Typo fixes

07_Kalman_Filter_Math.ipynb

@@ -534,20 +534,47 @@
     "\n",
     "$$x(t) = \\Phi(t-t_0)x(t_0)$$\n",
     "\n",
-    "In other words, we just want to compute the value of $x$ at time $t$ by multiplying its previous value by some matrix $\\Phi$. This is not trivial to do because the original equations do not include time \n",
+    "In other words, we just want to compute the value of $x$ at time $t$ by multiplying its previous value by some matrix $\\Phi$. This is not trivial to do because the original equations do not include time "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Taylor Series Expansion\n",
     "\n",
-    "Broadly speaking there are three ways to find $\\Phi$. The technique most often used with Kalman filters is to use a Taylor-series expansion: \n",
+    "Broadly speaking there are three ways to find $\\Phi$. The technique most often used with Kalman filters is to use a Taylor-series expansion. The Taylor series represents a function as an infinite sum of terms. The terms are linear, even for a nonlinear function, so we can express any arbitrary nonlinear function using linear algebra. The cost of this choice is that unless we use an infinite number of terms (and who has time for that these days) the value we compute will be approximate rather than exact.\n",
+    "\n",
+    "For the Kalman filter we will be using a form of the series that uses a matrix. But before we do that, let's work through a couple of examples with real functions since real functions are easier to plot and reason about. The Taylor series for either are nearly identical, so this is a good first step.$\n",
+    "For a real (or complex) function the Taylor series of a function $f(x)$ evaluated at $a$ is defined as\n",
+    "\n",
+    "$$ \n",
     "\n",
     "$$ \\Phi(t) = e^{\\mathbf{F}t} = \\mathbf{I} + \\mathbf{F}t  + \\frac{(\\mathbf{F}t)^2}{2!} + \\frac{(\\mathbf{F}t)^3}{3!} + ... $$\n",
     "\n",
-    "This is much easy to compute, and thus is the typical approach used in Kalman filter design when the filter is reasonably small. If you are wondering where $e$ came from, I point you to the Wikipedia article on the matrix exponential [1]. Here the important point is to recognize the very simple and regular form this equation takes. We will put this form into use in the next chapter, so I will not belabor its use here. \n",
+    "This is much easy to compute, and thus is the typical approach used in Kalman filter design when the filter is reasonably small. If you are wondering where $e$ came from, I point you to the Wikipedia article on the matrix exponential [1]. Here the important point is to recognize the very simple and regular form this equation takes. \n",
+    "\n",
+    "Before applying it to a Kalman filter, lets do the Taylor expansion of a math function since this is much easier to visualize than a matrix. I choose sin(x). The Taylor series for a real or complex function f(x) at "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Linear Time Invariant Theory\n",
     "\n",
     "*Linear Time Invariant Theory*, also known as LTI System Theory, gives us a way to find $\\Phi$ using the inverse Laplace transform. You are either nodding your head now, or completely lost. Don't worry, I will not be using the Laplace transform in this book except in this paragraph, as the computation is quite difficult to perform in practice. LTI system theory tells us that \n",
     "\n",
     "$$ \\Phi(t) = \\mathcal{L}^{-1}[(s\\mathbf{I} - \\mathbf{F})^{-1}]$$\n",
     "\n",
-    "I have no intention of going into this other than to say that the inverse Laplace transform converts a signal into the frequency (time) domain, but finding a solution to the equation above is non-trivial. If you are interested, the Wikipedia article on LTI system theory provides an introduction [2].\n",
-    "\n",
+    "I have no intention of going into this other than to say that the inverse Laplace transform converts a signal into the frequency (time) domain, but finding a solution to the equation above is non-trivial. If you are interested, the Wikipedia article on LTI system theory provides an introduction [2]."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Numerical Solutions\n",
     "Finally, there are numerical techniques to find $\\Phi$. As filters get larger finding analytical solutions becomes very tedious (though packages like SymPy make it easier). C. F. van Loan [3] has developed a technique that finds both $\\Phi$ and $Q$ numerically.\n",
     "\n",
     "I have implemented van Loan's method in `FilterPy`. You may use it as follows:\n",