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Unformatted text preview: AAE 567 Final Homework Spring 2011 You will need Matlab and Simulink. You work must be neat and easy to read. Clearly, identify your answers in a box. You will loose points for poorly written work. You must work alone. Exam is due in a box outside my oce ARMS 3201 by 4:00 pm Thursday May 5. NAME: 1 1 Problem Let x and y be the random variables which are uniform over in the triangle { ( x,y ) : 1 y x 2 } , that is, assume that the joint density function f x , y ( x,y ) is determined by f x , y ( x,y ) = c if 1 y x 2 = 0 otherwise where c is a constant. Find: The constant c . The conditional expectation b g ( y ) = E ( x  y = y ). Compute the error E ( x b g ( y )) 2 . Find a polynomial b p ( y ) = + y + y 2 of degree at most two which is the best approximation of x , that is, E ( x b p ( y )) 2 = inf { E ( x p ( y )) 2 : p is a polynomial of degree at most 2 } Compute the error E ( x b p ( y )) 2 . Can you think of any physical meaning behind your answer? 2 Problem Let and be two independent uniform random variables over the interval [0 , 2 ]. Consider the random process defined by y ( n ) = 2 cos( n + ) + 2 sin(2 n + ) 2 where n is an integer. Consider the prediction problem: 2 = min { E  y ( n ) 1 y ( n 1) 2 y ( n 2) 3 y ( n 3) 4 y ( n 4)  2 : j R } . The minimum is taken over all constants { j } 4 1 . Find the optimal constants { b j } 4 1 and the error such that 2 = E  y ( n ) 4 j =1 b j y ( n j )  2 . What happens if you choose solve the following optimization problem: 2 = min { E  y ( n ) j =1 j y ( n j )  2 } where > 4. Do you get a better result? 3 Problem Consider the discrete time system x ( n + 1) = Ax ( n ) + Bu ( n ) and y ( n ) = Cx ( n ) + Dv ( n ) where u and v are independent Gaussian white noise processes and the initial condi tion is x (0) = 0. The system is third order. Moreover, the matrices { A,B,C,D } and the output { y ( j ) } 10 are given on the course webpage in the file a567 data10.mat. Let M n = span { y ( j ) } n . Find the following estimates which are vectors in R 3 . P M 10 x (9). P M 10 x (10) P M 10 x (11) P M 10 x (13) 3 4 Problem Consider the state space system x ( n + 1) = Ax ( n ) + u ( n ) and y ( n ) = Cx ( n ) + v ( n ) (4.1) where u ( n ) and v ( n ) are mean zero Gaussian random process which are independent to the initial condition x (0). Moreover, assume that E u ( n ) v ( n ) [ u ( m ) v ( m ) ] = R 11 R 12 R 21 R 22 n m . Here = 1 and for k nonzero, k = 0. Let M n = span { y ( k ) } n and b x ( n ) = P M n 1 x ( n ) denote the optimal state estimate. Let Q n = E e x ( n ) e x ( n ) = E ( x ( n ) b x ( n )) ( x ( n ) b x ( n )) be the error covariance matrix. Find the Kalman filter for the state space system in (4.1). To be precise, find a recursive estimate for the optimal statein (4....
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 Fall '09

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