7. root locus analysis7. root locus analysis professor kyongsu yi ©2009 vdcl©2009 vdcl vehicle...

System ControlSystem Control

7. Root Locus Analysis

Professor Kyongsu Yi©2009 VDCL©2009 VDCL

Vehicle Dynamics and Control LaboratorySeoul National UniversitySeoul National University

Characteristic Equation

CK )(sGR C

)()()(

sAsBsG

)()( sBKsGKC)()(

)()(1

)(sBKsA

sBKsGK

sGKRC

C

C

C

C

Characteristic Equation : 0)()( sBKsA Cq 0)()( sBKsA C

Transient response

Characteristic roots, poles

22

Characteristic roots

1)(sG)(

)(bmss

sG

Characteristic Equation : 2 0Cms bs K

mKbb 42m

mKbbs C

24

0CK ,0bs

Im

C m

bKC 4

2

mbs

2

double roots x x x Re mbKC 4

2

mC 4 m2

bKC

2

jbmKbs C4 2

0CK

3

mC 4 j

mm 22 CK

3

Characteristic roots02 CKbsms

011 2

s

Kbsm

C

sm

1R m1

2sKbs C

RC

1 1 ( )0 1 0( )

B sm m A s

0)(1)( BA 0)()( sBm

sA

0)( sA

0s ; double roots

m1 0)(1)( sB

msA

0)(B

4

0)( sB

bKs b

4


0)()(1

sAsBKC

0)()( sBKsA C

0)(

)(

)()()( 1

n

i

m

ji

ps

zs

sAsBsG

)(1i

ips

0)()(11

m

jiC

m

ii zsKps

openloop poles openloop zeros

0CK 0)(1

m

iips

closed loop poles=open loop poles

1CK 0)(1

m

jizs

closed loop poles=open loop poles

m

jizs

sB 1)(

1)( l ti

5

n

ii

j

C psKsA1

)()()( real, negative

s : complex number5


1

2

1

2

j j x

j j y

x r e x e

y r e y e

y

xIm

2y r e y e21

Re

1 2( )1 2

jx y r r e

1 2( )1

2

jrx ey r

1 2

( )

yx y x y

x y x y

xxy y

x x y

x yy

m

)(npszs

ps

zsn

ii

m

jin

ii

ji

360180)()()(

)(

11

1

1

66

Characteristic roots( ) ( ) 0

( ) ( ) 0( )

C

c

A s K B s

A s K B sB s

1CK ( ) 0( ) 0( ) c

B s m rootsA s K n m rootsB s

( )B s

0)(

)(1

cn

m

ji

Kps

zs n-m=2 n-m=3

)(1

i

ips

01

1

c

n

i

ni

n

Ksps

1

1

cm

j

mi

m szs

01

1 1

c

mnn

i

m

jii

mn Kszps 1 1 i j

01 1

c

mnn

i

m

jii

Kmn

zps

mn

zpKs

n

i

m

jii

mncmn

1 111

)1(

77

n

mnj

mn e211

1)1(

complex real

Root – Locus Method

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

)(sB

)()()(

)()(1

)()(

sBKsAsBK

sAsBK

sAsBK

CR

closed-loop poles : (characteristic roots)

0)()( sBKsA

0)( sA ; open-loop poles ip)( ; p p p

0)( sB ; open-loop zeros iz

0)()(1

i

n

ipssA

1i

0)()(1

i

m

izssB Where, mn

0)()( i

m

i

nzsKps

88

)()(11

ii

ii

p


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>>12) K>>10)( sBK

0)()( sBKsA 0)()( sBKsA

( ) ( ) 0( )

A s K B sB s

(n-m)-poles m-poles

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

9

Root – Locus Methodn-m Poles

0)(

)(1

K

psm

i

n

i

)(1

zs ii

0

1

1

Ksps

m

ni

n

i

n

1

1

sps m

i

m

i

m

01

11

Kszps mn

i

m

ii

n

i

mn

11 ii

(approximation) 011

Kmn

zps

mn

i

m

ii

n

i

mni

m

ii

n

i Kzp

s

1

11 )(mn

mnmni

m

ii

n

i Kmn

zp

11

11 )1(

)21(1 Nie 3,2,1N

mnNimn e /)21(1

)1(

1010

real image


1 mn 1)1( 11

1111


iNiNi eee 222/)21(2

1

)1(

2 mn

1212


3 mn

iiNi eee ,)1( 33/)21(31

1313


4 mnii

ee 43

441

,)1(

1414

Root – Locus MethodExample) 3 mn n m poles

1515

Root – Locus MethodAngle condition

( ) ( ) 0A s K B s

1 1( ) ( ) 0

( )1 0

n m

i ii i

s p K s z

B sK

( )1 0( )

