# ex2s - ME 451 Examination#2 1 Consider the control system...

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ME 451 Examination #2 March 28, 2008

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1. Consider the control system shown below with C ( s ) = K and G ( s ) = 6 ( s + 1)( s + 2)( s + 3) . r ( t ) y ( t ) e ( t ) C ( s ) G ( s ) + - (a) [15 pts] Find the range of K for which the closed-loop system is stable. (b) [15 pts] Find the steady-state error for a unit step input when K = 1 and K = 12 . (a) 1 + C ( s ) G ( s ) = 0 1 + K 6 ( s + 1)( s + 2)( s + 3) = 0 ( s + 1)( s + 2)( s + 3) + 6 K = 0 s 3 + 6 s 2 + 11 s + 6 + 6 K = 0 s 3 1 11 s 2 6 6 + 6 K s 1 10 - K s 0 6 + 6 K 10 - K > 0 and 6 + 6 K > 0 ⇒ - 1 < K < 10 for closed-loop stability. (b) E ( s ) = 1 1 + C ( s ) G ( s ) R ( s ) E ( s ) = ( s + 1)( s + 2)( s + 3) ( s + 1)( s + 2)( s + 3) + 6 K 1 s When K = 1 , the closed-loop system is stable and thus e ss = lim s 0 sE ( s ) = 1 / 2 . When K = 12 , the closed-loop system is unstable and thus e ss = . 1
2. Consider the control system shown below with C ( s ) = K and G ( s ) = 1 s + 1 . r

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ex2s - ME 451 Examination#2 1 Consider the control system...

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