solmidterm09_mae143b

solmidterm09_mae143b - Midterm MAE143B Summer II 2009...

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Unformatted text preview: Midterm - MAE143B, Summer II 2009 SOLUTIONS Prof. R.A. de Callafon August 25, 2009 2:00pm-3:20pm, CSB 102 open-book and open-notes midterm exam use the available space to derive your results, attach extra paper if necessary no electronic equipment (calculator, phone, PDA, computer) allowed during exam During this midterm you will be working with a linear dynamic system characterized by the ODE d 2 dt 2 y ( t ) = 4 u ( t ) − 2 d dt u ( t ) − 2 d dt y ( t ) − 4 y ( t ) ( 1 ) 1. Using only ‘integrator blocks’ and ‘gain blocks’ draw a block diagram (of feedback and feedforward connections) to construct the linear dynamic system in (1). [10pts] Application of Laplace transform to (1) yields ( s 2 + 2 s + 4) y ( s ) = ( − 2 s + 4) u ( s ). This allows us to define ( s 2 + 2 s + 4)¯ y ( s ) = u ( s ) and y ( s ) = ( − 2 s + 4)¯ y ( s ). Going back to time domain via inverse Laplace transform we now have d 2 dt 2 ¯ y ( t ) = − 2 d dt ¯ y ( t ) − 4¯ y ( t ) + u ( t ) and y ( t ) = 4¯ y ( t ) − 2 d dt ¯ y ( t ) that can be represented by the following block diagram using ‘integrator’ R and ‘gain’ blocks: Z Z e H H H H H H- e ?- H H H H H H C C C C C O 6- y ( t ) u ( t ) ¯ y ( t ) d dt ¯ y ( t ) d 2 dt 2 ¯ y ( t ) − 2 4 − 2 − 4--- u- + + + + + 2. Show that the dynamic system given in (1) can be written in a transfer function model [10pts] y ( s ) = G ( s ) u ( s ) , G ( s ) = Kω 2 n ( s − a ) s 2 + 2 βω n s + ω 2 n , with β = 0 . 5 , ω n = 2 , K = − 1 / 2 , a = 2 (2) Application of Laplace transform to (1) yields ( s 2 + 2 s + 4) y ( s ) = ( − 2 s + 4) u ( s ). This allows us to write y ( s ) = G ( s ) u ( s ) , G ( s ) = ( − 2 s + 4) ( s 2 + 2 s + 4) = − 1 2 · ( s − 2) · 4 ( s 2 + 2 s + 4) allowing indeed G ( s ) to written as a standard 2nd order system with ω 2 n = 4 or ω n = 2, 2 βω n = 2 or β = 1 2 , K = − 1 2 and an extra Right Half Plane (RHP) zero at...
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This note was uploaded on 05/18/2010 for the course MAE 143B taught by Professor Bitmead during the Spring '10 term at San Diego.

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solmidterm09_mae143b - Midterm MAE143B Summer II 2009...

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