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# ps3 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Mechanical Engineering 2.160 Identification, Estimation, and Learning Spring 2006 Problem Set No. 3 Out: March 1, 2006 Due: March 8, 2006 For the first two problems below, hand calculation is recommended. Problem 1 Consider a simple scalar random process governed by the following state transition equation: x t 1 + = x t + w t where x t is state, and w t is a zero-mean process noise with Q > 0 t = E [ w w ] t s = s . 0 s t The output of the process is observed with a single sensor having the following properties: + y t = x t v t R > 0 = s t 0 s t v v E ] [ s t = w v E ] [ t s 0 t , s = We want to build a Kalman filter and examine its characteristics with respect to the state estimation error covariance and the sensor and process noise properties. (a). Write out all the recursive equations of Kalman filter for this scalar process. Simplify the equations as much as you can. (b). Given P = 100 , Q = 1, and R = 1, plot the values of a posteriori and a priori error 0 covariances, P , P 1|0 , P , P 2|1 , P , P 3|2 , P , " against time. Also plot the Kalman gain 0 1 2 3 K 1 , K , K , " . Repeat it for different values of Q and R, and discuss how the process

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ps3 - Massachusetts Institute of Technology Department of...

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