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hw7 - Homework 7 MAE143B Spring 2010 due Thursday May 27...

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Homework 7 - MAE143B Spring 2010: due Thursday May 27 Question 1: Frequency response Consider the linear system d n y t dt n + a 1 d n - 1 y t dt n - 1 + a 2 d n - 2 y t dt n - 2 + · · · + a n - 1 1 dy t dt + a n y t = b 1 d n - 1 u t dt n - 1 + b 2 d n - 2 u t dt n - 2 + · · · + b n - 1 1 du t dt + b n u t . Denote the Laplace transform of its solution as follows Y ( s ) = b ( s ) a ( s ) U ( s ) + x ( s ) a ( s ) , (1) where b ( s ) and a ( s ) are the numerator and denominator polynomials of the transfer function and x ( s ) denotes the polynomial reflecting the initial conditions. Part i: For the third-order system of this form, and assuming obvious notations, show that x ( s ) a ( s ) = s 2 y 0 + ( ˙ y 0 + a 1 y 0 ) s + (¨ y 0 + a 1 ˙ y 0 + a 2 y 0 ) s 3 + a 1 s 2 + a 2 s + a 3 = y 0 ( s 2 + a 1 s + a 2 ) + ˙ y 0 ( s + a 1 ) + ¨ y 0 s 3 + a 1 s 2 + a 2 s + a 3 . Part ii: Take u t = M sin( ωt ) so that U ( s ) = ω s 2 + ω 2 . Further, assume that a ( s ) = s 3 + a 1 s 2 + a 2 s + a 3 = ( s - p 1 )( s - p 2 )( s - p 3 ) , where these system poles, ( p 1 , p 2 , p 3 ) are distinct and in the open left-half complex plane. The partial fraction expansion of each term of ( 1 ) yields b ( s ) a ( s ) U ( s ) = w 0 s - + ¯ w 0 s
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