Orthogonal Projection and Kalman Filter

Abstract

The elementary approach to the derivation of the optimal Kalman filtering process discussed in Chapter 2 has the advantage that the optimal estimate ${\hat{\mathbf{x}}_k = \hat{\mathbf{x}}_{k|k} }$ of the state vector x _k is easily understood to be a least-squares estimate of x _k with the properties that (i) the transformation that yields $\hat{\mathbf{x}}_kk$ from the data ${ \overline{\mathbf{v}}_k = [\mathbf{v}_0^{\mathsf{T}} \cdots \mathbf{v}_k^{\mathsf{T}}]^{\mathsf{T}} }$ is linear, (ii) ${ \hat{\mathbf{x}}_k }$ is unbiased in the sense that ${E(\hat{\mathbf{x}}_k) = E(\mathbf{x}_k) }$, and (iii) it yields a minimum variance estimate with ${ (\mathit{Var} (\overline{\underline{\epsilon}}_{k,k}))^{-1} }$ as the optimal weight. The disadvantage of this elementary approach is that certain matrices must be assumed to be nonsingular. In this chapter, we will drop the nonsingularity assumptions and give a rigorous derivation of the Kalman filtering algorithm.