Proof reading Chapter 13
- Fxied some typos
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@@ -148,7 +148,7 @@
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"$$\\mathbf x(t_k) = \\mathbf F(\\Delta t)\\mathbf x(t_{k-1}) + \\mathbf B(t_k)\\mathbf u (t_k)$$\n",
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"\n",
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"Convention allows us to write $\\mathbf x(t_k)$ as $\\mathbf x_k$, which means the \n",
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"the value of $\\mathbf x$ at the k$^{th}$ value of $t$.\n",
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"the value of $\\mathbf x$ at the $k^{th}$ value of $t$.\n",
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"\n",
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"$$\\mathbf x_k = \\mathbf{Fx}_{k-1} + \\mathbf B_k\\mathbf u_k$$\n",
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"\n",
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@@ -157,10 +157,10 @@
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"Finding this matrix is often quite difficult. The equation $\\dot x = v$ is the simplest possible differential equation and we trivially integrate it as:\n",
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"\n",
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"$$ \\int\\limits_{x_{k-1}}^{x_k} \\mathrm{d}x = \\int\\limits_{0}^{\\Delta t} v\\, \\mathrm{d}t $$\n",
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"$$x_k-x_0 = v \\Delta t$$\n",
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"$$x_k = v \\Delta t + x_0$$\n",
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"$$x_k-x_{k-1} = v \\Delta t$$\n",
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"$$x_k = v \\Delta t + x_{k-1}$$\n",
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"\n",
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"This equation is *recursive*: we compute the value of $x$ at time $t$ based on its value at time $t-1$. This recursive form enables us to represent the system (process model) in the form required by the Kalman filter:\n",
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"This equation is *recursive*: we compute the value of $x$ at time $k$ based on its value at time $k-1$. This recursive form enables us to represent the system (process model) in the form required by the Kalman filter:\n",
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"\n",
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"$$\\begin{aligned}\n",
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"\\mathbf x_k &= \\mathbf{Fx}_{k-1} \\\\\n",
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@@ -200,24 +200,24 @@
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"\n",
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"We define two new variables:\n",
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"\n",
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"$$\\begin{aligned} x_1(u) &= x \\\\\n",
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"x_2(u) &= \\dot x\n",
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"$$\\begin{aligned} x_1(t) &= x \\\\\n",
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"x_2(t) &= \\dot x\n",
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"\\end{aligned}$$\n",
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"\n",
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"Now we will substitute these into the original equation and solve. The solution yields a set of first-order equations in terms of these new variables. It is conventional to drop the $(u)$ for notational convenience.\n",
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"Now we will substitute these into the original equation and solve. The solution yields a set of first-order equations in terms of these new variables. It is conventional to drop the $(t)$ for notational convenience.\n",
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"\n",
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"We know that $\\dot x_1 = x_2$ and that $\\dot x_2 = \\ddot{x}$. Therefore\n",
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"\n",
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"$$\\begin{aligned}\n",
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"\\dot x_2 &= \\ddot{x} \\\\\n",
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" &= 6\\dot x - 9x + t\\\\\n",
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" &= 6x_2-9x_1 + t\n",
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" &= 6\\dot x - 9x + u\\\\\n",
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" &= 6x_2-9x_1 + u\n",
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"\\end{aligned}$$\n",
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"\n",
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"Therefore our first-order system of equations is\n",
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"\n",
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"$$\\begin{aligned}\\dot x_1 &= x_2 \\\\\n",
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"\\dot x_2 &= 6x_2-9x_1 + t\\end{aligned}$$\n",
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"\\dot x_2 &= 6x_2-9x_1 + u\\end{aligned}$$\n",
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"\n",
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"If you practice this a bit you will become adept at it. Isolate the highest term, define a new variable and its derivatives, and then substitute."
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]
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@@ -989,7 +989,7 @@
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"\n",
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"while not correct, is often a useful approximation. If you do this for an important application you will have to perform quite a few studies to guarantee that your filter works in a variety of situations. \n",
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"\n",
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"If you do this, 'lower right term' means the most rapidly changing term for each variable. If the state is $x=\\begin{bmatrix}x & \\dot x & \\ddot{x} & y & \\dot{y} & \\ddot{y}\\end{bmatrix}^\\mathsf{T}$ Then Q will be 6x6; the elements for both $\\ddot{x}$ and $\\ddot{y}$ will have to be set to non-zero in $\\mathbf Q$."
