pset7_sol - 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4...

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ME3015 PROBLEM SET #7 (25 points). Solutions (1) (8 points) Obtain the unit-step response, x ( t ), of the following system. r 1 + s x 2 10 s e Solution The closed loop transfer function is ) 8730 . 8 )( 1270 . 1 ( ) 1 ( 10 10 10 ) 1 ( 10 10 ) 1 ( 1 10 ) 1 ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 2 2 2 + + + = + + + = + + + = + = s s s s s s s s s s s K s P s K s P s R s X For the unit step input, we have 8730 . 8 1455 . 1 1270 . 1 1455 . 0 1 1 ) 8730 . 8 )( 1270 . 1 ( ) 1 ( 10 ) ( + + + = + + + = s s s s s s s s X Hence t t e e t x 8730 . 8 1270 . 1 1455 . 1 1455 . 0 1 ) ( + = (ANS) (2) (8 points) Determine the value of k such that the damping ratio z is 0.5. Using MATLAB, plot the unit-step response (attach your MATLAB code). Obtain the maximum overshoot M p of this unit-step response. k 8 . 0 16 + s s 1 ) ( s R ) ( s C Solution The closed loop transfer function is
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16 ) 16 8 . 0 ( 16 ) ( ) ( 1 ) ( ) ( ) ( ) ( 2 + + + = + = s k s s K s P s K s P s R s C From the characteristic polynomial, 4 = n ω . Hence 2 . 0 16 8 . 0 4 5 . 0 2 2 = + = × × = k k n ςω (ANS) The maximum overshoot is 163 . 0 25 . 0 1 14 . 3 5 . 0 1 2 = = = × e e M p ς ςπ (ANS) MAT:AB code and unit-step response.
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Unformatted text preview: 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4 Step Response Time (sec) Amplitude (3) (9 points) Find the range of k such that the unity feedback system is stable. Use Routh-Hurwitz stability criteria. ) 5 )( 1 ( 1 ) ( + − + = s s s s s G Solution The characteristic equation is ) 5 ( 4 2 3 = + − + + K s K s s . Then the Routh array is K s K s K s K s : 4 / ) 20 3 ( : 4 : 5 1 : 1 2 3 − − For the stability condition, the Routh criterion gives 3 / 20 > K (ANS) num=16; den=[1 0.8+16*0.2 16]; step(tf(num,den)) grid...
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This note was uploaded on 02/08/2012 for the course ME 3015 taught by Professor Ueda during the Fall '08 term at Georgia Tech.

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pset7_sol - 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4...

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