Treatment of bias in recursive filtering

The problem of estimating the statexof a linear process in the presence of a constant but unknown bias vectorbis considered. This bias vector influences the dynamics and/or the observations. It is shown that the optimum estimate\hat{x}of the state can be expressed as\hat{x} = x + V_{x}\hat{b}(1) where\tilde{x}is the bias-free estimate, computed as if no bias were present,\hat{b}is the optimum estimate of the bias, and Vxis a matrix which can be interpreted as the ratio of the covariance of\tilde{x}and\hat{b}to the variance of\hat{b}. Moreover,\hat{b}can be computed in terms of the residuals in the bias-free estimate, and the matrix Vxdepends only on matrices which arise in the computation of the bias-free estimates. As a result, the computation of the optimum estimate\tilde{x}is effectively decoupled from the estimate of the bias\hat{b}, except for the final addition indicated by (1).