20112ee141_1_EE141_hw3_sol

20112ee141_1_EE141_hw3_sol - -pa 2 s 1 : p ( K-a 2 )-(...

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EE141 Principles of Feedback Control (Spring 2011) Solutions to Homework 3 Problem 3.42 (a) 1 + KG = 0 s 4 + 2 s 3 + 3 s 2 + 8 s + 8 = 0 The Routh array is where a = 2 * 3 - 8 2 = - 1 , b = 8 c = 8 a - 2 b a = 24 d = 8 2 sign changes in the ±rst column 2 roots in the RHP unstable. (b) 1 + KG = 0 s 3 + s 2 + 2 s + 8 = 0 The Routh array is 2 sign changes in the ±rst column 2 roots in the RHP unstable. Problem 3.43 (a) s 4 + 8 s 3 + 32 s 2 + 80 s + 100 = 0 1
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The Routh array is The frst column is positive All roots in the LHP. (c) s 4 + 2 s 3 + 7 s 2 - 2 s + 8 = 0 The Routh array is 2 sign changes in the frst column 2 roots in the RHP. Problem 3.46 The closed-loop transFer Function is Y R = K 0 ( s + z ) K ( s + p )( s 2 - a 2 )) 1 + K 0 ( s + z ) K ( s + p )( s 2 - a 2 )) = K 0 ( s + z ) Ks 3 + Kps 2 + ( K 0 - Ka 2 ) s + K 0 z - Kpa 2 The characteristic Function is Ks 3 + Kps 2 + ( K 0 - Ka 2 ) s + K 0 z - Kpa 2 = 0 s 3 + ps 2 + ( K 0 /K - a 2 ) s + K 0 /Kz - pa 2 = 0 Denote that ˆ K = K 0 /K , so the Routh array is s 3 : 1 ˆ K - a 2 s 2 : p ˆ Kz
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Unformatted text preview: -pa 2 s 1 : p ( K-a 2 )-( Kz-pa 2 ) p = K ( p-z ) p s : Kz-pa 2 2 For stability: 1) All coefcients o the characteristic polynomial should be positive: p > K-a 2 > K > a 2 ( K > 0) Kz-pa 2 > 2) All elements o the rst column should be positive: K ( p-z ) p > , since K > p > z Kz-pa 2 > , since K > z > pa 2 K p > ,K /K > a 2 , pa 2 K K < z < p NOTE: For the book edition with the block K ( s + z ) ( s + p ) instead o ( s + z ) K ( s + p ) , replace K /K by K K in the conditions or stability. 3...
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This note was uploaded on 07/02/2011 for the course EE 141 taught by Professor Balakrishnan during the Spring '07 term at UCLA.

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20112ee141_1_EE141_hw3_sol - -pa 2 s 1 : p ( K-a 2 )-(...

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