20075ee141_1_hw1_solutions

20075ee141_1_hw1_solutions - EE141 HW 1 Principles of...

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Unformatted text preview: EE141 HW 1 Principles of Feedback Control Instructor: Balakrishnan, A.V. Problem Set 1: Solutions 1. Find the initial and final values of the time function whose Laplace transform is: Y ( s ) = 2 s 2 + 7 s + 4 s ( s + 2)( s + 1) Check your results using the inverse Laplace transform (15 points). Solution Final value theorem: we have to check first the poles of sY ( s ) (i.e., if dy dt is Laplace transformable). Poles of sY ( s ) = {- 1 , 2 } < we can apply the theorem y ( ) = lim t y ( t ) = lim Re { s } sY ( s ) = lim Re { s } 2 s 2 + 7 s + 4 ( s + 2)( s + 1) = 2 Initial value theorem: it can be applied as the function is rational. y (0) = lim Re { s } sY ( s ) = lim Re { s } 2 s 2 + 7 s + 4 ( s + 2)( s + 1) = 2 If we find the inverse Laplace transformable we get with the same result: Y ( s ) = 2 s + 1 s + 1- 1 s + 2 y ( t ) = 2 + e- t- e- 2 t , t y (0) = 2 , y ( ) = 2 , as expected. 2. Find the responses of the systems governed by the following equations, using the Laplace transform. Determine the damping ratio for both systems (15 points each): (a) d 2 v dt 2 + 3 dv dt + 2 v ( t ) = u ( t ) where v (0) = 4 , v (0) =- 2 and u ( t ) = 1 for t . (b) d 2 v dt 2 + 4 dv dt + 4 v ( t ) = 3 du dt + 2 u ( t ) 1 given that v (0) = v (0) = 0 and u = e- 3 t for t ....
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20075ee141_1_hw1_solutions - EE141 HW 1 Principles of...

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