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
\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 V
x
is 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 V
x
depends 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).