State variable estimation using adaptive Kalman filter with robust smoothing

The development of a conventional Kalman filter is based on full knowledge of system parameters, noise statistics and deterministic forcing functions. This work addresses the problem of known system parameters and unknown noise statistics and deterministic forcing functions. Two concepts are investigated: 1) adaptive weight functions for the Kalman filter gain and error covariance matrices, where these weights are functions of sample means and variances of the innovations sequence; and 2) robust smoothing of the estimated state variables. The concepts presented relative to this particular problem address the limited class of linear system dynamics with associated linear measurements. Nonlinear system dynamics with associated linear or nonlinear measurements, however, are not precluded. The concepts apply to those cases where the observations made by a sensor are the variables to be estimated. An application to a simple linear system is presented; however, primary application would be to the estimation of position, velocity and acceleration for a maneuvering body in three dimensional space based on observed data collected by a remote sensor tracking the maneuvering body. Estimates of the state variables using the adaptive process for the simple linear system during the periods when the system is not being forced are relatively close to those of the conventional Kalman filter for congruent periods, but there is some increase in mean square error because the adaptive estimator is no longer optimal. During periods when the system is being forced a vast improvement, as compared with those estimates of the conventional Kalman filter, is realized with the adaptive gain, covariance weight, and associated robust smoothing procedure. The estimates derived with the adaptive procedure during the periods of system forcing do, however, contain a considerable level of mean-square error. This seems to be a prevailing shortfall of adaptive estimation procedures. The tradeoff is knowledge of the deterministic forcing functions versus high mean-square estimate error in the absence of that knowledge.