daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Fixed eqn alignment issues. overline to bar.

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Some equations used \\ without a gathered or aligned block.
They render fine in the notebook, not not in the PDF.

Also, switched back my ill chosen use of \overline for \bar.
This commit is contained in:
Roger Labbe
2016-01-24 14:08:07 -08:00
parent d5ea503fde
commit 71da22ad8c
9 changed files with 199 additions and 180 deletions
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15
Supporting_Notebooks/Iterative-Least-Squares-for-Sensor-Fusion.ipynb
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@@ -477,10 +477,11 @@
"\n",
"Taking the transpose of each side gives\n",
"\n",
"$${\\delta \\mathbf x} = ({{\\delta \\mathbf z^-}^\\mathsf{T}\\mathbf H(\\mathbf H^\\mathsf{T}\\mathbf H)^{-1}})^\\mathsf{T} \\\\\n",
"={{(\\mathbf H^\\mathsf{T}\\mathbf H)^{-1}}^T\\mathbf H^\\mathsf{T} \\delta \\mathbf z^-} \\\\\n",
"={{(\\mathbf H^\\mathsf{T}\\mathbf H)^{-1}}\\mathbf H^\\mathsf{T} \\delta \\mathbf z^-}\n",
"$$\n",
"$$\\begin{aligned}\n",
"{\\delta \\mathbf x} &= ({{\\delta \\mathbf z^-}^\\mathsf{T}\\mathbf H(\\mathbf H^\\mathsf{T}\\mathbf H)^{-1}})^\\mathsf{T} \\\\\n",
"&={{(\\mathbf H^\\mathsf{T}\\mathbf H)^{-1}}^T\\mathbf H^\\mathsf{T} \\delta \\mathbf z^-} \\\\\n",
"&={{(\\mathbf H^\\mathsf{T}\\mathbf H)^{-1}}\\mathbf H^\\mathsf{T} \\delta \\mathbf z^-}\n",
"\\end{aligned}$$\n",
"\n",
"For various reasons you may want to weigh some measurement more than others. We can do that with the equation\n",
"\n",
@@ -660,8 +661,10 @@
"source": [
"So let's think about this. The first iteration is essentially performing the computation that the linear Kalman filter computes during the update step:\n",
"\n",
"$$\\mathbf{y} = \\mathbf z - \\mathbf{Hx}\\\\\n",
"\\mathbf x = \\mathbf x + \\mathbf{Ky}$$\n",
"$$\\begin{aligned}\n",
"\\mathbf y &= \\mathbf z - \\mathbf{Hx}\\\\\n",
"\\mathbf x &= \\mathbf x + \\mathbf{Ky}\n",
"\\end{aligned}$$\n",
"\n",
"where the Kalman gain equals one. You can see that despite the very inaccurate initial guess (900, 90) the computed value for $\\mathbf x$, (805.4, 205.3), was very close to the actual value of (800, 200). However, it was not perfect. But after three iterations the ILS algorithm was able to find the exact answer. So hopefully it is clear why we use ILS instead of doing the sensor fusion with the Kalman filter - it gives a better result. Of course, we started with a very inaccurate guess; what if the guess was better?"
]