Kalman-and-Bayesian-Filters-in-Python/Appendix-B-Symbols-and-Notations.ipynb

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   "source": [
    "# Symbols and Notations\n",
    "\n",
    "Here is a collection of the notation used by various authors for the linear Kalman filter equations."
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    "## Labbe\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\overline{\\mathbf x} &= \\mathbf{Fx} + \\mathbf{Bu} \\\\\n",
    "\\overline{\\mathbf P} &=  \\mathbf{FPF}^\\mathsf{T} + \\mathbf Q \\\\ \\\\\n",
    "\\mathbf y &= \\mathbf z - \\mathbf{H}\\overline{\\mathbf x} \\\\\n",
    "\\mathbf S &= \\mathbf{H}\\overline{\\mathbf P}\\mathbf{H}^\\mathsf{T} + \\mathbf R \\\\\n",
    "\\mathbf K &= \\overline{\\mathbf P}\\mathbf{H}^\\mathsf{T}\\mathbf{S}^{-1} \\\\\n",
    "\\mathbf x  &= \\overline{\\mathbf x} +\\mathbf{Ky} \\\\\n",
    "\\mathbf P &= (\\mathbf{I}-\\mathbf{KH})\\overline{\\mathbf P}\n",
    "\\end{aligned}$$\n",
    "\n",
    "\n",
    "## Wikipedia\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat{\\mathbf x}_{k\\mid k-1} &= \\mathbf{F}_{k}\\hat{\\mathbf x}_{k-1\\mid k-1} + \\mathbf{B}_{k} \\mathbf{u}_{k} \\\\\n",
    "\\mathbf P_{k\\mid k-1} &=  \\mathbf{F}_{k} \\mathbf P_{k-1\\mid k-1} \\mathbf{F}_{k}^{\\textsf{T}} + \\mathbf Q_{k}\\\\\n",
    "\\tilde{\\mathbf{y}}_k &= \\mathbf{z}_k - \\mathbf{H}_k\\hat{\\mathbf x}_{k\\mid k-1} \\\\\n",
    "\\mathbf{S}_k &= \\mathbf{H}_k \\mathbf P_{k\\mid k-1} \\mathbf{H}_k^\\textsf{T} + \\mathbf{R}_k \\\\\n",
    "\\mathbf{K}_k &= \\mathbf P_{k\\mid k-1}\\mathbf{H}_k^\\textsf{T}\\mathbf{S}_k^{-1} \\\\\n",
    "\\hat{\\mathbf x}_{k\\mid k} &= \\hat{\\mathbf x}_{k\\mid k-1} + \\mathbf{K}_k\\tilde{\\mathbf{y}}_k \\\\\n",
    "\\mathbf P_{k|k} &= (I - \\mathbf{K}_k \\mathbf{H}_k) \\mathbf P_{k|k-1}\n",
    "\\end{aligned}$$\n",
    "\n",
    "## Brookner\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "X^*_{n+1,n} &= \\Phi X^*_{n,n} \\\\\n",
    "X^*_{n,n}  &= X^*_{n,n-1} +H_n(Y_n - MX^*_{n,n-1}) \\\\\n",
    "H_n &= S^*_{n,n-1}M^\\mathsf{T}[R_n + MS^*_{n,n-1}M^\\mathsf{T}]^{-1} \\\\\n",
    "S^*_{n,n-1} &= \\Phi S^*_{n-1,n-1}\\Phi^\\mathsf{T} + Q_n \\\\\n",
    "S^*_{n-1,n-1} &= (I-H_{n-1}M)S^*_{n-1,n-2}\n",
    "\\end{aligned}$$\n",
    "\n",
    "## Gelb\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\underline{\\hat{x}}_k(-) &= \\Phi_{k-1} \\underline{\\hat{x}}_{k-1}(+) \\\\\n",
    "\\underline{\\hat{x}}_k(+) &= \\underline{\\hat{x}}_k(-) +K_k[Z_k - H_k\\underline{\\hat{x}}_k(-)] \\\\\n",
    "K_k &= P_k(-)H_k^\\mathsf{T}[H_kP_k(-)H_k^\\mathsf{T} + R_k]^{-1} \\\\\n",
    "P_k(+) &=  \\Phi_{k-1} P_{k-1}(+)\\Phi_{k-1}^\\mathsf{T} + Q_{k-1} \\\\\n",
    "P_k(-) &= (I-K_kH_k)P_k(-)\n",
    "\\end{aligned}$$\n",
    "\n",
    "\n",
    "## Brown\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat{\\mathbf x}^-_{k+1} &= \\mathbf{\\phi}_{k}\\hat{\\mathbf x}_{k} \\\\\n",
    "\\hat{\\mathbf x}_k  &= \\hat{\\mathbf x}^-_k +\\mathbf{K}_k[\\mathbf{z}_k - \\mathbf{H}_k\\hat{\\mathbf{}x}^-_k] \\\\\n",
    "\\mathbf{K}_k &= \\mathbf P^-_k\\mathbf{H}_k^\\mathsf{T}[\\mathbf{H}_k\\mathbf P^-_k\\mathbf{H}_k^T + \\mathbf{R}_k]^{-1}\\\\\n",
    "\\mathbf P^-_{k+1} &=  \\mathbf{\\phi}_k \\mathbf P_k\\mathbf{\\phi}_k^\\mathsf{T} + \\mathbf Q_{k} \\\\\n",
    "\\mathbf P_k &= (\\mathbf{I}-\\mathbf{K}_k\\mathbf{H}_k)\\mathbf P^-_k\n",
    "\\end{aligned}$$\n",
    "\n",
    "\n",
    "## Zarchan\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat{x}_{k} &= \\Phi_{k}\\hat{x}_{k-1} + G_ku_{k-1} + K_k[z_k - H\\Phi_{k}\\hat{x}_{k-1} - HG_ku_{k-1} ] \\\\\n",
    "M_{k} &=  \\Phi_k P_{k-1}\\phi_k^\\mathsf{T} + Q_{k} \\\\\n",
    "K_k &= M_kH^\\mathsf{T}[HM_kH^\\mathsf{T} + R_k]^{-1}\\\\\n",
    "P_k &= (I-K_kH)M_k\n",
    "\\end{aligned}$$"
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