# HW6sol - P(s C 2(s C 1(s R(s Y(s U(s 1 Taking the Laplace...

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Unformatted text preview: P(s) C 2 (s) C 1 (s) + + R(s) Y(s) U(s) 1. Taking the Laplace transform of the differential equations m ¨ y m ( t )- d ( ˙ y M ( t )- ˙ y m ( t ))- k ( y M ( t )- y m ( t )) = M ¨ y M ( t ) + d ( ˙ y M ( t )- ˙ y m ( t )) + k ( y M ( t )- y m ( t )) = u ( t ) under the assumption of zero initial conditions yields ms 2 Y m ( s )- ds ( Y M ( s )- Y m ( s ))- k ( Y M ( s )- Y m ( s )) = Ms 2 Y M ( s ) + ds ( Y M ( s )- Y m ( s )) + k ( Y M ( s )- Y m ( s )) = U ( s ) Using the rearranged form Y m ( s )( ms 2 + ds + k ) = Y M ( s )( ds + k ) Y M ( s )( Ms 2 + ds + k ) = Y m ( s )( ds + k ) + U ( s ) the unnecessary variable is eliminated to yield Y m ( s )( ms 2 + ds + k ) = ds + k Ms 2 + ds + k ( Y m ( s )( ds + k ) + U ( s )) Collecting terms results in Y m ( s ) ms 2 + ds + k- ( ds + k ) 2 Ms 2 + ds + k = ds + k Ms 2 + ds + k U ( s ) which simplifies to Y m ( s ) ( ( Ms 2 + ds + k )( ms 2 + ds + k )- ( ds + k ) 2 ) = ( ds + k ) U ( s ) and Y m ( s ) ( Mms 4 + ( M + m ) s 2 ( ds + k ) ) = ( ds + k ) U ( s ) so the transfer function of the plant is P ( s ) = ds + k Mms 4 + ( M + m ) ds 3 + ( M + m ) ks 2 = . 001 s + 0 . 1 . 1 s 4 + 0 . 0011 s 3 + 0 . 11 s 2 = . 01 s + 1 s 4 + 0 . 011 s 3 + 1 . 1 s 2 2. Since2....
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HW6sol - P(s C 2(s C 1(s R(s Y(s U(s 1 Taking the Laplace...

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