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hw2 - energy consumed g a k z k u k = k a k-z k k 2 2 k u k...

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EE 292E Analysis and Control of Markov Chains Spring 2008 Prof. Ben Van Roy April 10, 2008 Homework Assignment 2: Due April 17 Tracking Consider an object taking a random walk in a three dimensional space: z k +1 = z k + w k where w k is an iid sequence of three-dimensional zero-mean unit-variance Gaussian random vectors. Our job is to manipulate a “vehicle” whose position evolves according to a k +1 = a k + v k , where the “velocity” is controlled according to v k +1 = v k + u k , and u k represents “force” we exert. The cost function balances proximity to the object against
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Unformatted text preview: energy consumed g ( a k , z k , u k ) = k a k-z k k 2 2 + k u k k 2 2 , and our objective is to optimize a sum over N periods: N-1 X k =0 g ( a k , z k , u k ) . Provide a formula for the optimal control action as a function of a k-z k and v k . Forecasts Do Problem 4.1 from the textbook. Exponentiated Costs Do Problem 4.2 from the textbook. The following integral may be useful: Z ∞ x =-∞ e-( ax 2 + bx + c ) dx = r π a e ( b 2-4 ac ) / 4 a for a > ....
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