This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: AAE 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 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 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 δ n − m . Here δ = 1 and for k nonzero, δ 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....
View
Full
Document
This document was uploaded on 01/15/2012.
 Fall '09

Click to edit the document details