Treatment of bias in recursive filtering

The problem of estimating the state of a linear process in the presence of a constant but unknown bias vector is considered. This bias vector influences the dynamics and/or the observations. It is shown that the optimum estimate of the state can be expressed as (1) where is the bias-free estimate, computed as if no bias were present, is the optimum estimate of the bias, and Vxis a matrix which can be interpreted as the ratio of the covariance of and to the variance of . Moreover, 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 is effectively decoupled from the estimate of the bias , except for the final addition indicated by (1).