hw2_sol - Question 1 The differential equation...

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Unformatted text preview: Question 1 The differential equation corresponding to the transfer function is ... y ( t ) + 2 . 4¨ y ( t ) + 25 . 84 ˙ y ( t ) + 50 . 08 y ( t ) = 4 ˙ u ( t ) + 4 u ( t ) . (1) Define a variable q ( t ) which satisfies ... q ( t ) + 2 . 4¨ q ( t ) + 25 . 84 ˙ q ( t ) + 50 . 08 y ( t ) = u ( t ) . (2) As is easily verified, this may be written as a vector, first-order differential equation; d dt ¨ q ( t ) ˙ q ( t ) q ( t ) = - 2 . 4- 25 . 84- 50 . 08 1 1 ¨ q ( t ) ˙ q ( t ) q ( t ) + 1 u ( t ) Note the following manipulations of (2): 4 * u ( t ) = 4 * d 3 dt 3 q ( t ) + 2 . 4 d 2 dt 2 q ( t ) + 25 . 84 d dt q ( t ) + 50 . 08 q ( t ) , (3) 4 * ˙ u ( t ) = 4 * d 4 dt 4 q ( t ) + 2 . 4 d 3 dt 3 q ( t ) + 25 . 84 d 2 dt 2 q ( t ) + 50 . 08 d dt q ( t ) , = 4 * d 3 dt 3 ˙ q ( t ) + 2 . 4 d 2 dt 2 ˙ q ( t ) + 25 . 84 d dt ˙ q ( t ) + 50 . 08 ˙ q ( t ) . (4) Adding expressions (3) and (4) we have 4 * ˙ u ( t )+4 * u ( t ) = d 3 dt 3 (4 ˙ q ( t )+4 q ( t ))+2 . 4 d 2 dt 2 (4 ˙ q ( t )+4 q ( t ))+25 . 84 d dt (4 ˙ q ( t )+4 q ( t ))+50 . 08(4 ˙ q ( t )+4 q ( t )) . (5) Equations (1) and (5) show the relation between y ( t ) and q ( t ); y ( t ) = 4 ˙ q ( t ) + 4 q ( t ) , = 4 4 ¨ q ( t ) ˙ q ( t ) q ( t ) ....
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This note was uploaded on 05/12/2010 for the course MAE 143B taught by Professor Bitmead during the Winter '10 term at San Diego.

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hw2_sol - Question 1 The differential equation...

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