homework2_10

# homework2_10 - A-AE 567 Final Homework Spring 2010 You will...

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A-AE 567 Final Homework Spring 2010 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. Due 4:00 pm Thursday May 6 at the box in my oﬃce Armstrong 3201. NAME: 1

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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 ) = α + β y + γ 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 θ and ψ be two independent uniform random variables over the interval [0 , 2 π ]. Consider the random process defined by y ( n ) = 2 cos( n + θ ) + 2 sin(2 n + ψ ) 2
where n is an integer. Consider the prediction problem: σ 2 = min { E | y ( n ) - α 1 y ( n - 1) - α 2 y ( n - 2) - α 3 y ( n - 3) - α 4 y ( n - 4) | 2 : α j R } . The minimum is taken over all constants { α j } 4 1 . Find the optimal constants { b α j } 4 1 and the error σ such that σ 2 = E | y ( n ) - 4 j =1 b α j y ( n - j ) | 2 . What happens if you choose solve the following optimization problem: σ 2 ν = min { E | y ( n ) - ν j =1 α j y ( n - j ) | 2 } where ν > 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 0 are given on the course webpage in the file a567 data10.mat. Let M n = span { y ( j ) } n 0 . 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

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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 δ n m . Here δ 0 = 1 and for k nonzero, δ k = 0. Let M n = span { y ( k ) } n 0 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 state b x ( n ) and a recursive formula for the error covariance Q n . (i) The recursion for the optimal state is b x ( n + 1) = (ii) The recursion for the error covariance is Q n +1 = (iii) Find b x ( n | n ) = P M n x ( n ) in term of b x ( n ) , Q n and the data.
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