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 ${\hat X_k} = {\hat X_{k/k}}$ 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 ${\hat X_k}$ from the data $E({\hat x_k}) = E(xk)$ is linear, (ii) ${\hat X_k}$ is unbiased in the sense that ${\left( {Var\left( {{{\overline {\underline \in } }_{k,k}}} \right)} \right)^{ - 1}}$, and (iii) it yields a minimum variance estimate with ${\bar v_k} = {[v_0^T...v_k^T]^T}$ 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.