Sol_HW4

# Sol_HW4 - 3.19 Find the transfer functions for the block...

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1 3.19 Find the transfer functions for the block diagrams in Fig. 3.54: Solution: Simplify the block diagram as above, 1 2 1 () () 1 Ys G Ts G Rs G == + + 3.22 Use block-diagram algebra to determine the transfer function between R(s) and Y(s) in Fig. 3.57. Solution: Move node A and close the loop: Add signal B, close loop and multiply before signal C. 1 1 1 G G +

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2 Move middle block N past summer. Now reverse order of summers and close each block separately. 3 3 12 3 1 1 1 2 11 12 3 1 () ( ) 1 (1 ) 1 feedforward feedback Y NG RN H GG G GH Y R GGH =+ + ++ + = +++ ±²³ ²´ µ²¶ ²·
3 3.24 For the unity feedback system shown in Fig. 3.59, specify the gain K of the proportional controller so that the output ( ) has an overshoot of no more than 10% in response to a uni yt t step. 2 22 2 Solution: () 2 2 , 21 (1) 2 In order to have an overshoot of no more than 10%, n nn n n Ys K Rs s s K s s K K ω ζω ζ == ++ + + = 2 /1 2 2 0.10 Solving for , (ln ) 0.591 (ln ) Using (1) and the solution for , 1 2.86 0 2.86 p p p Me M M K K πζ π −− =≤ =≥ + ∴≤≤ 3.28 The open-loop transfer function of a unity feedback system is

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## Sol_HW4 - 3.19 Find the transfer functions for the block...

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