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    1. Kalman Filter Tutorial (Cont'd...)

    1.3.1 Derivation of Kalman filter (cont'd...)

    Now, we have to compute a next priori estimate $\hat{X}_{k+1|k}$ with the help of a posteriori estimate $\hat{X}_{k|k}$. We could ignore process noise, $w_{k}$, because it has zero mean, i.e. \begin{equation}\label{3s} \hat{X}_{k+1|k}=\phi_{k}\hat{X}_{k|k}. \end{equation} Now, we have to calculate apriori error for $(k+1)_{th}$ step. So \begin{equation}\label{3t} e_{k+1|k}=\hat{X}_{k+1|k}-X_{k+1}=\phi_{k}\hat{X}_{k|k}-\phi_{k}X_{k}-w_{k}=\phi_{k}e_{k|k}-w_{k}. \end{equation} Now, we find apriori error covariance matrix $P_{k+1|k}$, assuming error $e_{k+1|k}$ has zero mean. i.e. \begin{equation}\label{3u} P_{k+1|k}=E[e_{k+1|k}e_{k+1|k}^{T}]. \end{equation} Now, substituting the value of $e_{k+1|k}$ into $P_{k+1|k}$, to obtain \begin{equation*} P_{k+1|k}=E[(\phi_{k}e_{k|k}-w_{k})(\phi_{k}e_{k|k}-w_{k})^{T}], \end{equation*} or, \begin{equation}\label{3v} \begin{split} P_{k+1|k} =E[\phi_{k}e_{k|k}e_{k|k}^{T}\phi_{k}^{T}-\phi_{k}e_{k|k}w_{k}^{T}-w_{k}e_{k|k}^{T}\phi_{k}^{T}+w_{k}w_{k}^{T}], \end{split} \end{equation} where, $e_{k|k}$ and $w_{k}$ are uncorrelated. So, second and third term of the above equation will be zero. i.e. \begin{equation*} P_{k+1|k}=\phi_{k}E[e_{k|k}e_{k|k}^{T}]\phi_{k}^{T}+E[w_{k}w_{k}^{T}]. \end{equation*} or, \begin{equation}\label{3w} P_{k+1|k}=\phi_{k}P_{k|k}\phi_{k}^{T}+Q_{k}. \end{equation}



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