Symbols and Notations¶

Here is a collection of the notation used by various authors for the linear Kalman filter equations.

Labbe¶

\begin{aligned} \overline{\mathbf x} &= \mathbf{Fx} + \mathbf{Bu} \\ \overline{\mathbf P} &= \mathbf{FPF}^\mathsf{T} + \mathbf Q \\ \\ \mathbf y &= \mathbf z - \mathbf{H}\overline{\mathbf x} \\ \mathbf S &= \mathbf{H}\overline{\mathbf P}\mathbf{H}^\mathsf{T} + \mathbf R \\ \mathbf K &= \overline{\mathbf P}\mathbf{H}^\mathsf{T}\mathbf{S}^{-1} \\ \mathbf x &= \overline{\mathbf x} +\mathbf{Ky} \\ \mathbf P &= (\mathbf{I}-\mathbf{KH})\overline{\mathbf P} \end{aligned}

Wikipedia¶

\begin{aligned} \hat{\mathbf x}_{k\mid k-1} &= \mathbf{F}_{k}\hat{\mathbf x}_{k-1\mid k-1} + \mathbf{B}_{k} \mathbf{u}_{k} \\ \mathbf P_{k\mid k-1} &= \mathbf{F}_{k} \mathbf P_{k-1\mid k-1} \mathbf{F}_{k}^{\textsf{T}} + \mathbf Q_{k}\\ \tilde{\mathbf{y}}_k &= \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf x}_{k\mid k-1} \\ \mathbf{S}_k &= \mathbf{H}_k \mathbf P_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k \\ \mathbf{K}_k &= \mathbf P_{k\mid k-1}\mathbf{H}_k^\textsf{T}\mathbf{S}_k^{-1} \\ \hat{\mathbf x}_{k\mid k} &= \hat{\mathbf x}_{k\mid k-1} + \mathbf{K}_k\tilde{\mathbf{y}}_k \\ \mathbf P_{k|k} &= (I - \mathbf{K}_k \mathbf{H}_k) \mathbf P_{k|k-1} \end{aligned}

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^\mathsf{T}[R_n + MS^*_{n,n-1}M^\mathsf{T}]^{-1} \\ S^*_{n,n-1} &= \Phi S^*_{n-1,n-1}\Phi^\mathsf{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^\mathsf{T}[H_kP_k(-)H_k^\mathsf{T} + R_k]^{-1} \\ P_k(+) &= \Phi_{k-1} P_{k-1}(+)\Phi_{k-1}^\mathsf{T} + Q_{k-1} \\ P_k(-) &= (I-K_kH_k)P_k(-) \end{aligned}

Brown¶

\begin{aligned} \hat{\mathbf x}^-_{k+1} &= \mathbf{\phi}_{k}\hat{\mathbf x}_{k} \\ \hat{\mathbf x}_k &= \hat{\mathbf x}^-_k +\mathbf{K}_k[\mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{}x}^-_k] \\ \mathbf{K}_k &= \mathbf P^-_k\mathbf{H}_k^\mathsf{T}[\mathbf{H}_k\mathbf P^-_k\mathbf{H}_k^T + \mathbf{R}_k]^{-1}\\ \mathbf P^-_{k+1} &= \mathbf{\phi}_k \mathbf P_k\mathbf{\phi}_k^\mathsf{T} + \mathbf 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^\mathsf{T} + Q_{k} \\ K_k &= M_kH^\mathsf{T}[HM_kH^\mathsf{T} + R_k]^{-1}\\ P_k &= (I-K_kH)M_k \end{aligned}