sol2 - EE292 Analysis & Control of Markov Chains Prof. Ben...

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EE292 Spring 2006 April 21, 2006 Prof. Ben Van Roy Homework Assignment 2 : Solutions 2 The system is a linear system of the type discussed in class. In particular, we have the system equation: x k +1 = Ax k + Bu k + ω k where x k = " a k - z k v k # ,A = " I I 0 I # ,B = " 0 I # ω k is an 6 -dimensional random vector, whose first three components are i.i.d N (0 , 1) , and whose remaining components are identically 0 . Our cost function is quadratic, and in the notation of the text, we have Q k = " I 0 0 0 # ,R k = I Eqs. 1 . 4 - 1 . 6 from the text give us the optimal control: μ * ( x k ) = L k x k where L k = - ( B 0 K k +1 B + I ) - 1 B 0 K k +1 A The K k are given by the usual Ricatti equation. Here: K N = 0; K k = A 0 ( K k +1 - K k +1 B ( B 0 K k +1 B + I ) - 1 B 0 K k +1 ) A + Q 4.1 The DP algorithm for our problem is: J N ( x N ,y N ) = x 0 N Q N x N J k ( x k ,y k ) = min u k E " x 0 k Q k x k + u 0 k R k u k + n X i =1 p k +1 i J k +1 ( x k +1 ,i ) ± ± y k # The (conditional) expectation over
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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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sol2 - EE292 Analysis & Control of Markov Chains Prof. Ben...

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