bound the sizes of the integer linear least-squares problems. Bounded problem sizes are achieved by treating integers from remote-in-time measurements as real-valued unknowns, which allows them to be dropped from explicit consideration. The resulting algorithms have been tested using a truth-model simulation. Their accuracies can be very near to optimal, and they reduce the computational costs of the filter and the smoother.