midterm05-sum-solution

# midterm05-sum-solution - MAE 143B Linear Control Prof. M....

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MAE 143B Linear Control Prof. M. Krstic MIDTERM August 15, 2005 NAME: Solution Open books and notes. Present your reasoning and calculations clearly. Random or inconsistent etchings will not be graded. Write only on the paper provided. If you run out of space for a given problem, continue on the pages at the end of the set and indicate “Continued on page X.” The problems are not ordered by diﬃculty. Total points: 30. Time: 6:30–7:50 1

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Problem 1. (7 points) Consider the system H 2 H 1 G 1 G 2 H 3 + - + θ + + R Y - (a) For H 1 = H 2 = 0, what is the transfer function from R to θ ? (b) H 1 6 = 0 ,H 2 6 = 0, what is the transfer function from R to Y ? Solution (a) Redraw the block diagram with H 1 = H 2 = 0, R as the input and θ as the output. Therefore, the transfer function from R to θ is H 2 H 1 G 1 G 2 H 3 θ + - R G 1 G 2 H 3 + - + θ + + R Y - θ R = G 1 1 + G 1 G 2 H 3 . The same result can be found by noting that Y R = G 1 G 2 1+ G 1 G 2 H 3 where Y = G 2 θ . 2
(b) The block diagram can be reduced in the following manner. (i) Move the G 2 block ahead of the pick-oﬀ point. T 2 Y R T 3 H 1 /G 2 H 2 /G 2 T 1 + + + - Y R H 1 H 2 G 1 H 3 1/G 2 + - + + + - Y R G 2 1/G 2 (ii) Combine the blocks in the lower left of the diagram in feedback T 2 Y R T 3 H 1 /G 2 H 2 /G 2 T 1 + + + - Y R H 1 H 2 G 1 H 3 1/G 2 + - + + + - Y R G 2 1/G 2 where T 1 = G 1 G 2

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## This note was uploaded on 06/02/2008 for the course MAE 143B taught by Professor Paoc.chau during the Spring '06 term at UCSD.

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midterm05-sum-solution - MAE 143B Linear Control Prof. M....

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