hw11_sol - MEEN 364 Fall 2004 Homework Set 11 Due 5:00 PM...

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MEEN 364 Homework Set 10 Fall 2004 November 11, 2004 1 Homework Set 11 – Due November 22, 2004 @ 5:00 PM 1. a) We are required to find V(s) / W(s) , taking a zero reference input V c . The block diagram then becomes: This is a closed-loop unity feedback system. Its transfer function is: 3 2 1 2 3 1 3 ( ) ( ) 1 m s k m V s ms k k m W s s mk s mk k s k m s + = = + + + + 2 b) If w is a unit ramp function, w(t) =t , its Laplace transform is W(s) = 1 / s 2 , and we can apply the final value theorem to find v ss : 2 2 0 0 3 1 2 1 ( ) 1 1 lim lim ( ) ss s s V s ms v s s W s s mk s mk k s k k = = = + + 2 c) To find the system type respect to reference, first we must find the transfer function between V c and V , considering w = 0: 1 2 3 1 2 2 1 2 3 1 3 ( ) ( ) 1 c k k m s mk s mk k V s k k m V s s mk s mk k s mk s + = = + + + + 2
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MEEN 364 Homework Set 10 Fall 2004 November 11, 2004 2 The error transfer function is: 2 3 1 2 2 2 3 1 2 3 1 ( ) ( ) 1 1 ( ) ( ) c c s mk s mk k E s V s V s V s s mk s mk k s mk s mk k + = = = + + + + 2 Now we evaluate the steady state error for unit step, ramp, and velocity inputs: Unit step: 2 3 2 0 0 3 1 2 ( ) 1 lim ( ) lim 0 ( ) ss c s s c s mk s E s e s V s s V s s mk s mk k s + = = = + + Unit ramp: 2 3 3 2 2 0 0 3 1 2 1 ( ) 1 lim ( ) lim ( ) ss c s s c s mk s k E s e s V s s V s s mk s mk k s k k + = = = + + 2 Therefore, the system is Type 1, the velocity constant is: 1 2 3 1 v
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