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"If you do this, 'lower right term' means the most rapidly changing term for each variable. If the state is $x=\\begin{bmatrix}x & \\dot x & \\ddot{x} & y & \\dot{y} & \\ddot{y}\\end{bmatrix}^\\mathsf{T}$ Then $\\mathbf Q$ will be 6x6; the elements for both $\\ddot{x}$ and $\\ddot{y}$ will have to be set to non-zero in $\\mathbf Q$."
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]
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},
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{
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@@ -1141,7 +1141,7 @@
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"\n",
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"$$\\mathbf P = \\mathbf{\\bar P} - \\mathbf{KH}\\mathbf{\\bar P} - \\mathbf{\\bar P}\\mathbf H^\\mathsf T \\mathbf K^\\mathsf T + \\mathbf K(\\mathbf H \\mathbf{\\bar P}\\mathbf H^\\mathsf T + \\mathbf R)\\mathbf K^\\mathsf T$$\n",
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"\n",
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"Now we need to the the derivative of the trace of $\\mathbf P$ with respect to $\\mathbf T$: $\\frac{d\\, trace(\\mathbf P)}{d\\mathbf K}$.\n",
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"Now we need to the the derivative of the trace of $\\mathbf P$ with respect to $\\mathbf K$: $\\frac{d\\, trace(\\mathbf P)}{d\\mathbf K}$.\n",
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"\n",
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"The derivative of the trace the first term with respect to $\\mathbf K$ is $0$, since it does not have $\\mathbf K$ in the expression.\n",
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"\n",
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@@ -1201,14 +1201,14 @@
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"execution_count": 13,
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"data": {
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truncated
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"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 900x400 with 1 Axes>"
|
||||
]
|
||||
@@ -1236,12 +1236,12 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 15,
|
||||
"execution_count": 14,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 900x400 with 1 Axes>"
|
||||
]
|
||||
@@ -1269,7 +1269,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 16,
|
||||
"execution_count": 15,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
@@ -1284,7 +1284,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"execution_count": 16,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
@@ -1312,12 +1312,12 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 18,
|
||||
"execution_count": 17,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 900x400 with 2 Axes>"
|
||||
]
|
||||
@@ -1339,7 +1339,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 19,
|
||||
"execution_count": 18,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
@@ -1420,7 +1420,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 20,
|
||||
"execution_count": 19,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
@@ -1456,7 +1456,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 21,
|
||||
"execution_count": 20,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
@@ -1468,7 +1468,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 900x400 with 1 Axes>"
|
||||
]
|
||||
@@ -1537,11 +1537,11 @@
|
||||
"\n",
|
||||
"In practice this is extremely reasonable, as many things have the *Markov property*. If you are driving in a parking lot, does your position in the next second depend on whether you pulled off the interstate or were creeping along on a dirt road one minute ago? No. Your position in the next second depends solely on your current position, speed, and control inputs, not on what happened a minute ago. Thus, cars have the Markov property, and we can make this simplification with no loss of precision or generality.\n",
|
||||
"\n",
|
||||
"The next simplification we make is do define the *measurement model* as depending on the current state $\\mathbf x_k$ with the conditional probability of the measurement given the current state: $P(\\mathbf z_t \\mid \\mathbf x_x)$. We write:\n",
|
||||
"The next simplification we make is do define the *measurement model* as depending on the current state $\\mathbf x_k$ with the conditional probability of the measurement given the current state: $P(\\mathbf z_k \\mid \\mathbf x_k)$. We write:\n",
|
||||
"\n",
|
||||
"$$\\mathbf z_k \\sim P(\\mathbf z_t \\mid \\mathbf x_x)$$\n",
|
||||
