In [1]:
#format the book
%matplotlib inline
from __future__ import division, print_function
import matplotlib.pyplot as plt
import book_format
book_format.load_style()
Out[1]:

Symbology

This is just notes at this point.

State

$x$ (Brookner, Zarchan, Brown)

$\underline{x}$ Gelb)

State at step n

$x_n$ (Brookner)

$x_k$ (Brown, Zarchan)

$\underline{x}_k$ (Gelb)

Prediction

$x^-$

$x_{n,n-1}$ (Brookner)

$x_{k+1,k}$

measurement

$x^*$

Y_n (Brookner)

control transition Matrix

$G$ (Zarchan)

Not used (Brookner)

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

$$ \begin{aligned} \hat{x}_{k} &= \Phi_{k}\hat{x}_{k-1} + G_ku_{k-1} + K_k[z_k - H\Phi_{k}\hat{x}_{k-1} - HG_ku_{k-1} ] \\ M_{k} &= \Phi_k P_{k-1}\phi_k^T + Q_{k} \\ K_k &= M_kH^T[HM_kH^T + R_k]^{-1}\\ P_k &= (I-K_kH)M_k \end{aligned}$$

Wikipedia

$$ \begin{aligned} \hat{\textbf{x}}_{k\mid k-1} &= \textbf{F}_{k}\hat{\textbf{x}}_{k-1\mid k-1} + \textbf{B}_{k} \textbf{u}_{k} \\ \textbf{P}_{k\mid k-1} &= \textbf{F}_{k} \textbf{P}_{k-1\mid k-1} \textbf{F}_{k}^{\text{T}} + \textbf{Q}_{k}\\ \tilde{\textbf{y}}_k &= \textbf{z}_k - \textbf{H}_k\hat{\textbf{x}}_{k\mid k-1} \\ \textbf{S}_k &= \textbf{H}_k \textbf{P}_{k\mid k-1} \textbf{H}_k^\text{T} + \textbf{R}_k \\ \textbf{K}_k &= \textbf{P}_{k\mid k-1}\textbf{H}_k^\text{T}\textbf{S}_k^{-1} \\ \hat{\textbf{x}}_{k\mid k} &= \hat{\textbf{x}}_{k\mid k-1} + \textbf{K}_k\tilde{\textbf{y}}_k \\ \textbf{P}_{k|k} &= (I - \textbf{K}_k \textbf{H}_k) \textbf{P}_{k|k-1} \end{aligned}$$

Labbe

$$ \begin{aligned} \hat{\textbf{x}}^-_{k+1} &= \mathbf{F}_{k}\hat{\textbf{x}}_{k} + \mathbf{B}_k\mathbf{u}_k \\ \textbf{P}^-_{k+1} &= \mathbf{F}_k \textbf{P}_k\mathbf{F}_k^T + \textbf{Q}_{k} \\ \textbf{y}_k &= \textbf{z}_k - \textbf{H}_k\hat{\textbf{}x}^-_k \\ \mathbf{S}_k &= \textbf{H}_k\textbf{P}^-_k\textbf{H}_k^T + \textbf{R}_k \\ \textbf{K}_k &= \textbf{P}^-_k\textbf{H}_k^T\mathbf{S}_k^{-1} \\ \hat{\textbf{x}}_k &= \hat{\textbf{x}}^-_k +\textbf{K}_k\textbf{y} \\ \mathbf{P}_k &= (\mathbf{I}-\mathbf{K}_k\mathbf{H}_k)\mathbf{P}^-_k \end{aligned}$$