Fixed explanation of time invarient systems.

07-Kalman-Filter-Math.ipynb

@@ -28,6 +28,7 @@
        "@import url('http://fonts.googleapis.com/css?family=Source+Code+Pro');\n",
        "@import url('http://fonts.googleapis.com/css?family=Vollkorn');\n",
        "@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
+       "@import url('http://fonts.googleapis.com/css?family=Fira_sans');\n",
        "\n",
        "    div.cell{\n",
        "        width: 900px;\n",
@@ -118,9 +119,9 @@
        "        white-space: nowrap;\n",
        "    }\n",
        "    div.text_cell_render{\n",
-       "        font-family: 'Arimo',verdana,arial,sans-serif;\n",
+       "        font-family: 'Fira sans', verdana,arial,sans-serif;\n",
        "        line-height: 125%;\n",
-       "        font-size: 120%;\n",
+       "        font-size: 115%;\n",
        "        text-align:justify;\n",
        "        text-justify:inter-word;\n",
        "    }\n",
@@ -586,9 +587,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtUAAAEWCAYAAACpJ2vsAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x6dd0e80>"
+       "<matplotlib.figure.Figure at 0x7941240>"
       ]
      },
      "metadata": {},
@@ -620,9 +621,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x436c9e8>"
+       "<matplotlib.figure.Figure at 0x43bc518>"
       ]
      },
      "metadata": {},
@@ -650,9 +651,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7ed9128>"
+       "<matplotlib.figure.Figure at 0x7f5f5c0>"
       ]
      },
      "metadata": {},
@@ -1022,9 +1023,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7ee7c50>"
+       "<matplotlib.figure.Figure at 0x8042e48>"
       ]
      },
      "metadata": {},
@@ -1034,7 +1035,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8012470>"
+       "<matplotlib.figure.Figure at 0x808fe10>"
       ]
      },
      "metadata": {},
@@ -1309,7 +1310,7 @@
     "\n",
     "We derived this equation in that chapter by using techniques that are much easier to understand. The advantage of the Taylor series expansion is that we can use it for any arbitrary set of differential equations which are time invariant. \n",
     "\n",
-    "Time invariant? This means equations that do not depend on time. For example, $f(x) = sin(t)t$ is not time invariant; the value will be different at different times due to the multiplication by t. On the other hand $f(x) = sin(t)*3$ is time invariant. $x(t)$ is dependent on time, but that is allowed. This is because to be time invariant it must be true that if $f(t) = x(t)$ then $f(t+\\Delta t) = x(t+\\Delta t)$. This will be true for the second equation, but not the first.\n",
+    "Time invariant? This means systems that do not depend on time. Suppose the input is $x(t) = \\sin(t)$, and the sytem is $f(x) = t\\times x(t) = t \\sin(t)$. This is not time invariant; the value will be different at different times due to the multiplication by t. On the other hand $f(x) = 3 \\sin(t)$ is time invariant. The input $x(t)$ is dependent on time, but that is allowed. This is because to be time invariant it must be true that if $f(t) = x(t)$ then $f(t+\\Delta t) = x(t+\\Delta t)$. In other words, the output is just time shifted, not altered. This will be true for the second equation, but not the first.\n",
     "\n",
     "However, we often use a Taylor expansion even when the equations are not time invariant. The answer will still be reasonably accurate so long as the time step is short and the system is nearly constant over that time step."
    ]
@@ -2005,7 +2006,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7fbc160>"
+       "<matplotlib.figure.Figure at 0x8042400>"
       ]
      },
      "metadata": {},
@@ -2038,7 +2039,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7fbc7f0>"
+       "<matplotlib.figure.Figure at 0x8df28d0>"
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      },
      "metadata": {},
@@ -2119,7 +2120,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8ddb940>"
+       "<matplotlib.figure.Figure at 0x8e504a8>"
       ]
      },
      "metadata": {},
@@ -2274,7 +2275,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9dbbc50>"
+       "<matplotlib.figure.Figure at 0x9e795f8>"
       ]
      },
      "metadata": {},
@@ -2337,7 +2338,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8d8b748>"
+       "<matplotlib.figure.Figure at 0x9eb0da0>"
       ]
      },
      "metadata": {},
@@ -2823,11 +2824,6 @@
     "\n",
     "$$\\mathbf{P}^- = \\mathbf{FPF}^\\mathsf{T}$$"
    ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": []
   }
  ],
  "metadata": {