lec24 - 24 APPENDIX 3: LQR VIA DYNAMIC PROGRAM MING There...

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24 APPENDIX 3: LQR VIA DYNAMIC PROGRAM- MING There are at least two conventional derivations for the LQR; we present here one based on dynamic programming , due to R. Bellman. The key observation is best given through a loose example. (Continued on next page)
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24.1 Example in the Case of Discrete States 131 A B B 1 B 2 3 1 C C 2 3 C D n = 3 s = 2 24.1 Example in the Case of Discrete States Suppose that we are driving from Point A to Point C, and we ask what is the shortest path in miles. If A and C represent Los Angeles and Boston, for example, there are many paths to choose from! Assume that one way or another we have found the best path, and that a Point B lies along this path, say Las Vegas. Let X be an arbitrary point east of Las Vegas. If we were to now solve the optimization problem for getting from only Las Vegas to Boston, this same arbitrary point X would be along the new optimal path as well. The point is a subtle one: the optimization problem from Las Vegas to Boston is easier than that from Los Angeles to Boston, and the idea is to use this property backwards through time to evolve the optimal path, beginning in Boston. Example: Nodal Travel. We now add some structure to the above experiment. Consider now traveling from point A (Los Angeles) to Point D (Boston). Suppose there are only three places to cross the Rocky Mountains, B 1 ,B 2 ,B 3 , and three places to cross the Mississippi River, C 1 ,C 2 ,C 3 . 3
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lec24 - 24 APPENDIX 3: LQR VIA DYNAMIC PROGRAM MING There...

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