"$$\\mathbf z_k \\sim P(\\mathbf z_k \\mid \\mathbf x_k)$$\n",
|
||||
"\n",
|
||||
"We have a recurrance now, so we need an initial condition to terminate it. Therefore we say that the initial distribution is the probablity of the state $\\mathbf x_0$:\n",
|
||||
"We have a recurrence now, so we need an initial condition to terminate it. Therefore we say that the initial distribution is the probablity of the state $\\mathbf x_0$:\n",
|
||||
"\n",
|
||||
"$$\\mathbf x_0 \\sim P(\\mathbf x_0)$$\n",
|
||||
"\n",
|
||||
@@ -1550,9 +1550,9 @@
|
||||
"\n",
|
||||
"The details of the mathematics for this computation varies based on the problem. The **Discrete Bayes** and **Univariate Kalman Filter** chapters gave two different formulations which you should have been able to reason through. The univariate Kalman filter assumes that for a scalar state both the noise and process are linear model are affected by zero-mean, uncorrelated Gaussian noise. \n",
|
||||
"\n",
|
||||
"The Multivariate Kalman filter make the same assumption but for states and measurements that are vectors, not scalars. Dr. Kalman was able to prove that if these assumptions hold true then the Kalman filter is *optimal* in a least squares sense. Colloquially this means there is no way to derive more information from the noisy measurements. In the remainder of the book I will present filters that relax the constraints on linearity and Gaussian noise.\n",
|
||||
"The Multivariate Kalman filter makes the same assumption but for states and measurements that are vectors, not scalars. Dr. Kalman was able to prove that if these assumptions hold true then the Kalman filter is *optimal* in a least squares sense. Colloquially this means there is no way to derive more information from the noisy measurements. In the remainder of the book I will present filters that relax the constraints on linearity and Gaussian noise.\n",
|
||||
"\n",
|
||||
"Before I go on, a few more words about statistical inversion. As Calvetti and Somersalo write in *Introduction to Bayesian Scientific Computing*, \"we adopt the Bayesian point of view: *randomness simply means lack of information*.\"[3] Our state parameterizes physical phenomena that we could in principle measure or compute: velocity, air drag, and so on. We lack enough information to compute or measure their value, so we opt to consider them as random variables. Strictly speaking they are not random, thus this is a subjective position. \n",
|
||||
"Before I go on, a few more words about statistical inversion. As Calvetti and Somersalo write in *Introduction to Bayesian Scientific Computing*, \"we adopt the Bayesian point of view: *randomness simply means lack of information*\"[3]. Our state parameterizes physical phenomena that we could in principle measure or compute: velocity, air drag, and so on. We lack enough information to compute or measure their value, so we opt to consider them as random variables. Strictly speaking they are not random, thus this is a subjective position. \n",
|
||||
"\n",
|
||||
"They devote a full chapter to this topic. I can spare a paragraph. Bayesian filters are possible because we ascribe statistical properties to unknown parameters. In the case of the Kalman filter we have closed-form solutions to find an optimal estimate. Other filters, such as the discrete Bayes filter or the particle filter which we cover in a later chapter, model the probability in a more ad-hoc, non-optimal manner. The power of our technique comes from treating lack of information as a random variable, describing that random variable as a probability distribution, and then using Bayes Theorem to solve the statistical inference problem."
|
||||
]
|
||||
@@ -1601,7 +1601,7 @@
|
||||
"\\frac{a}{a+b} &= \\frac{a/a}{(a+b)/a} = \\frac{1}{(a+b)/a} \\\\\n",
|
||||
" &= \\frac{1}{1 + \\frac{b}{a}} = \\frac{1}{\\frac{b}{b} + \\frac{b}{a}} \\\\\n",
|
||||
" &= \\frac{1}{b}\\frac{1}{\\frac{1}{b} + \\frac{1}{a}} \\\\\n",
|
||||
" &= \\frac{\\sigma^2_{x'}}{b}\n",
|
||||
" &= \\frac{\\sigma^2_{x}}{b}\n",
|
||||
" \\end{aligned}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
@@ -1610,7 +1610,7 @@
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\mu_{x}&=x+ \\frac{a}{a+b}(y-x) \\\\\n",
|
||||
"&= x + \\frac{\\sigma^2_{x'}}{b}(y-x) \\\\\n",
|
||||
"&= x + \\frac{\\sigma^2_{x}}{b}(y-x) \\\\\n",
|
||||
"&= x + g_n(y-x)\n",
|
||||
"\\end{aligned}\n",
|
||||
"$$\n",
|
||||
@@ -1667,7 +1667,7 @@
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.7.4"
|
||||
"version": "3.7.3"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
|
||||
Reference in new issue
Block a user