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rlabbe
GitHub Repository: rlabbe/Kalman-and-Bayesian-Filters-in-Python
 Path: blob/master/Appendix-B-Symbols-and-Notations.ipynb
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Kernel: Python 3

Symbols and Notations

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

Labbe

x‾=Fx+BuP‾=FPFT+Qy=z−Hx‾S=HP‾HT+RK=P‾HTS−1x=x‾+KyP=(I−KH)P‾\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

x^k∣k−1=Fkx^k−1∣k−1+BkukPk∣k−1=FkPk−1∣k−1FkT+Qky~k=zk−Hkx^k∣k−1Sk=HkPk∣k−1HkT+RkKk=Pk∣k−1HkTSk−1x^k∣k=x^k∣k−1+Kky~kPk∣k=(I−KkHk)Pk∣k−1\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

Xn+1,n∗=ΦXn,n∗Xn,n∗=Xn,n−1∗+Hn(Yn−MXn,n−1∗)Hn=Sn,n−1∗MT[Rn+MSn,n−1∗MT]−1Sn,n−1∗=ΦSn−1,n−1∗ΦT+QnSn−1,n−1∗=(I−Hn−1M)Sn−1,n−2∗\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

x^‾k(−)=Φk−1x^‾k−1(+)x^‾k(+)=x^‾k(−)+Kk[Zk−Hkx^‾k(−)]Kk=Pk(−)HkT[HkPk(−)HkT+Rk]−1Pk(+)=Φk−1Pk−1(+)Φk−1T+Qk−1Pk(−)=(I−KkHk)Pk(−)\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

x^k+1−=ϕkx^kx^k=x^k−+Kk[zk−Hkx^k−]Kk=Pk−HkT[HkPk−HkT+Rk]−1Pk+1−=ϕkPkϕkT+QkPk=(I−KkHk)Pk−\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

x^k=Φkx^k−1+Gkuk−1+Kk[zk−HΦkx^k−1−HGkuk−1]Mk=ΦkPk−1ϕkT+QkKk=MkHT[HMkHT+Rk]−1Pk=(I−KkH)Mk\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}