# ps5 - EE 482 PROBLEM SET 5 DUE 18 April 2008 Reading...

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Unformatted text preview: EE 482 PROBLEM SET 5 DUE: 18 April 2008 Reading assignment: Chapter 10, sections 10.7 and 10.8; Chapter 9 Problem 21: (20 points) Suppose that we suspect a linear relationship between the sequences { x ( k ) } and { y ( k ) } of the form y ( k ) = m x ( k ) + b, where m and b are unknown constants. Using N measurement points, { x ( i ) ,y ( i ) } for i = 1 ··· N , we wish to calculate the best estimate of the constants m and b in the sense of minimizing the cost function J ( m,b ) = N k =1 e ( k ) 2 , where e ( k ) = y ( k )- ˆ y ( k ) and ˆ y ( k ) = m x ( k ) + b is the estimate of y ( k ) for a given choice of m and b . If the value of J is minimized for m = ˆ m and b = ˆ b , then ˆ m and ˆ b are called the least-square estimates of m and b , respectively. The necessary conditions for ˆ m and ˆ b to minimize J ( m,b ) are ∂J ( m,b ) ∂m m = ˆ m = 0 ∂J ( m,b ) ∂b b = ˆ b = 0 . 1. (10 points) Solve for ˆ m and ˆ b in terms of the measurement points and place your answer in the form ˆ θ = A- 1 b where ˆ θ = ˆ m ˆ b A = a 2 × 2 matrix depending only on the values of x ( k ) b = a 2 × 1 column vector depending on both x ( k ) and y ( k ) . 2. (10 points) Find the least-squares estimate of ˆ b and ˆ m for the N = 20 data points ( x m ( k ) ,y m ( k )) stored in the MatLab file ps5 p21 data.mat that can be downloaded from the EE 482 web page. Write a MatLab m-file that: • Loads the measurement data from the file ps5 p21 data.mat. • Calculates the least-squares estimates of ˆ b and ˆ m using the equation derived in part 1. • In a single figure showing y versus x , plots the measurement data ( x m ( k ) ,y m ( k )) as series of open circles and the estimated line ˆ y ( k ) = ˆ m x m ( k ) + ˆ b. Include a copy of the m-file and figure along with your solutions. Make sure that your name appears at the top of the m-file. Problem 22: (20 points) Consider an n th order linear SISO system described by the transfer function G ( z ) = Y ( z ) U ( z ) = b m z m + b m- 1 z m- 1 + ··· + b z n + a n- 1 z n- 1 + ··· + a ( m ≤ n ) . The corresponding difference equation representation is y ( k ) =- a n- 1 y ( k- 1)- ···- a y ( k- n ) + b m u ( k + m- n ) + ··· + b u ( k- n ) . (1) Suppose we wish to estimate the parameters a ,a 1 , ··· a n- 1 and b ,b 1 , ··· ,b m given the sequences u ( k ) and y ( k ) for k = 1 ,...,N . As discussed in lecture, y ( k ) can be expressed as y ( k ) = φ T ( k ) θ where the regressor vector φ ( k ) is a n + m + 1 column vector φ ( k ) =...
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ps5 - EE 482 PROBLEM SET 5 DUE 18 April 2008 Reading...

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