INC 342 Lecture 1: Root Locus Dr. Benjamas Panomruttanarug - - PowerPoint PPT Presentation

INC 342

Lecture 1: Root Locus

• Dr. Benjamas Panomruttanarug

Benjamas.pan@kmutt.ac.th

DC motor

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TF of DC motor

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      

    

m b f a f a a a m f a a m a

K K K R s K L J R Js L K K Js R s L K s V s         

2

1 

6

10  J

6

10 

a

L

Approximate to the first order:

   

   

m b f a f a a a m a

K K K R s K L J R Js L K s V s     

2



   

) (

1

p s s K s V s

a

  

(No load)

Stability and step response

• What do you think about stability and step

response from the system?

• How can we improve step response?

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   

) (

1

p s s K s V s

a

  

Camera man Object Tracking using infrared

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Varying gain (K)

Varying K, closed‐loop poles are moving!!!

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Transient:

• K<25  overdamped
• K=25  critically damped
• K>25  underdamped
• Settling time remains the same

under underdamped responses. Stability:

• Root locus never crosses over

into the RHP, system is always stable.

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What is root locus and why is it needed?

• Fact I: poles of closed‐loop system are an important

key to describe a performance of the system (transient response, i.e. peak time, %overshoot, rise time), and stability of the system.

• Fact II: closed‐loop poles are changed when varying

gain.

• Implication: Root locus = paths of closed‐loop poles

as gain is varied.

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Concept of Root Locus

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Sketching Root Locus

• 1. Number of branches
• 2. Symmetry
• 3. Real‐axis segment
• 4. Starting and ending points
• 5. Behavior at infinity

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• 1. Number of branches

Number of branches = number of closed‐loop poles

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• 2. Symmetry

Root locus is symmetrical about the real axis

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• 3. Real‐axis segment

On the real axis, the root locus exists to the left of an odd number of real‐axis



180 ) 1 2 ( ) ( ) (    k s H s KG

• Sum of angles on the real axis is either 0 or 180

(complex poles and zeroes give a zero contribution).

• Left hand side of odd number of poles/zeros on the

real axis give 180 (path of root locus)

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Example

root locus on the real axis

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• 4. Starting and ending points

Root locus starts at finite/infinite poles of G(s)H(s) and ends at finite/infinite zeros of G(s)H(s)

) ( ) ( 1 ) ( ) ( s H s KG s KG s T  

closed‐loop transfer function

) ( ) ( ) ( s D s N s G

G G

 ) ( ) ( ) ( s D s N s H

H H



) ( ) ( ) ( ) ( ) ( ) ( ) ( s N s KN s D s D s D s KN s T

H G H G H G

 

K=0 (beginning) poles of T(s) are K=∞ (ending) poles of T(s) are

) ( ) ( s D s D

H G

) ( ) ( s N s KN

H G

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Example

• 5. Behavior at infinity

Root locus approaches asymptote as the locus approaches ∞, the asymptotes is given by zeros finite # poles finite # zeros finite poles finite   

 

a



... , 2 , 1 , zeros finite # poles finite # ) 1 2 (       k k

a

 

19

# of poles = # of zeroes has 3 finite poles at 0 ‐1 ‐2, and 3 infinite zeroes at infinity

) 2 )( 1 ( ) ( ) (    s s s K s H s KG

Rule of thumb

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Example

Sketch the root locus of the system

Example

Sketch root locus

3 4 1 4 ) 3 ( ) 4 2 1 (          

2 k for , 3 / 5 1 k for , k for , 3 / zeros finite # poles finite # ) 1 2 (               k

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