sol5 - EE292 Spring 2006 Analysis& Control of Markov...

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Unformatted text preview: EE292 Spring 2006 Analysis & Control of Markov Chains May 21, 2006 Prof. Ben Van Roy Homework Assignment 5 : Solutions 5.2 a) We have a linear quadratic problem with imperfect state information, where A k ,B b ,Q k , and R k are all 1 . Consequently, the optimal control law is given by: μ * k ( I k ) = L k E [ x k | I k ] where L k =- ( R k + B k K k +1 B b )- 1 B k K k +1 A k =- K k +1 1 + K k +1 with K N = 1 . Moreover the Ricatti equation may be simplified here to yield K k = 1 + 2 K k +1 1 + K k +1 Now, we claim that in fact E [ x k | I k ] = x k . To see this simply note that if z ∈ { 2- 1 / 4 , 2 + 1 / 4 } then it must be that x = 2 and if z ∈ {- 2- 1 / 4 ,- 2 + 1 / 4 } , then it must be that x =- 2 . Proceeding by induction, if we know x k- 1 perfectly given I k- 1 then with knowledge of u k- 1 and given z k , we know w k + v k perfectly since w k + v k = z k- x k- 1- u k- 1 But knowing w k + v k here is sufficient to know w k exactly since we have w k = 1 iff w k + v k ∈ { 3 / 4 , 5 / 4 } and w k =- 1 iff w k + v k ∈ {- 5 / 4 ,- 3 / 4 } . Consequently, we know x k = x k- 1 + u k- 1 + w k perfectly given I k ....
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This note was uploaded on 02/15/2011 for the course EECS 6.231 taught by Professor Bertsekas during the Spring '10 term at MIT.

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sol5 - EE292 Spring 2006 Analysis& Control of Markov...

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