Appendix_B_Symbols_and_Notations.ipynb

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   "source": [
    "#format the book\n",
    "%matplotlib inline\n",
    "from __future__ import division, print_function\n",
    "import matplotlib.pyplot as plt\n",
    "import book_format\n",
    "book_format.load_style()"
   ]
  },
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   "metadata": {},
   "source": [
    "# Symbology"
   ]
  },
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   "metadata": {},
   "source": [
    "This is just notes at this point. \n",
    "\n",
    "\n",
    "## State\n",
    "\n",
    "$x$ (Brookner, Zarchan, Brown)\n",
    "\n",
    "$\\underline{x}$ Gelb)\n",
    "\n",
    "## State at step n\n",
    "\n",
    "$x_n$ (Brookner)\n",
    "\n",
    "$x_k$ (Brown, Zarchan)\n",
    "\n",
    "$\\underline{x}_k$ (Gelb)\n",
    "\n",
    "\n",
    "\n",
    "## Prediction\n",
    "\n",
    "$x^-$\n",
    "\n",
    "$x_{n,n-1}$  (Brookner) \n",
    "\n",
    "$x_{k+1,k}$\n",
    "\n",
    "\n",
    "## measurement\n",
    "\n",
    "\n",
    "$x^*$\n",
    "\n",
    "\n",
    "\n",
    "Y_n (Brookner)\n",
    "\n",
    "##control transition Matrix\n",
    "\n",
    "$G$ (Zarchan)\n",
    "\n",
    "\n",
    "Not used (Brookner)"
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   "source": [
    "#Nomenclature\n",
    "\n",
    "\n",
    "## Equations\n",
    "### Brookner\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "X^*_{n+1,n} &= \\Phi X^*_{n,n} \\\\\n",
    "X^*_{n,n}  &= X^*_{n,n-1} +H_n(Y_n - MX^*_{n,n-1}) \\\\\n",
    "H_n &= S^*_{n,n-1}M^T[R_n + MS^*_{n,n-1}M^T]^{-1} \\\\\n",
    "S^*_{n,n-1} &= \\Phi S^*_{n-1,n-1}\\Phi^T + Q_n \\\\\n",
    "S^*_{n-1,n-1} &= (I-H_{n-1}M)S^*_{n-1,n-2}\n",
    "\\end{aligned}$$\n",
    "\n",
    "### Gelb\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\underline{\\hat{x}}_k(-) &= \\Phi_{k-1} \\underline{\\hat{x}}_{k-1}(+) \\\\\n",
    "\\underline{\\hat{x}}_k(+)  &= \\underline{\\hat{x}}_k(-) +K_k[Z_k - H_k\\underline{\\hat{x}}_k(-)] \\\\\n",
    "K_k &= P_k(-)H_k^T[H_kP_k(-)H_k^T + R_k]^{-1}\\\\\n",
    "P_k(+) &=  \\Phi_{k-1} P_{k-1}(+)\\Phi_{k-1}^T + Q_{k-1} \\\\\n",
    "P_k(-) &= (I-K_kH_k)P_k(-)\n",
    "\\end{aligned}$$\n",
    "\n",
    "\n",
    "### Brown\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat{\\textbf{x}}^-_{k+1} &= \\mathbf{\\phi}_{k}\\hat{\\textbf{x}}_{k} \\\\\n",
    "\\hat{\\textbf{x}}_k  &= \\hat{\\textbf{x}}^-_k +\\textbf{K}_k[\\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{}x}^-_k] \\\\\n",
    "\\textbf{K}_k &= \\textbf{P}^-_k\\textbf{H}_k^T[\\textbf{H}_k\\textbf{P}^-_k\\textbf{H}_k^T + \\textbf{R}_k]^{-1}\\\\\n",
    "\\textbf{P}^-_{k+1} &=  \\mathbf{\\phi}_k \\textbf{P}_k\\mathbf{\\phi}_k^T + \\textbf{Q}_{k} \\\\\n",
    "\\mathbf{P}_k &= (\\mathbf{I}-\\mathbf{K}_k\\mathbf{H}_k)\\mathbf{P}^-_k\n",
    "\\end{aligned}$$\n",
    "\n",
    "\n",
    "### Zarchan\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat{x}_{k} &= \\Phi_{k}\\hat{x}_{k-1} + G_ku_{k-1} + K_k[z_k - H\\Phi_{k}\\hat{x}_{k-1} - HG_ku_{k-1} ] \\\\\n",
    "M_{k} &=  \\Phi_k P_{k-1}\\phi_k^T + Q_{k} \\\\\n",
    "K_k &= M_kH^T[HM_kH^T + R_k]^{-1}\\\\\n",
    "P_k &= (I-K_kH)M_k\n",
    "\\end{aligned}$$"
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    "### Wikipedia\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat{\\textbf{x}}_{k\\mid k-1} &= \\textbf{F}_{k}\\hat{\\textbf{x}}_{k-1\\mid k-1} + \\textbf{B}_{k} \\textbf{u}_{k} \\\\\n",
    "\\textbf{P}_{k\\mid k-1} &=  \\textbf{F}_{k} \\textbf{P}_{k-1\\mid k-1} \\textbf{F}_{k}^{\\text{T}} + \\textbf{Q}_{k}\\\\\n",
    "\\tilde{\\textbf{y}}_k &= \\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{x}}_{k\\mid k-1} \\\\\n",
    "\\textbf{S}_k &= \\textbf{H}_k \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^\\text{T} + \\textbf{R}_k \\\\\n",
    "\\textbf{K}_k &= \\textbf{P}_{k\\mid k-1}\\textbf{H}_k^\\text{T}\\textbf{S}_k^{-1} \\\\\n",
    "\\hat{\\textbf{x}}_{k\\mid k} &= \\hat{\\textbf{x}}_{k\\mid k-1} + \\textbf{K}_k\\tilde{\\textbf{y}}_k \\\\\n",
    "\\textbf{P}_{k|k} &= (I - \\textbf{K}_k \\textbf{H}_k) \\textbf{P}_{k|k-1}\n",
    "\\end{aligned}$$"
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  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Labbe\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat{\\textbf{x}}^-_{k+1} &= \\mathbf{F}_{k}\\hat{\\textbf{x}}_{k} + \\mathbf{B}_k\\mathbf{u}_k \\\\\n",
    "\\textbf{P}^-_{k+1} &=  \\mathbf{F}_k \\textbf{P}_k\\mathbf{F}_k^T + \\textbf{Q}_{k} \\\\\n",
    "\\textbf{y}_k &= \\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{}x}^-_k \\\\\n",
    "\\mathbf{S}_k &= \\textbf{H}_k\\textbf{P}^-_k\\textbf{H}_k^T + \\textbf{R}_k \\\\\n",
    "\\textbf{K}_k &= \\textbf{P}^-_k\\textbf{H}_k^T\\mathbf{S}_k^{-1} \\\\\n",
    "\\hat{\\textbf{x}}_k  &= \\hat{\\textbf{x}}^-_k +\\textbf{K}_k\\textbf{y} \\\\\n",
    "\\mathbf{P}_k &= (\\mathbf{I}-\\mathbf{K}_k\\mathbf{H}_k)\\mathbf{P}^-_k\n",
    "\\end{aligned}$$"
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