7. Root Locus Analysis

System Control System Control

7. Root Locus Analysis

Professor Kyongsu Yi

©2009 VDCL

©2009 VDCL

Vehicle Dynamics and Control Laboratory Seoul National University

Seoul National University

Characteristic Equation

K

C

G (s )

R  C



) (

) ) (

( A s

s s B

G 

) ( )

( s K B s

G K C

) ( )

(

) ( )

( 1

) (

s B K s A

s B K s

G K

s G K R

C

C C C

C

 

 

Characteristic Equationq : AA((ss)) KK_CCBB((ss))00

Transient response

Characteristic roots, poles

Characteristic roots

) 1 (s

G( ) ( ) b ms s s

G  

Characteristic Equation : ms² bsK_C 0 mK b

b ² 4 m

mK b

s b ^C

2

4





0

KC b, 0

s 

Im

C m

K_C b 4

 2

 m

s b 2

 double roots x x x Re m K_C b

4

 2



C m

4 ^2m

K_C b

 2

 b mK b j

s 4 ^C  ²

 



0 KC

3

C m

 4 j

m

m 2

2 _

KC

Characteristic roots

2 bsK_C 0 ms

1 0

1 ₂ 

s K bs m

C

s m

R 

1

m 1

s2

K bs _C

R

C



1 1 ( )

0 1 0

( ) B s m   m A s 

0 ) 1 ( )

(  B

A( ) B(s)0 s m

A

0 ) (s  A

0

s ; double roots

 m 

1 1 ( ) 0

)

(  B s 

s m A

0 ) ( B(s)0 B

Characteristic roots

) 0 (

)

1 ( 

s A

s K_C B

0 ) ( )

(s  K B s 

A _C

0 ) (

) (

) (

) ) (

( ¹ 













 n

i m

j

i

p s

z s s

A s s B

G

) (

1



i

pi

s

0 ) (

) (

1 1













 

m j

i C

m i

i K s z

p s

openloop poles openloop zeros

0

KC ( ) 0

1







 m i

pi

s

closed loop poles=open loop poles

1

KC ( ) 0

1







 m j

zi

s

closed loop poles=open loop poles



 m 

j

zi

s s

B 1

) 1 (

)

( l ti

5











 _n

i

i j

C s p

K s

A

1

) ) (

( )

( real, negative

s : complex number

Characteristic roots

1

2

1

2

j j x

j j y

x r e x e

y r e y e









 

 



y

Im x

y r e

2

y e

2

1

Re

1 2

( )

1 2

x y    r r e

j ^{ }

1 2

( )

1 2

r

j

x e

y r

 



1 2

( ) y

x y x y

x y x y

  

    

x x

y y

x x y



    

x y



y

 

m

)



(

n p

s z

s p

s z

s n

i

i m

j n i

i

i j

i

360

180 )

( )

( )

(

) (

1 1

1

1        







 













Characteristic roots

( ) ( ) 0

( ) ( ) 0

( )

C

c

A s K B s A s K B s B s

 

 

 

 

 

1

KC

( ) 0

( ) 0

( )

^c

B s m roots

A s K n m roots B s

  

   

( )

B s

0 )

(

) (

1  







 n c m j

i

K p

s z

s n-m=2 n-m=3

) (

1





 i

pi

s

1 0

1



 





 



 









c n

i

n i n

K s

p

s 

1

1 



 



 







 c

m j

m i

m z s

s 

1 0

1 1









 



 

 ^{ }





ⁿ



^{n m c}

i

m j

i i

m

n p z s K

s 

1 1 

^{i j}

1 1  0











































 

 

c m n n

i

m j

i i

m K n

z p

s n m

z p

K s

n i

m j

i i

m cn m n















 

 

1 1 1 1

) 1 (

7



  

 



 



 

   

 ^{n m} e^{j n m n}

1 2 1

1 )

1 (

complex real

Root – Locus Method

• Graphically obtains closed-loop poles as function of a parameter in feedback system.

