daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Grammar fix.

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Roger Labbe 2018-01-31 07:25:23 -08:00
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08-Designing-Kalman-Filters.ipynb
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"source": [
"A straight line is the only possible answer. Furthermore, the answer is optimal. If I gave you more points you could use a least squares fit to find the best line, and the answer would still be optimal in a least squares sense.\n",
"\n",
"But suppose I told you to fit a higher order polynomial to those two points. There is now an infinite number of answers to the problem. For example, an infinite number of second order parabolas pass through those points. When the Kalman filter is of higher order than your physical process it also has an infinite number of solutions to choose from. The answer is not just non-optimal, it often diverges and never recovers. \n",
"But suppose I told you to fit a higher order polynomial to those two points. There are an infinite number of answers to the problem. For example, an infinite number of second order parabolas pass through those points. When the Kalman filter is of higher order than your physical process it also has an infinite number of solutions to choose from. The answer is not just non-optimal, it often diverges and never recovers. \n",
"\n",
"For best performance you need a filter whose order matches the system's order. In many cases that will be easy to do - if you are designing a Kalman filter to read the thermometer of a freezer it seems clear that a zero order filter is the right choice. But what order should we use if we are tracking a car? Order one will work well while the car is moving in a straight line at a constant speed, but cars turn, speed up, and slow down, in which case a second order filter will perform better. That is the problem addressed in the Adaptive Filters chapter. There we will learn how to design a filter that *adapts* to changing order in the tracked object's behavior.\n",
"For best performance you need a filter whose order matches the system's order. In many cases that will be easy to do - if you are designing a Kalman filter to read the thermometer of a freezer it seems clear that a zero order filter is the right choice. But what order should we use if we are tracking a car? Order one will work well while the car is moving in a straight line at a constant speed, but cars turn, speed up, and slow down, in which case a second order filter will perform better. That is the problem addressed in the Adaptive Filters chapter. There we will learn how to design a filter that adapts to changing order in the tracked object's behavior.\n",
"\n",
"With that said, a lower order filter can track a higher order process so long as you add enough process noise and you keep the discretization period small (100 samples a second are usually locally linear). The results will not be optimal, but they can still be very good, and I always reach for this tool first before trying an adaptive filter. Let's look at an example with acceleration. First, the simulation."
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