Optimal weighting of a priori statistics in linear estimation theory

摘要

Many scientific and engineering applications require the determination of the state of a dynamical system at some reference time. A common solution technique is to perform a Taylor series expansion about a nominal solution and compute linear corrections to this nominal by the use of a set of discrete observations of the dynamical system. In cases where the observational data set is ill conditioned, statistical information on the nominal solution is commonly incorporated into the solution process in order to provide stability in the computation of the linear corrections. This statistical information is generally the uncertainty in the nominal solution, known as the “a priori covariance.” This a priori information is typically weighted equally with the observational data in conventional minimum variance (or maximum likelihood) solution methods. Often the a priori information available for certain types of dynamical systems is inaccurate or generated by inadequate means. It has been shown that the accuracy of the estimates is highly dependent on the choice of the a priori covariance. Inaccurate a priori information can contribute to large errors in the estimates in ill-conditioned problems. Methods developed herein provide for the “optimal” weighting of the a priori covariance used in the solution of the linear correction to the nominal solution. These methods are demonstrated by their application to a space mechanics simulation in which a satellite's orbit must be precisely determined. Results are compared to those obtained by conventional minimum variance solution techniques in order to demonstrate the possible improvement in the estimate accuracy achievable by the use of such optimal weighting techniques.