) (s B

) ( )

(

) ( )

( ) 1 (

) (

) (

s B K s A

s B K s

A s K B

s A

s K B C

R





 









closed-loop poles : (characteristic roots) 0

) ( )

(s KB s  A

0 ) (s 

A( ) ; open-loop poles ; p p p pⁱ 0

) (s 

B ; open-loop zeros zⁱ 0

) (

) (

1  



 ⁿ _i

i

p s s

A

1 i

0 ) (

) (

1  



  i m

i

z s s

B Where, nm

0 ) (

)

(     

ⁿ(s p_i) K ^m(s z_i)

1

1 

 i

i i i

p

Root – Locus Method

1) K=0

: closed-loop poles = open-loop poles closed-loop characteristic eq. ;

 closed-loop poles are asymptotic to open-loop poles for small K

2) K>>1 2) K>>1

0 ) ( 

B s K

0 ) ( )

(s K B s A(s)KB(s)0 A

( ) ( ) 0 ( )

A s K B s

B s

 

  

 

 

(n-m)-poles m-poles

m closed-loop poles are asymptotic to open-loop zeros n-m poles

9

Root – Locus Method

n-m Poles

0 )

(

) (

1  





 K

p s

m

i n

i

) (

1



 s z_i

i

0

1

1  











 



 



 ^

 K

s p s

m

n i n i n

1 1







 



 



 ^

 p s

s _i ^m

m i m

1

0

1 1









 



 



   



^{ }



p z s K

s

_i ^{n m}

m i i n i m n

1

1





^{i i}

(approximation) ⁰

1

1  

















 



 









 K

m n

z p

s

m n i m i i n i





m n i

m i i n

i K

z p

s ^{ }    ^



 



 

 ^{1 1} ( ) ¹

m n 

m n m n i

m i i n

i K

m n

z p



 

   





 



 



1 1

1

1 ( 1)

 ) 2 1

1e^i( ^{ N}

 N 1,2,3

m n N m i

n^ e^{  }

 ^{(1 2 ) /}

1

) 1

( ^

real image

Root – Locus Method

1

m

n ( 1)¹ 1

1







11

Root – Locus Method

N i N i

i e e

e ^{(1 2 ) /2 2 2}

2 1

) 1 (

 



  



 

 

 

  





2

m n

Root – Locus Method

3

m

n ^

 i i

N

i e e

e ,

) 1

( ^{3 (1 2 ) /3 3}

1     



13

Root – Locus Method

4

m

n e ⁱ e ⁴ⁱ

3 4 4 1

, )

1 (



 _

 



Root – Locus Method

Example)

^{nm3 n m poles}

15

Root – Locus Method

Angle condition

( ) ( ) 0

A s  K B s 

1 1

( ) ( ) 0

1 ( ) 0

n m

i i

i i

s p K s z

K B s

 

     

1 ( ) 0

( )

( ) 1

( ) K B s

A s

B s

A K

  

 

1

( )

( ) 1

m i i

n

A s K

s z

K

 

  ( )

m

s z

 

1

( )

n i i

s p K

 

real negative

1

1 1

1

( )

( ) ( )

( )

( ) ( )

m n

i i

i i

n i i

i i

m n

s z

s z s p

s p



 



       

 

 

1 1

( ) ( )

180 360 360

180(1 2 ) 1, 2, 3

i i

i i

s z s p

N N

 

     

   

    

Root – Locus Method

3) Real axis

( ) 1

B s _{B s( )} ^{n n}

 

( ) ,

( )

A s  K

1 1

( ) ( ) ( ) 100(1 2 )

( ) i ⁱ i ⁱ

B s s Z s P N

A s  





 



   

 

17

Root – Locus Method

Example)

4 open-loop poles, 1 open loop zero Root Roci in the real axis

1

1 2

180

360 s z

s p s p

  

     



