Nomenclature¶
Equations¶
Brookner¶
$$
\begin{aligned}
X^*_{n+1,n} &= \Phi X^*_{n,n} \\
X^*_{n,n} &= X^*_{n,n-1} +H_n(Y_n - MX^*_{n,n-1}) \\
H_n &= S^*_{n,n-1}M^T[R_n + MS^*_{n,n-1}M^T]^{-1} \\
S^*_{n,n-1} &= \Phi S^*_{n-1,n-1}\Phi^T + Q_n \\
S^*_{n-1,n-1} &= (I-H_{n-1}M)S^*_{n-1,n-2}
\end{aligned}$$
Gelb¶
$$
\begin{aligned}
\underline{\hat{x}}_k(-) &= \Phi_{k-1} \underline{\hat{x}}_{k-1}(+) \\
\underline{\hat{x}}_k(+) &= \underline{\hat{x}}_k(-) +K_k[Z_k - H_k\underline{\hat{x}}_k(-)] \\
K_k &= P_k(-)H_k^T[H_kP_k(-)H_k^T + R_k]^{-1}\\
P_k(+) &= \Phi_{k-1} P_{k-1}(+)\Phi_{k-1}^T + Q_{k-1} \\
P_k(-) &= (I-K_kH_k)P_k(-)
\end{aligned}$$
Brown¶
$$
\begin{aligned}
\hat{\textbf{x}}^-_{k+1} &= \mathbf{\phi}_{k}\hat{\textbf{x}}_{k} \\
\hat{\textbf{x}}_k &= \hat{\textbf{x}}^-_k +\textbf{K}_k[\textbf{z}_k - \textbf{H}_k\hat{\textbf{}x}^-_k] \\
\textbf{K}_k &= \textbf{P}^-_k\textbf{H}_k^T[\textbf{H}_k\textbf{P}^-_k\textbf{H}_k^T + \textbf{R}_k]^{-1}\\
\textbf{P}^-_{k+1} &= \mathbf{\phi}_k \textbf{P}_k\mathbf{\phi}_k^T + \textbf{Q}_{k} \\
\mathbf{P}_k &= (\mathbf{I}-\mathbf{K}_k\mathbf{H}_k)\mathbf{P}^-_k
\end{aligned}$$
Zarchan¶
$$
\begin{aligned}
\hat{x}_{k} &= \Phi_{k}\hat{x}_{k-1} + G_ku_{k-1} + K_k[z_k - H\Phi_{k}\hat{x}_{k-1} - HG_ku_{k-1} ] \\
M_{k} &= \Phi_k P_{k-1}\phi_k^T + Q_{k} \\
K_k &= M_kH^T[HM_kH^T + R_k]^{-1}\\
P_k &= (I-K_kH)M_k
\end{aligned}$$