20075ee141_1_hw2_solutions

20075ee141_1_hw2_solutions - EE141 HW 2 Principles of...

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Unformatted text preview: EE141 HW 2 Principles of Feedback Control Instructor: Balakrishnan, A.V. Problem Set 2: Solutions 1. The transfer function of a typical tape-drive system is given by G ( s ) = K ( s + 4) s ( s + 0 . 5)( s + 1)( s 2 + 0 . 4 s + 4) where time is measured in miliseconds. Using Rouths stability criterion, determine the range of K for which this system is stable when a negative feedback with unity gain is applied to the tape-drive system. (15 points) Solution The closed loop transfer function is F ( s ) = G ( s ) / (1 + G ( s ) . Num { 1 + G ( s ) } = s 5 + 1 . 9 s 4 + 5 . 1 s 3 + 6 . 2 s 2 + (2 + K ) s + 4 K = 0 . We can directly construct the Routh array: s 5 1 . 5 . 1 2 + K s 4 1 . 9 6 . 2 4 K s 3 1 . 8 2- 1 . 1 K s 2 1 . 14( K + 3 . 63) 4 K s 1 c 1 s 4 K where c 1 =- ( K + 8 . 47)( K- . 78) . 91( K + 3 . 63) For stability, we need to satisfy the following conditions (all coefficients in the first column have to be positive). b 1 = K + 3 . 63 > K >- 3 . 63 c 1 > - 8 . 43 < K < . 78 d 1 = 4 K > K > Intersection of the 3 conditions gives us: < K < . 78 1 2. Find the unit step response for the following third-order system (15 points): H ( s ) = w 2 n p ( s + p )( s 2 + 2 w n s + w 2 n ) Solution Second-order system: H ( s ) = w 2 n p ( s + p...
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This note was uploaded on 03/18/2008 for the course EE 141 taught by Professor Balakrishnan during the Fall '07 term at UCLA.

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20075ee141_1_hw2_solutions - EE141 HW 2 Principles of...

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