 X

3 180

s p

   ^{ X X}

X O

4

4

0

( ) ( ) 360

s p

  

 





X

1

( _i) ( _i) 360

i

s z s p

       

; Segments which have an odd number of open loop poles & zeros lying to the right on the real axis become portions of the root locus

Root – Locus Method

Portions of the root locus

n-m=3

19

Root – Locus Method

4) Angle of departure

) 2 1 ( 180 )

( ) (

1 1

N p

s z

s _i

n i i m

i















 

) (

) (

) (

) (

)

(p  z  p  p  s p  p  p  p  p

(p₂  z₁)(p₂  p₁)(s p₂)(p₂  p₃)(p₂  p₄)



) (

) (

) (

) (

) 2 1 ( 180 )

(s p₂   N  p₂ z₁  p₂  p₁  p₂  p₃  p₂  p₄



Root – Locus Method

n=2, m=1 )

(s

R  C(s)

K )

(s

R  C(s)



^{( 1)}

1

p s s

z s





)

21

1 (

1 1

p s s

z K s



 

Root – Locus Method

5) Double root

s=b

Characteristic eq (s b)²( )0 Characteristic eq. (s b) ( )0





_sb 0 ds

d

At double root

0 ) ( )

(s KB s 

A ^

) (

) (

s B

s K A

( ) ( )

dA ( ) dB ( ) 0 ( ) ( ) ( )

( ) 0 dA s dB s

ds K ds

dA s A s dB s ds B s ds

 

 

( )

( ) ( )

( ) ( ) 0

ds B s ds

dA s dB s

B s A s

ds



ds



Root – Locus Method

• Positioning system

m

x

u m b:Lubrication oil

P control P control

Kc

) (s

R  X(s)



1

( )

s ms b

) (s U

) 0 (

1 1 

K_c b

) (ms b

c s

) (

)

(s s ms b

A   B(s)1 n=2 m=0 Open loop poles : 0,

m

 b

23

Root – Locus Method

I control

) (s

R  ₁ X(s)

( )

s ms b

) (s U s

K_i

1 1 0

( )

Ki

s s ms b

 





^{s s ms b(  )}

2

1 1 0

( )

Ki

s ms b

 

 n=3 m=0 Open loop poles :

m

 b 0 , 0

double loot

angle of departure

) 2 1 ( 180 )

( ) (

1 1

N p

s z

s _i

n i i m

i















 

) 2 1 ( 180 )

( ) (

)

(    N

_{( ) ( ) ( ) 180(1 2 )} (s p )90^

3 2

1 s p s p N

p

s      















90 ) (  ₁ 

 s p

Root – Locus Method

PI control

) (s

R  ₁ X(s)

( )

s ms b

) (s U



 



  s K T

i c

1 1



^{ i  ( )} ₀

) (

1

1 1 



 

b ms s s T

s K T

i i

c n=3 m=1 n-m=2

Open loop zero : T

1 Open loop pole : 0, 0,

m

 b Ti

25

Root – Locus Method

PID control

- Mass change, - PI, I gain variation )

(s

R  X(s)



1

( )

s ms b

) (s

1 U

(1 )

C d

i

K T S

T S

1

2

1

1 0

( )

i d

C

T s T s

K T b

    2 Open loop zeros

( )

C

T s

i

s ms



b

^{p p}

m

 b , 0 ,

3 Open loop poles : 0

Root – Locus Method

2

0

ms  bs  K 

1  1 bs  K

^C

 0

Example)

1

2

0

m s

 

27

Root – Locus Method

Minimum phase systems:

시스템의 모든 pole^과zero^가 LHP^{에 있는 경우}

Non minimum phase systems:

적어도 시스템의 한 개의pole^이나 zero^{가 평면의} RHP^{에 있는 경우}

(1 )

( ) ( 1)

K T s

d

G s s Ts

 



(1 )

1 0

( 1)

T sd

K s Ts

  



1 1

( 1) T s

d

s Ts K

 

