Nonlinear Estimation

2 Literature review on quadrature based nonlinear estimation (cont'd...)

2.6 General filtering algorithm (cont'd...)

$\bullet$ Calculate innovation covariance \begin{equation*} \begin{split} P_{y,k \vert k-1}=&\sum_{j=1}^{P_{s}}w_{s,j}(Y_{j,k \vert k-1}-\hat{Y}_{k \vert k-1}(Y_{j,k \vert k-1}-\hat{Y}_{k \vert k-1}^{T}+\hat{R}_{k} \end{split} \end{equation*} $\bullet$ Calculate cross-covariance \begin{equation*} \begin{split} P_{xy,k \vert k-1}=&\sum_{j=1}^{P_{s}}w_{s,j}(\tilde{x}_{j,k \vert k-1}-\hat{x}_{k \vert k-1}) (Y_{j,k \vert k-1}-\hat{Y}_{k \vert k-1})^{T} \end{split} \end{equation*} $\bullet$ Calculate Kalman gain \begin{equation*} K_{k}=P_{xy,k \vert k-1}P_{y,k \vert k-1}^{-1} \end{equation*} $\bullet$ Compute the posterior mean \begin{equation*} \hat{x}_{k \vert k}=\hat{x}_{k \vert k-1}+K_{k}(y_{k}-\hat{Y}_{k \vert k-1}) \end{equation*} $\bullet$ Compute the posterior covariance \begin{equation*} P_{x,k \vert k}=P_{x,k\vert k-1}-K_{k}P_{y,k \vert k-1}K_{k}^{T} \end{equation*} where $P_s$ is the number of quadrature points.

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