Treatment of Bias in Recursive Filtering

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