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

    1.3 Kalman Filter

    1.3.1 Derivation of Kalman filter

    We define a priori estimate error $e_{k|k-1}$ is \begin{equation}\label{3d} e_{k|k-1}=\hat{X}_{k|k-1}-X_{k}, \end{equation} and a posteriori estimate error $e_{k|k}$ is \begin{equation}\label{3e} e_{k|k}=\hat{X}_{k|k}-X_{k}. \end{equation} Apriori estimate error covariance $P_{k|k-1}$ is defined as \begin{equation}\label{3f} P_{k|k-1}=E[e_{k|k-1}e_{k|k-1}^{T}], \end{equation} and aposteriori estimate error covariance $P_{k|k}$ is defined as \begin{equation}\label{3g} P_{k|k}=E[e_{k|k}e_{k|k}^{T}]. \end{equation} Now, substituting error expression, apriori error covariance becomes \begin{equation}\label{3h} P_{k|k-1}=E[(\hat{X}_{k|k-1}-X_{k})(\hat{X}_{k|k-1}-X_{k})^{T}]. \end{equation} Now, the expression of the state estimator could be written as: \begin{equation}\label{3i} \hat{X}_{k|k}=\hat{X}_{k|k-1}+K_{k}(Y_{k}-H_{k}\hat{X}_{k|k-1}), \end{equation} where, $K_{k}$ is the blending factor, which is called as Kalman gain matrix.

    Now, to design Kalman filter we have to find $K_{k}$. From the above expression we could write, \begin{equation}\label{3j} P_{k|k}=E[(\hat{X}_{k|k}-X_{k})(\hat{X}_{k|k}-X_{k})^{T}]. \end{equation}

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