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Roger Labbe 2015-05-09 14:00:18 -07:00
parent ef0d770c08
commit 54bdf0e800
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06_Multivariate_Kalman_Filters.ipynb
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@ -2424,7 +2424,7 @@
"\n",
"We will cover this use case later, but for now passive tracking applications we set those terms to 0. In step 2 there was the unexplained term $\\mathbf{Bu}$ in equation (1):\n",
"\n",
"$$\\mathbf{x^-} = \\mathbf{Fx} + \\mathbf{Bu}$$.\n",
"$$\\mathbf{x^-} = \\mathbf{Fx} + \\mathbf{Bu}$$\n",
"\n",
"Here $\\mathbf{u}$ is the control input, and $\\mathbf{B}$ is its transfer function. For example, $\\mathbf{u}$ might be a voltage controlling how fast the wheel's motor turns, and multiplying by $\\mathbf{B}$ yields $[\\begin{smallmatrix}x\\\\\\dot{x}\\end{smallmatrix}]$. Since we do not need these terms we will set them both to zero and not concern ourselves with them for now.\n"
]
@ -3247,10 +3247,10 @@
"$$\n",
"\\begin{aligned}\n",
"x &= x + Ky \\\\\n",
" &= x + \\frac{P}{P+R}(z-x) \\\\\n",
" &= \\frac{P+R}{P+R}x + \\frac{Pz - Px}{P+R} \\\\\n",
" &= \\frac{Px + Rx + Pz - Px}{P+R} \\\\\n",
" &= \\frac{Pz + Rx}{P+R} \\quad \\blacksquare\n",
"&= x + \\frac{P}{P+R}(z-x) \\\\\n",
"&= \\frac{P+R}{P+R}x + \\frac{Pz - Px}{P+R} \\\\\n",
"&= \\frac{Px + Rx + Pz - Px}{P+R} \\\\\n",
"&= \\frac{Pz + Rx}{P+R} \\quad \\blacksquare\n",
"\\end{aligned}\n",
"$$"
]
@ -3264,14 +3264,14 @@
"$$ \n",
"\\begin{aligned}\n",
"P &= (1-K)P \\\\\n",
" &= (1-\\frac{P}{P+R})P \\\\\n",
" &= (\\frac{P+R}{P+R}-\\frac{P}{P+R})P \\\\\n",
" &= (\\frac{P+R-P}{P+R})P \\\\\n",
" &= \\frac{RP}{P+R}\\\\\n",
" &= \\frac{1}{\\frac{P+R}{RP}}\\\\\n",
" &= \\frac{1}{\\frac{R}{RP} + \\frac{P}{RP}} \\\\\n",
" &= \\frac{1}{\\frac{1}{P} + \\frac{1}{R}}\n",
" \\quad\\blacksquare\n",
"&= (1-\\frac{P}{P+R})P \\\\\n",
"&= (\\frac{P+R}{P+R}-\\frac{P}{P+R})P \\\\\n",
"&= (\\frac{P+R-P}{P+R})P \\\\\n",
"&= \\frac{RP}{P+R}\\\\\n",
"&= \\frac{1}{\\frac{P+R}{RP}}\\\\\n",
"&= \\frac{1}{\\frac{R}{RP} + \\frac{P}{RP}} \\\\\n",
"&= \\frac{1}{\\frac{1}{P} + \\frac{1}{R}}\n",
"\\quad\\blacksquare\n",
"\\end{aligned}\n",
"$$\n",
"\n",