( ) 1( )

B sKA s

B sA K

1

( )

( ) 1m

iin

A s K

s z

K

( )m

s z 1( )

n

ii

Ks p

real negative

1

1 1

1

( )( ) ( )

( )

( ) ( )

m nii

i in i ii

i

m n

s zs z s p

s p

1 1( ) ( )

180 360 360180(1 2 ) 1,2,3

i ii i

s z s p

N N

1616

Root – Locus Method3) Real axis

( ) 1B s ( ) n nB s ( ) ,( )A s K

1 1

( ) ( ) ( ) 100(1 2 )( ) i i

i i

B s s Z s P NA s

1717

Root – Locus MethodExample)

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 180s p XX

X

O

4

4

0

( ) ( ) 360

s p

X

1( ) ( ) 360i i

is 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

1818

Root – Locus MethodPortions of the root locus

n-m=3

1919


4) Angle of departure

)21(180)()(11

Npszs i

n

ii

m

i

)()()()()( pppppsppzp )()()()()( 423221212 pppppsppzp

)()()()()21(180)( 423212122 ppppppzpNps

2020


n=2, m=1

)(sR )(sCK

)(sR )(sC

)( 1

1

psszs

2121

)(1

1

1

psszsK


5) Double root s=b

Characteristic eq 0)()( 2 bsCharacteristic eq. 0) ()( bs

0 bsdsd

At double root

0)()( sBKsA )()(

sBsAK

( ) ( )dA dB( ) ( ) 0

( ) ( ) ( ) 0( )

dA s dB sKds ds

dA s A s dB sds B s ds

( )

( ) ( )( ) ( ) 0

ds B s dsdA s dB sB s A s

ds ds

2222

Root – Locus Method• Positioning system

m

x

u oiln Lubricatio:bm

P controlP control

cK)(sR )(sX

1

( )s ms b

)(sU

0)(

11 b

Kc )( bmssc

)()( bmsssA 1)( sBn=2 m=0

Open loop poles : 0, mb

2323

Root – Locus MethodI control

)(sR )(sX 1( )s ms b

)(sU

sKi

11 0( )

iKs s ms b

( )s ms bs

2

11 0( )iK

s ms b

n=3 m=0

Open loop poles : mb

0,0

double loot

angle of departure

)21(180)()(11

Npszs i

n

ii

m

i

)21(180)()()( N90)( ps

2424

)21(180)()()( 321 Npspsps

180)(2 1 ps

90)( 1 ps

Root – Locus MethodPI control

)(sR )(sX 1( )s ms b

)(sU

sTK

ic

11

( ) i

0)(

111

bmsssTsTK

i

ic n=3 m=1 n-m=2

Open loop zero :T1

Open loop pole : 0, 0, mb

iT

2525

Root – Locus MethodPID control

- Mass change, - PI, I gain variation

)(sR )(sX

1

( )s ms b

)(sU1(1 )C di

K T ST S

21 11 0( )

i dC

T s T sKT b

2 Open loop zeros( )C

iT s s ms b p p

mb

,0,03 Open loop poles :

2626


2 0ms bs K

11 0Cbs K

Example)

21 0m s

2727

Root – Locus MethodMinimum phase systems:

시스템의모든 pole과 zero가 LHP에있는경우

Non minimum phase systems:적어도시스템의한개의 pole이나 zero가평면의 RHP에있는경우

(1 )( )( 1)

dK T sG ss Ts

(1 )1 0( 1)

dT sKs Ts

1 1( 1)

dT ss Ts K

2828

7. root locus analysis7. root locus analysis professor kyongsu yi ©2009 vdcl©2009 vdcl vehicle...

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