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

    1.4 Properties of Kalman filter (cont'd...)

    Innovation sequence

    The Kalman filter innovation $Z_{k}$ is given by \begin{equation}\label{4a} Z_{k}=Y_{k}-H_{k}\hat{X}_{k|k-1}=H_{k}X_{k}+v_{k}-H_{k}\hat{X}_{k|k-1}=v_{k}-H_{k}e_{k|k-1}, \end{equation} and, its covariance is given by \begin{equation}\label{4b} E[Z_{k}Z_{k}^{T}]=R_{k}+H_{k}P_{k|k-1}H_{k}^{T}. \end{equation} Now, we write the innovation $Z_{k}$ in quadratic form $Z_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}Z_{k}$, and we take expectation of it, i.e. \begin{equation}\label{4c} E[Z_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}Z_{k}]=G, \end{equation} where, $G$ is a scalar quantity. Now, substituting the value of $Z_{k}=v_{k}-H_{k}e_{k|k-1}$ to the expression of $E[Z_{k}Z_{k}^{T}]$, \begin{equation}\label{4e} E[Z_{k}Z_{k}^{T}]=E[v_{k}v_{k}^{T}]+E[H_{k}e_{k|k-1}e_{k|k-1}^{T}H_{k}^{T}]-E[H_{k}e_{k|k-1}v_{k}^{T}]-E[v_{k}e_{k|k-1}^{T}H_{k}^{T}], \end{equation} or, \begin{equation} E[Z_{k}Z_{k}^{T}]=E[v_{k}v_{k}^{T}]+H_{k}E[e_{k|k-1}e_{k|k-1}^{T}]H_{k}^{T}-H_{k}E[e_{k|k-1}v_{k}^{T}]-E[v_{k}e_{k|k-1}^{T}]H_{k}^{T}, \end{equation} or, \begin{equation} E[Z_{k}Z_{k}^{T}]=R_{k}+H_{k}P_{k|k-1}H_{k}^{T}-H_{k}E[e_{k|k-1}v_{k}^{T}]-E[v_{k}e_{k|k-1}^{T}]H_{k}^{T}, \end{equation} where, $v_{k}$ and $e_{k|k-1}$ are uncorrelated. Now, substituting the value of $Z_k$ $G$ becomes, \begin{equation*} G=E[(v_{k}-H_{k}e_{k|k-1})^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}(v_{k}-H_{k}e_{k|k-1})], \end{equation*} or, \begin{equation} \begin{split} &G =E[v_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}v_{k}+e_{k|k-1}^{T}H_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}H_{k}e_{k|k-1} \\& \hspace{2cm}-2e_{k|k-1}^{T}H_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}v_{k}]. \end{split} \end{equation} or, \begin{equation}\label{4g} \begin{split} &G =E[v_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}v_{k}+e_{k|k-1}^{T}H_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}H_{k}e_{k|k-1}] \\& \end{split} \end{equation}

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