ps 5 soln - ECE 580 FEEDBACK CONTROL SYSTEMS (I) Fall, 2011...

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ECE 580 FEEDBACK CONTROL SYSTEMS (I) Fall, 2011 Solution to Problem Set #5 Problem 1: (From PS) Note: The solution gives more detail about the construction of the root locus than we covered in class. Your solutions do not need to give this detail. Matlab drawings of the root loci are sufficient. a) The loop transfer function for hands-off control is: L ho s ( ) = K 2 s + 1 ( ) s + 9 25 s + 0.03 ( ) s + 0.4 ( ) s 2 ! 0.36 s + 0.16 ( ) = K 2 25 s 2 + 25.75 s + 0.75 s 4 + 9.04 s 3 + 0.376 s 2 + 0.208 s + 0.576 = K 2 b s ( ) a s ( ) Thus, there are 4 branches starting at the loop transfer function poles p = ! 9 ! 0.4 0.18 ± j 0.357 " # $ $ % $ $ Two branches go to the two system zeros: z = ! 0.03 ! 1 " # $ % $ The real axis is on the root locus for: ! 0.03 < Re s ( ) < ! 0.4 ! 1 < Re s ( ) < ! 9 Two branches go to with asymptotes at angles ± ! and center of mass = " 4.01 . The angle of departure from the complex pole in the upper half-plane is: 3 = tan " 1 0.357 0.21 # $ % & ( + tan " 1 0.357 1.18 # $ % & ( " 90 ° " tan " 1 0.357 9.18 # $ % & ( " tan " 1 0.357 0.58 # $ % & ( + 180 ° # $ % & ( = 132 °
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The angle of departure from the complex pole in the lower half-plane is: !
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ps 5 soln - ECE 580 FEEDBACK CONTROL SYSTEMS (I) Fall, 2011...

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