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latexify g and h

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tv3141 2017-09-23 10:12:17 +01:00
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01-g-h-filter.ipynb
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@ -4026,7 +4026,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Choice of g and h"
"## Choice of $g$ and $h$"
]
},
{
@ -4341,14 +4341,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise: Varying g"
"## Exercise: Varying $g$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's look at the effect of varying g. Before you perform this exercise, recall that g is the scale factor for choosing between the measurement and prediction. What do you think the effect of a large value of g will be? A small value? \n",
"Now let's look at the effect of varying $g$. Before you perform this exercise, recall that $g$ is the scale factor for choosing between the measurement and prediction. What do you think the effect of a large value of $g$ will be? A small value?\n",
"\n",
"Now, let the `noise_factor=50` and `dx=5`. Plot the results of $g = 0.1\\mbox{, } 0.4,\\mbox{ and } 0.8$."
]
@ -4464,7 +4464,7 @@
"source": [
"Here we can see the effects of ignoring the signal. We not only filter out noise, but legitimate changes in the signal as well. \n",
"\n",
"Maybe we need a 'Goldilocks' filter, where g is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose g and h in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of g and h. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$."
"Maybe we need a 'Goldilocks' filter, where $g$ is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose $g$ and $h$ in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of $g$ and $h$. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$."
]
},
{
@ -4501,14 +4501,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Varying h"
"## Varying $h$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's leave g unchanged and investigate the effect of modifying h. We know that h affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n",
"Now let's leave $g$ unchanged and investigate the effect of modifying $h$. We know that $h$ affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n",
"\n",
"We will look at three examples. We have a noiseless measurement that slowly goes from 0 to 1 in 50 steps. Our first filter uses a nearly correct initial value for $\\dot{x}$ and a small $h$. You can see from the output that the filter output is very close to the signal. The second filter uses the very incorrect guess of $\\dot{x}=2$. Here we see the filter 'ringing' until it settles down and finds the signal. The third filter uses the same conditions but it now sets $h=0.5$. If you look at the amplitude of the ringing you can see that it is much smaller than in the second chart, but the frequency is greater. It also settles down a bit quicker than the second filter, though not by much."
]