daviddoji/Kalman-and-Bayesian-Filters-in-Python
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Fix a few typos

Browse Source
This commit is contained in:
Mana 2018-09-01 10:26:27 -04:00
parent 70858d0e34
commit f7e70354cb
1 changed files with 5 additions and 5 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

10
07-Kalman-Filter-Math.ipynb
View File

@ -1049,7 +1049,7 @@
"This allows us to define the covariance of our estimate, which is defined as the expected value of $\\mathbf{ee}^\\mathsf T$:\n",
"\n",
"$$\\begin{aligned}\n",
"P &= E[\\mathbf e, \\mathbf e^\\mathsf T] \\\\\n",
"P &= E[\\mathbf{ee}^\\mathsf T] \\\\\n",
"&= E[(\\mathbf x - \\mathbf{\\hat x})(\\mathbf x - \\mathbf{\\hat x})^\\mathsf T]\n",
"\\end{aligned}$$\n",
"\n",
@ -1103,11 +1103,11 @@
"\n",
"$$\\begin{aligned}\n",
"&= (\\mathbf I - \\mathbf{KH})\\mathbf{\\bar P}(\\mathbf I - \\mathbf{KH})^\\mathsf T + \\mathbf{KRK}^\\mathsf T\\\\\n",
"&= \\mathbf{\\bar P} - \\mathbf{KH}\\mathbf{\\bar P} - \\mathbf{\\bar PH}\\mathbf{K}^\\mathsf T + \\mathbf K(\\mathbf{HPH}^\\mathsf T + \\mathbf R)\\mathbf K^\\mathsf T \\\\\n",
"&= \\mathbf{\\bar P} - \\mathbf{KH}\\mathbf{\\bar P} - \\mathbf{\\bar PH}\\mathbf{K}^\\mathsf T + \\mathbf{\\bar P H^\\mathsf T}(\\mathbf{H \\bar P H}^\\mathsf T + \\mathbf R)^{-1}(\\mathbf{HPH}^\\mathsf T + \\mathbf R)\\mathbf K^\\mathsf T\\\\\n",
"&= \\mathbf{\\bar P} - \\mathbf{KH}\\mathbf{\\bar P} - \\mathbf{\\bar PH}\\mathbf{K}^\\mathsf T + \\mathbf{\\bar P H^\\mathsf T}\\mathbf K^\\mathsf T\\\\\n",
"&= \\mathbf{\\bar P} - \\mathbf{KH}\\mathbf{\\bar P} - \\mathbf{\\bar PH}^\\mathsf T\\mathbf{K}^\\mathsf T + \\mathbf K(\\mathbf{H \\bar P H}^\\mathsf T + \\mathbf R)\\mathbf K^\\mathsf T \\\\\n",
"&= \\mathbf{\\bar P} - \\mathbf{KH}\\mathbf{\\bar P} - \\mathbf{\\bar PH}^\\mathsf T\\mathbf{K}^\\mathsf T + \\mathbf{\\bar P H^\\mathsf T}(\\mathbf{H \\bar P H}^\\mathsf T + \\mathbf R)^{-1}(\\mathbf{H \\bar P H}^\\mathsf T + \\mathbf R)\\mathbf K^\\mathsf T\\\\\n",
"&= \\mathbf{\\bar P} - \\mathbf{KH}\\mathbf{\\bar P} - \\mathbf{\\bar PH}^\\mathsf T\\mathbf{K}^\\mathsf T + \\mathbf{\\bar P H^\\mathsf T}\\mathbf K^\\mathsf T\\\\\n",
"&= \\mathbf{\\bar P} - \\mathbf{KH}\\mathbf{\\bar P}\\\\\n",
"&= (\\mathbf I - \\mathbf{KH})\\mathbf P\n",
"&= (\\mathbf I - \\mathbf{KH})\\mathbf{\\bar P}\n",
"\\end{aligned}$$\n",
"\n",
"Therefore $\\mathbf P = (\\mathbf I - \\mathbf{KH})\\mathbf{\\bar P}$ is mathematically correct when the gain is optimal, but so is $(\\mathbf I - \\mathbf{KH})\\mathbf{\\bar P}(\\mathbf I - \\mathbf{KH})^\\mathsf T + \\mathbf{KRK}^\\mathsf T$. As we already discussed the latter is also correct when the gain is suboptimal, and it is also more numerically stable. Therefore I use this computation in FilterPy.\n",