Ill-Conditioned Covariance Matrices in the First-Order Two-Step Estimator
James L. Garrison, Penina Axelrad, N. Jeremy Kasdin · Journal of Guidance Control and Dynamics · 1998
The e rst-order two-step estimator is found to occasionally produce e rst-step covariance matrices with very low, sometimes negative, eigenvalues. These low eigenvalues can cause large errors or meaningless estimates. A single matrix is found, which isshown to havea rank equalto the difference between thenumber of e rst- and second-step states.Furthermore, itisdemonstratedthatthebasisofthecolumnspaceofthismatrixremainse xedoncethelarge initial state error has decreased. A test matrix containing the (constant) basis of this column space and the partial derivative matrix relating e rst and second step states is derived. This matrix numerically drops rank at the same locationsthat thee rst-step covariancedoes.A simple exampleproblem involving dynamicsdescribed by two states and a range measurement illustrates the cause of this anomaly and application of the aforementioned numerical test.Suggestedmodie cationstothee lterthatcanmitigatethenumericalproblemscausedbythisanomalyaregiven.