{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

problem_11_20

# problem_11_20 - Problem 11.20 of the Palm textbook Plant TF...

This preview shows pages 1–3. Sign up to view the full content.

Problem 11.20 of the Palm textbook Plant TF: G P s ( ) = 1 s 2 + 3 s + 2 Sketch the root locus. Design a PI controller that minimizes the dominant time constant with ! =1/ 2 . SOLUTION PI controller TF : G C s ( ) = K P + K I s = K P s + K I s Closed-loop characteristic equation : 0 = 1 + G C s ( ) G P s ( ) = 1 + K P s + K I s ! " # \$ % & 1 s 2 + 3 s + 2 ! " # \$ % & In order to produce a root locus plot, we need to have our CLCE in terms of a single parameter. For this problem, we have two parameters: K P and K I . Here we will write the CLCE in terms of K = K P and a = K I / K P (see footnote below) 1 : 0 = 1 + K s + a s s 2 + 3 s + 2 ( ) = 1 + K N s ( ) D s ( ) ! s s 2 + 3 s + 2 ( ) + K s + a ( ) = s 3 + 3 s 2 + 2 + K ( ) s + Ka = 0 1. Number of branches : max N P , N Z { } = max 3,1 { } = THREE branches 2. Start/end points : OL zeros : z 1 = ! a OL poles : p 1 = 0, p 2 = ! 1, p 3 = ! 2 Therefore, the RL START at poles p 1 , p 2 , p 3 . One RL branch ENDS at z 1 , and two RL branches end at ! . 1 If this problem were given to you on an exam as a root locus construction problem, we would likely give you a numerical value of K P , K I , K P / K I OR K I / K P , and ask you to construct the RL based on the remaining free parameter.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

problem_11_20 - Problem 11.20 of the Palm textbook Plant TF...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online